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t (e) i-1
'fit E
(2.88)
150
provid ed (2.89)
v(t,x) is We now choose 'io = v1(to,x0 ) for all i = 1, 2, ... , s. Since r 8 numbe e positiv contin uous and v(t, O) == 0, we can always find a == 6(t0 , e) such that
•
IIXoll < 13,
I d,v,(to,xo)
i-1
< !J..
(2.90)
implies hold simultaneously. From (2.90), we conclude that Xo E ~~
x(t; to, Xo)
C ~.
"'t
E
150, 'fiE
E
E,
(2.91)
sion is not and thus connective stability of x• = 0. Suppose that this conclu t0 such that for some (to,xo) E '5'X ~P and true, and that there exists t1 some E E E, we have x(t;t0 ,x0 ) C ~P• t E [to, It), but
>
llx(t,; to, Xo)il
(2.92)
= e.
Then we have
cflt(e) to +
T.
(2.100)
(2.100), there exists a sequence Let us suppose that cont radic tory to some E E E ther e to+ T, with t~~:-+ +oo ask~ oo, such that for {t~~:}, t~~: X ~.. and with the prop erty is a solu tion x(t; t0 , Xo) with (t0 , x 0 ) E '5" (2.101) = e.
>
llx(t~~:; to, xo)ll
This implies
·•: r '
'
f
I
Analysis: Connective Stability
88
J.
which is absurd. Therefore, x• = 0 is asymptotically connectively stable; and the proof of Theorem 2.5 is completed. 2.5, It is not difficUlt to show that under the conditions of Theorem p., 6, rs numbe the is, That . uniform asymptotic connective stability is also and 'T can be chosen independently of to. In applications, we are quite often interested in establishing global the connective stability. For this purpose we have to assume first that in on d bounde and ous, continu , defined equation (2.1), the functio nf(t,x) is and C3t" ~X E ) ,x (t all for exist 0 0 ~X~. so that solutions x(t; to,Xo) of (2.1) 'T E ~0 • From Theorem 2.5, we derive
Theorem 2.6. Suppose that all the conditions in Theorem 2.5 are valid for ~P
= C3t" and ct>t E %.,. Then asymptotic stability in the large of the solution ive = 0 of the differential equation (2.32) implies asymptotic connectthe
r• stability in the large of the equilibrium x• differential equation (2.1 ).
Proof.
Since cp(K) """) +oo as
-.
r """) +oo, we have
lim 6(to,e)
and -y
=
+oo for. all E E
= 0 of the system lii described by
=
(2.103)
+oo
E. This proves Theorem 2.6.
By imitating this section and using a number of results on vector one Liapunov functions outlined by Lakshmikantham and Leela (1969), y stabilit tive connec of can extend and generalize considerably the concept the apply to is r, presented here. Our interest in the next section, howeve concept to the stability analysis of large-scale dynamic systems.
2.5. LARGE-SCALE DYNAMIC SYSTEMS a We turn our attention now to dynamic systems that are composed of such number of interconnected subsystems. Mathematical descriptions of n equatio the to re structu more adding by systems are obtained
x = f(t,x)
(2.1)
and writing it in the form
x; = g;(t,x1) + h1(t,x),
i
= 1, 2, ... ' s.
(2.2)
the In this new description of the system ~. the functions g1(t, x1) represent interac the e describ h;(t,x) ns functio the s wherea , ~ isolated subsystems 1 tions among them.
89
Large-Scale Dynamic Systems
As demonstrated in Section 1.8, the concept of the vector Liapun ov
ic function enables us to determine the stability of a large-scale dynam their system from the stability of its subsystems and the nature of interactions (on the subsystem level), and the stability of the corresponding to use aggregate model (on the overall system level). Therefore, to be able h the concep t formalized by Theorems 2.5 and 2.6, we need first to establis the ct the stability of each subsystem when decoupled, and then constru spirit aggregate model involving the vector Liapunov function. This is the (1962), n Bellma and (1962) ov Matros by ced introdu s of stability analysi site and used ingeniously by Bailey (1966) to study the stability of compo systems. Our interest in this section is to develop a decomposition s aggregation method for connective-stability analysis of large-scale system of the in the contex t of vector Liapunov functions, and study the stability disconare ems~~ subsyst y whereb ations perturb ral system~ under structu ~. nected and again connected in various ways during the operation of each that obvious is it stable, tively connec be For the system ~ to the r conside us let re, Therefo . isolated when subsystem ~~ should be stable n equatio the by ith subsystem ~, described
X;
(2.3)
= 81 (t, X;),
and where x 1(t) E CiJl:" is the state of ~, and g1 : '3"x CiJl:" ~ CiJt"' is defined ~': E {x = CiJtp; and +oo) 1 [T, = " 3 ' with CiJI,P '3"X continuous on the domain (to,X;o) for "o 3 ' on (2.3) of ;o) x;(t;to,X s solution r \\x,\\ p;}. We conside E '3"x CiJtp~. We recall the requirement
ld E %, k = 1, 2, 3. Then the equilibrium x7 = 0 of the isolated • subsystem is asymptotically stable, and v1(t,x1) is a Liapunov function for ~1 be Proof. We immediately notice that the last inequality in (2.104) can rewritten as (2.105)
Analysis: Connective Stability
90
r.
where q,[;(p.) is the inverse function of ~;(0, that is, q,{;[~;(0] = It is easy to see that if ~ 1 (0 E :JC for E (0, to), and ~ 1 (to) = p.o, then q,{;(p.) is defined (at least) for p. E [0, JLo] and q,{; E %. Since cp 1 ~o cpfi E :1(, from (2.105) we conclude that~~ E X Therefore, the inequalities (2.104) can be rewritten as
r
~~(llx; II)
D+v1(t,x1)
< v;(t, x;) < ct>u(llx;ll), < -~1 [v1 (t,x1 )]
V(t,x 1) E 5'X 0tp;. (2.106)
Following the proof of Theorem 2.5 when s = 1, we consider the scalar differential equation (2.107) for (t0 ,r0 ) E '5'X 01,+ and t E '5'0 • Solutions of (2.107) are readily seen to be given as r(t; t0 , ro) = ql[cp(ro) - (t - to)],
(2.108)
on 5'0, where the function cp: gt+ --+ gt+ is determined as (2.109) and cp 1 is the inverse function of cp. Now, given any e > 0 we can choose 6(e) = e to establish the stability of the solution r* = 0 in (2.107). p, the choice y = p Furthermore, we can easily show that for any e t0 + 'T, where T(e) = t/l(p) - t/l(e). Conseyields r(t; t0 , r 0 ) < e for all t quently, r* = 0 is asymptotically stable. By Theorem 2.5 for s = 1, it follows that x! = 0 is also asymptotically stable, and Theorem 2.7 is proved.
From the proof of Theorem 2.7, we see that the inequalities (2.104) or, equivalently, (2.106) actually imply uniform asymptotic stability of x! = 0. The last argument in the proof of Theorem 2.7, which follows Equation (2.109), can be reinterpreted using the comparison functions, as shown by Hahn (1967). If the function [~ 1 (0r 1 is integrable, then from (2.108) and (2.109) it follows that r* = 0 is reached in finite time, which is incompatible with the Lipschitz condition. If [Cj)31 (0t 1 is not integrable on an interval containing zero, then -cp(r) is monotone increasing and unbounded as r tends tor* = 0. Therefore, -cp1 (r) is a function of class I; and we can write th{t- t0 )th(-cp(r 0 )], where t/IJ, th, E f. By using (2.108) as r(t; t0 ,ro) suitable notation, this last inequality can be rewritten as r(t; t0 , ro) cp1 (r 0 )th{t - t0 ) with 4>1 E :1(, which implies the asymptotic stability of r(t; to, ro), and the inequalities r* = 0. From v1o = ro, v1[t, x 1(t; t0 , x 10 )]
u(llxtOIDJo/t(t
/,[tf>t(ro)1f,{t- to)] (2.104), we get llx;(t;to,X;o)ll uality {2.79). Thus, the asym p- t0 )} lfJ(IIxtOII)lf(t- to), whi ch is the ineq . totic stability of x~ = 0 is established 2.7 and get a global result like that orem The nd exte to le simp It is now rem is alm ost auto mat ic: of The orem 2.6. The following theo
u(llx,ll) < v;(t, x;) < cf>u(llx,ll) (2.112) where tf>J., C/>2; E % and for each decoupled subsystem ~~ we have D+v,(t, x 1)(2.3)
< g [t,v (t,x 1
1
Vi= 1, 2, ... , s, V(t,x) E '!J"X ~,..
1)]
(2.113)
-
~~)and g(t,O) where the function g: '!J"X ~~ -4 ~is such that g E C('!rx X ~, -4 'ill!" are " 5 ' h,: ns ftmctio n == 0. Further suppose that the interconnectio i = 1, 2, .•. , all or ~m)f E h; such that h1 E C('!l"X t!it,), h1(t, 0) !!!! 0, and
s. n Then asymptotic stability of the solution r• = 0 of the equatio f
=
(2.32)
w{t,r),_
with W1(t,r)
= g1(t,r;) + K11i;(t, r),
i
=
1, 2, ... ' s,
(2.114)
1 is the inverse where 1i(t,r) = li,[t,lntf>I 1(r1),l12cf>{1 (r 1), ••• ,etrcf>{,(~;)], cp{,(l!) to v1(t, x 1), onding co"esp nt consta itz ftmction of tf>J.1(r,), and K; is the Lipsch the system of 0 = x• rium equilib the of y implies asymptotic connective stabilit ~for the model ate aggreg an nts ~; and the differential equation (2.32) represe ~ .i
.•
system~-
Proof.
Let us calculate the function D+v,(t, x 1)(u) as
D+v1(t, x,)(u) = lim sup _hl {v 1(t
+ h, x 1 + h[g1(t, x 1) + h1(t, x)]) -
v1(t, x1)}
11-+0+
1 =lim sup -h {v1[t +h,x1 + hg,(t, x,)]- v,(t,x1) IJ-oQ+
+ v1(t+ h,h[g1(t,x 1) + h1(t,x)]) - v1[t + h,x1 + hg1(t,x1)]}
(2.115)
< D+v,(t, x,)(l.3) + K;llh,(t, x)ll < g;[t, V;(t, X;)] + K;h;(t, v (t, X;)] 1
'Vi
=
1, 2, ... , s, 'V(t,x) E '5X 0t,.
and get the vector differential inequality
..:...
93
Large-Scale Dynamic Systems D+ v(t, x)
(2.87)
t; E :JC..,. Then asymptotic stability g!,P
es asymptotic connective r• = 0 of the aggregate representation !! impli 0 of the system l!i. stability in the large of the equilibrium x• =
um
The proof follows from Theorems 2.6 and 2.9. !! can be constructed for a There are several ways in which the aggregate connective stability that is of given system l!i. They depend on the kind of given for the interconnecinterest, and on the sort of constraints that are lish conditions under which tions. Let us first form an aggregate ~ and estab and the aggregate Ci imply asymptotic stability of each subsystem l!i1 system l!i. We consider global asymptotic connective stability of the overall t can be readily obtained stability, but the corresponding local resul so far in this section. This imitating the com~sponding theorems proved first by Grujic and Siljak construction of an aggregate Ci was proposed
Proof.
{1973b).
We first need the following:
tffi:" --,) ~ belongs to the class Definition 2.15. A continuous function h1 : '5' X '5' X ~ --,) ~such that ~ 116 ) if there exist bounded ./uitc'fit:~ns €u: 'l(t,x ) E crx~.
{2.116)
al interconnection matrix where eii are elements of an s X s fundament li E :JCfor allj = 1, 2, .•• , s.
etate matrix Furthermore, we define an s X s constant aggr
E
and
W = (w11) as (2.117)
where
~-·
"
=
{1,0,
i =j, i =I=},
(2.118)
ber, and the nonnegative is the Kronecker symbol, 14 is a positive num number a 11 is defined as
Analysis: Connectiv e Stability
94
au= max{O,~ ~(t,x)}.
(2.119)
1
;,;~
We also need the following definition advanced by McKenzie (see Newman, 1959): Definition 2.16. An s X s constant matrix W = (wii) is said to be quasidominant diagonal if there exist positive numbers ~ such that (2.120)
Let us prove the following (Siljak., 1975b):
Theorem 2.11. Suppose there exists a junction v:
~X
which is defined by
', 1
It'
.·
. .
99
La,rge-Scale Dynamic Systems
~~~;;-
{2.145) where au is again com pute d as in (2.119). Now, we can prove the following:
v: 5X ~--.~such that Theorem l.U. Suppose there exists a function upled subsystem~~ we have v E C 1(5X ~), v(t,O) = 0, and for each deco
cf:li;{IIxdl) 3,(llx; II) the interconnection functions where q,.,, cl>li E ~. ~ E %. Suppose also that 2, ... , s. Then the quasidomih,(t, x) belong to the class ~ 144>for all i = 1, egate matrix W = {w11) defined nant diagonal property (2.120) of the s X s aggr lity in the large of the equilibrium by {2.145) implies asymptotic connective stabi x• = 0 of the system ~. v (t,x ) alon g the g the total time derivative of the func tion 1 1
Proof. By takin inequalities (2.121), {2.144), and solutions of Equ ation {2.2) and using the (2.145), we obta in
) v,{t,x,)(2.2) = v,(t,xl)(2.3) +[g rad V;(t,x,)]fh,(t,X (2.146) I
3J(IIxJID
' 1l2;, 7131, 7141 are positive numbers. Then the equili exponentially stable in the large. Theo rem 2.13, but using the Proof. Following the steps of the proof of 1 (2.150) with II1 = 11ii 11u. inequalities (2.151), we obtain the inequality 1 •n1 = 712i 7131, which establishes Theorem 2.14. obtained by Krassovskii Theorem 2.14 is a slight modification of a result stability analysis of large(1959), which has been used effectively for the subsystems are linear or nonlinear of the scale systems when some (or all) 1 of estimates (2.151) in Lur'e-Postnikov type (Siljak, l'972a)., The use Section 1.8 of the precedconstructing aggregate models was illustrated in is presented next. ing chapter. A generalization of this construction e that the interconassum and (2.2) Let us now consider the system~ of ions specified by funct of class a to nections among the subsystems ~~ belong ~ ~"' belongs to the class Definition 2.18. A continuous junction h1: ~X ~ numbers ~ such that :JC(l.IS l) if there exist nonnegative llhi(t,x)il
•
0
(2.173}
E ~.
then it is obvious that the frequency condition (2.171) is equivalent to positivity of the polynomialp(w}, that is,
p(w} ~ Q
.Vw
E ~-
(2.174}
It is equally obvious that positivity of p(w} is equivalent to the nonexistence
>
0. Now, the existence of real zeros of p(w} of real zeros of p(w} and P2n can be established by a modified Routh algorithm (Siljak., 1969; Barnett and Siljak., 1977}. To see this, we note that zeros of p(w} are symmetrically distributed with respect to both the imaginary and the real axis of the wplane. We consider the new polynomial
(2.175} and conclude that the previous symmetry is preserved, but real zeros of p(w} (if there are any} become pure imaginary zeros of p(jw}. Consequently, if p(jw) has n zeros with positive real parts, there are no positive real zeros of 0. Therefore, we carry out the p(w}, and it is positive provided P2n computation in the modified Routh table (Siljak., 1969}:
>
Analysis: Connective Stability
108
t
(- 1 .P2.s
-p,
(-1),_ IPl(..-1)
1 (-l)"2n.Pln (-1).. - 2(n- 1)112(..-1)
wo
-2p,
Po (2.176)
Po
coefficients of the two where the first two rows are filled out by the s are computed as polynomials p(jw) and dp(jw) / dw, and the other entrie usual (e.g. Siljak, 1969; Barnett ~d Siljak, 1977). omial p(w) produces Now, the system ~ is absolutely stable if the polyn the first column of the Rout h table (2.176) so that (2.177) p,. 0, ,., (-1)"2 np,, ... ,Po] = n,
>
K[( -l)"p
sequence enclosed by where K denotes the numb er of sign changes in the condition (2.177) is the square brackets. It is clear that the algebraic ). equivalent to the Popov frequency condition (2.171 composed of intercons system scale largeof sis analy ity For the stabil lish the existence of nected Lur'e-Postnikov systems, it is impo rtant to estab used to aggregate the an appro priate Liapunov function, which can be It was shown by Siljak. prope rty of absolute stability of the system (2.166). ) can be used to and Sun {1972) that the conditions (2.171) or (2.177 Lur'e-Postnikov the of establish the existence of a scalar Liapunov function type,
r·,." cp(x) tfx,
V(x) = xT Hx + 9 )o
(2.178)
ntees the exponential which satisfies the inequalities (2.151) and thus guara property of absolute stability for the system~function V(x) along We comp ute the total time derivative V(x) of the solutions of the equations (2.166) to get (2.179) where
-G
= ATH + HA + liT,
-ylf2 / = -y
Hb + l(OAT + I)c,
= OcTb-
-r =
(~~:- 1 cp -
(2.180)
K- 1,
x)CJJ.
there exist a constant By a result of Yakubovich {1962), we have that tor I which satisfy the positive definite n X n matrix H and a constant n-vec
109
Large-Scale Dynamic Systems
Then, equations {2.180) if and only if the frequency condition (2.171) holds. > 0, fJ for that s from {2.178) and (2.179), we conclude that (2.171) implie the function v(x) satisfies the inequalities2 Am{H)IIxiF < V(x) < AJt(H)IIxll , {2.181) 'fixE~. V{x) < -Am{G)IIxW, II grad V(x)li
< 2;\M(J1) llxll
of the where Am and AM are the minimum and maximum eigenvalues indicated matrices and H = H + !rcfJccr. By substituting {2.182) v(x) = V11 2 (x) and using the relations ri(x)
grad v(x) = i v-l/2{x)grad V(x),
= ! v-112{x)V{x),
(2.183)
the inequalities (2.181) become
< v(x) < 1J21lxll, 'fix ri{x) < -1bllxll, llgrad v(x)ll < 7J4 7JJIIxll
E ~,
{2.184)
which are those of {2.151 ), where
If 8
7JJ
= ;\~ 2 {H),
7J4
=
7Jz ='XY,}(IJ.),
TJ3
= iAm(G);\:V112 (H),
{2.185)
;\;.li2(H);\M(11).
{2.184), but the < 0, then the function v(x) still satisfies the inequalitiesare reversed.
roles of the matrices Hand 11 in the arguments of {2.185) tems Let us now consider a system :!; composed of s interconnected subsys ons :;;, of Lur'e-Postnikov type, described by the equati X; = A;x,
•
~ evhu·· + b;cp,0cJ) + j-1
X;
=
cl X;,
i = 1, 2, ... , s. (2.186)
Popov For each decoupled subsystem {2.166) we assume that either the as Then, d. verifie is (2.177) condition {2.171) or the algebraic condition ons equati aic algebr the shown by Siljak and Sun {1972), we can solve n v {x) (2.180) and find a positive definite matrix H, such that the functio 1 :!;1• tems subsys led decoup the defined in {2.178) is a Liapunov function for ), {2.180 ons equati the The matrices G1, which were chosen in order to solve
r ·. Analysis: Connective Stability
110
the numbers in (2.185) and the matrices lit. f1. are then used to compute tion for the overall senta necessasary for constructing an aggregate repre stems are assumed subsy system :ii. Furthermore, the interactions among the to belong to the class X(l.23 )
1
I
= {hu(t,xJ): ilhu·(t,xJ)ii
< ~JilxJil V(t,xJ) E ~X~}
"
l
(2.24)
the s X s aggregate matrix Now, by using Theorem 2.16, we conclude that if W = (wu) defined by 2 2 (~) (2.187) 'Wu = -!XAI (.ti;)A,( G,)8u + eu~li')..;.l/ (1It)AM(fi;)X;,l/ m :ii is absolutely and satisfies the inequalities (2.132), then the syste 2.8 and the definition of exponentially connectively stable. From Definition m state x• = 0 of :ii is the class
.,.(Hi)
=
=
0, we have il
0.27,
G· I
J
0 = [0.305 0.35 •
{2.191)
= H, and >.,.(G1) = 0.35.
(2.192)
ml'''' 'F-''
Analysis: Connective Stability
Now the function v1(x 1) (2.151) for 'llli =
1 ~
112
0.52, T/21
=
= J-;112(x1) = (xT H,x1)112
0. 73, T/31
=
0.24,
T/4i
=
satisfies the inequalities
3.23,
i
=
1, 2, 3. (2.193)
Finally, from (2.187) we get the 3 X3 aggregate matrix
w=
-0.33 [
1.92~21
0 -0.33
0
1.9~32
1.9~13] 0 . -0.33
(2.194)
Applying the inequalities (2.132) to the matrix W, we obtain the condition on interactions (2.195) which guarantees the absolute connective stability of the system ~- The corresponding structural perturbations of ~ are shown on Figure 2.3(b); they in tum correspond to binary interconnection matrices E obtained from E of (2.189) when the unit elements are replaced by zeros. As a general comment concerning this secion, we can say that in a given system composed of a number of interconnected subsystems, the interactions among the subsystems can be viewed as perturbation terms in equations describing the subsystems. Then a large number of strong results from the theory of differential equations and inequalities (Lakshmikantham and Leela, 1969; Walter, 1970) become available for stability analysis of the overall system: Obviously, we have not exhausted all these results, but rather outlined those that we plan to use in the rest of this book. Our use of vector Liapunov functions was made predominantly with regard to the connective stability of composite dynamic systems. Therefore, we did not exploit all the rich properties of such functions, nor did we point out possible analogous developments in other kinds of stability, such as input-output stability, stability on a finite time interval, practical stability, etc. After Bellman (1962) and Matrosov (1962) introduce d the concept of vector Liapunov functions, Bailey (1966) proposed an efficient construction of such functions for composite systems. Bailey's results were significantly improved by Barbashin (1970), Thompson (1970), Michel (1970a, b, 1974), Weissenberger (1973), Grujic (1974, 1975b), LaSalle (1975), and many others, who considered a wide variety of stability properties of interconnected systems. A survey of these various results was given by Siljak (l972b), and more recently up-to-date reviews were presented by Michel (1974) and by Athans, Sandell, and Varaiya (1975). Not included in these surveys are importan t input-output stability results obtained by Porter and
1
113
Partial Connective Stability
Michel (1974), Callier, Chan, and Desoer (1976), Willems (1976a), Vidyasagar (1977a), Sundareshan and Vidyasagar (1977), and Moylan and Hill (1978). It should be noted that most of the reported results on input-output stability of composite systems, can be rewritten one way or another in terms of connective stability as shown by Willems (1976a), and Sundareshan and Vidyasagar (1977). This adds yet another important aspect to the stability study of large-scale interconnected systems based upon the decomposition principle and techniques. 2.6. PARTIAL CONNECTIVE STABD.JTY So far, we have required that the subsystems should be stable when isolated. This was a natural constraint, since we allowed the zero matrix to be an interconnection matrix, in which case all subsystems are decoupled from each other. As shown first by Grujic and Siljak (1973b), unstable subsystems may be permitted to be parts of a large composite system provided the stabilizing negative feedback is present at all times. Now, if we carefully choose interconnection matrices which do not remove the stabilizing feedback paths, we can use the results of Grujic (1974) and permit unstable subsystems, at the price of achieving only partial connective stability. Let us assume that a system ~ described by :X1
= g1(t,x1) + h;(t,x),
i
=
1, 2, ... , s,
(2.2)
contains k stable subsystems ~~ '(1 r= 1, 2, ... , k ), and s - k unstable subsystems~~ (i = k + l,k + 2, ... ,s). We assume that the stability property of each isolated subsystem ~1 is established by using a scalar function 1 v1: ':fX ~~~§],+such that v1 E C ('5"x ~"'), v1(t,O) = 0, and.
< v;(t,x,) < +u(llx,Ji), #L14>41(11xdl) < v;(t,x;)(2.3) < #Ltt/>l;(llx,ll) tf>li(llxdl)
'o'(t,x,) E '5"x ~\
(2.196)
where f/>!1 E :ICoo and f/>2;, 1/>31, f/> 41 E % Here
i i
= 1, 2, ... , k = k + 1, k + 2, ... , s
(stable~),
(unstable
~1 ).
(
2 197 ) .
That is, when p.1 = -1, the inequalities (2.196) are those of (2.104) for -2e21'Yv1(xt)- 2e21e22y8v2(x2) which is valid for all E E £.
V(t,x) E '!J"X~P'
129
Connective Instability
The inequalities (2.260) can be used to get the clift"erential inequality (2.261) ri ~ Wv, where v = (v 1 , v2)T is a vector Liapunov function and W is the 2 X 2 aggregate matrix
w=
J
-2{3 2 [ - 2y 1 - 2y8 .
- =[00] 11
£
(a)
[? b]
0
0
[gg) 0
0
(b)
FIGURE 1.6. System structure and perturbations.
(2.262)
Analysis: Connective Stability
130
The matrix W has negative off-diagonal elements, and it is possible to show (Theorem A.2) that it has all eigenvalues with positive real parts if (and only if) - W satisfies the conditions (2.132), that is, -1
+ 2y8 < 0,
1 - 2{3-y - 2y8
> 0.
(2.263)
Again, one can show (Theorem A.2) that the conditions (2.263) are equivalent to the quasidominant diagonal property (2.120) of the matrix W. The conditions (2.263) are sufficient for complete connective instability of the equilibrium x* = 0 of $ with respect to the region ~P = {x E ~: llxll 1}. The structural perturbations of the system structure shown in Figure 2.6(a) are listed in Figure 2.6(b). Again, we show only those perturbations which correspond to binary interconnection matrices. Under the conditions (2.263), not only is x* = 0 unstable for all E E E, but also every solution x(t; t0 , x 0 ) of the equations (2.256) which starts in the region ~P is increasing (leaving x* = 0) as fast as an exponential. To see this, we note that in the case of the system $, the last inequality (2.254) can be written as
FIGURE 2.7. Liapunov functions.
Analysis: Connective Stability
136
Let us first consider the function PJ {v) and form
{2.289)
iii{v)(2.ISS) = dTWv,
using {2.155). Since the matrix W is a Metzler matrix and satisfies the conditions {2.132), a positive vector d E ~~exists such that (2.290) for any positive vector c E lilt~. This fact, which is proved in Appendix (Theorem A.2), ensures that lit (v) satisfies the inequalities {2.285). According to (2.281) the estimate ~is determined by
min {d, v?} = l(i<s
'YI
(2.291)
where Jl are determined on the subsystem level. To consider the function v2 (v), we first note that from {2.273) and (2.287), the region~ (which is now assumed bounde d) is given by ~
= {v =
{v
< y} ~~: v < v = yd}.
E ~~: E
v(v)
{2.292)
We define the set of points~ as ~
= {v
E lilt~:
{2.293)
v(v) = y},
and from (2.292) get (2.294) where v1 = yd,. It follows readily that for v2 (v) = )'z, iil(v)cz.lss>
< max.{a,-1 :± Wuv1} l 0 and the case Wd = 0 is excluded.
137
Regions of Connective Stability
for any As in the case of the function P1(u ), stability of W implies that of the n solutio a as positive c E ~~. there exists a positive d E ~~ equation (2.298) d = -w-'c. Once a solution d
> 0 of (2.298) is found, we compute y 'Y2
2
from (2.281) as
1 V;0} . Dlln {d-1 = l din
Itqi, 1, 241-252.
Diamond, P. (1975}, "Stochastic Exponential Stability Concepts and Large-Scale Discrete Systems", International Journal of Control, 22, 141-145. Duhem, M.P. (1902), '"Surles conditions necessaires pour la stabilite de l'equilibre d'un sisteme visqueux", Comptes Rendus, 126, 939-941. Ertegov, V. D. (1970}, "On Stability of Solutions of Difference Equations.. (in Russian), Transactions of the Kazan Auiation Institute, 125, 14-19.
140
l f:
Analysis: Connective Stability
Fiedler, M., and Ptak, V. (1962}, "On Matrices with Nonpositive Off-Diagonal Elements and Principal Minors", CzechoslOIHlkian Mathematical Journal. 12, 382-400. Franklin, T. N. (1968}, Matrix Theory, Prentice-Hall, Englewood Cliffs, New Jersey. Gantmacher, F. R. (1960), The Theory of Matrices, Vols. I and II, Chelsea, New York. Grujic, Lj. T. (1973), "Uniform Asymptotic Stability of Discrete Large-Scale Systems", IEEE Transactions, SMC-3, 636-643. Grujic, Lj. T. (1974), "Stability Analy~ of Large-Scale Systems with Stable and Unstable Subsystems", International Journal of Control~ 20, 45~. Grujic, Lj. T. (1975a), "Uniform Practical and Finite-Time Stability of Large-Scale Systems", International Journal of Systems Science, 6, 181-195. Grujic, Lj. T. (1975b), "Non-Lyapunov Stability Analysis of Large-Scale Systems on Time-Varying Sets", International Journal of Control, 21, 401-415. Grujic, Lj. T., Gentina, J. C., and Borne, P. (1976}, "General Aggregation of LargeScale Systems by Vector Lyapunov Functions and Vector Norms", International Journal of Control, 24, 529-550. Grujic, Lj. T., and Siljak, D. D. (1973a), "On Stability of Discrete Composite Systems" IEEE Transactions, AC-18, 522-524. Grujic, Lj. T., and Siljak, D. D. (1973b), "Asymptotic Stability and Instability of · Large-Scale Systems", IEEE Transactions, AC18, 636-645. Grujic, Lj. T., and. Siljak, D. D. (1974), "Exponential Stability of Large-Scale Discrete Systems", International Journal of Control, 19, 481-491. Gunderson, R. W. (1970), "On a Stability Property of Krassovskii", International Journal of Non-Linear Mechanics, 5, 507-512. Gunderson, R. W. (1971), "A Stability Condition for Linear Comparison Systems", Quarterly of Applied Mathematics, 29, 327-328. Hahn, W. (1967), Stability of Motion, Springer, New York. Harrary, F., Norman, R. Z., and Cartwright, D. (1965), Structural Models: An Introduction to the Theory of Directed Graphs, Wiley, New York. Hale, J. K. (1969), Ordinary Differential Equations, Wiley, New York. Kamke, E. (1930), Differentialgleiclrungen reeler Funktionen, Akademische Verlagsgeselshaft, Leipzig, Germany. Kamke, E. (1932), "Zur Theorie der Systeme gewonlicher Dilferentialgleichungen. II", Acta mathematica, 58, 57-85. K.loeden, P. E. (1975), "Aggregation-Decomposition and Ultimate Boundedness", The Journal of the Australian Mathematical Society, 19, 249-258. K.loeden, P. E., Diamond, P. (1977), ••eonverse Theorems for Stochastic Exponential Stability", International Journal of Control, 25, 507-512. Krassovskii, N. N. (1959), Some Problems of the Theory of Stability of Motion (in Russian), Fizmatgiz, Moscow (English Translation: Stanford University Press, Palo Alto, California, 1963). Ladde, G. S. (1975a), "Systems of Dilferential Inequalities and Stochastic Dilferential Equations. II", Journal of Mathematical Physics, 16, 894--900.
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ions of La4de, G. S. (1975b), "Variatio nal Compari son Theorem and Perturbat 52, Society, tical Mathema American the of gs Proceedin , Nonlinea r Systems" 181-187. l Ladde, G. S. (1976), "Stability of Large-Scale Heredita ry Systems Under Structura Systems Scale Large on m Symposiu IFAC the of gs Perturbations", Proceedin Theory and Applications, G. Guardab assi and A. Locatelli (eds.), Udine, Italy, 215-226. l Ladde, G. S. (1977}, "Stability of Large-Scale Function al Systems Under Structura appear). (to Perturbations", ale Ladde, G. S., and Siljak, D. D. (1975), "Connective Stability of Large-Sc 713-721. 6, Science, Systems of Stochastic Systems", International Journal s", Lakshmikantham, V. (1974}, "On the Method of Vector Lyapunov Function System and Circuit on ce Conferen Allerton Annual Twelfth the of gs Proceedin Theory, University of Illinois, Urbana, Illinois, 71-76. ies", Lakshmikantham, V., and Leela, S. (1969}, "Differential and Integral Inequalit Vols. I and II, Academic, New York. s", Lakshmilcantham, V., and Leela, S. (1977}, "Cone-V alued Lyapunov Function 215-222. 1, ons, Applicati and Methods, Nonlinear Analysis, Theory, Annales Liapunov, A.M. (1907}, "Problem e general de 1a stabilite du mouvement", d in (Reprinte 203-474 9, de Ia facu/te des Sciences de l'universite de Toulouse, New , Princeton Press, ty Universi Princeton Annals of Matematics Studies, Jersey, Vol. 17, 1949.) of LaSalle, J. P. (1975}, "Vector Lyapuno v Function s", Bulletin of the Institute 139-150. Mathematics Academia Sinica, 3, (in Lur'e, A. I. (1951), Some Nonlinear Problems in the Theory of Automatic Control s Russian}, GOSTEH IZDAT, Moscow (English Translation: Her Majesty' Stationery Office, London, 1957}. , Matrosov, V. M. (1962}, "On the, Theory of Stability of Motion" (in Russian} • 992-100. 2.6, ka, Mekhahi i ika Matemat ya Prikladna , Matrosov, V. M. (1963}, "On the Theory of Stability of Motion" (in Russian} 22-33. 80, Institute, Transactions of the Kazan Aviation s in Matrosov, V. M. (1965}, "Develop ment of the Method of Liapunov Function ce Conferen All-Union Stability Theory" (in Russian}, Proceedings of the Second 112-125. s, in Theoretical and Applied Mechanic Matrosov, V. M. (1967}, "On Differential Equations and Inequalities with Discontinuous Right-Ha nd Sides" (in Russian}, Dif!erentsia/'nie Uravnenya, Part I:395-409, Part II: 839-848. r Matrosov, V. M. (1971}, "Vector Liapunov Function s in the Analysis of Nonlinea Intercon nected Systems", Instituto Nazionale di Alta Matematica: Symposia Mathematica, 6, 209-242. ectMatrosov, V. M. (1972a}, "Method of Vector Liapunov Function s of Interconn i ed Systems with Distribut ed Parameters (Survey}" (in Russian), Avtomati ka Telemekhanika, 33, 63-75. k Matrosov, V. M. (1972b), "Method of Veetor Liapunoy Function s in Feedbac Systems" (in Russian}, Automat ika i Telemekhanika, 33, 63-75.
r ! '
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McClamroch, N. H., and Ianculescu, G. D. (1975), "Global Stability of Two Interconnected Nonlinear Systems", IEEE Transactions, AC-20, 678-642. Michel, A. N. (1970a), "Quantitative Analysis of Simple and Interconnected Systems: Stability, Boundedness, and Trajectory Behavior", IEEE Transactions, CT-11, 292-301. Michel, A. N. (1970b), "Stability, Transient Behavior, and Trajectory Bounds of Interconnected Systems", International Journal of Control, 11, 703-715. Michel, A. N. (1974), "Stability Analysis of Interconnected Systems", SIAM Journal of Control, 12, 554-579. Michel, A. N. (1975a), "Stability Analysis of Stochastic Composite Systems", IEEE Transactions, AC-20, 246-250. Michel, A. N. (1975b), "Stability and Trajectory Behavior of Composite Systems", IEEE Transactions, CAS-22, 305-312. Michel, A. N., and Porter, D. W. (1972), "Stability Analysis of Composite Systems", IEEE Transactions, AC-17, 222-226. Michel, A. N., and Rasmussen, R. D. {1976), "Stability of Stochastic Composite Systems", IEEE Transactions, AC-21, 89-94. Montemayor, J. J., and Womack, B. F. (1975), "On a Conjecture by Siijak", IEEE Transactions, AC-20, 572-573. Moylan, P. J., and Hill, D. J. (1978), "Stability Criteria for Large-Scale Systems", IEEE Transactions, AC23 (to appear). Mwer, M. (1926), "Ober das Fundamentaltheorem in der Theorie der gewonlichen Differentialgleichungen", Mathematische Zeitschrift, 26, 619-645. Newman, P. K. (1959), "Some Notes on Stability Conditions", Reuiew of Economic Studies, 12, 1-9. Nyquist, H. (1932), "Regeneration Theory", Bell System Technical Journal, 11, 126-147. Piontkovskii, A. A., and Rutkovskaia, L. D. (1967), "Investigation of Certain Stability-Theory by the Liapunov Function Method" (in Russian), Avtomatika i Telemekhanika, 28, 23-31. Popov, V. M. (1973), Hyperstability of Control Systems, Springer, New York. Porter, D. W., and Michel, A. N. (1974), "Input-Output Stability of Time-Varying Nonlinear Multiloop Feedback Systems", IEEE Transactions, AC19, 422-427. Rasmussen, R. D., and Michel, A. N. (1976a), "On Vector Lyapunov Functions for Stochastic Dynamical Systems", IEEE Transactions, AC21, 250-254. Rasmussen, R. D., and Michel, A. N. (1976b), "Stability of Interconnected Dynamical Systems Described on Banach Spaces", IEEE Transactions, AC-21, 464-471. Siljak, D. D. (1969), Nonlinear Systems, Wiley, New York. Siljak, D. D. (1971), "On Large-Scale System Stability'', Proceedings of the Ninth Annual Allerton Conference on Circuit and System Theory, University of Illinois, Monticello, Illinois, 731-740. Siljak, D. D. (1972a), "Stability of Large-Scale Systems Under Structural Perturbations", IEEE Tran3actions, SMC-2, 657--663.
,1
.,j i
1
"··
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Proceedings of the Fifth
IFAC Congress, Paris, C-32:1-11. Large-Scale Systems Und er Structural Siljak, D. D. (1973); "On Stability of -3, 415-417. Perturbations", IEEE Transactions, SMC Competitive Equilibrium", Automaof lity Stabi ve necti "Con a), (l975 Siljak, D. D. tica, 11, 389-400. ms: Stability, Complexity, Reliability", Siljak, D. D. (1975b), "Large-Scale Syste
SA Ames Research Center SemiliLlr Proceedings of the Utah State Univer9ity-NA NAS A SP-371, Washington D.C. , Workshop on Large-Scale Dynamic Systems, 147-162. of Dynamic Systems", Nonlinear Analysis, Siljak, D. D. (l977a), "On Pure Structure Theory, Methods, and Applications, 1, 397-413. mic Systems", Proceedings of the IFAC Siljak, D. D. (1977b), "Vulnerability of Dyna Integrated Industrial Complexes, nt Workshop on Control and Manageme of
Toulouse, France, 133-144. onential Absolute Stability of Discrete Siljak, D. D., and Sun, C. K. (1971), "Exp ematik und Meclranik, 51, 271-275. Systems", Zeitschriftfiir Angewandte Math Exponential Absolute Stability", "On ~ Siljak, D. D., and Sun, C. K. (1972 . International Journal of Control, 16, 1000-1008 e-Scale Systems: Stability, "Larg ), (1976 B. M. evic, Vukc Siljak, D. D., and ute, 301, 49-69. Instit klin Fran Complexity, Reliability'', Journal of the of the Lur'e-Liapunov ion truct Cons "A ), (1970 S. Siljak, D. D., and Weissenberger, , 10, 455-456. itung ataverarbe Func tion" , Regelungstechnik und Process-D ity of Large-Scale tabil "Lr-S ), M. (1977 Sundareshan, M. K.; and Vidyasagar, IEEE Transacry", Theo ator Oper ive Posit Dynamical Systems: Criteria via tions, AC-22, 396-399. s, Monographic Matematycme, Vol. 43, Szaraki, J. (1965), Differential Inequalitie PWN , Warszawa, Poland. lity of Interconnected Systems", IEEE Thompson, W. E. (1970), "Exponential Stabi ' Transactions, AC-15, 504-506. ' "Stability of a Class of Interconnected ), (1972 E. H. ig, Koen and Thompson, W. E., 15, 751-763. ol, Systems", International Journal of Contr osov, V. M. (1972), "Me thod s of Matr and S., Vakhonina, G. S., Zemliakov, A. Func tions for Linear Systems" (in Construction of Quadratic Vector Liapunov 5-16. Russian), Avtomatika i Telemeklranika, 33, ria for Interconnected Systems", Crite Vidyasagar, M. (1977a), "Lr-Instability 15, 312-328. SIAM Journal of Control and Optimization, of Large-Scale Systems", IEEE bility Insta the "On b), Vidyasagar, M. (1977 Transactions, AC-22, 267-269. Inequalities, Springer, New York. Walter, W. (1970), Differential and Integral tions et des mega)ites diff'erentielles Waiewski, T. (1950), "Systemes des equa es et leurs applicationes", Annates ordinaires aux deuxiemes membres monoton 112-166. 23, , de Ia Societe Polonaise de mothematiques oximations for Relay-Control Appr dary Boun Weissenberger, S. (1966), "Stabilityion of Liapunov Functions", ASM E Systems via a Steepest-Ascent Construct ns: Journal of Basic Engineering, 88, 419-428.
Transactio
I'
!
144
Analysis: Connective Stability
ca, Weissenberger, S. (1973), "Stability Regions of Large-Scale Systems", Automati 9, 65~63. s Willems, J. C. (1976a), "Stability of Large Scale Interconnected Systems", Direction Control, lized Decentra and tion, Optimiza Person Many Systems, in Large-Scale Y. C. Ho and S. K. Mitter (eds.), Plenum Press, New York, 401-410. , Willems, J. C. (1976b), "Lyapunov Function s for Diagonally Dominant Systems" Automatica, 12, 519-523. Yakubovich, V. A. (1962), "Solution of Certain Special Matrix Inequalities Occurring in the Theory of Automatic Control" (in Russian), Dokladi Akademii Nauk SSSR, 143, 1304-1307. Y akubovich, V. A. (1964), ''The Method of Matrix Inequalities in the Stability Theory of Nonlinea r Control Systems" (in Russian), Avtomatika i Telemekhanika, 25, 1017-1029. (in Zemliakov, A. S. (1972), "On the Problem of Comparison System Construction" 46-54. 144, Institute, Aviation Russian), Transactions of the Kazan
3 SYNTHESIS
·-··· Decentralized Control Now that we have derived the conditions for connective stability of largescale systems, we may ask: Can we synthesize reliable complex systems by using feedback? A positive answer to this question is provided in this chapter, and we will present a decentralized multilevel scheme for synthesizing large-scale systems which are stable under structural perturbations, that is, connectively stable. By relatively simple exampl'e; 'we demonstrated in Section 1.10 that Simon's intuitive arguments about reliability of hierarchic structures are true. The objective of this chapter is to show rigorously that Simon's intuitive recipe can be used to construct dynamically reliable large-scale systems by hierarchic feedback control. Local feedback controllers are used to stabilize each subsystem when isolated from the rest of the system. Then, regarding the interactions among the subsystems as perturbations, a global controller is utilized to minimize the coupling effect of subsystem interconnections. Finally, connective stability of the overall system is established by testing for stability of the aggregate model as proposed in the preceding chapter. Prior to applying a decentralized feedback control, we have to make sure that the available inputs can influence (reach) each part of a large system and thus alter its performance according to the requirements. Since we propose to feed back the states and outputs, we also need to check that the state of each subsystem can be estimated from the outputs. These prelimi-
r I
i
146
Synthesis: Decentralized Control
therefore are form ulate d as nary consi derat ions are struc tural in natur e and in the framework of direc ted inpu t and outp ut reachability to be exam ined schemes which can be used graphs. We will also develop decentralization t or outp ut decentralized to trans form multivariable systems into inpu ral than simple perm utatio ns representations. Thes e schemes are more gene they are restricted to linea r of binar y matrices of direc ted graphs, but systems. economic systems {see Dece ntral izatio n arose in a natur al way in contr ol and d~cision ed Chap ter 4), and it was there that decentraliz ns (Arrow, 1964; izatio organ strategies were prop osed for team s and large the fact that upon based are Mars chak and Radner, 1971). Such strategies cted (for restri are ture struc the system infor matio n patte rn and contr ol stem subsy each that a way eithe r physical or economical reasons) in such r linea a lizing stabi lem of is contr olled by its own inpu t only. A prob rlyunde with e schem ol dyna mic system using a simple decentralized contr adde n (1969). This prob lem ing econ omic features was first studi ed by McF in contr ol literature by Lau, was later given much more general treat ment g and Davi son (1973), and Persiano, and Vara iya (1972), Aoki (1972), Wan and many other results Corf mat and Morse (1976). A survey of these by Sandell, Varaiya, and conc ernin g decentralized contr ol was provi ded Atha ns (1975). ol schemes, the scheme As distin ct from previous decentralized contr prese nted in this chap ter prop osed by Siljak and Vukcevic (1976a, c) and pulat ing only subsystem attem pts to stabilize a large linea r system by mani and estim ation problems is matrices. Thus , the dimensionality of contr ol the classical decomposition reduc ed in much the same way as it is by raic equa tions (e.g. Himelalgeb of techniques for solving a large numb er ems involving a large probl ing blau, 1973) and math emat ical prog ramm Besides a considerable . 1960) e, numb er of variables (e.g. Dant zig and Wolf scheme prod uces nted prese saving in numerical aspects of control, the tural pertu rbastrUc to ct systems which are dynamically reliable with respe inter actio ns the in ties neari tions and can tolerate a wide class of nonli ized in stabil be can ms syste amon g the subsystems. In fact, by this scheme, near nonli the of shape l actua cases where we have no infor matio n abou t the to able avail are ds their boun inter actio ns amon g the subsystems, and only ecise impr g datin in acco mmo the designer. Its reliability and its robustness features of decentralized ional addit two are ns knowledge of interactio ever a ques tion of centralcontr ol which must be taken into acco unt when olling a large-scale dyna mic ized vs. decentralized strategy appears in contr system.
Reachability, Vulnerability, Condensations
147
3.1. REACHABILITY, VULNERABILITY, AND CONDENSATIONS The behavior of a physical system can be altered efficiently by feedback control without changing the system itself. The principle of feedback is to choose inputs to the system as functions of its outputs so that the closedloop system accomplishes a desired controlled behavior. Before we can use this simple but powerful principle in a system design, we have to make sure that the inputs can "reach" each part (state) of the system, and that all parts of the system are "represented" by the outputs. These two inherent properties of dynamic systems were defined as input and output reachabi!ity (Siljak, 1977a, b). In this section, we use these results to formulate, study, and partially solve the problem of input and output reachability in the control and estimation of large-scale systems which are considered in the rest of this chapter. The material of this section will also be used in the following chapters for model building in such diverse fields as economics, space flight, ecology, and power systems. The formalization and study of input and output reachability, decentralized control and estimation schemes, canonical structures, and structural perturba tions are carried out in the natural framework of directed graphs (digraphs) and interconnection matrices. Only a bare minimum of notions and concepts from.digraph theory are defined here. For a deeper understanding of the structural analysis of dynamic systems outlined in this section, the books of Harary, Norman, and Cartwright (1965), Harary (1969), Deo (1974), and Berztiss (1975) are recommended. Let us consider a system ~ which is described by the equations
'
i
y
. d
f(t;x, u),
(3.1)
= g(t,x),
where x(t) E 0l? is the state, u(t) E 01,m is the input, and y(t) E tfJl! is the output of ~. The functions j: 0l X 0l? X 0l"' ~ 0l? and g: t!Jt X t!Jl!' ~ tfJl! are sufficiently smooth so that ~ represents a dynamic system (Kalman, Falb, and Arbib, 1969). With the dynamic system~ we associate a directed graph (digraph) (Siljak, 1977b) as the ordered pair 6j) = (V,R), where V = U U XU Y and U = {ut. Uz, ... , um}, X = {xi, x2, ... , Xn}, Y = {Yt.Y2· ... ,yf} are nonempty sets of input, state, and output points, respectively. R is a relation in V, that is, R is a set of ordered pairs which are the lines (uJ>xt), (x1 ,x1), or (x1 ,y;) joining the points of 6D. We make an importan t assumption about~ by requiring that 6j) does not contain Jines of the type (u1 , u1), (ui>y1), (x1, u1),
Synthesis: Decentralized Control
148
(Yi• x1), (yi> u1), and (yi,Yt). This requirement may seem to be overrestrictive, but in fact it is not, since it reflects the structure of what we ordinarily consider as a dynamic system :!i described by the equations (3.1). We merely assume that there are no lines joining the input points, no lines from the input points to the output points, etc. A convenient way to represent a digraph 6j) associated with :!i is to use interconnection matrices. We propose that the p X p interconnection matrix M = (mil), which we define as a composite matrix
M
=
E L
0 [F
0 0
0] 0 0
(3.2)
such that i, j = 1, 2, ... , p and p = n + m + /, be used to describe the basic structure of :!i. In (3.2), then X n state connection matrix E = (e11 ) is defined as a binary matrix with elements eu specified by
(x"x1) E R, (xi>x1) ~ R,
(3.3)
where i,j = 1, 2, ... , n. That is, e11 = 1 if x1 occurs inf,(t, x, u), and eu = 0 if x1 does not occur in j,(t,.x, u). Similarly, we define the n X m input connection matrix L = (Iii) as
(ubx;) E R, (u"x1) ~ R,
(3.4)
where i = 1, 2, ... , n and j = 1, 2, ... , m. In other words, 111 = 1 if u1 occurs in j,(t,x, u), and I;; = 0 if u1 does not occur in j,(t, x, u). Finally, the l X n output connection matrix F = (!11 ) is defined by
1, fu = { O,
(XJ>YI) E R,
(x"y1) ~ R,
(3.5)
where i = 1, 2, ... , l and j = 1, 2, ... , n. Again, /;; = 1 if x1 occurs in g,(t,x), and/11 = 0 if x1 does not occur in g1(t,x). If no component ui of the input vector u can influence a state x 1 either directly or via other states of~. then there is no way to alter the behavior of a; associated with the state x 1• Similarly, if a state x1 does not influence any component y, of the output vector y either directly or via other states of a;, then it is impossible to estimate the state Xr In order to express these
:Reachability, Vulnerability, Condensations
149
word "influence" by facts in the graph-theoretic terms, we replace the (Harary, Norman, phs "reach" and rely on the reachability concept of digra and Cartwright, 1965). of graphs, we need To define input and output reachability in terms graphs. We consider several well-known notions from the theory of directed •.. , vk} is specified v2, again the digraph 6j) = (V, R ), where the set V = {v1, of distinct points as V = U U X U Y and p = m + n + 1. If a collection ... , (vk-h vk) are v., v2, ... , vk together with the lines (v~o v2), (v2, v3), v ), ..• , (vk_ 1 ,vk)} is a placed in sequence, then the ordered set {(v1, v2), (v 2, 3 v if there is a path (directed) path from v1 to vk. Then v1 is reachable from 1 of points v1 reachable from v1 to v,. A reachable set V;(v1 ) of a point v1 is a set able set V;(Ji} of a set from v1• Carrying this a step further, we define a reach fore, V;(JJ) is JJ as a set of points v; reachable from any point v1 E JJ. There ofv consists set }J(v1) 1 the union of the sets V;(v1) for v1 E JJ. An antecedent JJ(V;) of set dent antece an arly, Simil able. reach is v of points v1 from which 1 able. V; is reach a set V; is a set of points v1 from which some point v, of ,b): 1977a k, (Silja ing follow Now, we can state the X U Y, R) is inputDefmidon 3.1. A system a; with a digraph GJ) = (U U reachable if X is a reachable set of U. The "directional dual" of Definition 1 is
X U Y, R) is outputDefinition 3.2. A system a; with a digraph 6Jl = (U U reachable if X is an antecedent set of Y. different sets, we By imitating Definitions 3.1 and 3.2, but otherwise using can formulate the following: ' ~ , X U Y, R) is inputDefinition 3.3. A system a; with a digraph 6j) = (U U antecedent set of Y. output-reachable if Y is a reachable set of U and U is an every point of a In Definitions 3.1-3, we ignore the trivial fact that re that a reachable digraph belongs to its reachable set. We also do not requi ermore, from the Furth s. point able reach set includes all corresponding is both input- and above definitions we can conclude that if a system but the converse is e, chabl ut-rea -outp output-reachable, then it is also input not true in general. matrix P = (p 11 ) We consider reachability of~ in terms of the p X p path d as define is and R) (V, which corresponds to the digraph 6Jl = Pu
=
1, { O,
there is a path from v1 to v~o there is no path from v1 to v,.
(3.6)
ISO
Synthesis: Decentralized Control
In this definition of P, the trivial paths of zero length are excluded (the length of a path being the number of lines in the path). To determine the path matrix for a given digraph, we need the following result which was obtained by Festinger, Schachter, and Back (1950):
Theorem 3.1. Let M = (mu) be the p X p interconnection matrix co"esponding to a digraph GD = (V,R), and let N = (nu) be the p Xp matrix such that N = M 11, where dE {1,2, ... ,p}. Then nu is the total number of distinct sequences (vi, ... ), ... , ( · · · , v1) of l~gth din GD. Proof. We can prove this theorem by induction, following Berztiss (1975). We show that the theorem is true ford= 1, and then show that it is true ford+ 1 whenever it is true for d. Then the theorem is true for any d. For d = 1, the theorem is actually the definition of the matrix M. Ford+ 1, m1kn~g = n~g if (vt.v1) is a line, and m~~cn~g = 0 if (vt.v 1) is not a line. The total number of sequences of length d + 1 having the form (vJo ••• ), ••• , ( • • ·, vk), (vk, v1) is equal to l:l- 1 m~~cnkft which is the ijth element of M 11+1• This proves Theorem 3.1.
The path matrix is calculated using the following:
Corollary 3.1. Let P = (pu) be a path matrix ofGD = (V,R), and Q = (qu) be a matrix defined as
Q = M + M 2 + ··· + M'. Then Pu
(3.7)
= 1 if and only if q!l + 0.
Once reachability is formulated in terms of the path matrix P, we can calculate P from a given M to determine the input and output reachability of~-
Let us note that M 11 can be written as
(3.8)
where again dE {1,2, ... ,p}. The matrix Q of (3.7) has the form
(3.9)
where
151
Reachability, Vulnerability, Condensations
= E + E 2 + · · · + E', 1 L0 = (I+ E + · · · + Er )L, ~ = F(l + E + · · · + Ert), 2 H 0 = F(l + E + · · · + Er )L. E0
(3.10)
We arrive immediately at the following (Siljak, 1977a,b): Theorem 3.2. A system S with an input-output connection matrix M defined 0 in (3.2) is input-reachable if and only if the matrix L o/(3.10) has no zero rows, it is output-reachable if and only if the matrix~ of(3.10) has no zero columns, 0 and it is input-output-reachable if and only if H o/(3.10) has neither zero rows nor zero columns. Proof. By constructing the path matrix P using Q of (3.9) and Corollary 3.1, the proof of Theorem 3.2 is automatic. To illustrate the application of Theorem 3.2, let us consider a system S with the digraph shown in Figure 3.1. The interconnection matrix M of (3.2) is given as Xt
M=
x2
0 1 0 0 0 0
XJ
X.
u y
1 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 ~ .1 0 0 0 o o' o ·o 0 0 1 0 0 0
Xt
x2 XJ
(3.11)
,
X.
u y
and the corresponding path matrix Pis calculated via the matrix Q of (3.9) as
X1 X2
P=
1 1 0 0 0 0
1 1 0 0 0 0
XJ
X.
1 1 1 1 0 1
1 1 1 0 0 l
u y 1 1 0 0 0 0
0 0 0 0 0 0
Xt
x2 XJ
(3.12)
X4 u y
By applying Theorem 3.2 to the matrix P of (3.12), we conclude that the
Synthesis: Decentralized Control
152
FIGURE3.1. Input- and output-unreachable system.
system 5i is neither input- nor output-reachable. Therefore, it cannot be input-output-reachable, which is confirmed by P. If we interchange the input and output in the digraph of Figure 3.1 to get the digraph shown in Figure 3.2, then we obtain a system which is both input- and output-reachable. The matrix M corresponding to the digraph of Figure 3.2 is
M=
Xt X2
X3, X4
0 1 1 0 0 0 0 0 0 0 0
0 0
0
u y
1 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0
Xt X2 X3
(3.13)
. X4
u y
which produces the path matrix Xi X2
X3
1 1 1 1 1 1 0 1 0 P= 0 0 1 0 0 0
X..
u y
1 1 0 1 1 0
x2
1 1 0
X3
0 1 0 0 0 0 0
Xt
X4
u y
(3.14)
Reachability, Vulnerability, Condensations
153
FIGURE 3.2. Input- and output-reachable system.
confirming the statement about the digraph of Figure 3.2. Since ~ is both input- and output-reachable, it is input-output-reachable, which is also confirmed by Pin (3.14) and verified by inspection of Figure 3.2. The computati on of the path matrix by generating powers of the interconne ction matrix, as performed above, is not a numerically attractive procedure for verifying reachability properties of the corresponding dynamic system. There are numerous algorithms developed to avoid various numerical difficulties in identifying the paths of a digraph, which started with the well-known Boolean representation algorithm of Warshall (1962) and culminated recently in tho depth-first search method of Tarjan (1972). A survey of these algorithms is g{ven by Bowie (1976}. Now, we tum our attention to partitions and condensations of digraphs. For either conceptual or numerical reasons, large-scale dynamic models in ecology, economics, and engineering may be considered as dynamic systems partitioned into interconne cted subsystems. In order to gain insight into the structure of such models we propose to investigate their condensation digraphs. These digraphs are obtained from the original ones by replacing the subgraphs corresponding to the subsystems by points, and joining the new points by lines which represent the interconnections among the subsystems. Let us consider again a system ~ described by the equations
i = f(t, x, u), y
= g(t,x),
(3.1)
where x(t} E ~ is the state, u(t) E ~ is the input, and y(t) E ~ is the
154
Synthesis: Decentralized Control
output of :!i. We assume that :!; is decomposed into s interconnected subsystems :;;, described by the equations
x, = ./;(t,;:c,u), y 1 = g1(t,x),
i
=
1, 2, ... , s,
(3.15)
where x 1(t) E ~ is the state, u1(t) E ~ is the input, and y; E ~, is the output of :!i;. We have ~-~X~X···X~, ~~~~
=
~X~X···X~,
~
=
~4 X~ X··· X~',
(3.16)
so that _ (XtT, XzT,
X -
T\T
••• , X, I ,
U = (u{', uf, •• •, uT)T, y
=
(3.17)
x 1) and (x1 ,y1) of 6j)*. In this way, the condensation 6j)* represents (uniquely) the structure of the composite system~ described by (3.15) with points of "D* standing for the subsystems and the lines of 6j)* standing for the interconnections among them. Another way to represent a partition of ~ is to use interconnection matrices in the same was as they were used to describe the original system :!i. Rewriting the matrix M of (3.2), but using different submatrices, we can define the p• X p• matrix
M* =
E* !
[r
OJ
L* 0 0, · 0 0
(3.18)
Reachability, Vulnerability, Condensations
155
which we associate with the condensation Gj)* in an obvious way. In (3.18), E*, L*, F* are s X s, s X r, q X s matrices, respectively, and p* == s + r + q. Now, input and output reachability of the condensation Gj)* can be determined by applying Theorem 3.2 to the matrix M* of (3.18). There are many different ways in which a dynamic system ~ and the corresponding digraph Gj) may be partitioned. In the pure theory of structures (Harary, Norman , and Cartwright, 1965), it is common to partition a digraph into its strong components so that each point of the condensation ·Gj)* corresponds to one-and only one strong compon ent of 6j).-a strong component of a digraph Gj) being a subgraph of Gj) in which every two points are mutually reachable. To be able to conclude input and output reachability of such a special condensation, let us denote by which is Gj), = (X, R,) the state-truncation of the digraph Gj) = (V, R), points output the and U points input the all g removin obtained from Gj) by GD; By Y. and U of points the to ed connect R in lines Y as well as all the the to nds correspo which Gj)* of ation condens d we denote the truncate truncation GD,. Now, we can prove the following: Theorem 3.3. Let the condensation GD; be constructed with respect to the strong components ojGD,. Then the digraph Gj) is input- (output-) reachable if and only if the condensation Gj)* is input- (output-) reachable. Proof. We prove only the input-reachability part of the theorem, since the output-reachability part is its directional dual. If Gj)* is input-reachable, then X* is a reachable set of U* and there is a path to each point of X* from a point of U*. Since GD; is a condensation with respect to strong components of GD, it follows from the reac.liabj.lity 9f components that there is a path to each point of X from a point of U. That is, X is a reachable set of U, and then X* is Gj) is input-reachable. Conversely, if Gj)* is not input-reachable, not a reachable set of U*, and there are points of X* that cannot be reached by any point of U*. Obviously, by the definition of condensation, those points of X that correspond to the unreachable points of X* cannot be reached by any point of U, and Gj) is not input-reachable. This proves Theorem 3.3.
With an abuse of Definitions 3.1 and 3.2, in the above proof we referred to input (output) reachability of the digraphs Gj) and GD*. This was done to avoid ambiguity arising from the fact that poth the digraph Gj) and its condensation Gj)* are related to the same dynamic system ~- The change in terminology should create no confusion, since input (output) reachability is defined unambiguously in terms of digraphs. Still another abuse of notation is committed in (3.17), where it would be appropriate to use starred notation on all components x 1, u1, y 1 of the vectors x, u, y, since they
156
Synthesis: Decentralized Control
represent a partition and condensation rather than the individual states. inputs, and outputs of the system ai. 'Ibis again should not pose any difficulty, since in the rest of this chapter, if not obvious or indicated to the contrary, all components of state, input, and output vectors correspond to partitions and condensations. Algorithms of Purdom (1970), Munro (1971), and Kevkorian (1975) ~an be used to compute the strongly connected components of the digraph 6D.,. Then the input and output reachability of a system a; can be determined by means of the appropriate condensation and Theorem 3.3. Partitions of mathematical models of dynamic processes in ecology, economics, and engineering are most often guided by the special structural properties of the models. Therefore, in general, partitions of the digraphs are not performed with respect to their strong components, and there is no reason why the corresponding condensations have to be constituted that way. Furthermore, it is also possible to alter our notion of subsystems and allow "overlapping" of subspaces ~in (3.16) so that, for example, the state vector x1c of the subsystem ai1c is formed pairwise as x1c = (xf,x})T, where x 1 and x1 are defined in (3.17). For purposes of control and estimation, it is desirable to obtain still another partition of a; into input- and outputreachable subsystems ai;; that is, it is advantageous to partition the corresponding digraph 6j) into input- and output-reachable components. After a short historical note, this topic is considered next in the context of Kalman's (1963) canonical structure of linear systems. E. F. Moore (1956) introduced the concept of reachability in the context of finite-state systems by formulating the notion of a strongly connected sequential machine. He wrote:
6f):
A machine~ will be said to be strongly connected if for any ordered pair (q1,q ) 1 of states of §, there exists a sequence of inputs which will take the machine from the state q1 to state qft
and thus laid a foundation for the concept of controllability of dynamic systems introduced later by Kalman (1963). While binary interconnection relationships in sequential machines lead naturally to connectedness and reachability considerations of the corresponding directed graphs, the general matrices of linear dynamic systems imposed a stronger rank condition for controllability which includes that of reachability. The same is true for Kalman's notion of observability, which is the dynamic system analog of Moore's notion of distinguishability of sequential machines. Although controllability and observability conditions are indispensable in certain basic problems of control ·and estimation of dynamic systems (Kalman, Falb, and Arbib, 1969), there are at least three good reasons why they may be replaced by the weaker reachability tests of the interconnec-
·~~
~
157
Reachability, Vulnerability, Condensations
tion matrices as proposed above. First, the conceptual significance of the difference between the controllability-observability rank condition and its reachability counterpart can be disputed on physical grounds. Cases where reachability tests succeed and the rank conditions fail can be .dismissed as unrealistic, since a slight perturbation of system parameters can restore the controllability and observability properties of the system. For example, consider a linear constant system
x =Ax.+ bu,
(3.19)
where
b=Cl
c=[~J.
(3.20)
It is a well-known fact (Kalman, 1963) that the system (3.19) is both uncontrollable and unobservable, because both matrices
(3.21)
have rank less than two. Now, from the digraph of the system shown in Figure 3.3, it can be concluded by inspection that both states x1 and x2 are directly accessible to the input u and the output y. It is also obvious that if the elements of the triple {A, b, c) are slightly perturbed, it is possible to recover both controllability ahcl pbse:rvability of the system (3.19). Therefore, looking from the side of the rank conditions for controllability and
u
y
FIGURE 3.3. System structure for the example.
Synthesis: Decentralized Control
158
observability, we cannot tell whether the failure of the system to satisfy the conditions is for structural reasons, or is due to a special choice of system parameters. To avoid this ambiguity in large-scale systems whc:re elimination of trivial cases can be quite costly, it is advantageous to use the ''pure" structural considerations of input and output reachability. The second reason for preferring reachability considerations is that the rank conditions are ill-posed numerical problems as compared to purely structural manipulations using binary interconnection matrices. For this reason, Lin (1974), Kevkorian (1975), and Shields and Pierson (1976) recommended a return to connectedness investigations via a new term of "structural controllability''. It is in the spirit of Moore's original investigations that we formulated the concepts of input and output reachability above and opened the possibility of using effective computing schemes (Bowie, 1976) from the pure theory of structures in determining the input and output reachability of dynamic systems. Finally, the· third reason for considering the reachability conditions more appeafuig is that they apply to linear and nonlinear systems alike. This provided a real possibility of formulating for the first time the pure canonical structure of dynamic systems (Siljak, 1977b). By imitating the canonical structure of linear systems defined by Kalman (1963) in the context of controllability and observability, but otherwise using input and output reachability, we present next the canonical structure of nonlinear dynamic systems. Let us consider again the dynamic system ai described by (3.1), together with its interconnection matrix M of (3.2). By permutation of rows and columns of M, it can be transformed into a matrix M which has the following form: iio i.to
En 0
M=
E31
0 0
Fi
i,. x....
u
y
Lt 0 0 0 E22 0 L3 E32 E33 E34 E42 0 E44 0 0 0 0 0 0 0 Fi 0 0 0 0
E12
0 0
0 0
i., X. i,.
x.... u y
The partition of the state vector i of the transformed system £, •Tf •T ·T XAt X• -_ (iT;,,X.to,Xia, into four components, which can be identified from the four subsystems with the following properties:
(3.22)
(3.23)
M in (3.22), represents
,,
Reachability, Vulnerability, Condensations
~... ~.m. ~ie•
~...
159
input-reachable and output-reachable, i,nput-unreachable and output-reachable, input-reachable and output-unreachable, input-unreachable and output-unreachable.
By using condensation, we can represent each component of x in (3.23) as a point of the condensation ®* as shown in Figure 3.4. This digraph represents the pure canonical structure of the dynamic system lii. Now, we turn our attention to the· decentralized systems, which are the main subject of this chapter. Decentralization is an effective way to cope with complexity and gross changes in the interactions of large-scale dynamic systems; thus decentralized systems, as well as control and estimation schemes which exploit decentralization in one form or another, have attracted rapidly increasing interest (see Sandell, Varaiya, and Athans, 1975). Nevertheless, it is only recently (Siljak and Vukcevic, l976c) that attempts have been made to formulate an appropriate definition of a decentralized system and produce effective decentralization schemes for linear dynamic systems. The definitions and reachability properties of decentralized systems are outlined here; a presentation of the decentralization techniques is postponed till the next section. Whether a system is decentralized or not is essentially a matter of its structure, and therefore decentrality of dynamic systems can be effectively defined in terms of digraphs. Intuitively, a system lii is an input-decentralized system if each subsystem lii1 has its own input u;. In order to put this intuitive notion of input decentralization into digraph-theoretic terms, we recall that ;
.
FIGURE 3.4. Canonical structure.
Synthesis: Decentralized Control
160
the point basis B of a digraph 6j) = (V, R) is a minimal subset of V from which all points of 6j) are reachable. Furthermore, a 1-basis is a minimal collection B1 of mutually nonadjacent points in 6j) such that every point of 6j) is either in B1 or adjacent to a point of BJ. To avoid trivial cases which are of no interest to us, we assume that no points in the subsets U and Y of V = U U X U Y are disconnected, that is, od(uk) =I= 0 for all k = 1, 2, ... , m, and id(yk) =I= 0 for all k = 1, 2, ... , l, where "od" and "id" stand for outdegree and indegree (Harary, Norman, and Cartwright, 1965). To be able to use the 1-basis in formulating definitions of decentralized systems, we form the input-truncated condensation "D: = (U* U X*, R!) from the condensation 6j)* of 6j) by removing all the output points Y; of Y*. Now we state Definition 3.4. A system §with an input-truncated condensation "Du* = (U* U X* , R!) is input-decentralized if and only if r = s and the set U* is a 1basis of "D: such that od(u;) = 1 for all i = 1, 2, ... , r. To provide a definition of an output-decentralized system, we need the notion of point contrabasis C of 6j) = (V, R), which is a minimal subset of V such that V is a set of points from which some point of C is reachable (Harary, Norman, and Cartwright, 1965). We also recall that 1-contrabasis of 6j) is a minimal collection C1 of mutually nonadjacent points such that every point of 6j) is either in C1 or adjacent to a point of C1• By "D; = (X* U Y*, R;) we denote the output-truncated condensation of "D. Then we have Definition 3.5. A system ~ with an output-truncated condensation "D; = (X* U Y*, R;) is output-decentralized if and only if q = s and the set Y is a 1-contrabasis of "Dy* such that id(y;) = 1 for all i = 1, 2, ... , q. It is simple to combine Definitions 3.4 and 3.5 and formulate a definition of input-output-decentralized systems. of an input-centralize d system is shown in Figure 3.5(a). The digraph The input-decentral ized version of the same system is given in Figure 3.5(b). Outputs are omitted in the digraphs, since they have no effect on the input-decentral ization property. Output-centrali zed and -decentralized systems can be represented by directional duals of the digraphs shown in Figure 3.5. A convenient way to characterize input- and output-decentra lized systems is to use the interconnection matrix M* defined in (3.18). Then a system~ with a condensation 6j)* is input-decentral ized if and only if in M* of (3.18), L* = I., where I. is the s x s identity matrix. Similarly, ~ is an output-decentra lized system if and only if F* = f..
6j):
Reachability, Vulnerability, Condensations
161
-·
g),:
u
FIGURE 3.5. (a) Input-centralized system. (b) Input-Qecentralized system.
the By using the methods of Kevkorian (1975), it is possible to permute 6j)* sation conden riate approp an choose and § system ar states of a nonline is in of the corresponding digraph 6j) so that the new transformed system 3.2, Section in shown M both. or form alized decentr either input- or outputand 1976c) ic, Vukcev and (Siljak better do to e in linear systems it is possibl use similarity transformations to produce input- and output-decentralized nary systems. Such decentralization procedures may be required prelimi ic dynam linear of ion estimat and ation, optimiz steps in the stabilization, this of rest the in red conside are which k, systems by decentralized feedbac chapter. F* By applying Theorem 3.2 to M* of (3.18), we conclude that if L* = eachautput-r input-o is § system lized = I., then the input-output-decentra the ble whenever each subsystem §/is ~npul-output-reachable regardless of trivial hat somew This "D*. of E* matrix n form of the state interconnectio result leads to an interesting conclusion: in decentralized systems inputn output reachability is invariant under perturbations of the interconnectio ic dynam of aspect bility vulnera the is This structure among the subsystems. systems, which is considered next. s, We argued in the previous chapter that large-scale dynamic system either nly commo quite ems, subsyst which are composed of interconnected are by design or fault do not stay "in one piece" during operation. They are ems subsyst of groups subject to structural perturbations whereby ictable unpred an in other disconnected from and again connected to each of way. In Chapter 1 and in the rest of this chapter, we consider the effects . systems c dynami of structural perturbations on the stability and optimality ating investig by In the rest of this section, we complement these results reachability under structural perturbations. In terms of digraphs (Harary, Norman, and Cartwright, 1965), disconand necting subsystems from each other is equivalent to "line removals",
162
Synthesis: Decentralized Control
disconnecting subsystems from the overall system is equivalent to ••point removals". These perturbations can be conveniently described by intercon. nection matrices (Siljak, 1975). For this purpose, the s X s fundamental interconnection matrix E* = (eij) is associated with a condensation iij)* = (U* u x• U Y*,R*) of :ii in (3.15) as follows:
-·
ey
= { 0,1,
(x1 , x,) E R.*, (x1, x,) f1; R*.
(3.24)
That is, eif = 1 if x1 occurs inf,(t,x,u), and eif = 0 if x1 does not occur in j,(t, x, u). Now, a structural perturbation is represented by the removal of a line (or a number of lines) of the condensation iiD* = (U* U X* U Y*, R.*) between points of x•. That results in a spanning subgraph GJ)* of iiD*, that is, a subgraph with the same set of points as 6i5*. All spanning subgraphs of iiD* obtained this way can be represented uniquely by an interconnection matrix E* which is obtained from £* as follows: eij = 0 in £*implies eif = 0 in E* for all i,j = 1, 2, ... , s; and a removal of a line (xi> x 1) of iiD* implies that = 1 in E* is replaced by eij = 0 in E*. The fact that an interconnection matrix E* is generated in this way by the fundamental interconnection matrix £* is denoted by E* E E*. Finally, without loss of generality, a point removal can be treated in the same way, as a special case of line removals. If a kth point of Gjj* is removed, then e~ = e~ = 0 for all i,j = 1, 2, ... , s. Structural perturbations are illustrated in Figure 3.6. The top digraph is the output truncation which represents the basic structure of the composite system :ii when all the inputs are removed, and which corresponds to the fundamental interconnection matrices
e;
6D;
£*=
0 0oo, 1] [ 1 0
F*
= [0 1 1].
(3.25)
1 0
The structural perturbations formed by line removals are represented by the digraphs GDy* below the digraph The digraph at the bottom of Figure 3.6 corresponds to E* = 0 and thus to the total disconnection of states x 1 , x2 , x 3• Inputs can be added to iiD; in an obvious way if input reachability is investigated in the presence of structural perturbations. On the basis of the above considerations, we introduce the following:
6D;.
Definition 3.6. A system ~ with a condensation 6i5* = (U* U X* U Y* , R.*) is connectively input- (output-) reachable if and .only if it is input- (output-) reachable for all interconnection matrices E* E E*.
163
t> 0
0
0> '·- ,
.
.
FIGURE 3.6. Structural perturbations.
Definition 3.6 parallels those of connective stability (Siljak, 1975) and connective suboptimality (Siljak and Sundareshan, 1976a, b). In the case of control systems, the three definitions complement each other in an important way discussed in the latter part of this chapter. Now we prove the following:
6D: = (U* x• ,R.:) is connectively input-reachable if and only if the set u• is a 1-basis
Theorem 3.4. A system ~ with an input-truncated condensation U
of
6fl: for E* =
0.
The "if'' part for E* = 0 follows directly from the definition of the 1-basis. ForE* =F 0, we get the corresponding digraph by adding lines to the one that corresponds toE* = 0. But it is obvious that there are no lines in any digraph whose addition can destroy its reachability property. For if (xJ>xt) is any line of GD:, any path in GD:- (x1 ,x1) is also in GD:. Proof.
T' ..
164
Synthesis: Decentralized Control
On the other hand, if U* is not a 1-basis forE* = 0, then there must be points of X* that are not reachable from U* for all E* E E*, and~ is not. connectively input-reachable. This proves the ..only if" part and thus Theor em 3.4. By the principle of duality of digraphs (Hara ry, 1969), it is possible to obtain an analog to Theor em 3.4 for connective outpu t reachability of~: Theorem 3.5. A system ~ with an output truncated conden = (X* U Y* ,R.;) is conrrectively output-reachable if and only if thesation set Y* is a 1-contrabasis oj6'D; forE* = 0.
iiD;
Proof. The theorem is an obvious dual of Theor em 3.4. The result of Theorems 3.4 and 3.5 is intuitively clear: If we want reachability to be preserved under structural pertur bations, we shoul d check the ..worst case", that of E* = 0. It is interesting to compare this statem ent with the connective stability of competitive dynamic systems considered in Chapt er 2, where the worst case was E* = E". That is, stability is improved by reducing the numb er of interc onnections amon g the subsystems, but reachability may be destroyed. Anoth er straightforward but nevertheless impor tant conclu sion coming from Theorems 3.4 and 3.5 is that for reachability to be invulnerable to struct ural pertur bation s, each subsystem shoul d have its own input and outpu t or share them with other subsystems. In other words , if a subsystem is reacha ble from the input only throug h anoth er subsystem, then it is liable to becom e input-unreachable due to structural perturbation s. Similarly, it can becom e outpu t-unre achab le if its state is represented at the outpu t only by an adjace nt subsystem. If a system is under struct ural perturbations, then to preserve reachability, each subsystem when isolated should be input-output-reachable. This is a symmetric situat ion to that of the connective stability of competitive dynamic systems, where each subsystem was stable when isolated. The symmetry is not complete, since reachability of each isolated subsystem is a necessary and sufficient condi tion for connective reachability, while stability of each isolated subsy stem was only a necessary condi tion for connective stability of large-scale systems. In Figur e 3.6, the digraph 6D; represents an outpu t-reac hable system, and so do the three digraphs immediately below The remaining four digraphs describe outpu t-unre achab le systems. By the principle of duality, input reachability under struct ural pertur bation s can be visualized using the same Figur e 3.6. Finally, one can generalize the above connective-reacha bility concept by defining partial connective reachability with respect to a pair of interconnection matrices ('E*, £• ), where a ·fixed matrix £• E 1!* takes the role of
6D;.
165
Decentralization
E* = 0. Again, it is intuitively clear that if U* is a 1-basis for :£•, this being the worst case, then ~ is partially connectively input-state-reachable for all E* E E*- E*, where by the difference£*- E* we mean all interconnection matrices generated by£* which have the unit elements corresponding w t•. It is of interest to observe that input- and output-decentralized systems are invulnerable with respect to structural perturbations. That is, we have the following corollary to the Theorems 4 and 5: CoroUary 3.2. A system ~ is connectively input- (output-) reachable input- (output-) decentralized.
if it is
This corollary announces a strong advantage of decentralized systems over centralized ones, where common inputs are shared by the subsystems. This fact has not been stated yet in the open literature on control systems, but is probably appreciated intuitively by the authors of the many recent new results on decentralized control (Sandell, Varaiya, and Athans, 1975). Further use of digraphs in studying vulnerability of dynamic systems is a wide open field. One way to approach problems that arise in this context, is to apply the results obtained in vulnerability studies of communication nets (Boesch and Thomas, 1970). These results, however, would have to be modified in an essential way in order to reflect the inherent structural properties of dynamic systems as defined above. Another possible aid in reliability studies of large control systems and breakdown phenomena, is the fault tree analysis (see Barlow, Fussell, and Singpurwalla, 1975). This other approach could shed some.Iight on the intricate interplay between the components reliability and the r·eliability of the overall dynamic system. 3.2. DECENTRALIZATION Despite the effective solution of many important problems in system theory by decentralized control, our knowledge of how to decentralize a dynamic system has remained superficial. Recently, an effective decentral.ization procedure was proposed (Siljak and Vukcevic, 1976a, c), which is a preliminary step in a multilevel control and estimation scheme for largescale linear systems. The procedure yields a number of subsystems that have either decentralized inputs, or decentralized outputs, or both. The process of decentralization is carried out on the subsystem level, and it does not require a test for controllability and observability of the overall system. Let us consider a system ~ described by the linear differential equation i
= Az + Bu,
(3.26)
Synthesis: Decentralized Control
166
where z(t) E ~ is the state of the system, u(t) E ~ is the input to the system, and A and Jj are constant n X n and n X s matrices. We decompose~ into r dynamic elements
,
I AJIIlz9 + Jjpu, + q-1
ip = Apzp
p
=
1, 2, ... , r,
(3.27)
q+p
where Zp (t) E ~and z = (z[, zf, ... , z!Y. n = I;-1 1,. such that all pairs (AP,JjP) are controllable, that is, the lp x Ips matrix Jj [JjP 1I A PP
Jj I PPI
1 A2 1
...
I
1
ArljjP] P
(3.28)
has rank equal to lp (for this well-known result see Chen, 1970}. By using the linear transforma tion proposed by Luenberge r (1967} (see also Chen, 1970}, we can write the elements (3.27} as r
-
•
Zp =.Ap!p
-
-
l: .AJIIlzq + Bpu, + q=l
p = 1, 2, ... , r,
(3.29)
q.;.p
such that the matrices
and b!
JJP have the following form:
E ~.-, lp = If-1 np;. The linear nonsingular transformation (3.31}
is defined by
QP = where bf E %-, i (3.29} we have
=
[
~ I • ] • ~ I • • b(, ... ,.A';1- b(; ... ;bf, ... ,A';'- bf,
(3.32}
1, 2, ... , s, are the columns of the matrix JjP' Then in
Due to (3.30), we can decompose the state Zp of each transforme d element (3.29) as zp = (z'ft,z}l., ... ,z~)T, zp;(t) E ~.-. so that with each zp; we
167
Decentralization
associate the vector "bi and the component U; E ~ of the input vector u E '1Jt'. Now, we group the 'lp~'s of each of p elements which correspond to the same input u;, and form the ith subsystem with the state x1(t) E ~. . -T)T and n; = ""' -T •.. ,Zrt ""''-1 np~. This process of groupmg zu,z'Jj, sueh that X; = (-T intercons of yields finally the representation of the system :ii as composed nected subsystems :ii1 described by equations 8
i 1 = A 1x 1 + ~ Aiixl J=l
+ b1u,
i = 1, 2, ... ' s.
(3.34)
•
)-Jol
To compute the matrices A 1, Ag, and the vector b1 from A;,, Apq, B,. let us denote by z = (lf, zf. ... , z'!Y and x == (xf, xf, ... , xiY the state vectors of the overall systems corresponding to (3.29) and (3.34), respectively. Then the grouping process described above is carried out by the nonsingular linear transformation X=
(3.35)
PZ,
where the permutation matrix P has the block form P = (P{, Pl, ... , P.T)T, and the ith block P1 E '1JI!'fx. is defined by
P;
~ [.:.
0 0 0
----------
...
lu
0
0 0 ... 121 0
0 0
. .. 0
0
.. . . . . ........ .... 0~
_...
lrt
~
(r- 1)s + 1
i- 1
.:. R·(336)
..._.... ._ s-i
where 4; is the np; X np; identity matrix, and the zero matrices in (3.36) have the appropriate dimensions. Now we write (3.29) as
z = Az+ Bu,
(3.37)
where
A=
[A,~:
1;2
Ari
A,2
A2
....
~-] A2r A,
'
H-[!J
(3.38)
r '
Synthesis: Decentralized Control
168
and apply (3.35) to (3.37), to get
x =Ax+ Bu,
(3.39)
with B
=
(3.40)
PB,
and
b:] br
0 ]
b;
b, '
=
[ i~
'
(3.41)
where b1 E ~· and n = ~t- 1 n1• Finally, we identify the overall system (3.39) ass interconnected input-decentralized subsystems described by the equations (3.34). It is important to note that all steps but the last in the input-decentralization scheme are performed on the subsystem level. The last step, which involves the transformation (3.35) and is performed on the overall system level, consists of regrouping the components of the state vector and does not require the matrix inversion. That is, for the permutation matrix P in (3.35), p-l = PT. For the output decentralization, we consider the lin~ system§ as
i y
= Az + Bu, = Cz,
(3.42)
where the output y(t) E ~ and C is an m X n constant matrix. By the output-decentralization scheme, we get the system (3.42) as m interconnected subsysteiill) m
~ Aux1 + B;u, x1 = A 1x 1 + }=! J+i
i
= 1, 2, ... , m,
(3.43)
where y 1(t) E '8t is a scalar output of the ith subsystem, and c1 is a constant n,-vector. To obtain (3.43), we form the dual of (3.42) as
i y
= Rz + (;Tu, = BTz,
(3.44)
169
J)ecentralization
where for convenience we have kept the same notation z(t), u(t), y(t) but recall that now z(t) E ~. u(t) E ~. y(t) E tBt'. Applying the input-decentralization scheme to (3.44), we get
i,
..
~ AJx1 + c,u" = ATx, + J-1 J ..i
i
=
1, 2, ... , m,
(3.45)
where c1(t) E t!il!", and the B/s are s X n, matrices obtained from b in the course of the input decentralization of (3.44). It remains to note that (3.45) is the dual of (3.43), and conclude that (3.43) is the output-decentralization version of (3.42). To illustrate the decentralization procedure, let us consider the system
i=
1 3 4 0 5 2 2 4 0 2 0 2 2 2 3
5 6
4 1 0 6 z+ 1 6 0
1 2 0 4 2 0 u, 0 2
(3.46)
3
which is of the form (3.26) with n = 5, s = 2. The system (3.46) is decomposed into two (r = 2) dynamic elements
(3.47)
so that /1 = 2, /2 = 3. The labeled digraph of the decomposed system (3.47) is shown in Figure 3.7. Since both pairs (At. ..81) and (..42 , ..82 ) in (3.47) are controllable, we can use the transformation (3.31):
6 OJ
10 2 7 3 to get (3.42) as
(3.48)
Synthesis: Decentralized Control
170
.
A,
u
FIGURE 3.7. Input-centralized system.
[~u] Zt2
50 86.50 22.5 0J[z 2t] + [10 01][u"'], = [• 4J[ zu] + [11. 2 :Zn 1.25 14.75 2.75 0 5 Zt2
] + [~:~~ ~:~!][:zll 21] 2][: ~~~~ ~ 12 [zz:....: ] = [001 ~~~ 0.18 0.09 -9.8 2 -6.3 6 Zn +
3 49 ( · )
G~][:J.
) with The matrices IJ, and 1J2 have the form {3.30
h: =
1,
b~ = 1,
~
= 1
(3.50)
Therefore, n, = nu + n21 = 3, and nu = 1, n, 2 = 1, n21 = 2, n22 = 1. lization scheme produces two n2 = n, 2 + n22 = 2, and the input-decentra ciated with the inpu t components subsystems of second and third order asso s are formed by regrouping the u, and u2, respectively. The two subsystem components of
171
Stabilization
:!i = (:r?;,:r&l,
?2 =
(:rft,zll)r
(3.51)
in (3.49) using the permutation matrices P1 and P2 deftned by (3.36) as
pi =
[~I ~ ~~ ~
p2 =
0 [0
~2
J ~I J
0 0 0 0 In ;
~2
= 1,
= 1,
/21 = [ ~ ~} 122
=
(
3 52 . )
1.
The two subsystems have the states
(3.53) and have the representation (3.34), which is
x. = .
x2
=
[o.~5 0.18
[ 4 11.50 86.50] 22.50] X2 + [1] 1 Ut. -4.09 Xt + 8.91 -0.82 0 0 0.36 3.27 8.36 1
2.75 0.18 -6.36
[ 5
J + [0.180 X
2
1.25 14.75 0 -9.82
J+ Xt
(3.54)
[1] 2 1 U •
The two interconnect ed subsystems (3.54) have an input-decent ralized digraph as shown in Figure 3.8. Now, several remarks concerning the decentralization scheme are in order. Although to perform the decentralization procedure we do not need controllability and observability ~tests of the overall systems, the procedure implicitly depends on the system c6ntrdllability and observability. This fact is particularly important in the next section when we propose to stabilize a linear composite system by decentralized control after testing for controllability of the subsystems only. To avoid futile attempts to stabilize a system which is controllable piece by piece but is uncontrollable as a whole, we should test for input reachability of the system first, using the results of Section 3.1. This way, we still avoid testing for controllability of the overall system, which is an ill-posed numerical problem, especially when too many variables are involved, but differs from the input-reacha bility property only in "physically unrealistic" cases. 3.3. STABILIZA TION
Once a large-scale system is given in the input-decentralized form, either as a result of input decentralization or by being identified as such through physical considerations, we propose here to stabilize the system by a
Synthesis: Decentralized Control
172
.41
FIGURE 3.8. Input-decentralized system. multilevel control scheme based upon the decomposition-aggregation stability analysis presented in Chapter 2. In the scheme, local controllers are used to stabilize each subsystem when decoupled , while global controllers are applied to reduce the effect of interconne ctions among the subsystems. The local controllers provide a desired degree of stability for each subsystem separately , and can be designed by any of the classical techniques, such as pole shifting by state feedback (Chen, 1970), root-locus method (Thaler and Brown, 1960), parameter plane method (Mitrovic, 1959; Siljak, 1969), etc. After the subsystems are stabilized, an aggregate model is used to deduce stability of the whole system. To ~tabilize the system~ given as
•
:i1 = A1x1 + ~ Ayxi J-1
+ btUt. '
i
=
1, 2, ... ' s,
(3.34)
we apply the decentraliz ed multilevel control
Ut(t) = r4(t) + uf(t),
(3.55)
where ul(t) is the local control law chosen as
ul =
-k{x~>
(3.56)
-r· .
,,
;.t::-=.~
173
Stabilization
with a constant vector k, E ~"'.and uf(t) is the global control law chosen as (3.57) where ku E ~J are constant vectors. By substituting the control (3.55) into (3.34), we get the closed-loop system as
x1 =
(A,- b,kl)x, +
3
'
~
(Au- b,klf)x1 ,
j=l
=
i
1, 2,
0
..
,
s. (3.58)
Since each pair (A 1, b1) is controllable, a simple choice of k 1 can be always made to place the eigenvalues of A 1 - b, kl at any desired distinct locations 0, q -- 1, 2, ... ,n;- p, (-1 i _j . i -op+t . i ... , -a,i + o9 •••• , -a.,_, -Jw,. -a1i +-JWi, [n;/2]). Then each uncoupled subsystem p and 0
>
<
...•,· . ..· .,,,.'.·
:.:-~
·:.:
Synthesis: Decentralized Control
174
A1
wl. -a{ -of = diag {[. -w{
J
[-~ , · .. , -w~
J
1 _~ _, } w~ ,-a,._ t. ... ,-o;,. -ai
.
(
3.63)
unov function v1: ~-+ ~+• For the system (3.62) we choose the Liap
(3.64) where (3.65)
AT 11, +/! A,= -G, and
IJ, = 8,1,,
(}, = 28,diag{a{,of, ... ,ai,a~,aJ.+~o ... ,af.,-1 }
(3.66)
and I, is the n1 X n1 identity In (3.66), 81 0 is an arbi trary cons tant (3.66) provides the exact tion func matrix. Tha t the chosen Liapunov n in Section 6.5. estimate of 'IIi defined by (3.60) is show lving the vector Liapunov funcinvo The aggregate comparison system tion v: '8l!' -+ ~~. (3.67)
>
(3.58): is obta ined for the transformed system
• "' (A 11 x,. = A, x, + J-1 ~
~
- r.r) b, "u xi>
i = 1, 2, ... ' s,
(3.68)
1 1 = k{7], and using the Liapunov where ..411 = r,- A 11 7j, S, = r,- b, k{ g the aggregation meth od pres ente d functions v1(x1) defined in (3.64). Usin · e model (f, as in Cha pter 2, we construct the aggregat (3.69) JVv,
ri
i=l ~ d;lwul,
j
= I, 2, ... , s,
i ..j
(3.139)
where the d;'s are positive numbers. Apparently, we can make the matrix w satisfy the conditions (3.139) if we can increase the diagonal elements w.u sufficiently while keeping the off-diagonal elements wu bounded. This is exactly the case with the class of systems under consideration. We notice that the diagonal elements (i = j ), W;; =-a-&;,
{3.140)
depend linearily on the adjustable parameter a. The off-diagonal elements
(i =I= j ), wu = ~u(a),
(3.141)
are bounded functions of a. To see this, we note that the elements a'-'afq of the matrices R/ 1 AuR1 either are zero for p < q due to (3.124), or are bounded for p :> q due to nonpositive powers of a. We have (3.142)
190
Synthesis: Decentralized Control
where the matrix Du = (djq) is defined by djq = aj. when p = q, and djq = 0 when p ¥= q. From (3.137) and (3.142), we define Du = 1;-1 Du 1; and conclude from (3.143) that the off-diagonal elements wu are bou nded in a. Therefore, for the selected class of dyna mic systems' we can always choose a sufficiently large param~ter a, and use local linear feedback control to stabilize the systems. Fro m (3.12 6), we see that by increasing the value of a, we move the subsystem eige nvalues away from the origin, thus increasing the degree of exponential stability of each subsystem. This, however, requires an increase of the loca l feedback gains in the course of stabilization. Let us illustrate the local stabilization procedure using the following example:
.X=
0 0 0 -2 -1 4 0 5 6
0 1 -1 0 0
2 3 2 0 -3
0 4
0 0 0 0 x+ 0 u. 0 0 -2 0
(3.144)
The eigenvalues of the system matrix A corresponding to (3.144) are A1 = 1.7244, A2 = 5.1042, A = -1.2633 3 , ~.s = -4.2 826 ±jl. 775 5, and the system (3.144) is unstable. The system (3.144) can be decomposed
XJ = [
= [
as
~ ~ ~ ]xi+[~ ~]X2 + [~]UJ,
-2 -1 Ji:2
(3.145)
-1
2 1
(3.146a)
1
~3 ~2 ]x2 + [~ ~ ~]x1 + [~]u2.
(3.146b)
The eigenvalues of the subsystem (3.14 6a) are moved from
AI = -1.3532, to the new locations
Ab = 0.1766 ±j1. 202 8
(3.147)
Stabilization
A} =
191
-al =
M =-a!=
-1,
-2,
M = -a!=
-3 (3.148)
by applying the local control (3.34) and
kt = (4, 10, 5).
{3.149)
Similarly, the eigenvalues of the subsystem (3.146b) are changed from
Ab =
-1
+ jl.4142
(3.150)
to A} =
M=
-af = -1,
-a~
=
-2
(3.151)
by applying the local control (3.34) and k{ = (-1, 1).
(3.152)
Referring to {3.129), we see that in (3.148) and {3.151), the parameter a= 1. We construct the transformation matrices Rh R 2, 1i, 1i for a > 1 as
~- [~ R2
=
0 0OJ, a
t,-
0 a2
[~ ~].
[~I
1 72= -1 A
[
I I]
-2 -3 , 4 9
(3.153)
~2].
The numbers wh -&2 are both set equal to one. Then, the aggregation matrix of (3.69) defined by (3.140) is given as {3.154) which for a = 1 takes the form
-1
Hi"= [ 12.2936 where -
~12 =
1/2
AM
('II A
-I
Al2
...,
12),
and A12, A 21 are specified in (3.146).
17.0011] -1 '
(3.155)
Synthesis: Decentralized Control
192
lities It is obvious that the matrix Win (3.155) does not satisfy the inequa (3.72). From (3.146) a:i:td (3.142), we find that
Do -
n
G
D, = [
~~
n
(3.156)
a = 25, we and for a> 15, we have ~12 ~ 32.55, ~21 ~ 18.98. Thus, for have the aggregate matrix
-=[
w
-25 32.55] 18.98 -25 '
(3.157)
is stable. The which satisfies the conditions (3.72) and the overall system are system corresponding eigenvalues of the overall closed-loop ;\2,3 = -25.95 99 ±j3.52 19, XI = -36.0364, (3.158) 4. j6.047 ± 13 -68.52 A...s = f.t and f. 2 For the chosen value of a = 25, we have the eigenvalue sets defined in (3.126) given as
e.,= {-aol ,-aaL -aaD = {-25,- 50,-7 5}, e2 = {-aor ,-aoD =
(3.159)
{-25,- 50}.
of (3.159) are The locations of the subsystem eigenvalues specified by f.t. ~ and achieved by the local state-variable feedback defined by (3.34)
k{ = (93748, 6874, 149),
kl =
(3.160)
(1247, 73).
use of local The gains in (3.160) are relatively high, which is due to the ng global applyi controllers only. The gains can be considerably reduced by of this ing controllers in the multilevel scheme outlined at the beginn section. ted in this Applications of the decentralized stabilization scheme presen er 6, and Chapt section to spacecraft control systems are outlined in er 7. Chapt in applications to interconnected power systems are given 3.4. ESTIM ATIO N which was The scheme for multilevel stabilization by state feedback ption that assum the upon presented in the preceding section was based
193
Estimation
each state of all the subsystems can be read out as outputs. In large dynamic systems, this assumption cannot be expected to hold even, if the subsystems are simple. Therefore, we should be able to build the state estimator whose task it is to use the knowledge of each subsystem (its inputoutput equations) and its actual input and output, and produce a good estimate of the unknown present state of the overall system. If a system is stabilized as "one piece", then standard design procedures can be used to build an asymptotic state estimator for "one shot" determination of the system state, as reviewed by Chen (1970). Such procedures require that the observability test be applied to the overall system, which for high-dimen~ sional systems can be a costly, complicated endeavor whose final outcome is unreliable due to errors in computations. The question then is: Can we build a state estimator for a large dynamic system in the spirit of decomposition principle by building low-order estimators for each subsystem separately? An affirmative answer to this question has been provided only recently (Siljak and VukCevic, 1977c, 1978), and a multilevel estimation scheme is available for state estimation of large-scale systems. One of the most pleasing facts about the scheme is that the stabilization of the error between' the real state and the estimated state is accomplished by the same decentralized control method developed in the preceding section. Let us consider the large-scale linear system
+
Az bu, y = Cz,
i =
(3.42)
where z E tire', u E C!Jt', y E C!Jt•. We assume that the system (3.42) is given in the output-decentralized form either as a product of the outputdecentralized scheme, or by being recognized as such directly from physical considerations. That is, we consider (3.42) as X;= A;x;
.. Auxi + B;u,
+~
J-1
j+i
y;
i
=
1, 2, ••• , m,
(3.43)
= clx;,
where each pair (A~o
en is observable (Chen, 1970), that is
rank[c1
:
ATcT: ···:
1 (AT)"'- cT]==n,·.
(3.161)
Comparing (3.161) with (3.78), we conclude that (A;, cT) is observable if and only if the pair (AT,c 1) is controllable. This fact is nee~ below. In order to estimate the states X;(t), we construct subsystem observers of the form
194
Synthesis: Decentralized Control
.
.
x1 = F,x1 + g1y 1 + l; Fgx1 + J-l
· I""'
.
l:
1-l
ggy1 + B,u,
i = 1, 2, ... , m, (3.1 62 )
I""'
where the matrices and vectors F,, Fg, g;, gg are to be determined so that (3.162) are the identity observers for the subsystems {3.43). For the error of estimation, W;
=
(3.163)
X;-X;,
we subtract (3.162) from (3.43) to obtain
We choose
Fi =A,- g,cT,
Fu = Au - ggcJ,
(3.165)
and from {3.164) we get the equations describing the error between the real state and the estimated state as
w1 ==
F,w1 +
"' l:
j=l
FqwJ>
i = 1, 2, •.• , m.
(3.166)
l""i
In order to obtain the asymptotic estimator from (3.166), we stabilize the dual of {3.166), :. W;
=
rT W; c;
+~ """"' 1-l
rT Wj> Cjl
i
= I, 2, ... , m.
(3.167)
J""i
Using (3.165) and (3.167), we write i
=
1, 2, ... , m.
(3.168)
We recognize immediately the important fact that the equations (3.168) have the same form as (3.58) and the pairs (AT. c1) are all controllable. Therefore, we can select g,, gg by the hierarchic scheme to stabilize (3.168) as we used to select k;, ku to stabilize {3.58). Then, all 'IV; will approach zero exponentially; and regardless of how large is the discrepancy between x;o and X;o. each estimate x1 will rapidly approach the corresponding state vector X;.
Estimation
195
'·
u
FIGURE 3.11. Subsystem estimator. The final form of the estimator obtained after the appropriate choice of g;, 8v. can be given as
i
= 1, 2, ... , m.
The "wiring diagram" of the ith estimator for the ith plant (3.43) can be drawn by inspection as shown in Figure 3.12. It is. important to note that the outlined hierarchic construction of estimators, which results into the interconnected estimators (3.169), reduces the dimensionally of the estimation problem for the overall system (3.42). Instead of constructing one estimator for the nth-order system (3.42), we propose to design m estimators for n1th-order subsystems. Now we are in a position to specify the regulator for each subsystem, as a system comprising the state estimator and the control law. When the plant of ith subsystem is described by the equations ·
Synthesis: Decentralized Control
196
X
'i
~
,+,..~
.
X
II
y
I I
.• lh
I
Joi
plant
r---------~-------1
I
I I
I
;ttl estimator
I I I I I I I
I
-
..
XI
II I I
L-.-
r -k,
jfh regulator
1
I
I I I I
I
~------------------~
FIGURE 3.13. Subsystem regulator. i;
=
A;x;
+
•
~ Aux1 + b1u;, J-1 J+l
i = 1, 2, ... , s,
(3.170)
the block diagram of the ith regulator is as shown in Figure 3.13, where a suitable notation is introduced for the input and output signals (see Figure 3.12). In the scheme of Figure 3.13, only the local control law u1 = r;- k{x;
(3.171)
is used, where r;(t) is a reference input for the ith subsystem. It is of interest to demonstrate the fact that the separation property (Chen, 1970) holds for the arrangement of Figure 3.13, and that we can use the control u;
= r;- ktx,
(3.172)
where we have replaced the real state x 1 by the estimated state x, instead of (3.171).
197
Estimation
We assume that stabilization of .the estimator is accomplished by choosing the g;'s only, and that the gy's in (3.164) are all zero. Now, by substituting (3.172) into (3.170), and using (3.164), the subsystem of Figure 3.13 can be described by
x, 0
1_11_ ___ _A!_
=
g1cT
•••
0
o_ _
-b,kt
...: ____ I AI
0
- gl
cr +
b~k{
- - - - - - A~
I
T I
g,c,
---
A,- g,c'[
I
+ b,k,T
(3.173)
b1r1
+
:X,
b,li -b-:r~-
b,lj By applying the following transformation to Equation (3.173), XI
x, WI
=
we get
I I
0
I
x,
~---
I
w,
XI
I
I
I 1 -I I
XI
:X,
(3.174)
198
Synthesis: Decentralized Control
w. 0 x.
=
0
A,- g,c'!
~
(3.175)
+
b,l;
0 Since (3.175) can be split into two independent sets of equations, the separation property is established. From (3.175) we see that as far as stability is concerned there is no difference between using the estimat ed state x and the real state x. The stabilization scheme of the previous section can be used twice to independently stabilize the tw.o matrices in the diagonal blocks on the right-hand side of (3.175), by applying the state feedback. This fact about the multilevel estimation is of interest in a multilevel optimization proposed in the next section, for which the availability of all the states is absolutely essential. Some comments can be made about the multilevel estimation scheme presented in this section. First, it is immediately possible to reduce the order of the estimator for each subsystem as outlined by Chen (1970). Second ly, as in the case of multilevel stabilization, it is possible to select the class of large-scale systems for which the subsystem asymptotic estimators can always be built by local design only. As expected, this class is the dual of the system defined by (3.34) and (3.123). Thirdly, it is obvious that the separation property, established above for locai stabilization only, can be directly extended to include the global controllers. Fourthly, we have
OptimiZation
199
considered systems without noise. If there is noise associated with subsystems, subsystem estimators can be suitably modified (e.g., Anderson and Moore, 1971) to include noise in the decentralized estimator for the overall system. Finally, it should be mentioned that the asymptotic estimators produced by the proposed multilevel scheme are highly reliable under structural perturbations and can tolerate a wide range of nonlinearities in coupling among the subsystems. This important robustness of decentralized estimators was first observed by Weissenberger (1976) and later used by Siljak and Vukeevic (1978). 3.5. OPTIMIZATION
By a simple example in Section 1.10, we demonstrated a possibility that optimal systems may become unstable if subjected to structural perturbations. Intuitively, the higher the degree of cooperation among different parts of the system (subsystems), the higher the efficiency of the overall system. Increased cooperation, however, means increased interdependence among the subsystems, which in turn may jeopardize the functioning of the overall system when some of the subsystems cease to participate. Since for a proper operation of large-scale systems, it is essential that structural changes do not cause a breakdown, a trade-off between optimality and dynamic reliability should be established. Therefore, the control schemes for large systems should be designed to ensure dynamic reliability despite a possible deterioration of the optimal performance. In this section, we will outline a multilevel control scheme proposed by Siljak and Sundareshan (1976a, b), which inherently incorporates the desired trade-off. It is assumed that a large system is decomposed into a number of subsystems which are optimized by local feedback controllers with respect to a local performance index when ignoring the interactions among the subsystems. That is, each subsystem optimizes its performance as if it were decoupled from the rest of the system. This strategy is shown to result in a reliable system when the interconnections are suitably limited. When the locally optimal subsystems are interconnected, the interconnections act as perturbations causing a degradation of the system performance and thus a suboptimal performance of the overall system. For this reason, a suboptimality index is defined which measures the performance deterioration and represents the trade-off between optimality and reliability in large-scale dynamic systems. Unless the subsystems are weakly coupled (Kokotovic, Perkins, Cruz, D'Ans, 1969; Kokotovic and Singh, 1971), the local optimizations may be in conflict with each other, producing a poor or even unstable overall
200
Synthesis: Decentralized Control
system. To reduce the conflict among the goals of the subsystems, we introduce the global controllers, whose function is to decrease the effect of some (or all) interactions among the subsystems using the partial (or total) information available on the subsystem level. Therefore, the interactions among the subsystems are necessarily treated as perturbation terms, thus deliberately ignoring their possible beneficial effects. Such an approach, however, opens a real possibility for treating nonlinearities in the interconnections as well as in the subsystems and establishing the robustness of the overall system. It should be noted, .however, that at present we do not know how to attach cost to global control. If this aspect is critical, global control being optional may be entirely excluded, and the proposed decentralized optimization can be carried out by using only local feedback control associated with each subsystem. The suboptimality design due to Siljak and Sundareshan (l976a), which will be presented in this section, uses the results of Popov (1960) and Rissanen (1966) obtained in the context of perturbed optimal systems. A similar approach has also been used by Bailey and Ramapriyan (1973) and by Weissenberger (1974). Neutralization of the interconnection effects by the global control is motivated by the work of Johnson (1971) concerning disturbance rejection in linear systems. Besides the reliability aspect, the proposed multilevel scheme offers advantages in the following situations: (1) When the individual subsystems have no information about the actual shape of interactions except that they are bounded, suboptimality and stability of the overall system can be accomplished using local controllers only. (2) When a system is too large, a straightforward optimization is either uneconomical because excessive computer time is required, or impossible because excessive computer storage is needed to complete the optimization. (3) When the state of the overall system is not accessible for direct measurement, and a single observer is not feasible, the proposed multilevel optimization scheme can accommodate the decentralized estimators described in the preceding section. Let us consider a system
~
described by the equation X
= j(t, X, u),
(3.1)
where again x(t) E· tgt• is the state of the system§ and u(t) E tgtm is the input at time t E ~ The function j: '8" X '!R? X tgtm ~ l!il? is continuous on a
Optimization
201
bounded domain GD in 5"x I!R? and is locally Lipschitzian with respect to x in GD, so that for every fixed input function u(t), a unique solution x(t; t0 , x0 ) exists for all initial conditions (t0 , x 0 ) E GD and all t E '5j. The symbol '5j represents the closed time interval [to,td, and GD = {(t,x) E '5"X I!R:': to< t t" llxll p}, where 0 p +oo. We assume that the systemS> can be decomposed into s interconnected subsystems Si1 described by the equations
Since suboptimality in the system is a result of the presence of interconnections, the index e depends on the size of h,(t,x), and the following problem is of interest:
Problem 3.1. Establish conditions on h;(t, x) to guarantee a prescribed value of the suboptima/ity index e. A solution to Problem 3.1, which will be given later, involves only bounds on the norms of the interconnection functions h1(t, x). Therefore, the results obtained are valid for a class of h1(t,x) and thus do not depend on the actual form of these nonlinear functions. This robustness is of major importanc e for modeling uncertainties and possible variations in the shape of nonlinear interconnections during operation (Weissenberger, 1976). It is important to observe now that the suboptimal performance of the system has resulted from the use only of controllers associate!i locally with each individual subsystem. The suboptimality index e is a measure of the performance degradation, which is directly proportional to the size of interconnections. Hence, an improvement in the system performance is possible if llh1(t,x)ll can be reduced. We propose to accomplish this by using additionai·control functions that neutralize the effect of interconnections. These functions are generated by a global controller on a higher hierarchical level using the states of the subsystems. If, however, some of the subsystem states are not available to the global controller, it can perform its function partially or be excluded entirely. Since the global control functions are introduced to modify the existing interconnections h1(t,x), the effective interconnections can be represented by h,(t, x, uf), where uf(t) E t3t"" is the global control component applied to the subsystem ~1• Now, the interconnected subsystems are described by
X;
= g;(t, X;, uf) + h,(t, X, uf),
i
=
1, 2, ... ' s,
(3.188)
which is an obvious modification of the equations (3.176). With this modification, the index e becomes a function of ilh(t,x,u')ll. where h: ~X~ X~-+~ is h = (hr,hf,. ~.,hiland u' E ~ is u' = [(uff, (uf)r, ... ,(uf)T]T, and we solve the following:
204
Synthesis: Decentralized Control
Problem 3.2. Find a control law (3.189) for which
e0 = inf{e[Jih(t,x,ug)ll]} u•(t)
'V(t, x)
E 6j).
(3.190)
It is impor tant to note that the choice of global control to neutralize the interconnections using available subsystem states does not disturb the reliability of the system accomplished by the propo sed use of local controllers. Therefore, a solution to Problem 3.2, outlin ed subsequently, mitigates suboptimality and at the same time preserves the reliability of the closed-loop system. Now that Problems 3.1 and 3.2 are precisely formulated, we proceed to solve them by decentralized optimization propo sed by Siljak and Sundareshan (1976a). A solution to Problem 3.1 may be obtain ed by using the classical Hamilton-Jacobi theory. Since in our optimization proce dure we chose the local control laws to optimize the decoupled subsy stems, the optimal indices satisfy the corresponding Hamilton-Jacobi equati ons. When the subsystems are interconnected, the equations are not satisfied by the respective performance indices, and the overall system is not optimal. However, a majorization procedure is possible to provid e an estimate of the performance deviation from the optim um due to intera ctions. We assume that the optimal index V;(t, x;) is a function V;: 'iii, X 01!' ~'iii,+ which belongs to the class C 2 in both arguments, and satisfies the Hamilton-Jacobi equat ion (e.g. Ande rson and Moore, 1971)
av;~/;) +[gra d V;(t,x;)fg;[t,x;,k/(t,x;)]
+ L;[i,x;,k/(t,x;)] = 0, i
= l, 2, ... , s,
and V;[t~,x;(ti)] = P[t~,x;(t1 )], [t~,x;(t1 )] E 6j). Let us define the functions V: 'iii, X 01!' ~ 'iil+, P: 'iii, X 'iiln X 'iilm ~ 'iil+ as
V(t,x) =
•
~ i=l
P(t,x) =
•
~ i=l
L(t, x, u1)
= ~• i=l
'V(t,x)
E 6j),
(3.191) ~ 'iil+,
L: 'iii, X 01!'
V;(t,x;), P;(t,x;), L;(t, X;, uf),
(3.192)
Optimization
205
and k 1 : iffi.X ~-+ C!il!" as k 1 = [(k{)r, (k1f, ... ,(k1)ry , where the k/(t,x;) 's are defined in (3.178). Now we provide a solution to Problem 3.1 by the following: Theorem 3.9. Let the interactions h;(t,x) in (3.187) satisfy the constra int
[grad V(t,x}Vh(t,x)
~(Q8j.Qe •... ,Q~) ~(QoJ.Qv •... ,Q,y) (4.3) for all Q01, Qv • ... , Q.1 satisfying the budget constraint (4.2). Under the appropriate assumptions (the utility function is sufficiently smooth and strictly quasiconcave, the consumer's indifference curves do not intersect any axis, and no consumer has a bliss point) this utility maximization problem can be solved by standard optimization techniques (lntriligator,
1971). The excess demand of the jth consumer for the ith commodity is defined as
Fu = Q!i- Q0 • (4.4) Then the utility function ~ of (4.1) can be expressed in terms of Fy as ~(Fo1 + Qo1, Fi1 + Qv •... , F,.1 + Q.J, and the Lagrangian for this problem can be formulated as
•
J) = ~(Fo1 + Qoi,Fv + QlJ• ... ,F,y + Q,y) +A~ P;Fij, ;-o
(4.5)
where A is the Lagrange multiplier for the jth consumer. Now the necessary conditions for maximizing consumer's utility subject to the budget constraint are
aJJ a~ aFv = aFv + A~ = o, av.
a
•
{=~ P1F!i
1\
1=0
=
.·
0,
i = 0, 1, ... , n.
(4.6)
Economics: Competitive Equilibrium
224
Under certain well-known conditions [the corresponding bordered Hessian determinants alternate in sign (Quirk and Saposnik, 1968)], we can apply the implicit-function theorem to the equations (4.6), and assert that there is a regular constrained maximum of the utility function. In a sufficiently small neighborhood of the utility-maximizing point, we can use one equation of (4.6) to eliminate A and solve the remaining n for the excess demands as functions of prices: j
=
I, 2, ... , n.
(4.7)
Now, the aggregate excess demand for the ith commodity is defined as m
F;(Po, P~o ... , P.)
=
~
j-1
Fu(Po, P" ... , P.),
(4.8)
where m is the number of consumers. That is, the aggregate excess demand function for the ith commodity is obtained by summing up the individual excess demand functions of the m consumers. The fact that we eliminated A from the first n equations of (4.6) means that the implicitly defined quantities are invariant under the multiplication of prices by a positive constant. That is, i =I, 2, ... , n,
(4.9)
for any positive number K, and the aggregate excess demand functions are homogeneous of degree zero in prices. Therefore, we can choose arbitrarily a commodity (say commodity 0) to act as a medium of exchange, and we refer to that commodity as the numeraire. Prices of commodities other than the numeraire can be considered as "normalized prices" p 1 = P/P0 , i = I, 2, ... , n. Then the aggregate excess demand function for the ith commodity in terms of the normalized prices is defined as m
.MP~oP2. · · · ,p.)
=
~ fu(Pt.P2• · · · ,p.),
i = 0, I, ... , n, (4.10)
j=l
where fu(P~op2 , ••• ,p.) is obviously the excess-demand function of thejth consumer for the ith commodity with normalized prices as independent variables. The market system described by the aggregate excess demand functions (4.10) is truly a decentralized system. Each economic agent takes prices as given and maximizes his utility with respect to his own budget constraint. The question of fundamental interest is: Are the desired actions of economic agents mutually compatible for some set of admissible prices?
225
A Dynamic Model
This is the question of the existence of equilibrium prices~, pt, ... , ~ at which the aggregate excess demand functions are equal to zero, and the independent decisions of economic agents can be carried out simultaneously. Since our interest is centered on the stability problem of competitive equilibrium, we shall assume that the system satisfies the conditions which ensure the existence of the equilibrium prices. That is, we assume that there is a set of (normalized) prices pt, pt, ... , p! such that the aggregate excess demand functions become simultaneQusly zero, that is,
.MPI* ,pj, ... ,p!) = 0,
i
=
1, 2, ... , n.
(4.11)
The aggregate excess demand functions and the existence of the equilibrium prices are two basic ingredients in setting up a stage for the analysis of price adjustment processes in dynamic models of the market systems. Our sketchy description of these two entities is all we can do here. For a detailed treatment of both subjects, the reader is referred to the books by Quirk and Saposnik (1968), Nikaido (1969), and Arrow and Hahn (1971). We should also note that we have used the neoclassical Hicks-Samuelson analysis to derive the excess demand functions of a market. The reason for this choice of economic environment is that it is more suitable for stability considerations of competitive equilibrium than the alternative set-theoretic approach, which is more convenient for equilibrium-existence problems. 4.2. A DYNAMIC MODEL
The ability of a competitive economy to allocate resources efficiently to producers and distribute the products among consumers would be of little use if the competitive equilibrium were not stable. The mere existence (and even uniqueness) of competitive equilibrium explains nothing about how the equilibrium is achieved if the market is not already in it. Even if the market is at equilibrium, any changes in conditions and parameters (such as tastes of consumers, technology, resources, weather, etc.) would move the market away from the equilibrium. Being aware of this problem, Walras (1874) suggested that the market competitive mechanism works as a large computing machine which solves the numerous market equations for the equilibrium on the basis of the law of supply and demand and an iterative procedure which he called tdtonnement. Assume, as Walras did, that an initial set of prices is not at the equilibrium and that for some commodities demand exceeds supplies, whereas for others supply exceeds demand. Considering commodities in a definite order, we can adjust the price of the first commodity so that supply equals demand. This we do on the basis of
226
Economics: Competitive Equilibrium
the law of supply and demand: We raise the price if demand exceeds supply and lower it if supply exceeds demand. When we repeat the procedure for the second comrilodity, we generally destroy the partial equilibrium of the first commodity market. Therefore, at the time we adjust the last commo-: dity market, all prices of the preceeding markets will be at nonequilibrium values. Walras argued, however, that adjustments of partial equilibria are most significant, since the demand and supply for any commodity depend on its own price much more than on the prices of other commodities. Therefore, adjusting partial equilibria recursively, we move the overall market cloSer and closer to its equilibrium. To aid in understanding the iterative process of the price adjustments, we may imagine an auctioneer who at the beginning of the marketing day announces a set of prices to the market participants and then collects transaction offers from them. If the offers do not match, the announced prices are not the equilibrium prices, and he calls another set according to a rule based upon the law of supply and demand. The auctioneer raises the price of a commodity for which demand exceeds supply and decreases it in the opposite case. No transactions are allowed to take place at the nonequilibrium set of prices. Following the arguments of Walras, we conclude that the auctioneer may never be able to get the transaction offers to match, but applying his rule he will drive the economy closer and closer to the equilibrium. Although intuitively appealing, the arguments of Walras concern.irig the "price mechanism" in the working of a decentralized economy cannot be accepted as rigorous. This fact motivated Hicks (1939) to use the shapes of the supply and demand functions in the neighborhood of equilibrium and derive stability conditions for convergence of the price adjustment process to the equilibrium. He proposed a set of determinantal inequalities as conditions for a linear market model to be stable, and those conditions will be used throughout this exposition. The stability analyses carried out by Walras, Hicks, and some others (see Negishi, 1962) were purely static and did not correspond to an explicit dynamic model of the market This situation was rectified by Samuelson (1947), who introduced time into competitive-equilibrium models in an essential way by identifying tatonnements as solutions of the system of differential equations
1,' = k;fi(PI,P,., • · • ,p,.),
i
=
1, 2, ... , n,
(4.12)
where p1 = p1(t),fi(PI,fJ2, ... ,p,), and k1 > 0 are the price, the excess demand function, and the coefficient of price adjustment of the ith commodity, respectively. We note immediately that the system of ditreren-
A Dynamic Model
227
tial equations (4.12) is a· ..dynamic representation,. of the classical law of supply and demand, since /;(PJ.,Pz, ••• ,p,) 0 implies a iise in the price p,, and /;(PJ.,Pl, ... ;p,) 0 implies a fall in the price p,. Therefore, we expect the solutions p 1(t), i = 1, 2, ..• , n, to mimic the price adjustment process which we believe goes on in actual markets. There are several comments that we should make at this time. First, we should mention that the existen~ of solutions of (4.12) for all times is obviously a prerequisite for the possibility of tatonnement and its stability properties. Furthermore, it would be equally awkward if the uniqueness of solutions were· to fail, so that for given initial prices p 10 = p 1(t0 ), i = 1, 2, ... , n, there were a number of possible solutions p1(t), i = 1, 2, ... , n, for t ~ to. It is logical, therefore, to assume that excess demand functions fi(PJ.,Pl, •.. ,p,) are sufficiently smooth to guarantee the existence and uniqueness of solutions to the system of equations (4.12). This assumption would be in agreement with the utility-maximization process which produced the excess demand functions in the previous section, because the implicit-function theorem asserts the continuity of these functions and the existence of continuous first partial derivatives. This fact allows us also to take for granted that the solutions of (4.12) are continuous with respect to initial conditions to. P10• i = 1, 2, ... , n. It is also possible (Arrow and Hahn, 1971) to make an analysis in the case when prices are announced by the auctioneer in terms of the numeraire which is used as a unit of accounting and transaction. Then we consider a dynamic model given as
dPo
tit- 0' dP; dt
=
K;F,(Po,Pt. ... ,P.),
(4.13)
i = 1, 2, ... , n,
where the functions Fj(P0 ,P11 ••• ,P.) are defined by (4.8). Finally, we should make some comments on the role of time in the model (4.12). Obviously, in postulating the model and auctioneer's rules, we assumed that when prices change they do so in continuous time. This assumption is made for convenience, and it is fairly clear that a tatonneTMnt process can also be described in terms of difference equations and other types of functional equations. It is only for descriptive reasons that the tdtonnement process was formulated as if it were taking place in real time. The process could equally well be envisioned as an iterative one, whose steps have no direct relation to real time. Our primary interest in the following consideration of competitive equilibrium is to remove the assumptions that tastes. of consumers, technology, weather, etc., remain constant during the price adjustment process,
228
Economics: Competitive Equilibrium
and thus treat cases when, in general, the basic underlying nature of the economy is not assumed to be stationary. This generalization of competitive-equilibrium ·models reqwres that time appear explicitly in aggregate excess demand functions on the right side of Equation (4.12). That is, we consider a model of a competitive market economy to be given as a system of differential equations
i
=
1, 2, •.. , n,
(4.14)
where again p1 .... p1(t), i = 1, 2, ... , n, are prices of the n commodities. For convenience, the coefficients of the speeds of adjustment are assumed equal to unity for all commodities (that is, k 1 = 1, i = 1, 2, ... , n). The timet in the excess-demand functionsjj(t,Pt,.Pl •... ,p,), which is the same as that appearing implicitly in the derivative dpjdt, will be used to describe various nonstationary phenomena that may take place in the economy, such as shifts in excess-demand schedules, inaccuracy of the t4tonnement process, reduction in the number of commodities during adjustment of prices, etc. All these situations will be discussed separately in the course of this exposition. It should be noted that nonstationary models-that is, models which involve time explicitly as that of (4.14)-were introduced in economic analysis by Samuelson (1947) to discuss the moving equilibrium for price. Considerations of nonstationary linear models were made later by Arrow (1966) to study multiple markets with steadily increasing demands. An extensive study of nonstationary nonlinear models of the general type (4.14) were initiated recently by Siljak (1975c, 1976a). These results will be outlined in the following sections, except for the competitive analysis of nonnormalized processes, which can be found in Siljak (1976b). 4.3. CONNECTIVE STABILITY: LINEAR CASE The objective of this section is to introduce the concept of connective stability in competitive equilibrium analysis via linear constant models. These models can be obtained from the general nonlinear stationary models formulated in the preceeding section by applying linearization. Let us consider a stationary version of the market model (4.14) in the vector form
p = j(p),
(4.15)
where p E ~ is the price vector and j: ~ -+ ~ is the vector of aggregate excess-demand functions. We assume thatj(p) has continuous first deriva-
Connective Stability: Linear;Case
tives and that f(p*)
229
= 0, so that p* is an equilibrium price of the market
(4.15). Expanding f(p) about p* in a Taylor series and substituting p = p* + q, we obtain (4.15) as
q = Aq + b + g(q ),
(4.16)
where A = [a}i(p*)japi] is a constant n X n Jacobian matrix, b = f(p'*), and g(q) denotes the higher-order terms in the Taylor series expansion. To study the local properties of the nonlinear model (4.15) in the neighborhood of the equilibrium p*, we can use the linear model jJ
= Ap + b,
(4.17)
which is obtained from (4.16) when the higher-order terms of the Taylor expansion are neglected and q is replaced by p. We use the model (4.17) to introduce the basic idea of structural perturbations and connective stability. We assume that all the commodities are substitutes, so that rise in the price on any market decreases the demand in that market, but increases the demand in some (or all) other markets. This amounts to saying that the elements of the matrix A = (ail) in (4.17) are such that i =j, i -=F j.
0,
(4.18)
Since A has nonnegative off-diagonal elements, it is a Metzler matrix (Arrow, 1966; Newman, 1959). The gross-substitute case is of special importance in the context of the model (4.17), and various rich properties of the related Metzler matrices, which we are going to use throughout this exposition, are listed in the Appendix. We also assume for a moment that 0), by which we mean b; 0, the vector b in (4.17) is nonnegative (b i = 1, 2, ... , n. As shown by Arrow (1966), solutions p(t; t0 ,p0 ) are nonnegative whenever they "start nonnegative". That is, for all initial conditions t0 , p 0 0, we have
>
>
>
p(t; to,Po)
> 0,
t
> to.
(4.19)
The equilibrium price of the market is a constant solution of Equation (4.17) determined by the algebraic equation Ap
+b =
0.
(4.20)
If det A -=F 0, the equilibrium p* is a constant vector given as
p* = -A-1 b,
which is the unique solution of (4.20).
(4.21)
EcOnomics: Competitive Equilibrium
230
By using a simple transformation q
= p- p*,
(4.22)
q=
(4.23)
(4.17) can be rewritten as Aq,
and stability of the price equilibriump* of the market model (4.17) implies and is implied by stability of the equilibrium q* = 0 of (4.23). Stability of q* = 0 in (4.23) is equivalent to stability of A (eigenvalues of A all have negative real parts). It is well known (Newman, 1959) that a Metzler matrix is stable if and only if it is a Hicks matrix, that is, all its odd-order principal minors are negative and all its even-order principal minors are positive (Theorem A.2). That we can use only leading principal minors to test for stability of a Metzler matrix is also well known (Gantmacher, 1960), and the conditions
a2k
>O,
k
=
1, 2, ... , n,
(4.24)
are both necessary and sufficient for stability of A (Theorem A.9). These conditions were obtained independently by Kotelyanskii (1952), using the results of Sevastyanov (1951), and by Hawkins and Simon (1949). For a Metzler matrix A with negative diagonal elements as in (4.18), the conditions (4.24) are equivalent (Newman, 1959) to its quasidominant diagonal property, which amounts to saying that there exist numbers dt O,j = 1, 2, ... , n, such that
>
n
dtlaiil
> ~ dtlaul, i-=1
j
=
1, 2, ... , n.
(4.25)
t+j
The last inequality in (4.24) implies that det A =I= 0, and stability of A implies uniqueness of the equilibrium price p* given by (4.21). Furthermore, 1 it can also be shown that stability of A is equivalent to saying that A- is 1 cannot have a zero row (it is nonpositive element by element. Since Anonsingular), nonpositivity of A-1 in (4.21) means that for any positive b 0) we have a positive equilibrium p*. (that is, b Now, if A =. (av) satisfies (4.25) and is a stable matrix, then the system (4.17) would remain stable even if some of the nonzero off-diagonal elements au (i =I= j) were to go to zero. This fact is obvious from (4.25).
>
Connective Stability: Linear Case
231
Furthermore, since the perturbed matrix is still a stable Metzler matrix, the price adjustment process is nonnegative, and so is the equilibrium price of the system (4.17). A systematic investigation of such· structural perturbations of the matrix A in (4.17) can be carried out in the framework of connective stability, which is considered next. Let us reconsider the system (4.17) in the form
p = Ap + b, where p(t) is again the price vector,· and matrix with elements defined as
(4.26)
A= (a;;) is the i =}, i-=Fj.
n X n constant
(4.27)
Here a;, a;; # 0 are numbers, and e;; are binary elements of the n x n fundamental interconnection matrix E = (e;;) defined as
e!J = {
1,
aiJ
o,
-=F 0,
(4.28)
a;;= 0.
ByE = (eli) we denote a constant n X n interconnection matrix which is obtained from the fundamental interconnection matrix E = (eli) by replacing the unit elements eli with numbers eli such that i,j
= l, 2, ... , n,
(4.29)
while the zero elements e!l remain the zero elements eli in the matrix E. As in the preceeding chapters, by E E Ewe denote the fact that E is generated by E. We also recall that E E E, since we can choose E = E. We note immediately that for each E E E we have a different system (4.17), which we denote by :ii(4.I7)• since the constant n X n matrix A =(ali) is now defined as i =j,
i
~j.
(4.30)
The underlying idea of the concept of connective stability is the possibility of establishing the stability of a class of systems :ii(4.l7) obtained for E E E, by proving the stability of one member of that class which corresponds to E, namely, the system represented by (4.26). Thus, we need the following: Definition 4.1. The system :ii(4.17) is connectively stable if the equilibrium p• = -A- 1b is stable in the sense of liapunov for all E E E.
Economics: Competitive Equilibrium
232
ctive stability Comparing Definition 4.1 with the definitions of conne we say that "the formulated in Chapter 2, we note that in Definition 4.1 equilibrium p* is system ~(4• 1 7) is connectively stable" instead of "the se in the system becau connectively stable". This modification is necessary p• which is, in brium (4.17), for each E E Ewe have a distinct equili general, different from that of (4.26), (4.31) stability of each However, our stability conditions will establish the equilibrium p* by showing the stability of p* only. of ~4. 17)> let us Before we engage in analyzing the connective stability the strength of ing chang of consider the effect on the equilibrium price p* nts eu of the eleme the by red interactions between individual markets measu and (4.30), (4.29), (4.27), from interconnection matrix E. We note first that we have (4.32)
; 0). It is a where the inequality is taken element by element (A -Ato, E
E E,
(4.35)
whenever Po
An interesting interpretation of Theorems 4.1 and 4.2 can be given in the case where the kth commodity "disappears" from the market. When the price of a commodity becomes negative, then the agent selling that commodity has to give up at the same time units of some other commodity. If the agent can "throw away" commodities without using up units of other commodities, then he will prefer this option of "free disposal", and negative prices cannot arise. A reduction of the number of commodities was considered by Arrow and Hurwicz (1962) in the context of the nonlinear model (4.15). Under certain conditions imposed on the excess demand functions of free commodities, they concluded that properties of the whole market remains valid for the reduced market. Our interest here is in considering linear models in which a reduction of the commodity space can take place and extending the conclusion of Arrow and Hurwicz to include stability. By using interconnection matrices, we can describe a structural perturbation caused by the disappearance of the kth commodity by setting eik
=
e19
= 0,
i,j=1,2, ... ,n.
(4.38)
From Theorem 4.1, we conclude that the equilibrium price p* of the reduced market will be always smaller than the equilibrium price p* of the original stable market system. Theorem 4.2 tells us that price adjustment process p(t; to,Po) will have the same property for the two market systems. The immediate question is then: can we prove a similar result for stability properties of the whole and the reduced markets? To this effect, we prove the following:
234
EcOnomics: Competitive Equilibrium
'Theorem 4.3. The system ~4.l7) is connectively asymptoti if and only if the Metzler matrix A is a Hicks matrix.cally stable in the
large
Proof. As in the proo f of Theo rem 4.1, we use the fact (see Appendix) that for Metzler matrices A, X the ineq ualit y (4.32 ) implies that A is a stable matr ix (that is, a Hicks matrix) if and only if A is. This proves the "if' part of the theorem. To establish the "onl y if" part, we notic e that if A is not a Hicks matrix, then it is not a stable matrix, and therefore the system (4.26) corresponding to E E E is not a _stable syste m. Therefore, ~4.l7) is not a stable system for all E E E, and the proo f of Theo rem 4.3 is complete. An immediate conclusion is that in case of the structural pertu rbati on (4.38), stability of the original mark et system implies stability of the reduced mark et system. It is obvious that this conclusio n holds even when a num ber of commodities disap pear from a given mark et system. This fact was observed by Lang e (1945) when he establishe d the notio n of total stability. In the connective-stability framework, we are not restricted to the case (4.38), but can treat any case described byE E E. Furth ermo re, we .can exten d our results to nonl inear models as well, thus giving considerable generality to the structural pertu rbati ons of the multiple market systems.
4.4. CON NEC I'IVE STABILITY: NON LINE AR CAS E In this section, we show that a stable comp etitive equilibrium is robu st and can tolerate not only the struc tural pertu rbati ons, but also a wide range of nonlinearities in the interactions amo ng the individual markets. This task however, requires a more refined analysis. Let us consider a mark et desc ribed by nonl inea r differential equations (4.14) written in the vector form
p = j(p) ,
(4.39) where p(t) E ~is the price vector; the exce ss-demand func tion/ : ~ --+ ~ is defined, boun ded, and continuous on the dom ain ~. so that solutions p(t; to,Po) of (4.39) exist for all initial condition s (t0 ,Po) E 5"X ~ and t E '50. We recall that ~is the time interval {'r, + oo), where T is a num ber or the sym bol- co, and~ is the semi-infin ite time valu e [t0 ,+oo ). Since we would like to use the effective diag onal-dominance conditions (4.25) for nonl inea r models, it is convenien t to consider (4.39) in the form
p = A(p)p,
(4.40)
where p(t) E ~. and in the excess-demand func tion A(p) p the matrix func tion A: ~ --+ ~nz is defined, boun ded, and continuous on ~. so that
)
r .
.
~
~~/\:
~>-'~
Connective Stability: Nonlinear Case
235
solutions p(t; tG,Po) of (4.40) exist for all (to,Po) E '!ix '3t" and all t E '!i'Go It is obvious that we can always choose the n X n matrix A·= [a (p)] as a 0 diagona l matrix, A(p) = diag{!I (p)/Pt,f 2(p)/PJ , ... ,J,.(p)jp,.}, but this choice is by no means unique. For example,
0 -3IPIo I][PI]
f(p) = [-2pdP JIJ = [-2IP21 -3P2IPI I
P2
(4.41)
where sgn 0 = 0. We assume that there exists an equilibr ium price p* 0 of (4.39), so that f(p*) = 0 for all t E ~ To conside r a model with the equilibrium at the origin, we use the transfor mation q = p - p• in (4.39), and get q = f(q + ~ ). Denotin g g(q) = f(q + p* ), we get the system (4.39) as q = g(q), where the equilibrium p* 0 of (4.39) is represented ·by the equilibrium q* = 0, sinceg(O ) = f(p*) 0. With the system q = g(q) we associate the system q = B(q)q in the same way as before. For convenience, we replace q by p and B by A, and conside r again the system (4.40). It is importa nt to note, however, that in this new interpre tation of (4.40) we have to allow the price p to be negative, with the understa nding that in the original system this means that p(t; tG,Po) is below the equilibrium price p* 0. In order to establish conditio ns for asympto tic connective stability of the market (4.40), we define the coefficients av(P) of the matrix A(p) as
>
>
=
>
ao(P) = {-cp,(p ) + eucpu(p), evcpiJ(p),
i =j, i-:;. j,
(4.42)
where cp,(p), cpu(P) E C('3t") are nonline ar function s which represen t the nonline ar interdep endence among the individu al markets. In (4.42), the elements ev of the intercon nections matrices E E E are assumed to be constan t. We further assume that there exist number s a ~ 0, a;> a; ~ 0 such 1 0 that
fPt(P)IPtl ~ a,q,,(lp,l), 'Vp
i,j = 1, 2, ... , n,
E '31,".
(4.43)
Here q,,: tilt+ ~ tilt+ are compar ison function s of the class %: q,,a) E C(tilt+ ); I/>1(0) = 0; and 1/>;(S,.) I/>;(S2) for all !I, S2: 0 ~ SI S2 + oo (see Definition 2.11).
>
i,j
= 1, 2, ... , n,
'ripE~.
(4.51)
>
where again a 1 a 11 0, aii 0. The conditions (4.51) follow from those of (4.43) when the comparison functions cf>J(jp1 1) are chosen as jp1 j. We want to show that stability of the constant Metzler matrix A is both necessary and sufficient for stability of the equilibrium price p* = 0 of the market {4.40) for arbitrary functions cp1(p), cpu(P) which belong to the classes ell 1, ell11, and for arbitrary matrices E generated by the matrix E. The fact that the system is asymptotically stable in the large for all cp1 E ell~o cpii E ell 11 we acknowledge by the term "absolutely stable" (see Definition 2.8). We can do even better, as follows: Definition 4.3. The equilibrium price p* = 0 of the market (4.40) is absolutely, exponentially, and connectively stable in the large if there exist two positive numbers II and 'IT independent of the initial conditions (t0 ,p0 ), such that
llp(t; to,.Po)ll ~ IIIIPoll exp[-'tT{t - to)]
'Vt E
'50
(4.52)
239
Connective Stability: Nonlinear Case
for all (t0 ,p0 ) E 5X ~.all cp1 E cPi> cp11 E cP 11, and all E E E.
To establish this kind of stability, we can use the following:
Theorem 4.5. The equilibrium price p* = 0 of the market (4.40) is absolutely, exponentially, and connectively stable in the large constant Metzler matrix A= (a11 ) is a Hicks matrix.
if and only if then X
n
The proof of the "if'' part of the theorem follows the proof of Theorem 4.4. We again choose P(p) as in (4.43), and as in (4.45) obtain the expression
Proof.
n
n
J-1
~;)
D+v(p)(4.40} i-J ~ iaui.
(4.61)
I+}
which can be accomplished by altering suitably the measurement of commodities. From (4.61), we see that stability of the market is assured when the price of any given commodity is more affected by the changes in its own price than by total absolute change in prices of other commodities. It is believed that this justifies the intuitive argument that Walras (1874) used to establish convergence of the tdtonnement process in his original investigations. A number of extensions and applications of the results presented in this section are possible in various other areas and models. A competitive
Nonstationary Models: Moving Equilibrium
241
analysis of Richardson's (1960) model of the arms race was developed by Siljak (1976b, 1977b) to study how formations of alliances and neutral countries affect the equilibrium and stability of the armament processes. Applications of the obtained results to pharmacokinetics models (Bellman, 1962) and compartmental systems have been proposed by Ladde (1976b, c). Further possibilities of using these results are in studying interactions in social groups along the lines of Simon (1957) and Sandberg (1974), as well as in the analysis of certain electronic circuits initiated by Sandberg (1969). 4.5. NONSTATIONARY MODELS: MOVING EQUlllBR IUM
In initiating the dynamic analysis of competitive equilibrium, Samuelson
(1947) assumed that both supply and demand functions are explicit
functions of time. Consequently, the price at which supply equals demand and the excess demand is zero becomes a function of time. For this price to be an equilibrium, it must be constant (equilibria are constant solutions of the corresponding differential equations). This case takes place only if the effect of time in the market is restricted to changes in the slopes of supply and demand characteristics. The situation in which shifts in demand cause a time variation in the price of zero excess demand may be termed a "moving equilibrium for price" (Samuelson, 1947). Samuelson used some simple examples to examine whether the adjustment process diverges from, follows, or reaches moving equilibrium, and thus initiated a stability analysis of competitive equilibrium under shifts in excess demand. Steady upward shifts in time of demand functions on some or all of the interrelated marketS were considered by Arrow (1966). He assumed that the demand is shifting upward in time and that the supply curve may do the same, but never more rapidly than the demand; he then showed that the prices rise at a rate that approaches a limiting value. The positive linear time function was chosen to represent trends in an otherwise stable, linear, and constant market model. In this section, we will consider the shifts in demand and supply functions which have no specified sign or form except that they are bounded. We will show that in stable market systems under bounded shifts the role of the equilibrium is played by a compact region, and prices on all the markets are ultimately bounded globally with respect to that region. That is, all prices reach the region in a finite time, and once in the region, they stay there for all future times. This property of the price adjustment process will be established for a nonlinear and time-varying model studied in the preceeding section. We will provide an upper estimate of the above mentioned region by means of the same Liapunov function used to
,,,, ·_,:.
242
Economics: Competitive Equih"brium
\
~
;~
determine global stability properties of the model. The estimate is directly proportional to the size of the shifts in the excess-demand functions. Furthermore, the estimate of the regio n is invariant under structural perturbations,. and the adjustment process is again exponential-prices on all the markets reach the region faster than exponentially despite structural changes in the models. We continue to consider the price adjustme nt model of type (4.40), jJ = A(t,p)p + b(t,p), (4.62) where a function b: 5"x ~-+~is added on the right side of (4.62). The function b(t,p) has components of the form
b,(t,p) = l,(t)l[!,(t,p), (4.63) where 11(t) are components of the inte rconnection vector l(t) = [11 (t), 12 (t), ... , ln(t))T, such that l1(t) E [0, 1] for all t E 5: Similarly, as in the case of the matrix E, we define the binary vect or i E ~~as there is a demand shift on the ith market: l[!,(t,p) ;;& 0, there is no demand shift on the ith market: 1[11(t,p) a 0, .(4.64) and denote by l(t) E i all vectors obta ined from 1 by replacing unit elements with the corresponding functions 11(t). In (4.63), the functions 1[11(t,p) E C(5" X ~) satisfy the conditions
- { 1, I,= 0,
(4.65) where the {1,'s are nonnegative numbers. Furt hermore, we define a constant vector b E ~~ as
b, = i,p,. With the system (4.62) we associate a com pact region ~ =
,.,.}
'IT,
j
=
I, 2, ... , n, {4.96)
We shall also assume that b(t) is a bounded function, that is, sup,e!l"llb{t)ll = ~. where ~ is a positive number. Now we establish the following:
Theorem 4.8. A system (4.95) is convergent if A(t,p) is everywhere a negative quasidominant diagonal matrix. Proof. The proof of Theorem 4.8 follows directly from the proofs of Theorems 4.6 and 4. 7.
In the context of competitive equilibrium, Theorem 4.8 states that if the matrix A(t,p) is dominant diagonal and if the shifts in excess demand b(t) are bounded, then there is a price adjustmen t process p*(t) which is bounded and globally stable. That is, all prices tend to that process as the time progresses. To express the above result in terms of the Jacobian, we can use either
248
EcOnomics: Competitive Equih'brium
Corollary 4.2. A syst~ (4.95) is conv ergent if J(t,p) is everywhere a negative diagonal matrix.
~idominant
or Corollary 4.3. A system (4.95) is convergent
(4.85).
if J,(t,p) satisfies the inequality
By choosing
A(t,p) ...;
fo
1
J(t, p.p) dp.
(4.97)
and using the Liapunov function P(p) == l:~-• d,lp1 l, Corollary 4;2 follows from Theorem 4.8. By using the func tion P(p) = illplf, Corollary 4.3 is a consequence of Theorem 4.7. We can now turn our atte ntio n to the case of a linear constant mar ket with time-dependent shifts,
p = Ap
+ b(t), (4.98) which was considered by Arrow {1966). We assumed that A is ann Xn constant Metzler matrix and b(t) = ct, where c E ~ is a constant vector. Und er the assumption that A is a stable matrix, Arrow showed that the actual price p(t) is always und er the mar ket clearing price ("movin g equilibrium" in previous terminology of Samuelson, 1947) p 0 (t) defined by 0 = Ap0 + b(t), that is, p 0 = - A- 1 b(t). Furthermore, the difference p 0 p approaches a limit which decreas es as the speeds of adjustments on the different markets increase. If we consider bou nde d shifts such that b(t) E C(~) and SUPres-llb(t)l l = p, where p is a positive number, we conclude from Corollary 4.2 that stability of A implies that the mar ket (4.98) is convergent. This is bec ause for Metzler matrices stability imp lies and is implied by the negativ e quasi dominant diagonal pro pert y of A. Furthermore, the limiting proc ess p• (t) is determined by (4.99) In the context of the mar ket mod el (4.98) it would be of interest to determine conditions under which prices on all markets approach the mar ket clearing price p'(t). Tha t wou ld take place if the difference
q(t) = p•(t) - p0 (t)
(4.100)
app roac hed zero as the time progress es. To obta in the desired conditio ns, let us use the fact that p• = Ap• + b(t) and p'(t) = - A-• b(t) to get from
Nonstationary Models: Moving Equilibrium q(t)
= p* (t)
-
249
p0 (t) the following equation: q = Aq + c(t),
(4.101)
where c(t) = A- 1b(t). Now we prove the following: Theorem 4.9. The actual price p(t; t0 ,p0 ) converges to the clearing price p 0 (t) = -A- 1b(t) of the market (4.98) if A is a negative quasidominant diagonal matrix, c(t) is bounded, and lim1_. 00 c(t) = 0. Proof. Applying a result of Hahn (1967) to (4.101) and having in mind that llc(t)ll ~ K. we can say that for each e > 0 we can find a t1 such that llb(t)ll e fort > tl. Also, for to t ~ tl' llc(t)ll ~ Then we recall that
r.
a~o
I'Pu(t,p)l
< au,
l1/111 (t,p)l
< fJu
'r:J(t,p)
E '5'X ~.
(4.109) In (4.109), the functions eg, lu: ~-+ [0, 1] are elements of then X n interconnection matrices E = (eu), L = (lu). The elements e11 measure the strength of the deterministic interactions among prices ,in the market, whereas the elements /0 measure the infiuence of the external stochastic disturbance on the market. Again, we need the notion of the fundamental n X n interconnection matrices E = (~IJ) and L = (lu) defined by
1,
eu= { 0,
'Pu(t,p) 'Pu(t,p)
;;E
=
0, 0,
lu =
{1,0,
l[lu(t,p) ;;E o, (4.11 0) 1/lu(t,p) = 0.
Therefore, the matrix pair ( E, L) represent the basic structure of the market system (4.106), and any pair of interconnection matrices (E, L) can be generated from the pair (E;L) by replacing the unit elements of (E,L) by corresponding elements eu, 111 of (E, L). This fact is denoted by (E,L) E
(E,L).
252
Economics: Competitive Equilibrium
Stoc
hast ic stability of the equi libri um price p* = 0 of the mark et (4.106) is a convergence of the solution process p(t; t0 ,Po) start ing at time t0 and an initial price Po = p(t0 ) towa rd the equilibri um. The convergence is measured in terms of "stoc hast ic closeness" (e.g., in
the mea n, almo st sure, in probability, etc.), which in turn generates vario us noti ons of stochastic stability. In considerations of the mark et mod el (4.106), we are interested in establishing cond ition s for globally expo nent ial and connective stability in the mea n (Siljak, 197 7a)- that is, cond ition s unde r which the expected value of the dista nce betw een the price adju stme nt process p(t; t0 ,Po) and the equi libri um price p* = 0, which is deno ted by ®{ilp(t; to,Po)ll}, tends to zero expo nent ially as time increases for all initial data (t0 ,Po) E ~X~ and all inter conn ectio n matrices (E,L ) E (E,Z:). Mor e precisely, we state the following:
Definition 4.6. The equilibrium price p* = 0 of the market (4.1 06) is globally and exponentially connectively stable in the mean if there exist two positive numbers II and 'IT, independent of the initial conditions (to,Po). such that ®{llp(t; to,Po)ll}
is negative definite. Let us calculate fP{p)(4.106>as
=
±
J-1
2aDpJ
±±
±
+ }=1 Pi i-t 2aup; + 1=1 \)=1 I~ huPJ)
(4.116)
2 •
I+}
The negative dominant diagonal property of C in (4.113) is equivalent to 2ajj
+ bjj < 0, j
= 1, 2, ... , n, (4.11 7)
where 7T is a positive number. From (4.116) and (4.117), we get the inequality
eP(p)(4.t06>
< -wv(p)
'r:J(t,p) E
~X~.
'r:J(E,L) E (E,L.)
(4.118)
By applying Ladde's (1975) stochastic comparison principle to (4.118), we obtain
&{v[p(t; to,Po)]}
'r:J(E,L)
< v(Po)exp[-7t(t- to)] E
(E,L)
'r:Jt E
~.
'r:J(to,Po) E
~X~.
(4.119)
From (4.114) and (4.119), we get (4.111) with II = 1, '1T = !w. The proof of Theorem 4.10 is complete. We note immediately that due to the nonnegativity of the matrix B, it follows from (4.117) that the "random shocks are likely to introduce instability into the system", as observed by Turnovsky and Weintraub (1971) on a linear constant model of multiple markets. Theorem 4.10 extends this observation to nonlinear nonstationary market models. From
EcOnomics: Competitive Equilibrium
254
(4.117) we conclude that the matrix A+ AT should be sufficiently negative dominant diagonal to offset positivity of the matrix B. We also mention the fact that the negative dominant diagonal property is more conservative than the usual quasidomina nt diagonal property used in the previous sections to establish the stability of deterministic market models. This is so because we were limited in choice of Liapunov .functions that are twice differentiable, and had to choose the function (4.114) instead of (4.44). Stochastic-instability consideraao ns of competitive equilibrium were carried out by Siljak (1977a) using the stability framework of Theorem 4.10 and instability analysis of deterministic market models of (Siljak, 1976a). The stochastic-instability conditions are important in that they can provide a necessity part which is missing in our sufficient stability conditions. Two distinct sources of instability were exposed (Siljak, 1977a): The existence of strongly inferior goods for which the law of supply and demand does not hold [this is known as the Giffen paradox (Arrow and Hahn, 1971)], and the presence of random disturbances. Similar instability results have been obtained for stochastic ecological models and are presented in Section 5.9. The results outlined in this section can be improved in a number of different ways. It is possible to relax the constraints on the interactions among the individual markets and establish the weaker global asymptotic property of stochastic stability as shown in Section 5.10. In the same section, it will be shown how to construct stochastic hierarchic models of ecosystems which in tum can be used to represent markets of composite commodities in stochastic environmen t along the lines of deterministic hierarchic models considered in Section 4.8 of this chapter. Finally, it would be of interest to try various other kinds of stochastic stability (Tumovsky and Weintraub, 1971) in the context of diagonal dominance, and provide a less conservative stability criterion. This could open up a real possibility of including price expectations (Arrow and Nerlove, 1958; Arrow and Hurwicz, 1962; Tumovsky and Weintraub, 1971) in an essential way in our nonlinear and nonstationar y competitive-equilibrium models. 4.7. DISCRETE MODELS In constructing a dynamic model from static equations describing an equilibrium situation instead of differential equations one can use difference equations (Solow, 1952; Arrow and Hahn, 1971). In order to give a brief account of how discrete models arise in competitive equilibrium analysis, let us consider the algebraic equations ft
x;
=
~ au x1
J-1
+ b,
i
= 1, 2, ..• , n.
(4.120)
Discrete Models
255
If x 1 is the national income of the ith country (or region) of ann-country system, a11 the marginal propensity of thejth country to import from the ith country, a;; the marginal propensity of the ith country to consume domestic commodities, and b1 the autonomous expenditure in the ith country, then the equations (4.120) represent the internationa l trade system of Metzler (1950). On the other hand, if x 1 is interpreted as the output of the ith industry, au as the input of the ith commodity per unit output of the jth commodity, and b1 as the amount of the ith commodity in the "bill of commodities", then (4.120) is a Leontief (1936, 1948) open-end inputoutput system. Finally, when Xt is the price of the ith commodity, (4.120) can be used to describe the competitive-equilibrium condition discussed in Section 4.1. The linear equation system (4.120) is easily seen to be the static solution of the linear difference-equation system x(t
+ 1)
= Ax(t)
'+ b,
(4.121)
which is the dynamic model of Solow (1952) written in vector notation with time t taking integer values. The equilibrium solution x• of (4.121) for which x*(t + 1) = x*(t) for all tis obtained from (4.120) as x* =(I- Af 1b.
(4.122)
For x• to be economically meaningful, we must have x ;;;... 0, that is, all components Xt of the vector x should be nonnegative. Furthermore, in order to have a workable system (4.121), we require that the equilibrium be asymptotically stable. It turns out that stability of x• implies existence and nonnegativity of(/- Af 1• Thus, stability of x* and nonnegativity of b (that is, b ;) 0), imply x* ;) 0. It was shown by Metzler (1950) that if ail ;;a: 0, then the.necessary and sufficient condition for stability of x• in (4.121) is that A - I is a Hicks matrix, that is, I - A has all its principal minors positive. It is a well-known fact (e.g. Hahn, 1967) that stability of x• in (4.121) is equivalent to the condition that the eigenvalues of A are less than one in absolute value, that is, JA,(A)I 1, i = 1, 2, ... , n. Now, Metzler's result is an application of the classical Perron-Frobenius theorem (Gantmache r, 1960; Seneta, 1973): If A is a nonnegative matrix (that is, aii ;;a: 0 for all i,j), then there exists an eigenvalue Ap(A) of A, the "Perron root of A", such that Ap(A) ;) 0 and IAt(A)i ~ Ap(A) for all i = 1, 2, ... , n. We see that I- A has all offdiagonal elements nonpositive (that is, au 0, i =I= A and from the Appendix we recall that positivity of principal minors of I - A is equivalent to the existence of a positive eigenvalue Am(/- A) of I - A such that the real part of any eigenvalue of I - A is at least Am(/ - A). Now, Ap(A) = 1
to)]
't/tk E ~
(4.126)
for all (to,Xo) E 5'x ~. This kind of stability for the system (4.124) can be established by the following Grujic-Siljak (1973) discrete version of the classical K.rassovskii (1959) theorem established for continuous systems:
Theorem 4.11. The equilibrium x• = 0 of the system (4.124) is exponentially stable in the large if and only if there exists a continuous junction v: 5'X ~ --+ '!lt+ and numbers T/I > 0, 112 > 0, 0 < T/J < 1 such that
TJdlxW
where p. [XM(H)X;;; 1(H)]11 2, and AM(H) and Am(H) are the maximum and the minimum eigenvalue of H, respectively.
258
Economics: Competitive Equilibrium
Using {4.126), from {4.128), we get P(1t; x) ;) xT(It; It, x)Hx(lt; It, x) ;) A,.{H) llxiF
{4.130)
and
The choice "lt = A,.{H),
'112 = miJlAM(H)
{4.132)
implies the first two inequalities of {4.127). Making use of the solution property {4.125), we derive v[tt+t, x{lt+t; lo, Xo)]
=
v{lk+t, x[lt+J; It, x(lti to,xo)]} t+•
A community is connectively stable interconnection matrices.
if
it is stable for all admissible
By "admissible" interconnect ion matrices E we mean such as are generated by the fundamental interconnect ion matrix E of the community. In the context of connective stability, for example, one can consider
274
Ecology: Multispecies Communities
structural perturbations whereby a certain speci es (or group of species) is disconnected from and again connected with the rest of the community. Such was the case with a community of marine invertebrates reported by Paine (1966), where removal of a species caused the community to become unstable and shrink from a fifteen to an eight species community. The structural perturbations of this kind can be readi ly described by interconnection matrices. Removal (or destruction) of the kth species from the community (5.1) is represented by e;k = 0,
i = 1, 2, ... , n,
e~g
j = 1, 2, ... , n.
= 0,
(5.4)
If we show that the community (5.1) is conn ectively stable, then the struc
tural perturbation (5.4) cannot produce instability of the equilibrium. Structural perturbations that change the number of species were analytically considered by MacArthur (1970) assuming a special symmetric form of the community matrix. Not only do we not requi re this special symmetric form of A, but also we will waive the assumption of constancy of the system (5.1), thus allowing for time-varying structural perturbations which take place at disequilibrium populations and give rise to a time-dependent community matrix. Let us now present the conditions for the connectiv e stability property of the community (5.1). We recall that an n X n matri x A = (au) was called quasidominant diagonal by McKenzie (1966) if there exists an n-vector d = {d.,d2 , ••• ,dn} with positive components (d 0) such that
>
n
a)iaill
> i-1 ~ d;jaul,
j
= 1, 2, ... , n.
i+J
(5.5)
As shown by McKenzie, for a negative diago nal matrix. A, the inequalities (5.5) are sufficient for stability (all eigenvalue s of A have negative real parts; Theorem A.ll) . That is, if the community matrix A is negative and quasidominant diagonal, the community (5.1) is globally asymptotically stable. The condition (5.5) includes the usual diagonal dominance n
!aiii >~!au!. i-1
I+J
j = 1,2, ...
,n,
(5.6)
as a special case when d = {1, 1, ... , 1}. Altho ugh the condition (5.6) is easier to interpret in the context of the system (5.1), it is more restrictive than that of (5.5). It should be noted, however, that for large classes of
Linear Constant Models
275
,,,... __
nonlinear time-varying models we shall be able to replace the co~dition (5.5) with readily interpretable inequalities involving only a,·s. Now, if we establish (5.5) for a given E, then it will be satisfie d for any E generated by that E. To see this, we use (5.2) to rewrite (5.5) as ~l-a1 +
•
e.ua.ul > I-I l: d,!evaul.
j
=
I, 2, ... , n.
(5.7)
l+j
Obviously, if (5.7) holds for E, it will hold for any E which is obtain ed from Eby replacing unit elements of E by· the numbers eii which lie between zero and one as defined in (5.3). Specifically, (5.7) will hold in case of the structural perturbation (5.4), and removal of the kth species from a stable community described by (5.1) will not cause instability. From the above analysis, it is possible to draw conclusions about the tolerance of stability for increasing complexity of the system: So long as the strength of interactions does not violate the inequalities (5.5), the stability of the community is assured. This conclusion is compatible with the empirical evidence noticed by Margalef (1968): "It seems that species that interact feebly with others do so with a great number of other species. Conversely, species with strong interactions are often part of a system with a small number of species ... ", and it does not contradict the statement of Levins (1968) that "there is often a limit to the complexity of systems". Since the conditions (5.5) are only sufficient for stability, they cannot be used to test for instability and verify entirely the Gardner-Ash by and May experiments. The conditions (5.5), however, are both necessary and sufficient for stability of (5.1) if the interactions obey the rules "friends of friends are friends, enemies of enemies are friends, and friends of enemies and enemies of friends are enemies". This situation is impor tant in the analysis of substitute-complement multiple markets of c~mmod ities or services and is known as the Morishima case (see Siljak, 1975a, 1977a).In the context of the community model (5.1), it means that the comm unity is composed of species in either competitive or symbiotic mutual relatio nships according to the rules given above. A special case of the Morish ima matrix (Definition A.13) is that of the Metzler matrix (Definition A.lO) used in Chapter 4 to represent the case when all commodities are substit utes. A community model (5.1) described by a Metzler matrix is of little significance in population dynamics, since it represents a rare "gross mutualism" among species. It should be noted, however, that in the case of the Metzler matrix A, the conditions (5.5) are necessary and sufficient for stability (Theorem A.IO), and if they were violated, the community would suddenly become unstable. This conclusion certainly agrees With the May's (1973a) speculation that "predator-prey bonds are more common than symbiotic
., ' ')
..;-
.....•......
276
Ecology: Multispecies CommUDities
ones", and via the Morishima matrix the conclusion can be extended to his stateme nt that "compe tition or mutuali sm between two species is less conducive to overall web stability than is a predato r-prey relationship". · Finally, in the context of the model (5.1) we should mention that if the commun ity matrix A is negative diagonal, then the conditions (5.5) imply that it is D-stable, that is, the product DA is stable for any positive diagonal matrix D = diag{d11 ,d22 , ••• ,d..,} in which dtt 0 for all i and d11 = 0 for all i .P. j. This fact demons trates that stability is highly reliable in communities descn'bed by negative quasido minant diagona l matrices, and that they are stable under arbitrar y "speeds of adjustm ents" of populations in the community. Unfortu nately, neither D-stability nor the closely related qualitative stability which was applied to populat ion dynatnics by May (1973a) can be used for other than linear constan t models. This is not true, however, for the connective-stability concept outlined in Chapter 2, which can be used to establish the stability of nonline ar time-varying model ecosystems at the expense of more refined analysis (~iljak, 1975a).
>
5.1. NONLINEAR MODELS: LINEAR IZATIO N Interact ion in multispecies communities is a highly nonlinea r and nonstationary affair. For this reason, it is not surprising that the first populati on model of Lotka (1925) and Volterra (1926) was a system of nonlinea r differential equations. In Section 1.6, where the model was considered, it was shown that there are two populat ion equilibria. This fact is good enough to discourage any attempt to approximate large fluctuations of populations by means of a linear model which is limited to a single equilibrium state. Nevertheless, local behavior of the commun ity in the neighbo rhood of the equilibrium populati ons, can be successfully predicte d by standard linearization techniques (Allen, 1975; May, 1973a). A simple demons tration of this fact in the case of the Lotka-Volterra model, was given in Section 1.6. To describe a general linearization mechanism, let us assume that a commun ity is described by a nonauto nomous differential equation
x = j(t,x),
(5.8)
where x = {x~ox2, ... ,x,.} is again the populati on vector and the function f(t, x) has continuo us first derivatives. Then we set
x(t)
= x* (t) + y(t),
(5.9)
where x*(t) is a solution of Equatio n (5.8), whose neighbo rhood is of our concern. As shown at the end of Section 2.2, we can assume without loss of
;
277
Nonlinear Models: Linearization
generality that x• (t) is an equilibrium, that is, a constant solution of Equation (5.8) such _that
f(t, x*) = 0
for all t.
(5.10)
Expanding j(t, x) about x = x• in a Taylor series expansion and using the transformation (5.9), we get
y = A(t)y + g(t,y),
(5.11)
I,._,..
(5.12)
where
A(t)
= aj(t, x)
ax
is the well-known n X n Jacobian matrix, g(t,y) represents the higher-order terms in the Taylor series expansion, and g(t,O) = 0, ag(t,O)jay = 0. Now, for sufficiently small initial values x0 = x(to), the solutions x(t; t0 , x 0 ) of the nonlinear equation (5.8) can be studied by considering the corresponding solutions of the linear equation
x=
A(t)x,
(5.13)
which is obtained from (5.11) by removing the higher-order terms and replacing y by x. When we have an autonomous equation
x = j(x),
(5.14)
the above linearization process yields the linear constant system
x =Ax,
(5.15)
where A is ann X n constant Jacobian matrix defined by surpressing tin
(5.12). To illustrate the power and limitation of the linearization, let us consider the well-known Verhulst-Pearllogistic equation (Pielou, 1969),
x = (a -
j3x)x,
(5.16)
which describes the growth of a population x(t) in a restricted environment. 0, and set We assume that a, fJ
>
=0 states xt =
(a- {Jx)x
(5.17)
0 ~d xi = a/{J. Since at to determine the two equilibrium least one member of the community is needed for it to grow, the growth
278
Ecology: Multispecies Communities
rate is zero at xi = 0. For a small initial population x 0 = x(O), we linearize Equation (5.16) about xi = 0 and get the linear equation
x =ax,
(5.18)
which has the solution
x(t)
=
x0 eat.
(5.19)
Therefore, for a small population, there is no interference among its members, and the population grows in a Malthusian manner at the exponential rate a. Since the environment is restricted, the population growth is limited by a shortage of resources, and it reaches the saturation level xi = a/ f3 set by the carrying capacity of the environment. If the initial population x 0 is larger than the saturation level xi = 0, the population will decrease, approaching the equilibrium xi asymptotically from above. To see this, we use the transformation (5.20)
x =xi+ y and rewrite (5.16) as
y
=
(-a- f3y)y.
The linearized version of (5.21) around y•
y
=
=
(5.21) 0 is
-ay,
(5.22)
which has the solution
y(t) = Yo e-at.
(5.23)
Both predictions made by linearization-that the equilibrium xi = 0 is unstable, and that the equilibrium xi = a/f3 is stable-can be confirmed by solving the original logistic equation to get
x(t)
=
~
[1 + ( a/f3x~ )e-« Xo
r
1
1
(5.24)
Two typical solutions (5.24) are shown in Figure 5.2. Although the linearization predicted the stability properties of the two equilibria, in each case the predictions were correct only locally, in the immediate neighborhood of the equilibrium states. For large deviations from either of the equilibrium states, which exceed the distance a//3 between them, the predictions are ,clearly inconsistent. Therefore, for large perturbations of populations and global analysis of community stability, nonlinear analysis is necessary.
279
Nonlinear Matrix Models x(t)
t FIGURE 5.2. Logistic curves.
5.3. NONLINEAR MATRIX MODELS There are several ways to bring the stability analysis of the preceeding section closer to reality, and a step in that direction is to consider the nonlinear matrix model .i
= A(t, x)x.
(5.25)
The model (5.25) is an obvious generalization of the linear constant linear model .i
(5.26)
=Ax,
in which the constant elements iiy of the n X n matrix A by nonlinear time-dependent functions
au = au(t,x).
=
(iiy) are replaced (5.27)
Equation (5.26) is the same as (5.1), and the bar on A is used to distinguish it from A(t,x), since we will often omit the arguments (t,x) of A for simplicity. Stability analysis cannot be carried out in algebraic terms for the nonlinear matrix model (5.25) as it can for the linear model (5.26), and we need to use the Liapunov direct method outlined in Chapter 2. It is, however, of fundamental importance to note that we shall be able to deduce various stability properties of the nonlinear time-varying model (5.25) from stability properties of the linear constant model (5.26) established by simple algebraic criteria involving only the elements iiu of the matrix A in (5.26). It
Ecology: Multispecies Communities
280
is fair to say, at the same time, that this powerful result can be accomplished only at the expense of a more refined analysis. • In (5.25), x(t) E ~"is again the population vector, and then X n matrix function A: ~X ~ ~ ~ is defined, continuous and bounde d on ~X ~. so that the solutions x(t; t0 , x 0 ) of (5.25) exist for all initial conditions (t0 , x 0 ) E ~X ~ and t E ~- The symbol '5" represents the time interval (T,+oo), where Tis a number or the symbol -oo, and~ is the semi-infinite time interval [t0, +oo ). In the following analysis, we will consider stability of the equilibrium populat ion x* = 0 of the model (5.25). If A(t, x*)x* = 0 for all t E ~and x* =F 0 is of interest, then we can define the nonlinear matrix function B(t,y)y A(t,x* + y)(x* + y) and consider the equation j = B(t,y)y instead of (5.25), where y* = 0 represents the equilibrium x* under investigation. Although it is always possible in principle to use the transformation x = x* + y introduced in (5.9) and place the equilibrium at the origin of ome to handle. ~. the new transformed equation is usually more cumbers in In order to include the connective property of stability the analysis, we write the elements a11 (t, x) of A(t, x),
=
i =j, i =F j,
(5.28)
where the functions cp1(t,x), CJ>u(t,x) E C(5"x ~). In (5.28), we consider the elements eu = e;(t) of then X n interconnection matrix E = (e 11 ) as functions of time such that eu(t) E C(~) and restricted only by the condition e;;(t) E [0, 1]. This represents a major improvement over the use of E in Section 5.1, since structural perturbations can take place at the disequilibrium populations without any prescribed pattern -the elements e11 (t) may be arbitrary functions of time without any statistical description. This is remarkable in that we allow the interactions among the species in a community to vary arbitrarily in strength during the populat ion adjustment process without any a priori specification of such variations except that they are bounded. To derive necessary and sufficient conditions for connective stability of the equilibrium x* = 0 of (5.25), we need further specifications of the elements au(t,x) of the matrix A. We assume that the functions cp1(t,x), CJ>u(t, x) in (5.28) are bounde d functions on ~X ~ and that there exist 0 such that 0, au positive numbers a,
>
a1 = ~ cp1(t, x), and a1 >ail.
>
au =
suplcpu(t,x)l, !l)dl'
i,j
==
1, 2, ... , n,
(5.29)
2Sl
Nonlinear Matrix Models
the syste m (5.25) to be absolute Since we wan t the conn ectiv e stabi lity of conn ectiv e stabi lity of x* = 0 in (Def initio n 2.8), we need to estab lish the x) satisfying the cond ition s (5.25) for any set of func tions !p;(t, x ), 'Pii(t, classes of functions: (5.29). For this purpose, we define the following
= {~P1 (t,x): !p;(t,x) ;;;> a1}, ~ii = {cpy(t, x): I'Pii(t, x)l < aii}, ~~
(5.30 )
au ;;;. 0. As in . the case of the elements eofof the wher e a11 ;;;. 0, a 1 the to specify the precise shap inter conn ectio n matrix, we do not have in the com mun ity so long as the nonl inea r inter actio ns amo ng the species stic assu mpti on, since mos t of the inter actio ns are boun ded. This is a reali by Pielo u (1969), Goel, Mait ra, a.nd fami liar mod els of ecosystems surv eyed this prop erty. In particular, the Mon troll (1971), and May (1973) have ratio n in preda~or attac k capa bilrestrictions (5.30) inclu de the cases of satu and its "swi tchin g" mec hani sm. ity, chan ges in pred ator searc hing beha vior, de a.ny sign patte rn of the inter acFurt herm ore, the cond ition s (5.30) inclu (com petit ive-p reda tor-s ymb iotic -sations, thus allowing for the "mix ed" chan ges of a pred ator to a prey of prop hitic ) communities, as well as for cons train ts in (5.30) are illus trate d anot her species over a time inter val. The in Figu re 5.3. ity is stable, we wou ld also like Besides conc ludin g mere ly that a com mun popu latio ns to their equi libri um to estim ate the rate of conv erge nce of ed in this secti on imply exponenvalues. The stability cond ition s to be deriv the equi libri um faster than tial stability, that is, the popu latio ns appr oach an expo nent ial (Def initio n 2.7). e stability discussed abov e, we Com binin g the two aspe cts of conn ectiv com e up with the following:
>
system (5.25) is absolutely and Definition 5.1. The equilibrium x• = 0 of the two positive numbers II and 7T exponentially connectively stable if there exist that independent of the initial conditions (t0 , x0 ) such (5.31) 'tit E '5Q llx(t; to,xo)ll < IIIIxollexp[-7r(t- to)] E ~ii• and all interconnection for all (to, xo) E I!T X ~. all 'P1 E ~~. 'Pii matrices E E E. by Defi nitio n 5.1, we deno te To estab lish the kind of stabi lity specified with elem ents by A= (a0 ) the cons tant n X n matr ix i =j, i
* j,
(5.32 )
EcOlogy: Multispecies Communities
282
0
W#~.d'#.&'.d'£&%2
'/&/.£££#~
au
FIGURE 5.3. Nonlinear constraints.
where the eii's take on values 1 or 0 according to the fundamental interconnection matrix E. Since the elements au of X are such that
-{ 0,
aii
0,
i =j, i .f=j,
(5.33)
X is a Metzler matrix regardless of the sign of interactions in the original
nonlinear matrix function A(t,x). The Metzler matrix was introduced in mathematical economics and was used considerably in Chapter 4. We recall
283
Nonlinear Matrix Models
that a Metzler matrix A is stable if and only if it is quasidominant diagonal (Theorem A.10), that is, satisfies the conditions (5.5). This is equivalent to saying that A is a Hicks matrix, that is, the sign of its kth-order principal minor is (-1)k. This is also a well-known result in economics (see Chapter 4). The Hicksian property of a Metzler matrix ~ and thus stability, is equivalent to the Sevastyanov-Kotelyanskii inequalities (Theorem A.9)
(-1t
au
a,2
a,k
a21
an
a2k
ak,
ak2
akk
>O,
k
=
1, 2, ... , n,
(5.34)
whereby only the signs of the leading principal minors of A need be tested for stability. For these properties of Metzler matrices see the Appendix. We prove the following: The equilibrium population x• = 0 of the community (5.25) is absolutely, exponentially, and connectively stable if and only if the n X n constant matrix A defined by (5.29) and (5.32) is a quasidominant diagonal matrix.
Theorem 5.1.
Proof. To prove the "if" part of Theorem 5.1, let us consider a decrescent, positive definite, and radially unbounded function v: '31!' -+ lilt+, n
v(x) = ~ d1!x 1!,
(5.35)
i=l
which was used by Rosenbrock (1963), as a candidate for Liapunov's function. Here d = (d1 , d2 , ••• , dnf is a constant, as yet unspecified n· 0). vector with positive components (d
>
We calculate the derivative n+v[x(t)] of the function v(x) with respect to (5.25). Since the derivative of !x 1(t)! need not exist at a point where x 1(t) = 0, it is necessary to use the right-hand derivative n+!x1(t)l. For this purpose, a functional a1 is defined as
>
0, if X;(t) if X;(t) = 0 0, if X;(t)
0,
or if x 1(t) = 0 and 1(t) and .i1 = 0, or if x 1(t) = 0 and i 1(t)
(5.36)
< 0,
Ecology: Multispecies Communities
284
n+v(x)
=
n
~ i-1
d1a1i 1
n
n
= ~ d1 a1 ~ i=l
OijXj
j=l
(5.37)
Since X is quasidominant diagonal, there exist a vector d 0 such that number '1T
>
n
la.ul - ~-~ ~
1-1 ;.,j
ddaul
> 'TT,
j = 1, 2, · · ·,
>0
n.
and a
(5.38)
From (5.37) and (5.38), we get the differential inequality
'Vt E '5;
'f/p E '3\,+,
(5.39)
valid 'Vrp1(t,x) E II>;, 'Vtpy(t,x) E ll>ii, 'VE E E. By integrating (5.39) we get
P[x(t)]
< P(x )exp[-'TT(t- t
0 )]
0
'Vt E
~,
'V(t0 ,x0 ) E '5'X 'ffi!'. (5.40)
Using the well-known relationship between the Euclidean and absolutevalue norms, llxll lxl ni1 2 IIxll, we can rewrite (5.40) as
x(c + ±b x 1
1
J-1
11 1 )
=
0.
(5.55)
From (5.55), we immediately conclude that the ongm x = 0 of the population space is an equilibrium for the community (5.54). This equilibrium is not interesting, so we assume that x ¥= 0. In that case, (5.55) becomes a matrix equation
c+Bx=O,
(5.56)
where B = (b 11 ) is ann X n constant matrix and c = {c~oc2, ... ,c.} is annvector.
Ecology: Multispecies Communities
288
We assume that there exists an equilibrium population x• positive solution
> 0 as
a
(5.57) of Equation (5.56). This assumption is consistent with consideration of community stability, but it is not simple to express the conditions for its validity in terms of the B. In the special case when the matrix B has all offdiagonal elements nonnegative (that is, it is a Metzler matrix), then as shown in the preceeding chapter, stability of B implies (and is implied by) 0 exists. In other words, it is possible to show (see the fact that x* Appendix) that for a Metzler matrix B, the diagonal-dominance condition (5.5) is equivalent to saying that - B- 1 is nonnegative element by element (Theorem A.2 and 4). Since B- 1 cannot have a row of zeros, positivity of c implies positivity of x•. Nonnegativity of the off-diagonal elements of B implies a "gross mutualism" among the species in the community, which is of limited interest in population ecology. Considerations of population equilibria in the context of general interactions among species can be found in the book by Goel, Maitra, and Montroll (1971). One can certainly benefit from the similar considerations in the framework of the general economic equilibrium theory discussed in Chapter 4. 0, we use the standard transformaTo investigate the stability of x* tion x = x• + y of (5.9) and rewrite the equations (4.54) as
>
>
n
j; = (y;
+ x!) L huYJ>
i = 1, 2, ... , n.
(5.58)
j=l
The equilibrium y• = 0 of (5.58) corresponds now to the equilibrium 0 of (5.54), and we can readily identify the model (5.58) as our general x• matrix model (5.25), where the n X n community matrix A(y) has the coefficients au(Y) specified as
>
aii ( Y) =
-b;;(y; + x!), • { eifbu(Y; +X;),
i =j, i :Fj.
(5.59)
Therefore, the nonlinear functions of (5.28) are
i,j = 1, 2, ... , n,
(5.60) i :F j.
We should note here that the functions 0, are satisfied on the open region Yo= {y e ~= IY,I < fJ,xr, i = 1,2, ... ,n}
(5.62)
defined by (5.45) when~ and x are replaced by Yo andy. In (5.62), numbers that measure the size of the region Yo, and are such that 0
< fJ, < 1,
=
i
1, 2, ... , n.
From the diagram of the function y 1 + · calculate the numbers a.1, a.lj as
fJ1 are (5.63)
xr given in Figure 5.4, we i,j
=
1, 2, ... , n. (5.64)
Now an estimate Y for a stability region Y of the ecomodel (5.58) can be obtained by imbedding a Liapunov function n
PJ(y) YJ+
=
~
I= I
d;IYd
(5.65)
xt 2xt
r-----------1
I
I
I I I
I
I (I +,B,lAj* 'I ( ~------'----'--I-------.F--------1S~ Jj
I~ I I I I I I I I I
I
I I
I I I
I I I I
I
I I I
I
I I I I I I
*)
+AJ
~----{-----l--------~ inf
lly{t; to,Yoll
i =j,
(5.83)
i =I= j,
>
>
and that there exist numbers a 1 0, au 0 such that a1 a 11 and the constraints (5.29) are satisfied by the functions q~1 (t, x), fPu(t, x). Then we specify the elements aii of the matrix X= (au) by the corresponding matrix
E.
We prove the following:
Theorem 5.4. The equilibrium population x*
= 0 of the community (5.25) is completely and exponentially connectively unstable if the n X n constant matrix -X= (-aii) defined by (5.29) and (5.32) is a quasidominant diagonal matrix.
296
Ecology: Multispecies Communities
Proof. Let us consider again the function ll(x) of (5.35), and let us compute the function D+P(x) with respect to (5.25) as
D+v(x)
= =
•
~ i=l
•
I
i-1
=
•
~
J-1
d1 a1 i 1 d,a,
• I
J=l
a11 x1
d,;ajxJaii
• • + ~ x1 ~ d1 a;aii J-1
(5.84)
1""1
'""1
Since -A is a Metzler matrix and has a quasidominant diagonal, there exist a vector d 0 and a number 7T 0 such that
>
>
j
=
I, 2, ... , n.
(5.85)
Therefore, from (5.84) and (5.85) we get
Vt E ~0 , Vv E ~+• VE E E
(5.86)
Integrating (5.86) and manipulating the result as we did (5.39), we get the inequality (5.82) with (5.87) where dm = min;d1, dM = ln.a.Xjd1• This completes the proof of Theorem 5.4. It is of interest to note that the quasidominant diagonal property of the matrix -A can be a necessary and sufficient condition if the instability in Theorem 5.4 is required to be absolute as in Theorem 5.1. If instability is going to hold for all qJ1(t, x) E CI> 1~ 'Pu(t, x) E CI>p, it must hold also for 'Pt(t, x) = a;, 'Pu(t, x) = -a,. which makes the system (5.25) with A(t, x) = A unstable, since -A is a quasidominant diagonal matrix. Theorem 5.4 has little application to population models, since it implies that none of the species in the community is density-dependent. That is, the diagonal elements a11 (t, x) of the matrix A(t, x) are all positive. There is a possibility of relaxing the positivity conditions placed on all a11's and requiring only that one of the populations be not densitydependent. Then, however, we require positivity of all off-diagonal coefficients ay for that population. More precisely, we have the following:
Environmental Fluctuations
297
Theorem 5.5. The equilibrium population x* = 0 of the community (5.25) is completely connectively unstable if for some i E {1, 2, ... , n} there exist numbers a, > 0, au :> 0 such that the coefficients au(t, x) of the n X n matrix A(t,x) defined by (5.84) are such that the conditions
cp1(t,x)x,
:>
a,x, cpq(t,x)x1
a,x, + ~ eu(t}aulxJI
~
a,x1
(5.89)
i-1
'V(t,x) E '3"X A, 'VE E E.
By integrating the last inequality, we get
lx,(t; to, xo)l ~ x,oexp[a;(t- to)}
'Vt
E
'50, 'V(to,Xo) E '3"XA,
'VEE E,
(5.90)
>
where x 10 = x1(t0 ) 0. The inequality (5.90) implies complete connective instability of the equilibrium x* = 0 in (5.25), and the proof of Theorem 5.5 is completed. 5.7. ENVIRONMENTAL FLUCTUATIONS The ecomodels considered so far have described only the internal interactions among the species in a community which is "closed". That is, there were no terms in the models that would explicitly reflect influences of the environment on the affairs that go on in the community. Among other things, such terms may represent changes in the climate conditions, pollution effects, application of pesticides, etc. A problem of interest in this context is to find out how much of the environmental fluctuations can be tolerated by stable communities. Again, a good place to start is a linear model
x =Ax+ b(t,x),
(5.91)
where A = (au) is an n X n constant matrix and b(t,x) is continuous for small llxll and t E 5o. and b(t, 0) = 0 for all t E '5" so that x• = 0 is the
298
Ecology: Multispecies Communities
equilibrium of (5.91). As expected, the nature of solutions of Equation {5.91) depends to a great extent on the properties of the linear model
x =Ax,
{5.1)
especially when the perturbation function b{t,x) is dominated by the function Ax. There are numerous results {e.g. Coddington and Levinson, 1955; Hale, 1969) which relate the stability of the system {5.1) to that of {5.91), and they can be traced back to Liapunovs work. The basic result is that if the matrix A has all eigenvalues with negative real parts and llb{t, x)ll
=
o{llxll)
{5.92)
as llxll ~ 0 uniformly in t E '5o {that is, llb(t, x)ll/llxll ~ 0 uniformly in t with llxll ~ 0), then the equilibrium x* = 0 of {5.91) is exponentially stable. This result can be established by using Liapunov's direct method. We recall from Section 1.8 that when the linear system {5.1) is asymptotically stable, there is a Liapunov function P{x)
= xTHx,
{5.93)
with the estimates 11tllxll ' P{x) ' '112llxll, P{x)(s.J) '
-'ll311xll,
(5.94)
llgrad P{x)ll ' 'IJ4, where the 11/s are positive numbers calculated using the symmetric positive definite matrices H and G of the Liapunov matrix equation {Section 1.8)
ATH +HA =-G.
(5.95)
Taking the total time derivative v{x)(5.91 > of P{x) along the solutions of the perturbed Equation {5.91), we get P(X)(5.9J)
=
P(X)(S.l) +[grad P{x)fb{t,x)
'
-'ll311xll + 'll4llb{t, x)ll
'
-'ll311xll + 'll4o{llxll).
Therefore, for sufficiently smallllxll, we can find a number 11 from {5.96) we obtain
(5.96)
> 0 such that (5.97)
and asymptotic stability of x*
=
0 in (5.91) follows. When we assume that
,,:.
r' Environmental Fluctuations
299
llb(t,x)ll 0 can be chosen arbitrarily small. This is because from (5.107) we get
11[x(t)]
< 7T-tp + ll{x0)exp[-'1T(t- t 0)]
'lit E ~o.
(5.108)
which is valid for all to E ~ x 0 E ~·. and all E E E, I E l. Imitating the development of {5.40) in the proof of Theorem 5.1, we obtain the inequality {5.104) with {5.109) and 7T given by the quasidominant diagonal condition on A,
Ecology: Multispecies Communities
302
1-x,*
x1
I I I IL ______-x* J ________ _ FIGURE 5.7. Ultimate boundedness region .
• ialil - ~- 1 ~ d;ja11 1 > 'TT, i=1
j = 1, 2, ... , n.
(5.38)
i+j
The region ~ ::J ~is now given by Equation (5.105). Since (5.108) is valid for all E E E, I E l, so are II, 'TT, p., p, and thus the region ~ This proves Theorem 5.6. It is now simple to compute from (5.107) an upper estimate t1 =
t0
+ 'TT-ltn[e- 1(w0 - ,B)]
(5.110)
of the time necessary for the population process to enter the computed region i For the case n = 2, the two regions~ and~ are shown in Figure 5.7. The case of a linear system
x=
Ax + b(t)
(5.111)
is of special interest. We assume that A is a constant negative diagonal matrix, and that b(t) has the property that for some t 1 E '5"0
b(t)
=
c
(5.112)
where c E tBt~ is a constant vector. Then if the matrix A is a quasidominant diagonal matrix, we have (5.113) and the convergence is exponential. That is, the population processes track the equilibrium levels determined by the environmental fluctuations. 5.8. STOCHASTIC MODELS: STABILITY The effects of unpredictable changes in weather, resources, etc., on multispecies communities cannot be satisfactorily estimated from the deterministic models. This fact was recognized long ago by several authors
Stochastic Models: Stability
3-03
(Erlih and Birch, 1967; Levins, 1969; Lewontin and Cohen, 1969), who proposed stochastic models as better descriptions of the real community environment. Stability as a central aspect of stochastic multispecies models was considered by May (1973a, b). In order to contrast the two kinds of models, May showed how various notions of stability are related in deterministic and stochastic environments. The mathematical apparatus that May applies in his studies of random disturbances in otherwise linear constant multispecies models is the Fokker-Planck-Kolmogorov equation, also called simply the diffusion equation. Since our analysis in this section goes beyond linear constant models, ·the intractability of the diffusion equation makes the Liapunov method via Ito's calculus a much more attractive framework for investigating the nonlinear phenomena in stochastic environments. This other approach was developed by Ladde and Silj ak (1976a, b), and the results are the subject of this section. Our stability criterion is again the diagonal-dominance property of the community matrix. It will be shown that this property guarantees stability under both structural and random perturbations. Moreover, it is an ideal mechanism for establishing the trade-off between the degree of community stability and the size of environmental stochastic fluctuations that can be absorbed by stable communities. Before we start our analysis of community models in the framework of the Ito differential equations and Liapunov direct method, it is of interest to list the appropriate references. Besides the book by Kushner (1967) on this subject, there is a survey paper on stochastic stability by Kozin (1969). These references should be supplemented by the original articles written by Bertram and Sarachik (1959), Kats and Krasovskii (1960), Bunke (1963), Khas'minskii (1962), and Kats (1964). Introductory tutorial papers on lto differential equations as models of physical systems have been written by Mortensen (1969) and Papanicolaou (1973). The results presented in tllis section are based on the work of Ladde, Lakshmikantham, and Liu (1973) and of Ladde (1974, 1975), devoted to the development of the stochastic comparison principle. Let us start our stochastic stability analysis with a linear constant equation of the Ito type, (5.114) = Axdt + Bxdz, where x = x(t) is ann-vector x = {x1 , x 2 , ••• , xn} the components of which represent the populations. Then X n matrix A = (au) is a constant commudx
nity matrix that reflects the interactions among the populations in the community. Here z = z(t) is a scalar function, representing the random environmental fluctuations, which is a normalized Wiener process with (5.115)
Ecology: Multispecies Communities
304
where & denotes expectation, that is, averaging across the statistical ensemble. The constant n X n matrix B = (b 11 ) is the diffusion community matrix which specifies how the random variable z(t) influences the community. Stochastic stability of the equilibrium x• = 0 of the community model (5.114) means convergence to equilibrium of the solution process x(t; t0 , x 0 ) starting at time to and the initial population vector x0 = x(t0 ). The convergence is measured in terms of "stochastic closeness" (e.g., in the mean, almost sure, in probability, etc.), which, in tum, generates various notion of stochastic stability. In case of the linear model under consideration, we are interested in establishing conditions for global asymptotic stability in the mean (Ladde and Siljak, 1976c)-that is, conditions under which the expected value of the distance between the solution process and the equilibrium &{\lx(t; t 0 ,x0 )11} tends to zero as t ~ oo for all initial data (to, xo). We assume that each population in the community (5.H4) is densitydependent, which is a realistic assumption (Tanner, 1966). This means that all diagonal elements a11 of the community matrix A are negative. We make no assumption on the off-diagonal elements a11 of A, thus allowing for "mixed" (competitive-predator-symbiotic-saprophitic) interactions among species in the community. The coefficients d11 of the community diffusion matrix B can have arbitrary signs, which allows a good deal of freedom in the stochastic interactions among species and their environment. To establish stochastic stability of the equilibrium of the chosen model, we will use the Liapunov direct method and the comparison principle. We propose the function n
v(x)
=
~
(5.116)
x'f
i=-1
as a candidate for Liapunov's function for the system (5.114). Using Ito's calculus we examine the expression
av(x)
ev(x) =-a-Ax+ X
I ~
2 iJ£..J
1
a2 v(x)
-X,.aaX1s 11 (x),
(5.117)
where av/ aX = (av/ aXi> avj aX2, . . . , av/ aXn) is the gradient Of P(X), a2 vjax 1 ax1 is the (i,j)th element of the Hessian matrix related to v(x), and the su's are the elements of then X n matrix S = BxxT BT. To establish the stability of the equilibrium, we observe that v(x) is a positive definite function, and demonstrate that ev(x) is negative definite. Let us calculate ev(x) as
ev(x)
= =
±2x/ ±a x ) + ±{±b11x1) 1-1 \i-1 1 1-1 \J~1
2
11
~. 2a11 x] + ~. x1 ~. 2a11 x; + ~. (~, b;;xJY·
(5.118)
3{)5
Stochastic Models: Stability
Let us define the elements of the matrices X= (ay), B
_ {-laiil, ay = lay I,
i=j, i =I= j,
(by) as
=
•
by= lbyl ~ lbikl
(5.119)
k-1
and use the inequality 2lx1xjl ~ xt + xJ to rewrite (5.118) as
eP(x)
~
f
[(ail+
;=I
f
ay) + (ail+
'/;)
f Qj;) + f by]xJ.
'i+)
(5.120)
•-I
Our central interest is to estimate how much of random perturbations can be absorbed by the deterministic stable version of the model (5.114). Therefore, we assume that the community matrix A is stable. Since A is negative diagonal, in view of relations (5.119) and (5.120), we assume su.ch property of A by the diagonal-dominance conditions
•
•
ail + ~ ay ~ -'IT.,
ail + ~ Qj; ~ -'IT"
i,j
i=l
I= I i ..j
= 1, 2, ... , n, (5.121)
i ..j
where 'ITc and 'IT, are positive numbers. If the diffusion matrix B is zero [which implies that the model (5.114) ignores the random disturbances] and all by's are zero due to (5.119), then
b(x)
~
-('ITc + 'IT,)P(x).
(5.122)
Integrating the inequality (5.122) and taking into account the definition (5.116) of the function P(x), we arrive at the inequality
llx(t; to,xoll ~ llxoll exp[-H'ITc + 'IT,)(t- to)],
t ;> t0 ,
(5.123)
which establishes the global exponential stability of the equilibrium. We would like to obtain conditions on the matrices A and B which would guarantee stability in the stochastic sense (in the mean), and thus establish the tolerance of random fluctuations by stable community models. The conditions are expressed in terms of the matrix A+ _AT+ B. We first require that this matrix be negative diagonal, which amounts to j
=
1, 2, ... , n.
(5.124}
Then stability of the model (5.114) is established by the diagonal-dominance property
•
2ail + bil + ~ (ay + lij; +by) ~ -'IT, i=l
j = 1, 2, ... 'n, (5.125)
i ..j
of the matrix A+ _AT+ B. We observe that the stability conditions (5.124) and (5.125) are expressed explicitly in terms of the elements ay, bu of the model matrices A, B, since they are simply given in terms of their absolute v~lllP~
Ecology: Multispecies Communities
306
From (5.120) and (5.125), we get (5.126) ell(x) < -'ITP(x). Imitating in stochastic terms the development that led from (5.122) to (5.123), we obtain ~{llx(t;to,xo)ll}
< llxollexp[-i'll'(t- to)],
t;) to,
(5.127)
which is a stochastic version of (5.123). The inequality (5.127) says that the expected value of the distance between the equilibrium x = 0 and the solution process x(t; t0 ,Xo) decreases faster than an exponential. The exponential decrement 'IT can be determined directly from the community matrices A and Bas specified in the inequalities (5.125). It is important to note that the algebraic conditions (5.124) and (5.125) imply stability for a range of values of C/11 and b11• The off-diagonal elements u11 (i + j) and all the elements bu can have any values (including zero) smaller than those for which the conditions (5.124) and (5.125) hold. In particular, (5.124) and (5.125) imply stability when all b;/s are zero (that is, B = B = 0), in which case the model ignores the random disturbances of the environment. This confirms our earlier assumption that the deterministic community matrix A satisfies the conditions (5.121) and that the deterministic part of the model is stable. From the above analysis, we conclude that if we ignore the nature of the interactions among the species in a community, then the smaller the magnitude of the interactions, the better the chances for an increase of community stability. To consider large v~tions of population size and broaden the type of interactions among species to include phenomena such as predator switching, resource limitations, saturation of predator attack capacity, and the like, it is imperative to widen the scope of stochastic s~bility analysis and incorporate nonlinear time-varying community models. The fact that our stability conditions obtained for linear constant models are insensitive to magnitude variations in both deterministic and stochastic interactions, and that the chosen Liapunov function tolerates such variations, makes it possible for such variations to be time- and state-dependent. This leads to the following nonlinear time-varying stochastic model: dx ~
,.
=
A(t,x)xdt
+ B(t,x)xdz.
(5.128)
Again, as in (5.114), x(t) E ~ill!' is the population vector, and z(t) E ~is a random variable. The community matrices A(t, x), B(t, x) are now n X n matrix functions A, B: ~X ~ill!' 4 ~ which are smooth enough that the solution process x(t; t0 , x0) of (5.128) exists for all initial conditions (t 0 , xo) E ~X ~ill!' and all t E ~The symbol~ represents the time interval (T, +co),
Stochastic Models: Stability
where ,. is a number or -oo, and '50 is the semi-infinite time interval [to, +oo). In the following analysis, we will consider stochastic stability of the equilibrium population x• = 0 of the model (5.128). If A(t,x*)x* = 0, B(t,x*)x* = 0 Vt E ~and x* + 0 is of interest, then we can define the nonlinear matrix functions A(t,y)y == A(t,y + x*)(y + x*), B(t,y)y = B(t, y + x*)(y + x*) and consider the equation dy = A(t,y)ydt + B(t,y)ydz instead of (5.128), where y• = 0 represents the equilibrium x* under investigation. In order to include the connective property of stochastic stability, we write the elements au= au(t,x),
b11
=
(5.129)
b11 (t,x)
of the matrices A(t,x), B(t,x) as au(t, x)
= { -!p;(t, x) + eu(t)'Pu(t, x), e11 (t)cp!J(t, x),
bu(t, x)
i =}, i ¥=},
(5.130)
= lu(t)t/Ju(t, x),
where the functions 'Pt. 'P!I• 1[;11 E C('5 X ~). In (5.130), eu = eii(t) and /ii = lu(t) are elements of the n X n interconnection matrices E = (eu) and L = (/11 ) which are defined and continuous on the time interval '5 with values in [0, 1]. The interconnection matrices reflect structural changes in both the deterministic and the stochastic interactions among species in the community. In particular, a disconnection of a trophic link between ith and jth species in the community is represented by eu = ej1 = 0 for all i, j. Such structural perturbations may occur independently in the stochastic interconnections involving the elements 111• Therefore, a wide variety of situations can take. place as an interplay among deterministic and stochastic interconnections, and they can be conveniently described by various forms of interconnection matrices E and L. It is important to note that community stability will be established for arbitrary forms of the functions e11 (t), 111 (t) E [0, 1]. This fact implies a high degree of reliability of the stability properties of communities for which our stability conditions hold. Now we introduce the notion of stochastic connective stability as follows:
Defmition 5.5. The equilibrium x* = 0 of the system (5.128) is stochastically connectively stable in the mean if and only if it is stable in the mean for all interconnection matrices E(t) E K, L(t) E "l.
Ecology: Multispecies Communities
308
To derive sufficient conditions for stochastic connective stability, we need to impose certain bounds on the coefficients of the matrices A(t, x) and B(t,x). We assume that the functions in (5.130) satisfy the constraints ~ qJ1(t, x)
=
a;,
supi!J! 9 (t,x)l
=
ay,
/3 9
~
supll{ly(t,x)l ~
~
=
f3!i,
(5.131)
i, j = 1, 2, ... , n,
>
aii, for some numbers a!i ~ 0, a 1 the following classes of functions:
q,;
=
0. The constraints (5.131) specify
{!Jl1(t, x): qJ;(t, x) ~ a 1},
< a9}, ll{ly(t,x)l < /3y}.
(5.132)
q,!i = {!Jly(t, x): I!Jly(t, x)l it!i =
{1{19 (t,x):
The classes of function q,1, q,v include nonlinear interaction s among species such as saturation of predator attack capacities and death among predators and prey, as well as predator switching, nonlinearit y in the food supply, etc. Moreover, they include the changing of a species from predator to prey of another species over a finite time interval. The class of functions itii allows for a possibility that the interaction s of the community with the random environme nt are not known precisely, but are specified only by their magnitude . It should be noted that our stability conditions will assure that the solution process approaches the equilibrium faster than an exponential. This additional property of stability provides an estimate of transient process in the community. More precisely, we are going to establish stability as defined by the following: The equilibrium x* = 0 of the system (5.128) is stochastically, exponentially, and connectively stable in the mean if and only if there exists two positive numbers IT and 'TT independent of initial conditions (t0 , x0 ) such that
Definition 5.6.
f9{11x(t; to, xo)ll}
< ITIIxo II exp[-'TT(t -
to)]
'Vt E
for all (t0 , x 0 ) E '5'X ~and all interconnection matrices E(t) E
~
(5.133)
E, L(t)
E
L.
Actually, the stability conditions to be derived will assure the validity of the inequality (5.133) for all qJ; E q,~> !py E q,y, 1{19 E i'u, and thus add the "absolute" aspect to Definition 5.6 as in Definition 5.1. In order to establish stability as given in Definition 5.6, we denote A= (aii) and 1i = (Iiu) the constant n X n matrices with elements
r 309
Stochastic Models: Stability
av
=
{-a.;+ e;;a;;, eiiaii,
i =j, i
=F j,
where a;, av, f3u are as in (5.131). Here ev and lu are elements of the n x n fundamental interconnection matrices E and L. The matrices E and L are binary matrices in which each element is equal to 1 if there is an interaction between the corresponding species, or 0 if there is none. As in the linear constant case, we assume that each species is in the 0), and moreover we have "stabilized" form (a.u
The equilibrium x* = 0 of the system (5.128) is stochastically and completely connectively unstable in the mean if the n X n constant matrix -(A+ _AT) is dominant diagonal.
Theorem 5.8.
Proof.
We consider again the function P(x) of (5.142) and compute
(5.152)
313
Stochastic Models: Instability
Since -(A+ AT) is dominant diagonal, we can use the conditions (5.121) to rewrite (5.152) as Vt E 5(,,
V(to,x 0 ) E ~X~.
(5.153)
The inequality (5.153) is valid for all interconnection matrices E(t) E E, L(t) E L. By applying again Ladde's (1974) stochastic comparison principle, we get from (5.153) &{!ix(t; to,xo)ll}
> llxollexp[!('IT, + 'IT,)(t- to)]
Vt E ~o. (5.154)
which establishes Theorem 5.8. From (5.154), it follows that instability under the conditions of Theorem 5.8 is exponential. Furthermore, it is easy to show that the inequality (5.155) is valid for all nonlinearities q;;, q;lj, 1/;lj that belong to classes of functions defined in (5.132), and that the instability is also absolute. Theorem 5.7 is over restrictive, in the sense that the negative-diagonal property of -(A+ AT) specified by a; >a;; > 0 implies that all species are unstable if disconnected from the community. We can relax this restriction and ask that only one species be unstable when disconnected, at the expense of specifying the sign of interactions between the species and the community. The equilibrium x• = 0 of the system (5.128) is stochastically and completely connectively unstable in the mean if for some i = 1, 2, ... , n, 0, a;> 0, such that the coefficients alj(t,x) of the there exist numbers alj n X n matrix A(t, x) defined by (5.151) satisfy the conditions
Theorem 5.9.
>
IPt(t, x)
> a;,
Vt E ~.
Vx E ~.
(5.155)
Proof. Let us consider the Liapunov function of (5.137) as v;(x;) =
xt.
(5.156)
Computing tv;(x;) with respect to (5.128), we get
> 2a;v;(xt) + 2 ~• ey{t)aljvAx1)
(5.157)
j=l
Vt E
~0 ,
Vx E
~
for any interconnection matrices E(t) E E, L(t) E r: By followin.g the same argument as in Theorem 5.8, we obtain from (5.157)
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Ei::ology: Multispecies Communities
Vt E 5(,, where x 10
= x 1(t 0 )
(5.158)
¥= 0. This proves Theo rem 5.9.
If we dispose of completely connective instability and ask simply that the system (5.128) be unstable for funda menta l interc onnec tion matrices, then we can relax the conditions of Theo rem 5.9. Let us assume that for some i = l, 2, ... , n the functions in (5.130) satisfy the follow ing cpnstraints: cpt(t, x) n
+ (L -ly)i};p = (M" - Mix),
!yO + (I" - I,)# = (~ - M2y),
(6.14)
L~ + (!y- !x)04> = (Mz - M3,). It is now necessary to evaluate the various torques. Since the internal torques on the reaction wheels are small, it may be assumed that these are proportional to the control signals actuating the wheels. Hence,
Mx =
-Klul,
=
-K2u2,
M2y
(6.15)
M3z = -K3U3, where K~o K2, and K3 are the drive-motor constants [the negative signs in (6.15) merely indicate the directions of these torques]. The external torques acting on the body of the LST are mainly due to environmental disturbance forces and are composed of gravity-gradient, magnetic, aerodynamic, and solar-pressure torques. The latter two will be negligibly small compared to the others and will usually be accounted for in control-system designs by treating them as equivalent zero-mean station-
Large Space Telescope: A Model
333
be represented as purely deterministic signals involving a constant term and a sinusoidal function of time with twice the orb~tal frequency. Hence, following the analysis of Schiehlen (1973), the external torques can be obtained as M, = hu + Y12cos(wt + x) + s1}-t,,
+ Y22cos(wt + x) + s2}4, M. = {'y31 + y32cos(wt + x) + s3}-',
(6.16)
My = hz1
where 'YIJ• i = 1, 2, 3, are constants that can be determined from the inertia components /Jt> I, I., the magnitude of the LST dipole moment, and the earth's magnetic field intensity; and s;, i = 1, 2, 3, are white-noise processes characterizing the aerodynamic and solar-pressure torques. Substitution of (6.15) and (6.16) in (6.14) and further simplification results in the following system of equations:
4> + a1~
=
f31u1 + M,, (6.17)
~ + a3tPB = /33u3 + M., where a1 = (I.- 4)/IJ az = (4- 1.)/I, a3 = (4- /,)/-', PI = K·JIJ P2 = KJ4, /33 = K3/I., and M,, Myo M. are the external disturbance torques given by (6.16). It is now simple to obtain a state-space representation of the LST by choosing the state vector X=
(q,,,P,O,O,I/J,~)T,
(6.18)
which results in the time-invariant model .i =Ax+ h(x)
+ Bu + FM,
(6.19)
where
A=
0 0 0 0 0 0 0
B=
131 0 0 0 0
0 0 0 0 0 0 0 0 f3z 0 0
0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 ' 0
/33
0 0 0 0 0 0
0 0 0 0 '
1 0
0
h(x)
=
-a~~ 0
-az#' 0
-a3tP0 0 0 1 0 0 0 F= ·o 1 0 0 0 0
(6.20)
0 0 0 0 0 1
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Engineering: Spacecraft Control Systems
The diagonal structure of the matrices A, B, and F permits us to partition the state vector as X --
x
( T , XzT , 1
X3T)T ,
(6.21}
where
X1 -_[Xu] -_ [cf>J . , ~ cf>
[9]
X2 -_ [X21] -_ 9· , ~
_
X3 - [X31] -_ [t/J] · . ~ t/1 (6.22}
With this, (2.18} can be described as a set of interconnected subsystems, .X;
=
A;x;
+ h;(x) + b1u1 + ./;d1,
i
= 1, 2, 3,
(6.23}
where
i = 1, 2, 3,
h2(x} =
h3(x} = [
0 -a3X12Xn
0 [ -a2XnX12
J ,
(6.24}
J ,
with d1 = Mx, d2 = My, and d3 = M. the external .disturbances. It may be observed that when h1(x} == 0, i = 1, 2, 3, (6.23} represents three decoupled subsystems which describe the motions of the spacecraft along the three axes. However, h;(x} are not zero and constitute the interconnections among the subsystems, thus making an analysis based on the smaller-dimensional decoupled subsystems alone inaccurate. The system represented by (6.23} is driven by the disturbance forces d1 in addition to the control signals u1• However, these external disturbances can be completely canceled by constructing a disturbance-accommodating controller as described by Schiehlen (1973}. This involves the determination of a suitable differential-equation model for the disturbances and, after augmenting the disturbance variables with the state variables of the system, designing a feedback controller that counteracts the disturbance forces by feeding back the estimated disturbance variables. Although this analysis is conducted for a single-axis model of the LST (only for the pitch motion control) by Schiehlen (1973}, a straightforward extension that uses three separate disturbance-accommodating controllers can be obtained for the
;.
' . .
t~~ [O,
k
=
1, 2, ..• , s,
(6.41)
-'ITk
if and only if the matrix W does. These inequalities applied to W determine the constants vt, ~, ••. , v, in (6.35). It is possible to calculate these constants recursively. To see this, we note that the kth leading principal k X k submatrix Wt can be expressed as
I
- [ g{ Wt=-1•
OJ[ 1
Wk-1
0
0 ][J wkk - g{ m:•.~t 0
(6.42)
Jfj;:.•.Jk] 1
.
Therefore, the kth leading principle minor of W is det
Wt =
(det Wk-l)(wkk- g{Wt:\jk).
(6.43)
For the inequalities (6.43) to be satisfied by W, it is necessary and sufficient that k = 1, 2, ... ' s.
(6.44)
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Engineering: Spacecraft Control Systems
From (6.39), we have
Jl
=
<E1k. ~ •...• Elk).
(6.45)
and from (6.39) and (6.44), we get the constants v1 as
[- {1,k +
k 1,
k
=
s,
* s.
(6.46)
Once the constants v1 are calculated by (6.46), the region ~ of (6.36) is determined. Now it remains to imbed a Liapunov function v: ~ ~ gt+ inside the region ~ and determine an estimate of the stability region as ~= {x E ~= P{x)
< y}.
(6.47)
Here we choose (6.48) where d1 are positive numbers, and v1(x1) = Hx;ll. Following Section 2.9, we calculate the positive constant yin (6.47) using (6.46) and y =min d-v· ; J ,
where the positive vector dT
=
(dt. d2 ,
i
=
••• ,
1, 2, ... ' s,
(6.49)
d.) is computed by
(6.50)
>
which is obtained for p = y'2 a1• The corresponding matrix Win (6.37} is
(6.60}
340
Engineering: Spacecraft Control Systems
w~[-r
(6.61)
From (6.46) and (6.61), we get (6.62) Choosing v1 = 112 = v3 = v, and using (6.60) and (6.62), we compute 0.584. Selecting v = 0.574, cr1 = 10, and c = (1, 1, lf, we further compute from (6.50) the vector
v
0 such that
>
I
lw.ul - dr 1 I-I ~ d;lwul > 'IT,
j
=
l, 2, · · ·• s,
(6.97)
;,.j
that is, W is a quasidom inant diagonal matrix. In order to study the effect of structural parameters eii on the overall system stability, it is more convenient to use the diagonal-dominance conditions (6.97) than the Sevastyanov-Kotelyanskii determinantal inequalities (6.95). Instead of the quasidominant-diagonal property (6.97), we will apply the weaker but simpler dominant-diagonal conditions I
lwiil > i-1 ~ wii,
j
=
I, 2, ... , s,
(6.98)
i+j
which are obtained from (6.97) when the d;'s are all set equal to one. By using the definition (6.94) of the wg's and (6.88), we can rewrite (6.98) in terms of the matrices 11;, G1 as
1 Am(Gj)
2 AM(~)
~
AM(Jl;)
> >.!{2(~) i~·~ eii~ii>.!{2(H;)'
j
= I, 2, ... , s.
(6.99)
From (6.99), it is obvious that our ability to determine stability by the decomposition-aggregation method depends crucially on the choice of the matrix G1 0 and the corresponding solution 11; 0 of the Liapunov matrix equation (6.86) for each free subsystem (6.83). It is also clear from (6.98) and (6.99) that the best estimates of the values of the structural parameters eu· are provided by the optimal aggregate matrix W 0 which is the solution of the following:
>
>
Problem 6.1. Find: subject to :
'v'W
AT 11;
+ ll;A;
= -G;,
i
=
I, 2, ... , s.
Here the matrix inequality is taken element by element. Furtherm ore, if the decomposition (6.82) is performed with "the least violence" done to the system a>,. as proposed by Steward (1965), the interactio n matrices Au are sparse, and their respective norms ~i are small. From (6.99), we conclude tliat..the largest estimates for the parameters eil are
~
347
Maximization of Structural Parameters
obtained for the pair of matrices (Gt.H;) with the largest ratio Am(G;)/A.M(H;). Therefore, we replace Problem 6.1 with the following:
Problem 6.2. Find:
subject to :
which should be solved for each of the s free subsystems ~i of (6.83) separately. As a byproduct of the solution, we obtain the Liapunov function (6.84) which provides the exact value of the degree of exponential stability for each isolated subsystem ~1• To solve Problem 6.2, we assume that all eigenvalues of the matrix A 1 are known numerically and all of them are distinct. This would be a prohibitive assumption for the overall system~. but since the decomposition-aggregation method (Section 2.5) requires such assumptions for the low-order subsystems, it is quite realistic for them. Furthermore, in the proposed multilevel stabilization scheme (Section 3.3), such an assumption can readily be realized using local linear feedback controls and any of the conventional techniques such as pole assignment, the root-locus method, the parameter plane method, etc. To solve Problem 6.2, we use again the transformation introduced in Section 3.3, (6.30)
where T; is a nonsingular constant n X n matrix. This produces the free subsystem~~ in the form (6.100)
with A;
= r;-t A 1 r; having the following canonical quasidiagonal form: •
A1
11t
Wj
J [--w~ -a~J, -aJ.+t, ... , -a:..-p}
i I = diag { [ -wl -af , ... ,
-a;,
11p i
Wp i
.
.
(6.32)
>
0 q = 1,2, ... ,n1 - p, and 0
(6.106) which provides the exact estimate of the degree of exponential stability, 7Tp = ait. Furthermore, the solution· ii;0 = ()I; yields the lowest possible Value for the ratiO AM(it)/'N/, 2 (ii;)"A.!/, 2 (~) Which appears On the right-hand side of the inequalities (6099)0 Therefore, the solution ( GP, il; 0 ) of Problem
Maximization of Structural Parameters
349
6.3 in the transformed space provides the best aggregate matrix W0 required by Problem 6.1. Consequently, the best estimates for the structural parameters e9 are obtained by using the inequalities (6.99) in the transformed space, 0
ak
> i=l ~ e9 ~9 , •
-
j = I, 2, ... , s,
(6.107)
x-
1 provided~~~~ ~j (i,j = 1,2, ,s) and~}= A~2 (.AJAy), Ay = Ay'Ij. Maximization of the interconnection parameters e9 can be now performed in the mathematical-programming format (Kuhn and Tucker, 1951) for each subsystem separately. Let us define the s-vectors e1 = (ev, e21 , • •• , e,1)T, b1 = (~v.~21 , •.. ,~~)T, and the positive number a1 = a/.t + e, where e 0 is an arbitrarily small number. Then, we can state the following vector maximization problem for the jth subsystem: 0
0
0
>
Problem 6.4. Maximize:
e1
subject to:
That is, we are interested in finding an s-vector eJ constrained by a1 - bJ e1 ~ 0, e1 ~ 0 such that e1 ~ eJ for no e1 satisfying the constraints (the vector inequalities taken element by element). A Pareto-optimal solution to Problem 6.4 can be obtained using the results of Kuhn and Tucker (1951), and DaCunha and Polak (1967). If weights can be assigned to the components of the interconnection vector e1 by choosing an s-vector c1 = (cv, c21 , .•. , c~)T such that cii ;> 0 (i = 1,2, ... ,s) and ~:- 1 c9 = 1, then Problem 6.4 can be reformulated as a linear-programming problem:
Problem 6.5. Maximize:
cJ e1
subject to:
which can be solved by known techniques (e.g. Dantzig, 1963; Zukhovitskii and Avdeyeva, 1966). To establish stability of the large-scale system ~ by the method outlined, we rely entirely on the stability properties of the free subsystems ~1 • If active feedback elements are available, we can use them in a multilevel stabilization scheme as proposed in Section 3.3, and stabilize unstable large-scale systems. Furthermore, the scheme can be used to increase the values of the
350
Engineering: Spacecraft Control Systems
interconnection parameters eu. The local controllers for the decoupled subsystems can be used to raise the level of o/.t, while the global controllers can be applied to minimize some (or all) numbers ~ in the inequalities (6.95). This control strategy generally leads to the satisfaction of the stability conditions (6.95) with higher values of the interconnection parameters eiJ without a decrease in the degree 'IT of exponential stability of the overall system. So far, it has been shown how the multilevel stability analysis can be directed towards obtaining the maximum estimates of the interconnection parameters. These estimates can be further improved by introducing either output or state feedback. The output-feedback scheme (Siljak, 1975) is a straightforward application of the decomposition-aggregation method. We consider a linear system
x =Ax+ Bu,
y = Cx,
(6.108)
where x E 'i.it" is the state of the system; u E ~ is the control; y E 'iR!' is the output; and A, B, and Care constant n X n, n X m, and p X n matrices. The linear feedback
u(y)
=
-Ky
(6.109)
is introduced directly into the system (6.108), where K is a constant m matrix, and we get X= Acx,
X
p
(6.110)
where the closed-loop system matrix is Ac = A - BKC. Then the system (6.110) is decomposed into s dynamic subsystems (6.82), and the stabilization proceeds in pretty much the sameEay as the stability analysis outlined in this section. The only difference is n the freedom provided by the gain matrix K of the linear control law (6.1 The elements of K can be chosen on the subsystem level to produce an opt a egate matrix and the largest estimates of the structural system parameters. The application of this approach is presented in the following section. The above stabilization scheme is basically a trial-and-error procedure, since there is no systematic way of choosing the elements of the gain matrix K. The only advantage that the decomposition-aggregation scheme has over the conventional techniques is that the stabilization is performed "piece by piece" and the structural parameters appear explicitly in the aggregate model. It is possible to retain this advantage and improve the search for appropriate feedback gains a great deal, if one rises the multilevel control scheme based upon the state feedback, as outlined in Section 3.3
Maximization of Structural Parameters
351
Let us consider a linear constant dynamic system §> described by i =Ax+ Bu,
(6.111)
where again x E ~ is the state of the system, u E ~ is the control, and A and B are constant n X n and n X m matrices. In order to stabilize the system, it is decomposed into s dynamic subsystems Si1 represented by the equations i1
where x 1 E
• = A 1x 1 + j=l ~ eiiAvx1 + B1 ut,
~"'
i
=
I, 2, ... , s,
(6.112)
is the state of §>1, so that ~
=
~xgr>x
.. ·x~·;
(6.113)
u1 E ~· is the decentralized control, so that ~m
01:"' X
=
~m,
X··· X C!R!"';
(6.114)
A 1, Av, B1 are constant matrices of appropriate dimension; and the pairs (At.B1) are controllable for all i = 1, 2, ... , s. The control functions u1 : ~ ~ ~· are chosen as u;(x) = ui(xt) + uf(x),
(6.115)
where the local control uf(x1) and the global control uf(x) are linear functions of the states x 1 and x, I
uf(x;)
= - K1 x 1,
uf(x)
= -
~
j~l
evKiix1 •
(6.116)
By substituting (6.II6) into (6.ll2) and using the transformation (6.30), we get the closed-loop system as
.i, = Mx, +
•
~ j-1
evAijx1,
i
=
I, 2, ... , s,
where the quasidiagonal matrix A~ and the matrix
i, j
=
(6.117)
Aij are given as (6.118)
1, 2, ... ' s.
The gain matrices K 1 can be chosen to fix the eigenvalues of the Afs and get a sufficiently high degree of stability 'Ill of the subsystems. AB suggested in Section 3.3, this can be accomplished by the pole-shifting technique, or by solution of the optimal-linear-regulator problem as in Section 3.5.
1
.~ [(AijfAij] and obtain the optimal aggregate matrix WO as formulated in Problem 6.1. If rank 11i = mi, then .11] = (BJ 11 111i> and Kii is simply 1 calculated as
t
(6.120) The design process can be entirely and effectively computerized as shown by Siljak, Sundareshan, and Vuk.cevic (1975). 6.6. STAB ILIZA TION OF THE SKYLAB
To provide an artificial-gravity environment, NASA initiated and conducted a study to determine the fe¢bility-~inning the Skylab (Seltzer, Justice, Patel, and Schweitzer, 1972; Seltzer, Patel, and Schwe itzer, 1973). In spinning the spacecraft, it is necessary to point the solar panels toward the sun, which requires the vehicle to spin about a princip al axis of intermediate moment of inertia. Since such spin cannot be achiev ed without stabilization, it was proposed to establish passive stability by deploying masses either on cables or extendable booms attached to the Skylab as shown on Figure 6.2. Such configuration has the principal axis of maximum moment of inertia pointing (in the same direction as the solar panels ) to the sun. In order to inertially fix the axis in presence of disturbance torques, attitude-control torques must be applied to the vehicle, which depend on error signals that are proportional to the angle between the princip al 3-axis and the solar vector. Sun sensors and rate gyros on the present Skylab can
353
Stabilization of the Skylab
SERVICE MOOULE
TIPMASS--
\ \ _ COMMAND MODULE
FIGURE 6.2. Skylab.
readily provide the control signals ~. q>2, wh and w 2 shown on the simplified model of the spinning Skylab, which consists of a core mass with two tip masses connected to it by flexible massless beams lying in two different planes as shown in Figure 6.3. The angular-velocity vector of the vehicle may be written in body-fixed coordinates 1, 2, 3 as [WJ,w2,w3 + oy, where lw;J « 1 (i = 1,2,3) represent small perturbations about the steady-state velocity 0. Small displacements of the two tip masses m from the steady state are denoted by ut (i = 1, 2, 3; k = I, II). The rotational dynamics of the Skylab may be represented by a set of nine differential equations written in terms of w; and ut. It is possible to reduce the set of nine equations to siX by using the substitution U; = ul - u11, where u; now represents the skew-symmetric mode of the elastic deformation and hence causes angular motion about the vehicle's steady-state attitude. Since the stability of rotational motion will be of interest, only the skew-symmetric mode is considered. The linearized equations of motion are wobble motion: ~ w1
+ (13 - 12)0w2 + mi2(u3 + 0 2u3) - mi;(20uJ + u2 - 0 2u2) = 71, (~ - 13)0w1 + 12 w2 + mi;(u1 - 0 2ut - 20&.h) = 'Ji, 2mr2(w1 + Ow2) + mii3 + d3 u3 + (k3 + m0 2)u3 = 0; (6.121)
Engineering: Spacecraft Control Systems
354
Axis of instantaneous \angular momentum
\
\
3
m
FIGURE 6.3. Simplified model.
spin motion:
/3 W3 -
= 1), I - 2mS2ri2 = 0, + d2ri2 + (k2- mS2 2)u2 (6.122) = 0.
I;(ul - 2S2Lh)
2ml3(Slwl + w2) - 2mi; W] + mul + dt Ut + kl 2ml3(-wJ + Slw2)- 4mi;Slw3 + 2mSlu1 + mu
The spm velocity and its perturbation w~ are controlled separately and are not considered here. Consequently, \Ve ~sume that w3 = w3 = 0 (Seltzer, Patel, Schweitzer, 1973). The linear control is postulated (Seltzer, Justice, Patel, Schweitzer, 1972) as T
= a.p + {Jw,
(6.123)
where T = ('11, 'Jl, 13f is the vector of control torques; .p = (2,4>Jf is the vector of angular rotations; w = (w~o w2 , w3 + Slf is the vector of angular velocities; a, {J are 3 X 3 matrices
Stabilization of the Sk.ylab
au
a
355
a12 0] 0 , 0 0
= a21 an [
0
(6.124)
and the kinematic relations are
w
-1
0
=[ ~
~1
(6.125)
The control law in this study is chosen as all other a 11 = 0;
Pu = 1i 08, /333 = -.li Op;
all other {311 = 0;
(6.126)
so that the normalized control torques v = [vt. v2, v3 JT = [1i'/li 0 2, 12/li 0 2, ~/li 0 2 JT are Vt
=
(e + 8~- &/>1,
(6.127)
Referring to Equations (6.121) and (6.122), the control torque 1j' is used to stabilize the wobble motion, and the torque ~ is used to stabilize the spin motion. In (6.127), e, 8, p are control parameters to be selected in the stabilization process. Upon introducing these transformations in addition to the dimensionless variables and constants as defined in Table 6.1, the linearized equations of motion become wobble motion:
cf>'l - (1 + K1)cf>2 - Kt cf>I - y(p.') + p.3 ) + ~y(2p.l + p.'2- JL2) + (e + 8~ - 8cf>l (1
=
0,
(6.128)
+ K1 )ct>l + K2 acf>l + acf>'2 - h(J.I.'I - P.t - 2p.2) = 0, -cp'l - cf>t + p.') + ~J p.) + (ai + l)p.J = 0;
spin motion:
Pcf>'3 + y(p.'; - 2p.2) + pcpJ = 0, -~cpl- ~cf>'2 ~cf>'l
- 2~cf>2 - ~cf>I
+ cf>'3 + p.'l + ~tJLI + atJLt- 2JL2 + ~cf>l = 0, + 2cf>J + 2p.l + p.'2 + ~2 JL2 + (ai - l)JL2 = 0.
An important feature of these equations is that ·when
~ =
(6.129) 0 (that is,
13 = 0), they become uncoupled into two sets of equations: the wobble
356
~e~g: Spa~tConuolSym~
motion described by (6.128) and the spin motion described by (6.129). This suggests that the influence of the asymmetry in the arrangements of the booms (:£3 + o o~ E+ 0) can be treated as the structural parameter between the two motions, so that I~ I = eu = e21 and E2 = eu = e • In the 22 decomposition-aggregation analysis each motion represents a subsystem. The state-space representation of the overall system (6.128}-(6.129) is obtained as
x'(T)_= Ax(T), where the state 11-vector x(T) is chosen as
(6.130)
(6.131) and the 11 x 11 matrix Ac is given in (6.132) -see pages 358 and 359. The system of equations (6.130) can be .decomposed into two interconnected subsystems described by
wobble motion: (6.133)
spin motion: (6.134) where the state vectors x(T), x1(T), x2(T) of the system (6.130) and the two subsystems are
The following identity relationships we e used in order to get the subsystem matrices A 1 and A 2 independent f the coupling parameter E: 1
cx(l - 'Yl) -
e'Y e
e'Y
1
cx(1 - y3) + cx(l - 'Y3)(cx(1 - 'Y3) - eyJ'
1 1 -1---y---E -2 = -1---y
+ (1
E2 y
(6.136)
- y)(1 - y- e'Yr
The structural configuration of the system (6.130), as composed of the two subsystems (6.133) and (6.134) and the interconnections between them through the coupling parameter E. is represented by the direct graph in Figure 6.4(a}. It is obvious that the digraph of Figure 6.4(a) becomes that
Stabilization of the Skylab
357
eA21 <el (a)
8
8 (b)
lim
e2A11 <e>
2 lim e A <el e-01) 22
e..... OD
C8
@) (c)
FIGURE 6.4. Structural decomposition.
of Figure 6.4(b) when~ = 0. When~~ CXl, the digraph of Figure 6.4(a) is again decomposed into the two separate digraphs shown in Fig. 6.4(c), because the interconnection matrices ~A,2 (~), ~A 21 (~) .and €2Au(0, An(O become zero and constant matrices, respectively, as seen in (6.132) or (6.133}-(6.134). The subsequent stability analysis shows that the free subsystems (6.133) and (6.134) (~ = 0) are stable. It is easy to check, however, that the decoupled subsystems in Figure 6.4(c) are unstable. Therefore, our main objective is to determine the best estimate of the maximum allowable value of~ which lies between the two extremes~ = 0 and~ = oo, and for which the overall system of Figure 6.4(a) is stable. On the basis of the Skylab physical characteristics given in Table 62, Au(~) (i = 1,2) of (6.132) can be made independent of~ and denoted by Au. This is achieved by neglecting the term = 8.5 X 10-4 in comparison with the terms 1 - y = 0.803 and a(l - y3 ) = 5.33. After this simplification, the numbers ~12 and El 1 are the norms of the coupling matrices A12 andA 2" which are computed in (6.89). The subsystem
e
e
··~
p•= 0 0 0 K, + y- ~ly 1- y- ey 0 1 + K, + y- ~2y 1- y- ~ly
0 0 0 0 -K1 -2y) 1- y- ~ly
W
__l
0 0 0
0 1 0
0 0
1 + K1 ya~ 8 1- y- ~ly 1- y- ~ly 1- y1 ~- ~ly (1-. 3 )K2a + ~ 1 y (1 - YJ)(1 + K1 ) + ~ 1 y 0 0 a(1 - y3) - ~ 1 Y a(1 - Yl) - E1Y e+6 1 + Kl8 ' yai -(ai + 1) 1- y- ~ly 1- y1- y- E1Y 1- y- ~ly
b 3aCK1 + 1) a(1 - Y3) - ~ 2 Y 0 ~a(K1 + 1) a(1 - Y3) - ~ 1 Y 0 He+ 6) 1- y- ~ly
0
b1(2a- 1 - K,) a(1 - y,) - ~ly 0 E(2a- 1- K 1) a(1 - y3) - ~ 1 y 0
E-ya~
~8
0 0 0
1- y-
~ly
1- y-
~ly
~1 y
ey
0 0 1 y.1.3 1- y-
~ly
0 ~ly ~ly.
-4 y43 3 1- y- ~ly
r.
0
0
0
0
0
0
'!'!
0
~
0
w- Kj- 2y) 1- y-
~ly
E-y.1.3 1- y- ey
~·
g ~ (')
g :g
-
f
0 0 0
0 0 0
0 0 0
2(y 1- 'Y- ~2y
0
0
fie
P[a(l - 'Y3) - ~ 2 y] 2(y 1- 'Y- ~2y p
apy3
--p- .B[a(1- 'Y3)- ~2y] 0 ap .B[a(1 - y3) - ~ 2 y]
0 2(1- r) 1- 'Y- ~2y
_ E'r(1 - 'Y3 + af) a(1 - 'Y3)- ~2y
0
EYai
Ey~
a(1 - y3) - ~2 1 0
2
Y3(afa + ~ Y) a(1 - 'Y3) - ~2y 0 ala+ ~2y a(1 - 'Y3) - ~2y 0
0
0 0 0
ll'Y341
0 0 0 Ey~
1- 'Y- ~2y
1- 'Y- ~2y
0
0
EYai
l - 'Y- ~2y
~y42
l - y-
a(1 - 'Y3) - ~2y
0
0
1
0
0
0
2
1141 a(1 - Y3) - ~2y
0 -2
~2y
0 (1 - y)(af- I)+ ~ 2 y 1- 'Y -~2y
-
(1- y)~ 1- 'Y- ~2y
(6.132)
360
Engineering: Spacecraft Control Systems
Table 6.1. Nomeadature for the Skylab Principal moment of inertia of body, ith coordinate: li* + 2.mJ22, fz*, I,* + 2mi;2 , respectively Principal moment of inertia of rigid core body about ith body-fixed coordinate Stiffness coefficient characterizing nomotating-boom stiffness Tip mass Applied torque about ith coordinate Time Differentiation with respect to real time t Skew-symmetric mode of elastic deformations Displacement of kth tip mass from spinning steady state in ith direction
k; m
7i t
0 = d/dt ul=u/-uJI uf
(k =I, D) Perturbation (about spinning steady state) of angular velocity about ith coordinate Angular rotation about ith coordinate Steady-state boom dimension in 2-axis direction from center of mass to tip mass Asymmetry in setting of booms Ratios of inertia (12 - 13 )/li and (13 - li)/lz, respectively
WI
I]
x.,Kz 11
1+
Ki
I,.
=l-Kz=:4
y = 2fmi/li A;- d,jmfl. I';=
u;/212
.r; = k,/mfl 2 E= r,jrz T
0 P1
=
Ot
= w,jfl.
(') = d/dT
Y3 = 2mi;2j/,
/3 ... 1,/Ji
Ratio of inertia
~
Dimensionless inertia ratio Dimensionless damping ratio General skew-symmetric coordinate Dimensionless natural frequency coefficient of boom Dimensionless length ratio Dimensionless time Steady-state spin rate about 3-axis Dimensionless wobble ratio (i = l, 2, 3) Differentiation with respect to T Dimensionless inertia ratio Dimensionless inertia ratio
l
Liapunov functions v1 and v2 are chosen as in {6.106), and the aggregate comparison syste~, where the aggregate matrix 'W is given by
€12i~IAM(,&)
_! A,(GI) + €2AM(HI') W =
2 AM(~)
[
A..(~)'
td~IAM(.82)
~2 (~)A~ 2 (R2)
~2(~)A~2(.02)
_! A,{G2) + ~2AM(H2) 2 AM(R2)
(6.137)
.
A..(~)
In (6.137), A.. and AM denote the minimum and maximum eigenvalues of the indicated matrices, respectively. From {6.95), we have that
w11 < 0,
det
W>
0
{6.138)
r
References
~ = 1.25 X 1Q6kg m2 2 / 2 = 6.90 X 106 kg m 2 6 / 3 - 7.10 X 10 kg m I;= 0
12
361
m = k1 = k2 = ~ =
227kg k 3 = 146N/m 7.4 X 104N/m d3 = 0.04(~ m)l/2 dz = 0.04(klm)112 D = 0.06s- 1
.
= 23.3m
are necessary and sufficient for stability of JV in (6.137), and sufficient for stability of the overall system. After several trial-and-error steps, the control parameters are chosen as E =
25,
8
=
-14,
p
= 0.5,
(6.139)
and the aggregate matrix is obtained as __ [-7.279 x 10-4 + 42.337le 2.26251~1 W1.39441€1 -19.321XIO"""+Il.515QE 2
J •
(6.140) The conditions (6.138) yield the best estimate of the parameter € as
Je 1 < 6.584 x 10-4 •
(6.141)
This concludes the design of the Skylab control system.
REFERENCES DaCunha, N. 0., and Polak, E. (1967), "Constrained Minimization Under VectorValued Criteria in Finite-Dimensional Spaces", JOUTIIQ] of Mathematical Analysis and Applications, 19, 103-124. Dantzig, G. B. (1963), "Linear Programming and Extensions", Princeton University Press, Princeton, New Jersey. Kuhn, H. W., and Tucker, A. W. (1951), "Nonlinear Programming", Proceedin~:s of the Second Berkeley Symposium on Mathematical Statistics and Probability, J. Neyman (ed.), University of California Press, Berkeley, California, 481-492. O'Dell, C. R. (1973), "Optical Space Astronomy and Goals of the Large Scale Telescope", .Astronautics and Aeronautics, 11, 22-27. Schiehlen, W. 0. (1973), "A Fine Pointing System for the Large Space Telescope.., N.AS.A Report, No. N.AS.A TN D-7500, National Aeronautics and Space Administration, Washington, D.C. Seltzer, S. M., Justice, D. W., Patel, J. S., and Schweitzer, G. (1972}, "Stabilizing a Spinning Skylab", Proceedings of the Fifth IFAC World Congress, Paris, 17.2:1-7.
362
Engineering: Spacecraft Control Systf;ms
Seltzer, S. M., Patel, J. S., and Schweitzer, G. (1973), "Attitude Control of a Spinning Flexible Spacecraft", Computers and Electrical Engineering, 1,
323-339.
Seltzer, S. M., and Siljak, D. D. (lm), "Absolute Stability Analysis of Attitude Control Systems for Large Boosters", J0U111Ql of S]IQCtlcraft and Rockets, 9,
506-510.
Siljak, D. D. (1969), "Nonlinear Systems", Wiley, New York. Siljak, D. D. (1975), "Stabilization of Large-Scale Systems: A Spinning Flexible Spacecraft", Proceedings of the Sixth JFAC World Congress, Boston, Massachusetts, 35.1:1-10. (See also: Automatita, 12, 1976, 309-320). Siljak, D. D., Sundareshan, M. K., and Vuk~c, M., B. (1975), "A Multilevel Control System for the Large Space Telescope", NASA Contract Report, No. NAS 8-27799, University of Santa Clara, Santa Clara, California. Siljak, D. D., and Vuk~c; M. B. (1976), "Multilevel Control of Large-Scale Systems: Decentralization, Stabilization, Estimation, and Reliability", LargeScale Dynamical Systems, R. Saeks (ed.), Point Lobos Press, Los Angeles, California, 34-57. Siljak, D. D., and VukeeviC, M. B. (1977), "Decentrally Stabilizable Linear and Bilinear Large-Scale Systems'', International Journal of Control, 26,289-305. Steward, D. V. (1965), "Partitioning and Tearing Systems of Equations", SIAM Journal of Numerical Analysis, 2, 345-365.
Weissenberger, S. (1973), "Stability Regions of Large-Scale Systems", Automatica,
9, 653-663.
Zukhovitskii, S. 1., and Avdeyeva, L. I. (1966), "Linear and Convex Programmi~', Saunders, Philadelphia, Pennsylvania.
7 ENGINEERING Power Systems In this chapter, we consider two distinct problems in power systems: transient stability and automatic generation control. Both problems are formulated, analyzed, and resolved by partitioning appropriate powersystem models into interconnected subsystems. In the case of transient stability, a multimachine system is decomposed into two-machine subsystems, whereas an automatic generation control system is of decentralized type, with each subsystem representing an area and all its tie lines originating from that area. In both situations, such decomposition is only possible if overlapping is allowed among the subsystems. In the transientstability partition, all subsystems overlap the reference machine. In automatic generation control system, tie lines are common parts of the subsystems. Overlapping of subsystems in both cases is a necessity dictated by physical characteristics of the power-system models. It should be pointed out, however, that the fact that overlapping is permitted in our methods of analysis shows a considerable flexibility of our decomposition and decentralization approach. Although we plan to describe both problems and the corresponding power system models in some detail, it may not be enough for a reader who is not familiar with the subject. For further reading-advanced as well as tutorial-the following references are recommended. For transient-stability analysis, the book by Anderson and Fouad {1977), as well as the papers by Willems (1971) and Fouad {1975) are a good place to start. For automatic generation control, the books by Kirchmayer {1959) and Cohn {1966), the 363
364
Engineering: Power Systems
ERDA conference proceedings edited by Fink and Carlsen (1975), as well as the papers by Elgerd and Fosha (1970), talovic (1972), Ewart (1975), and Reddoch (1975) are recommended.
7.1. TRANSIENT STABILITY Before we engage in the problem of model specifications for multimachine power systems, let us briefly review the physical problem of transient stability. Assume that a power system is in its steady-state operation and the mechanical input power to the generators is equal to the electrical power delivered to the network. When a large disturbance occurs (such as a short circuit, sudden loss of a large load, etc.), the steady-state operation is perturbed. Unless the fault is cleared before a certain maximum time (the critical clearing time), a loss of synchronism may take place, so that the system is not capable of recovering steady-state operation. The transientstability problem consists in finding out whether or not the system motion converges to a steady state after clearing the fault. If at the time the fault is cleared the state of the system is within the stability region of the postfault equilibrium (the post-fault steady state), convergence to steady-state operation takes place. Therefore, solutions of the transient-stability problem have two distinct phases. The first phase consists in keeping track of the system motion during the disturbance, whereas in the second phase one determines the stability of the post-fault equilibrium state. Both solution phases can be carried out by standard numerical techniques for integration of the system equations of motion. These techniques become increasingly unattractive as the size of multimachine power systems becomes large, because of the excessive computer time and memory required. It is for this reason that in the second phase .of the solution, numerical techniques can be advantageously replaced by the direct method of Liapunov. /~~ Initial applications of Liapun.ov's method to transient-stability analysis were made in the 1960s by Gle~(-J966), and by El-Abiad and Nagapan (1966), and later developed by a large number of authors, as surveyed by Willems (1971), Ribbens-Pavella (1971), and Fouad (1975); Since power systems are never asymptotically stable in the large, Liapunov's method is used to estimate the region of attraction of the post-fault equilibrium state. Once a multimachine system was recognized as a multinonlinear Lur'ePostnikov system by Willems (1971), the method of Popov (1973) and the Kalman-Yakubovich lemma in matrix form (Kalman, 1963; Popov, 1973) were available for systematic construction of Liapunov functions (Willems,
Transient Stability
3~5
1970, 1971; Henner, 1974). These functions were used to estimate stability regions by extending the procedures of Walker and McClamroch (1967) and Weissenberger (1968), which were originally applied to Lur'e-Postnikov systems with a single nonlinearity. As the size of the system increases, these direct methods lose much of their appeal, mainly because of the conceptual and numerical difficulties involved in considering a large nwp.ber of unstable equilibrium states of the system. In 1975, Pai and Narayana (1975) made an attempt to apply the BellmanMatrosov concept of the vector Liapunov function, as proposed in the decomposition-aggregation method developed by Siljak (1972) and Grujic and Siljak (1973), to the transient-stability analysis of multimachine systems. After the method was applied to construct an aggregate model involving the vector Liapunov function, a single Liapunov function was used to estimate the stability region by the procedure proposed by Weissenberger (1973). The main drawback of the Pai-Narayana approach is in that ann-machine system with uniform damping is decomposed into n(n - 1)/2 second-order subsystems, so that the most significant characteristic of the decomposition-aggregation method, that of reducing the dimensionality of stability problems, is lost. In addition to this main drawback, there were some unresolved details of varied significance that prevented computation of valid estimates of the stability regions. Nevertheless, the new approach of Pai and Narayana is quite promising, and in the first part of this chapter we will present the works of Jocic, Ribbens-Pavella, and Siljak (1977), and Jocic and Siljak (1977), which is in the same direction but eliminates all major difficulties encountered in the Pai-Narayana approach. In particular, it reduces the order of the aggregate model of an n-machine system to n - 1. As compared to the existing direct methods (Willems, 1971; Fouad, 1975) for estimating stability regions of multimachine power systems, the follow" ing are the advantages of the method proposed by Jocic, Ribbens-Pavella, and Siljak (1977), and Jocic and Siljak (1977): (1) The method is rigorous and can be carried out systematically with parametrically simple and explicit estimates of the stability region obtained as the end result. (2) In the course of transient-stability analysis, decomposition can be used to take advantage of (or obtain information about) the special structural features of the power system. (3) The method opens up a real possibility for more refined models of the subsystems to be included in the analysis of large power systems.
366
Engineering: Power Systems
(4) In particul ar, the transfer conduct ances can readily be incorpo rated in the analysis, a feature which is missing in almost all previous considerations. The plan of our exposition is as follows: In the next section, we will formula te a model of an n-machi ne power system, and then perform a pairwise decomp osition of the model into n - I intercon nected secondorder subsystems. Since each subsyste m is of the Lur'e-Po stnikov type with one nonlinea rity, a simple analysis in Section 7.3 produce s estimates of the stability regions for each free (disconn ected) subsystem. Finally, in Section 7.4, we construc t an aggregate model involvin g a vector Liapuno v function , and use the subsystem-region estimates to determi ne an estimate for the stability region of the overall system.
7.2. A MODEL AND DECOM POSmO N We consider ann-ma chine power system in which the absolute motion of the ith machine is describe d by the equation
i where Pc
=
~j_, E,~ Yqcos(B, -
=
1, 2, ... , n,
(7.1)
81 - Bu) and
81 is the absolute rotor angle, ~
is the inertia coefficient, D 1 is the damping coefficient, l!t is the mechanical power delivered to the ith machine, I:t is the electrical power delivered by the ith machine, E1 is the internal voltage, ~ Yu is the modulus of the transfer dmittan ce between the ith and jth machines, Bu is the phase angle of transfer /dmitta nce between the ith and the jth machines. / .. In Equatio n {7.1), it is assumed that M;, l!t, and E are constan t for all 1 machine s. In addition , we ~ume uniform damping , that is,
D'='
~
"•
i
= 1, 2, •.. , n.
(7.2)
With the above assumptions, we must refer the system motion to the motion of one of the machine s (say the nth) taken arbitrari ly as the compari son machine and choose the state vector x E C8lft(Yt2) = 0, and we cann ot establish global asymptotic ~ bility of x~ = 0 in (7.15). The nonlinearity !p1(y 1) belongs to the class
and the stability analysis is restricted to a finite region of the state space, as considered by Walk er and McO amro ch (1967 ) and Weissenberger (1968). Following their analysis, we start with the Lur'e -Postnikov type of Liapunov func tion (2.178), defined here by (7.18) where H; is a cons tant matrix and tis a scala r. As is well know n (e.g. Siljak., 1969), the Popov condition
is necessary and sufficient for the existence of a function V,(x,) such that its time derivative with respect to (7.15) is (7.20) where
Subsystem Analysis
371
-g,g,T - AT H; + H,A~o
-y,l/2g, .... H; b; + HtAT + I )ch
(7.21)
= !tcTb,- ~-t, -II= [~- CJ~t(Yt)- Yt]cpt(y,);
-y,
1
H; is a constant, symmetric, and positive definite matrix; I is the identity matrix; g is a constant vector; and 'Yo II are positive scalars. Therefore, we have V;(x,), -V;(xt)(1.ts> 0 for Yt E (yii,Yt2).
>
Now, to estimate the stability region corresponding to xi = 0, we follow the procedure of Walker and McClam.roch (1967). We first conclude that the Popov condition (7.19) for the system (7.15) and JC; = +oo, aJJ.t + (At - 1)w2 (ap; _ w2)2 + A2w2
>0
't/w
> 0,
(7.22)
> 1. Taking At > 1, we can factor the left side of the
is satisfied for At
inequality (7.22) as
(ap.;)l/2 + jw(At, - 1)1/2 (ap.;)l/2 - jw{A~ - 1)1/2 )z A = 2 . (- A + }w 2 . ) -ap. - J.w{- A - JW . ) . (7.23) w + w -ap.; + }W 1
ap; + (A!t - 1)w2
(ap.
1-
Then the vector g1 in (7.21) can be computed from the identity
.T(A·- "wi)- 1h·
gl
J
I
I
2 = (ap.;)l/ + jw{A~- l)l/ -aJJ.t+jw{-A+jw
2
)
(7.24)
as
(7.25) By solving the first equation in (7.21) with g1 of (7.25), we get the matrix 1ft as
1
[t
J
1 1ft = 2 1 A + atp; - 2[ap.,(~ - 1)]1/2 · We can choose now a V;(x1) in the form
V;(x;) = H!ixlt +
(7.26)
= 0 and obtain from (7.18) the Liapunov function
2x,tX;2
+ Axn) + r,p., !oxn [sin(y; +at)- sin al!.]dy,. (7.27)
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Engineering: Power Systems
Compu ting the time derivative of ~(x;) with respect to (7.15) and noting that y; = x;2, we get
(7.28) where tp1(xn) is defined by (7.11). Since V;(x1)! . > is zero at the limits 1 15 of the interval [Y;~>Yi2] in (7.17), in order to have ~(x1 )(7.IS) 0 we must reduce the interval to [y/ 1 ,.Y/2 ]. The choice of the reduced interval is conside red next. We can always select a positive numbe r e1 so that the inequality
..!j - 1 0
J.
0 f;Jl.;
(7.32)
>
Since ~(x;), - V;(x;){7.1s) 0 is assured for all y1 E [y/1 ,yh], we can procee d to calculate an estimate for the stability region for each subsyst em (7.15) using the function ~(x 1 ) in (7.27). We compute first
k = 1, 2,
(7.33)
and note that the minimum is at the point X;m = yikll;- 1 cJcT n-l C; on the hyperp lane cT x; = Yik at which grad ~(x;) is orthogonal to the hyperp lane. Therefore, Pile = Y;k2 ( C;T 11;-1 C; )-I
+ ~ )o(Yik qJ; ( Y1 ) dy;.
On the basis of the new limits Ytk> k ~ = mink-1,2Pik as
=
(7.34)
1, 2, we use (7.34) to find
Subsystem Analysis
373
An estimate cX1 of the stability region ~ containing the origin (xi E cX1) is the connected open region contained in 'X 1 (eX; 2~t~~ [IM,-
1
-
(7.50)
O~nisin 8?..\
Mi-1 jA;,cos
i = 1, 2, ... , n - 1.
(7.51) For a given !i, a suitable choice for tt(O
< tt < cos 8B) in (7.30) is such that (7.52)
and Am(G1) = e1p.1 for G1 defined in (7.32). Now , we can express (7.51) in terms of e1 as e1
AM(~ )[IM-I > 2 Ji;~ Am (Jl.) n
-
1 \A M; ;,
~I.. cos 01n Ism w 0
i
=
1, 2, ... , n - 1. (7.53)
From (7.53) we can conclude that the smaller the values of Ali, the easier it is to come up with suitable numb ers e • This, in tum, means that the 1 decomposition of the powe r system model shou ld be perfo rmed in such a way that the resulting subsystems are weakly couplec;l. This conclusion confirms the general prope rty of decomposit ion princi:p:~ as applied to stability analysis of dynamic systems. To determine an estimate ~ of the overall stabi lity region ~ using the aggregate model, we follow the general analy sis of Section 2.8. We first choose the overall Liapu nov function
(7.54) . where d = (d~od2, ... ,d,_ 1f is a vector with posit ive components (that is, d 0). We recall that for any given positive cons tant vector c E ~t" 1 , we can comp ute a vecto r d as ·
>
(7.55)
I
I
377
Stability Region
We choose c = (e1,e2, •.. ,en-dT and compute the vector d from (7.55). Then we calculate v1 = (V; 0 )112 from (7 .35) and determine
°
Y2
=
. vP
Dllil
1
j-1 j..i
i = 1, 2, ... 's,
(7.77)
where X;= [(x;- x 1,.)r, v1 - v1,.,6.p.t]r,
W; =
W;- W~u
(7.78)
and
A
A;=
[
A, dl •
ali~ mo
j=l j+l
0 0 0
~}
. [B,v'] 0 '
bt
=
0
0
A..= [ u
00 0 -aumJ 0
~] (7.79)
384
Engineering: Power Systems
with .X;o = (- x!., v; 0 - Vtss, 0 )r, where the subscript "ss" designates a steady-state value of the indicated variable. From (7.79), we see there is an overlapping of the subsystem state vectors x;(t) due to the fact that each subsystem in (i79) incorporates the corresponding tie-line models. Therefore, the subsystem state vector contains all accessible states from the individual area it represents. Such definition of a system is quite natural. Subsystems without overlapping would lead to a complete separation of an interconnected power system, which would aggravate the problem of decentralized AGC regulator design.
7.8. REGULATOR DESIGN Having in mind the properties (1)-(3) outlined in Section 7.6, we require that the AGC regulator should be of a decentralized type with linear constant feedback. Using the complete state feedback applied to the decomposed system model (7.77), we choose the local control for the ith area as (7.80) where (7.81) is an (n; + 2 )-vector of feedback gains associated with the ith area. In (7.81), is the n; vector of proportional control gains, is the integral control gain, kn is the proportional control gain for the net interchange load deviation. By substituting the local control (7.80) into (7.77), we get the closed-loop interconnected system as kp;
k1;
i = l, 2, ... 's.
(7.82)
A choice of the gain vector i.:; which produces a closed-loop system (7.82) with a satisfactory degree of exponential stability can be made by using the stabilization procedures of Chapter 3. Since results are already available for the power-system load and frequency control of an individual area by utilizing the optimum linear regulator with proportional-plus-integral feedback (Calovic, 1972), we choose k;'s in (7.82) to optimize each subsystem locally as decoupled (see Section 3.5). With this choice of gains k , we 1
3ti5
Regulator Design
establish stability of the overall closed-loop system (7 .82), applying eith.er the vector Liapunov function or the suboptim al design of Section 3.5. With each decoupled subsystem i
=
(7.83)
1, 2, ... , s,
we associate a quadratic performa nce index
~(t0 , x10 , l\11) =
i"" elfl(x[ Qx + ~lilT 1 1
(7.84)
lli;) dt,
>
0) and;. where Q1 is a symmetric nonnegative definite matrix (Q1 = QT onal quasidiag a as Q matrix the choose We 1 0). (P; number is a positive matrix
>
Q; =
QPi [
0 0]
0
qli
0
0
(7.85)
0 , qn
>
>
>
0 are 0, qTi 0 is a constant n, X n; matrix and qPi where QPI = Q}; degree the of measure a is which number positive a is w scalars. In (7 .84), of exponent ial stability. It is a well-known fact (e.g. Anderson and Moore, 1971) that under the assumpti on that the pair (A 1, b1) is completely controllable, there is a unique optimal gain vector kP for the feedback control law in (7 .80), which is given as •OT
k1
o1 P1, = f; -ttT•
(7.86)
where F1 is an (n1 + 2) X (n 1 + 2) symmetric positive definite matrix which is the solution of the algebraic Riccati equation
(A;• + wl; )TP1 + P(. 1 A 1 + wl;) Here l; is the (n 1 + 2) X (n; ith subsystem is
; + Q1 =· 0. P1 b;o; Tp. - 1i·-···t
(7.87)
+ 2) identity matrix. The optimal control for the (7 .88)
which results in the optimal cost (7.89) It is equally well known (e.g. Anderson and Moore, 1971) that if Q1 can be factored as Q1= M,M,T, where M, is an (n1+ 2) X (n1 + 2) matrix, so
386
Engineering: Power Systems
that the pair (A 1, M;) is completely obse rvable, then each closed-loop subsystem without interactions, i = 1, 2, ... 's, (7.90) is globally exponentially stable. That is, there exists a positive number Il 1 such that the solution x;(t; t0 ,x; ) satisfies the inequality 0 iix;(t; to,x;o)ll 0 (au < 0) for all i E N then A is domina (negative) diago~l. 1~; A generalization of Definition A.l is the following (McKenzie, Newman, 1959): ominant Defiaitlon A.l. An n X n matrix A == (a,) is said to be quasid either that diagonal if there exist positive numbers ~. j E N, such dtlaul
•
l: ~laul > J-1
ViE N
(A.2)
Vj EN.
(A.3)
}+I
or ~laJJI
•
> l:I d,laul I•
i+j
is true, Again, if all au's are positive (negative) and either (A.2) or (A.3) for all 1 = d (A.2), in If al. 1 diagon then A is quasidominant positive (negative) (A.3) and (A.2) that later shown be i E N, then it reduces to (A.l). It will in true not is (A.l) for ent statem are equivalent, whereas an analogous general. We also need poDefinition A.3. Ann X n matrix A = (au) is said to be reducible (decom that such N) C (M N of M sable) if there exists a nonvoid proper subset said to be a11 = 0 for i E M, j E N - M. Otherwise the matrix A is irreducible (indecomposable). al row and In other words, a matrix A is called reducible if by identic column transpositions it can be brought into the form
Atz]
Au [ 0 A:z:z ' matrix A where Au and A:z:z are square matrices. Note that an irreducible cannot have a zero row or column. Finally, we state the following: (positive) Definition A.4. Ann X n matrix A = (a11 ) is said to be nonnegative 0) for all i,j E N. 0 (au if au 0) we denote that the matrix A is nonnegative (positive). 0 (A By A e) if Similarly, an n-vector x = (x~ox2 , ••• ,x.)T is nonnegative (positiv 0. > B Ameans x > 0 (x 1 > 0) for all i E N. Furthermore, B >A
>
>
>
1
>
396
Appendix: Matrices
The following is the important Perron-Fro benius theorem: Theorem A.l. A nonnegative n X n matrix A always has a nonnegative eigenvalue A.. (A), the "Perron root of A ", such that IA; (A) I A.. (A) for all i E N. If A is irreducible, then the Perron root A.p(A) is positive and simple, and the corresponding eigenvector can be chosen as a positive vector.
0 for all i E •
b;;> ~ lbvl
N, we have B E
'En. and (A.4) can
'Vi EN.
(A.5)
j=l j ..i
By Definition A.l, the matrix B = AD is diagonal dominant. (4) ~ (5). Let us first establish_ that if B is ~ do~ant ~a~onal ~trix, 1, where I 1s the n X n 1dent1ty matnx and Q 1s the then a(I- Q- 1 B) 1 diagonal of B. To see this, let Ak = Ak(I - Q- B). Then there exists a 1 vector x =I= 0 such that AkX = x- Q- Bx. Let also lx 11 = ma:KjeN!x1 1 0. Then
•
"A1x 1 = ~ wii 1 w11 x1
(A.6)
J-1
J+i
and
!Xk!Jx;!
< (~ lwiiJ-'Jwol)!x;J < lxd.
(A.7)
J+i
Therefore,
>
>
0. For this purpose, we define ann X n matrix B = (b 11 ) as i,j EM, i,j (!£ M, i,j (!£ M,
i =},
(All)
i :Fj.
Obviously, B E em.. and B ;>A. From what was just established above, this implies that det B > 0 and all real eig_envalues of B are positive. Positivity of ~e au's for i (!£ M, and the fact that det B is the product of det A(M) and a1/s for i ~ M, imply that det A(M) > 0. Since M is arbitrary, (8) is established. (8) ~ (9) is trivial. (9) ~ (10). We start by showing that for ann X n matrix A = (a 11 ) the sequence of principal minors det A(Mt), where M .= 1, 2, ... , i, is positive if and only if there exist a lower triangular matrix U and an upper triangular matrix V, both with positive diagona l elements, such that A = UV. The "if'' part is established by induction. For n = 1, the hypothesis follows trivially. We assume that the hypothesis is true for n- 1 and show that it is also true for n. We write
A= [An-t a b a""
J
(A12)
and assume An-t = OfJ, where 0 (P) is a lower (upper) triangular matrix with positive diagonal elements. From (A.l2), we have _
a""
_
_
detA
1 bA,._, a - det A_, > 0.
(A.13)
The choice (A.l4) establishes the "if" part of the statement. The "only if" part follows immediately by observing that the principa l minors det A(Mt) are equal to the product of the first i diagonal elements of U and V. We also need to show that if A E ~ and A = UV as above, then j U, V E em. To see this, let U = (u,) and V = (v;;), so that uu = 0 fori n inductio use again and v11 = 0 fori> j with uu, v11 > 0 for i,j E N. We and show that u11 "' 0, v11 "' 0, i + j. For i + j = 3, we have tlll = un v12 and a21 = ~~ Vtt. and~~ "' 0, v12 "' 0. We assume that ukl "' 0, vld "' 0 for i + j, where i + j > 3, i + j. If i j, then from k +I
i =j, i =F j,
(Al7)
for all i,j E N. Defmition A.tO (Arrow, 1966). An n X n matrix A matrix if au> 0, i =F j, i,j E N.
= (au)
is a Metzler
403
Appendix: Matrices
Obviously, Definition A.l implies Definition A.2, but not vice versa. As we will see shortly, a necessary condition for stability of Metzler matrices is negativity of the diagonal elements au of A. Thus, with respect to stability, the two definitions are equivalent. We also note that according to Definition A.1 0, if A is a Metzler matrix, then -A E '!){, and vice versa. This simple fact opens up a possibility of establishing numerous properties of Metzler matrices using the properties of ~matrices listed in Theorem A.2. Since our interest is predominantly in stability, we need
Definition A.ll. Ann X n matrix A is called stable (or Hurwitz) ifRe A;(A) < 0 for all i E N. We also recall (Newman, 1959)
Definition A.ll. A matrix A is called a Hicks matrix if all odd-order principal minors of A are negative and all even-order principal minors of A are positive. Then we have the fundamental result of Metzler (1945):
Theorem A.8. A Metzler matrix A is stable if and only
if it is Hicks.
Proof. By noting that -A E ~the theorem follows from (8) and (13) of Theorem A.2.
It is of interest to note that to test for stability of a Metzler matrix one does not need to test all principal minors as required by Theorem A.8, but only the leading principal minors of A. This result was proved· by Kotelyanskii (1952) using the results of Sevastyanov (1951). Thus, we have Theorem A.'J. Ann X n Metzler matrix. A = (au) is stable if and only
alk
Proof.
>0
Vk EN.
if
(A. IS)
For the proof of this theorem see Gantmacher (1960).
It is obvious that Kotelyanskii's result (Theorem A.9) can now be used to conclude that (8) of Theorem A.2 can be replaced by: All leading principal minors of A are positive. In mathematical economics this result is known as the Hawkins-Simon conditions (see Nikaido, 1968). Another fundamental result used throughout this book, which is due to McKenzie (1966), is the following:
404
Appendix: Matrices
Theorem A.lO. A Metzler matrix A is stable if and only if it is quasidominant negative diagonal. Proof. A.2.
The theorem follows immediately from (3) and (12) of Theorem
We can dispense with the Metzler structure of A and still use the diagonal dominance to conclude stability of A. For this we need McKenzie's "diagonal form" of an n X n matrix A = (a 11 ) with a negative diagonal (a 11 0, i E N ), which is defined ·as the n X n matrix B = (b ) with 11
~!alii J-I
ViE N
(A.20)
i+i
or n
la.ul > I-I ~ la11i
'Vj EN
(A.21)
I+J
implies stability of A. This follows from above choosing d1 = 1, i E N in (A.2) and (A.3). When a matrix is not Metzler, we still can establish stability by quasidomina nce conditions. However, the conditions are only sufficient. We can recover the necessity part even if some of the off-diagonal elements of a matrix are negative, so long as the matrix is of Morishima's (1952) type. Definidon A.13. A matrix A is called a Morishima matrix if A is i"educible matrix and can be permuted into the form
405
Appendix: Matrices
where A 11
~
0, A 22
~
0 are square matrices and A 12
~
0, A21
~
0.
Bassett, Habibagahi, and Quirk (1967) proved the following: ima matrix Theorem A.12. Let B = A - al, where A is an n X n Morish if A is a only and if matrix stable a is A Then ali for all i E N. and a " matrix. al quasidominant negative diagon review For more results on diagonal-dominance conditions, one should as the well as (1968), ik the books by Varga (1962) and by Quirk and Saposn ens Beauw and (1974), n papers by McKenzie (1966), Pearce (1974), Johnso
>
(1976).
es the There is anothe r interesting proble m in this context, which combin t this presen To ces. ~matri properties of nonnegative, Metzler, and s. eration consid proble m we need some preliminary We first recall the well-known (Bellman, 1960)
Definition A.14. A real n X n matrix A is called positive definite > 0 for all n-vectors x .P. 0.
if xrAx
tric and By writing A = B + C, where B = (A + AT)/2 is the symme xTCx that show can we A, of part ric ymmet C = (A - AT)/2 the skew-s A ore, Theref Bx. xT = Ax xT that so 0, = Cx xr = (xT Cx)T = - xr Cx and is 2 AT)/ + (A = B part tric symme its if only is positive definite if and positive definite. 1960): We also recall the classical Liapun ov result (see Gantm acher, definite Theorem A.13. A matrix A is stable if and only if for any positive that such H matrix tric symmetric matrix G there is a positive definite symme ATH+ HA =-G.
(A.22)
1973) it In the stability analysis of large-scale systems (Grujic and Siljak, ov Liapun the of H n solutio the is of interes t to find conditions under which all are ts elemen its is, that , matrix matrix equati on (A.22) is a positive Siljak and Grujic by d resolve ly partial positive numbers. This proble m was (1973) as follows: e and Theorem A.14. If a matrix A is Metzler and stable, then for any positiv definite positive positive definite symmetric matrix G there is a positive and ov matrix symmetric matrix H as a solution of the corresponding Liapun equation.
406
Appendix: Matrices
Proof. As is well known (e.g. Gantmacher , 1960), Liapunov's matrix equation (A.22) can be rewritten as a linear matrix equation Bh
=
-g,
(A.23)
where B
n2 X
=
AT®J+ l®A
(A.24)
n2
is an matrix with ® denoting the Kronecker product; H = (h11 , h12, ... , hhl, h21> ••• , h,.f, g = (gu, g12, .•. , g11, g21> ... , g,.)r; and A = (au), H = (hu), G = (gu) are all n X n matrices. The matrix B is stable, and by construction it is also Metzler. By (2) of Theorem A.2, we conclude that for any g 0 we have h 0. That His positive definite follows from Theorem A.l3. This completes the proof of Theorem A.14.
>
>
It is fairly obvious that an "only if" part can be proved by Theorem A.l4. It is more important, however, to try to dispense with the Metzlerian structure of A. To this effect, the following conjecture was stated by Siljak
(1972).
Conjecture. If a matrix A is stable, then there is a positive definite symmetric matrix G such that the matrix H as a solution of the corresponding Liapunov matrix equation is a positive and positive definite symmetric matrix. In a private communication to the author, Professor V. M. Popov from the University of Florida pointed out that some additional assumptions are needed regarding the matrix A, since otherwise
A=
[1 -2] 3 -4
(A.25)
is a counterexample to the conjecture. It is of interest to point out that AT of (A.25) satisfies the conjecture. Conditions on A for the conjecture to be true, as well as some important generalizations, are presented by Womack and Montemayo r (1975), Montemayo r and Womack (1976), and Datta (1977).
REFERENCES Arrow, K. J. (1966), "Price Quantity Adjustments in Multiple Markets with Rising Demands", Proceedings of the Symposium on M athetMtica/ Methods in the Social Sciences, K. J. Arrow, S. Karlin, and P. Suppes (eds.), Stanford University Press, Stanford, California, 3-15.
References Arrow, K. J., and McManus, M. (1958}, "A Note on Dynamic Stability", EcOMm· etrica, 26, 448-454. Bassett, L., Habibagahi, H., and Quirk, J. (1967), "Qualitative Economics and Morishima Matrices", Econometrica, 35, 221-233. Beauwens, R. (1976}, "Semistrict Diagonal Dominance", SIAM Journal of Numerical AM/ysis, 13, 109-ll2. Bel1man, R. (1960}, Introduction to Matrix Ano/ysis, McGraw-Hill, New York. Datta, B. N. (1977}, "Matrices Satisfying Siljak's Conjecture", IEEE Transactions, AC-22, 132-133. Debreu, G., and Herstein, I. N. (1953}, "Nonnegative Square Matrices", Econometrica, 21, 597-6crl. Enthoven, A. C., and Arrow, K. J. (1956}, "A Theorem on Expectations and the Stability of Equilibrium", Econometrica, 24, 28s-293. Fan, K. (1957}, "Inequalities for the Sum of Two ~Matrices", Inequalities. 0. Shisba (ed}, Academic, New York, 105-117. Fan, K. (1958), "Topological Proofs for Certain Theorems on Matrices With NonNegative Elements", Monatshejtejflr Mathematik, 62,219-237. Fan, K., and Householder, A. S. (1959}, "A Note Concerning Positive Matrices and ~Matrices", Monatsheftejur Mathematik, 63, 265-270. Fan, K. (1960}, '"Note on '!JR,Matrices", Quarterly Journal of Mathematics, 11, 43-49. Fiedler, M., and Ptcik, V. (1962}, "On Matrices with Non-Positive Off-Diagonal Elements and Positive Principal Minors", Czeclws/oiiOkitln Mathematical Journo/, 12, 382-400. Fiedler, M., and PtAk, V. (1966}, "Some Generalizations of Positive Definitness and Monotonici.ty", Numerische Mathematik, 9, 163-172. Gantmacher, F. R. {1960}, The Theory of Matrices, Vols. I and II, Chelsea, New York. Grujic, Lj. T., and Siljak, D. D. (1973}, "Asymptotic Stability and Instability of Large-Scale Systems", IEEE Transactions, AC-18, 636-645. Hawkins, D., and Simon, H. (1949}, "Note: Some Conditions of Macroeconomic Stability", Econometrica, 17, 53-56. Johnson, C. R. {1974}, "Sufficient Conditions for iSj)..Stability", Journal of Ecotll)mic Theory, 9, 53-62. Kotelyanskii, I. N. (1952}, "On Some Properties of Matrices with Positive Elements" (in Russian), Mathematicheski Sbornik, 31, 497-506. Maybee, J. S. {1976}, "Some Aspects and Solutions of '1J%.Matrices", SIAM J0&1171a/ of Applied Mathematics, 31, 397-410. McKenzie, L. (1966), "Matrices with Dominant Diagonals and Economic Theory", Proceedings of the Symposium on Mathematical Methods in the Social Sctence.s, K. J. Arrow, S. Karlin, and P. Suppes (eds.), Stanford University Press, Stanford, California, 47-62. Metzler, L. A. (1945), "Stability of Multiple Markets: The Hicks Conditions", Econometrica, 13, 277-292. Montemayor, J. J., and Womack, F. B. (1976), "More on a Conjecture by Siljak", IEEE Transactions, AC-21, 805-806.
408
Appendix: Matrices
Morishima, M. (1952), "On the Laws of Change of the Price-System in an Economy which Contains Complementary Commodities", Osaka Economic Papers, 1, 101-113. Morishima, M. (1964), Equilibrium, Stability, and Growth, Clarendon, Oxford, England. Mosak, J. L. (1944), General Equilibrium Theory in International Trade, Cowles Commission Monograph 7, Principia, Bloomington, Indiana. Newman, P. K. (1959), "Some Notes on Stability Conditions", Review of Economic Studies, 72, 1-9. Nikaido, H. (1968), Convex Structures and Economic Theory, Academic, New York. Ostrowski, A. (1937), "Uber die Determinanten mit iiberwiegender Hauptdiagonale", Commentarii Mathematici Helvetici, 10, 69-96. Ostrowski, A. (1956), "Determinanten mit iiberwiegender Hauptdiagonale und die absolute Konvergenz von linearen lterationsprozessen", Commentarii Mathematici Helvetici, 30, 175-210. Pearce, I. F. (1974), "Matrices with Dominant Diagonal Blocks", Journal of Economic Theory, 9, 159-170. Quirk, J., and Saposnik, R. (1968), Introduction to General Equilibrium Theory and Welfare Economics, McGraw-Hill, New York. Sandberg, I. W., and Wilson, A. N. Jr. (1969), "Some Theorems on Properties of DC Equations of Nonlinear Networks", The Bell System Technical Journal, 48, 1-34. Seneta, E. (1973), Non-Negative Matrices, Wiley, New York. Sevastyanov, B. A. (1951), ''Theory of Branching Stochastic Processes" {in Russian), Uspekhi Matematicheskih Nauk, 6, 47-99. Shisha, 0. (ed.) (1970), "Inequalities", Vols. I and II, Academic, New York. Siljak, D. D. (1972), "Stability of Large-Scale Systems", Proceedings of the Fifth IFAC Congress, Part IV, Paris, C-32: 1-11. Uekawa, Y., Kemp, M. C., and Wegge, L. L. (1972), "'P- and ~Matrices, Minkowski- and Metzler-Matrices, and Generalizations of the Stolper-Samuelsen and Samuelson-Rybczynski Theorems", Journal of International Economics, 3, 53-76. Varga, R. S. {1962), "Matrix Iterative Analysis", Prentice-Hall, Englewood Cliffs, New Jersey. Willems, J. C. (1976), "Lyapunov Functions for Diagonally Dominant Systems", Automatica, 12, 519-523. Womack, B. F., and Montemayor, J. J. (1975), "On a Conjecture by Siljak", IEEE Transactions, AC-20, 512-513.
INDEX
Absolute stability, 106, 238 AGC regulator, 387 Aggregate excess demand, 224 Aggregate matrix, 36, 375, 389 optimal, 346 Aggregate model, 36, 92, 174 Aggregate model of the economy, 262 Aggregation, 32, 261 Antece dent set, 149 Area control, 380 Area control concept (ACC), 383 Area control error (ACE), 382 Area model, 381 Arms race, 18, 241 Lotka-Volterra model, 287 Arrow -Hahn conjecture, 237 Asymptotic stability, 89 connective, 70, 86, 92 connective in the large, 71 in the large, 91 Auctioneer, 226 Automatic generation control (AGC), 379
decentralized model, 381 dynamically reliable, 389 Basic structure, 148 market, 251 Bilinear systems, 335 Bounded perturbations, 299 Breakdown, 20 Budget constraint, 223 . Canonical structure, 159 Centralized design, 380 ·centralized system input, 43 output, 48 Chemical reaction, 4 Clearing time of fault, 364 Commodity, 3 complements, 23 composite, 260 consumed, 223 free disposal of, 233 substitutes, 23 409
410 Community of species, 272 aggregate model, 317 complexity of, 316 connectively stable, 273 density dependent, 272 matrix of, 272 mixed, 281 stabilized form of, 317 stable, 295 stochastic models of, 303 structure, 273, 315 subcommunity, 317 Companio n matrix, 177 Comparison function, 83 Comparison principle, 16, 38, 78 stochastic, 253 strict, 75 vector, 38 Competition, 5 among species, 272 Competitive equilibrium, 4, 219 dynamic model of, 226 Complements, 23 Complete instability, 127 Complexity, 14, 269, 275 Complexity of ecosystem, 316, 321 Complexity vs. stability, l, 269 Composite commodities, 260 Condensation, 154 Connective reachability, 162 Connective stability, 18, 49, 69 Consumers, 220 Continuous function, 65 Control, 43 decentralized, 172 function, 43 global, 55, 173, 203, 342 law, 201 local, 55, 172, 187, 335, 384 multilevel, 55, 172 optimal, 50, 20 l power area, 380 proportional-plus-integral, 386 Control function, 43 Controllability, 157, 177 Convergent system, 244 Cooperation, 23
Index Decentralized estimator, 195 Decentralized regulator, 196, 381 Decentralized system, 145 input, 44, 159 market, 224 output, 48, 160 Decomposition, 25 LST model, 334 mathematical, 25 physical, 25 principle, 25, 376 with overlapping, 363 Decomposition-aggregation analysis, 32, 38, 344 Decoupled subsystem, 26, 72 Decrescent, 84 Degree of stability, 173 Demand, 220, 222 Demand shift, 242 Depth-first search, 153 Inagonal dominance, 240, 270,27~ 305 discrete version of, 256 ecosystems, 321 matrix, 13 Diagonal-dominant matrix, 13 Differential inequality, 14 Digraph, 6, 67, 147 arcs of, 67 branches of, 67 condensation of, 154 connectively reachable, 162 directed lines of, 5 · food web, 316 fundamental, 68 input-truncation, 160 lines of, 6, 67 nodes of, 67 output-truncation, 160 point basis of, 160 point contrabasis of, 160 points of, 5, 67 state-truncation of, 155 strong components of, 155 vertices of, 67 weighted, 67 Dini derivatives, 65
Index
Disconnection, 8 Discrete time system, 256 D-stability, 276, 402 Dynamic reliability, 20, 49, 52 Dynamic system, 147 Economic dispatch (ED), 379 Economy, 220 aggregate model, 262 Ecosystems, 316 hierarchic models, 315 limitation of resources, 287 matrix model, 279 resilience of, 285 robustness of, 285 stability, 295 stability, regions of, 285 structure of, 315 Eigenvalue, 394 Electronic circuits, 241 Equations of motion, 28, 330 Large Space Telescope, 329 Skylab, 353 Equilibrium, l 0, 73 competiive, 219 moving, 241 population, 22, 276, 280, 287 post-fault, 364 price, 220 Equivalencing, 379 Estimator, 195 asymptotic, 195 decentralized, 195 separation property, 195 subsystem, 195 Euclidean norm, 65 Euclidean space, 64 Excess demand, 222 Expectation, 251 Exponential stability, 13 connective, 71, 103, 281 in the large, 100, 101 Feedback, 176 output, 180, 350 state, 176, 384
411
Fixed interconnection matrix, 114 Food web, 315, 316 Free disposal of commodities, 233 Free system, 5 Frequency deviation, 382 Function, 65 comparison, 83 continuous, 65 decrescent, 84 demand, 224 excess demand, 222 Liapunov, 11, 89 Lipschitzian, 80 Lur'e-Postnikov, 108 negative definite, 84 perturbation, 298 positive definite, 84 quasimonotone, 75 radially unbounded, 85 supply, 222 transfer, 105 utility, 223 vector Liapunov, 86, 174 Volterra, 293 Fundament al digraph, 68 Fundament al interconnection matrix, 8, 66, 273 Generalized matrix inverse, 175, 352 Giffen paradox, 254 Global control, 56, 173, 203, 342 cost of, 211 Goods, 260 Graph, 5 Gross-mutualism, 288 Gross-substitute case, 229 Hamilton-Jacobi equation, 205 Hawkins-Simon conditions, 230, 403 Hessian matrix, 253, 304 Hicks conditions, 21, 4-l, 96 Hicks matrix, 403 Hierarchic models, 49 economics,260 ecosystems, 315 Hurwitz matrix, 403
··~
412
Inclusion principle, 121 Income vector, 256 Indecomposable matrix, 395 hidex of suboptimality, 202 Initial-value problem, 75 Input, 147 Input-decentralized system, 44, 159 Input-output model, 256 Input-output reachability, 149 Input reachability, 149 Input-truncation digraph, 160 Instability complete, 114 complete connective, 126, 127, 295 global, 114 stochastic, 313 Interacting species, 272, 287 Interaction, 6 (See also Interconnection) Interactions in social groups, 241 Interconnected subsystems, 26, 154 Interconnection, 6 bilinear, 335 bounds of, 336 competitive, 3, 272 constraints, 54, 288, 374 deterministic, 307,318 mixed, 281, 304 physical, 66 populations, 272 predator-prey, 272 saprophytic, 272 stochastic, 307, 318 symbiotic, 272 Interconnection matrix, 6, 66, 148, 273 constant, 231 fixed, 114 fundamental, 8, 66, 273 nonlinear, 19 time-varying, 19 International trade system, 256 Irreducible matrix, 395 Isolated subsystem, 89 Ito differential equation, 250, 303 Jacobian matrix, 229, 244, 277
Kalman construction of Liapunov function, 369 Kronecker product, 406 Kronecker symbol, 93 Lagrange multiplier, 223 Lagrangian, 223 Large-scale system, 2, 88 Large Space Telescope, 339 control system of, 340 equations of motion, 329 model of, 33~ Law of composition of goods, 260 Law of supply and demand, 225 Leading principal minors, 96 Liapunov direct method, 10 Liapunov function, 11, 86 Liapunov matrix equation, 405 Liapunov stability, 70 Liapunov theorem, 89 Liapunov vector function, 32, 86, 174 Linearization, 277 Linear system, 25, 207 constant, 25, 96 time-varying, 33 Line removal of digraph, 6 Lipschitz condition, 15, 80 constant, 80 local, 80 Load and frequency control (LFC), 379 Local control, 55, 172, 187, 335, 342 autonomous AGC, -380 Local output, 48 Local stabilization, 190 Loop,6 Lotka-Volterra model, 21, 287, 293 Lur'e-Postnikov system, 105, 369 mu1tinonlinear, 367 Lur'e-Postnikov type Liapunov function, 108, 370 Malthusian growth, 22, 278 Mapping, 65 Market, 3, 222 gross-substitute, 229 mixed,239
Index
413
Moving equilibrium, 241 reduced, 233 Multilevel control, 55, 172 stochastic model, 250 Multilevel stabilization, 186 structure of, 251 Multimachine systems, 366, 368 Mathematical programming, 349 disturbance of, 364 Matrix, 3, 394 equivalencing of, 379 adjacency, 6 nonuniform damping, 368 aggregate, 36, 345 post-fault equilibrium, 364 community, 272 stability region, 377 companion, 177 three-machine system, 378 decomposable, 395 transient stability, 364 diagonal, 188, 404 uniform damping, 366 dominant diagonal, 274, 394 fixed interconnection, 114 Negative definite function, 84 fundamental interconnection, 8, 66, Negative dominant diagonal matrix, 252 273 Nonlinearity, 55, 106, 370 generalized inverse of, 175, 352 sector of, 106 Hessian, 253, 304 Norm, 10 Hicks, 230, 282, 403 Euclidean, 10, 239 Hurwitz, 403 absolute-value, 239 interconnection, 6, 66, 148, 273 N onnalized prices, 224 Jacobian, 229, 244, 277 Numeraire, 224 leading principal minors of, 96 Metzler, 3, 229, 275, 282, 402 Observability, 157 Minkowski, 396 Occurrence matrix, 66 '!JRrmatrix, 58, 400 Optimal control, 50 Morishima, 275, 404 LST, 340 nonnegative, 395 power area, 386 occurrence, 66 trajectories, 201 path, 149 Optimization, 200 Perron root of, 396 local, 200, 386 t31-matrix, 401 LST control, 341 .: · ~matrix, 402 multilevel, 207 positive, 395 Output, 147 positive definite, 405 Output-decentralized system, 48, 160 principal submatrix, 118, 394 Output feedback, 180, 350 quasidominant diagonal, 94 Output reachability, 149 reducible, 395 Output-truncation digraph, 160 stable, 403 Output vector, 256 V andermonde, 188 Overlapping of subsystems, 69, 156, 363 McKenzie diagonal form, 404 Metzler matrix, 402 Parameter plane method, 172, 347 time-varying, 75 Parasite-host system, 23 Minkowski matrix, 396 Pareto-optimal, 349 '!J!L..matrix, 400 Partition, 153 Modified Routh table, 108 Path, 149 Morishima matrix, 275, 404
414
Path matrix, 149 Performance index, 50, 201, 341, 385 Perron-Frobenius theorem, 396 Perron root, 255, 396 Perturbation function, 297 Perturbations, 63 bounded, 299 constraints, 300 environmental, 297 forcing function, 63 initial condition, 63 parameter, 63 principal structural, 7, 118 random, 250 structural, 6 ~-matrix, 401 ~matrix, 402 Point basis, 160 Point contrabasis, 160 Point removal, 7 Pole assignment, 347 Pole-shifting method, 172 Popov counterexample, 406 Popov criterion, 107, 370 Population equlibrium, 22, 276, 280, 287 Population vector, 272, 276 Positive definite function, 84 Power area control concept (ACC), 383 control error (ACE), 382 exchange, 382 local control, 384 Power area control, 380 Power exchange, 382 Power system, 381 connectively stable, 389 decentralized, 383 dynamically reliable, 389 interconnected, 383 stability, 388 Predation, 272 Predator attack capability, 281 Predator-prey model, 21, 29 interaction, 272, 275 Predator searching behavior, 281 Predator switching, 281
Index
Price, 3 equilibrium of, 225 negative, 233 normalized, 124 vector, 3, 228 Price equilibrium, 225 Principal minor, 396 Principal submatrix, 118, 394 Producers, 220 Quasidominant diagonal matrix, 395 Quasimonotonous functions, 75 Radially unbounded function, 85 Random disturbance, 303 Random variable, 250 Reachability, 147 connective, 163 input, 149 input-output, 149 output, 149 partially connective, 164 set, 149 Reachable set, 149 Reference input, 196 Regulator, 196 AGC, 387 decentralized, 196, 381 linear quadratic, 207 subsystem, 196 Reliability, 20 dynamic, 20, 49, 52, 213 ecomodels, 285 principle of, 2 Reliability principle, 2 Resilience, 285 Riccati equation, 208, 341, 385 Robustness, 203, 213 of competitive equilibrium, 221 of control, 203 of ecosystems, 285 Root-locus method, 172, 180, 347 Routh table, 108 Saprophytism, 272 Separation property, 196
Index
415
Sevastyanov-Kotelyanskii conditions, 230, 403
Siljak conjecture, 406 Single-valued mapping, 65 Skylab, 353 equations of motion, 355 nomenclature for, 360 physical characteristics of, 361 structural decomposition, 357 Social groups, 241 Solution, 75 constant, 10, 73 equilibrium, 73 existence of, 80 fixed, 73
maximal,77 minimal, 79, 130 process, 251 uniqueness, 80 Spanning subgraph, 68 Species, 272 crowding, 287 destruction, 274 gross mutualism, 288 interacting, 272, 287 predator-prey, 21 removal, 274 Spectral radius, 396 Speed of adjustment, 240, 276 Spin motion, 354 Stability, 10 absolute, 73, 106, 238 absolutely connective, 72, 96, 103 asymptotic, 89 asymptotically connective, 70, 86, 92 asymptotically connective in the large, 71, 88, 93, 99 asymptotic in the large, 91 connective, 18, 49, 69 connective in the mean, 252 degree of, 173 D-stability, 276 exponential, 13 exponentially connective, 71, 103, 281
exponential in the large, 100, 101
exponentially connective in the large, 102, 104 in the mean, 304 partially connective, ll3, ll5 principally connective, 119, 120 region of, 133, 290, 338 stochastic, 252, 303, 3ll total, 234 transient, 364 uniformly connective, 70 Stability region, 133, 290, 338 ecosystems, 285 LST, 340 multimachine systems, 377 Stabilization, 54, 175, 185 bilinear systems, 335 Large Space Telescope, 338 local, 190 multilevel, 186 Stable matrix, 403 State, 3 subystem, 69 system, 69 State feedback, 176, 384 State-truncation digraph, 155 Stochastic instability, 313 Stochastic stability, 252, 303, 311 Strong components, 155 Structural parameters, 344, 350 maximization of, 349 Structural perturbation, 6, 68, Ill, 162, 202
partial, 118 principal, 118, 294 Structure, 5 basic, 148 canonical, 159 ecosystems, 316 parameters of, 343 perturbations of, 6, 68, 162, 163, 202 trophic web, 315 vulnerability of, 161, 269 Subeconomy, 261 Subgraph, 68 Suboptimality, 202 connective, 206, 208 index of, 202
416
Substitutes, 3, 23 Substitution effect, 3 Subsystem decoupled, 26, 72 estimator, 195interconnected, 26, 334 mathematical, 68 overlapping, 68, 156, 363, 380 regulator, 196 unstable, 113 weakly coupled, 199 Subsystem regulator, 196 Supply, 220 Symbiosis, 272 System, 66 bilinear, 335 closed-loop, 173, 384 connectively stable, 231 convergent, 244 discrete, 256 dynamic, 147 environment, 8 free, 5 input-centralized, 43 input-decentralized, 44, 159 large-scale, 2, 88 linear, 96 linear constant. 25, 96 linear time-varying, 33 Lur'e-Postnikov, 105, 367 multimachine, 364 output-centralized, 48 output-decentralized, 48, 160 reliable, 2, 52
Index
robust. 54 stabilization of, 175 n.tonnement, 220, 256 Tearing, 25 Time interval, 65 Transfer conductances, 366 Transfer function, 105 Transient stability, 364 · Transpose, 65 Trophic web, 315 Ultimate boundedness, 242 connective, 301 region of, 300 Uniform damping, 366 Utility function, 223 Vandermonde matrix, 188 Vector Liapunov function, 32, 86, 174 Vector maximization, 349 Vector norm, 65 Verhulst-Pearllogistic equation, 277 Volterra function, 293 Vulnerability, 161, 269 Walrasian competitive economy, 220 Weakly coupled subsystems, 199 Weighted digraph, 67 Wiener process, 250, 303, 317 Wobble motion, 353 Zero interconnection matrix, 8, 53, 67 Zero interconnection vector, 9
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,, I
CATALOG OF DOVER BOOKS
Engineeri ng DE RE METALUCA, Georgius Agricola. The famous Hoover translation of greatest treatise on technological chemistry, engineering, geology, mining of early mod0..486-60006-8 em times (1556). All289 original woodcuts. 638pp. 6'% x 11. FUNDAMENTA lS OF ASTRODYNAMICS, Roger Bate et a!. Modem approach developed by U.S. Air Force Academy. Designed as a first course. Problems, exer0-486-60061-0 cises. Numerous illustrations. 455pp. 5% x 8'h. DYNAMICS OF FLUIDS IN POROUS MEDIA, Jacob Bear. For advanced students of ground water hydrology, soil mechanics and physics, drainage and irrigation engineering and more. 335 illustrations. Exercises, with answers. 784pp. 6'k x 9%. 0-486-65675-6 THEORY OF VISCOELASTIC ITY (Second Edition), Richard M. Christensen. Complete consistent description of the linear theory of the viscoelastic behavior of materials. Problem-solving techniques discussed. 1982 edition. 29 figures. 0-486-42880-X xiv+364pp. 6'k x 9'4. MECHANICS,] . P. Den Hartog. A classic introductory text or refresher. Hundreds of applications and design problems illuminate fundamentals of trusses, loaded 0-486-60754-2 beams and cables, etc. 334 answered problems. 462pp. 5% x 8'h. MECHANICAL VIBRATIONS, J. P. Den Hartog. Classic textbook offers lucid explanations and illustrative models, applying theories of vibrations to a variety of practical industrial engineering problems. Numerous figures. 233 problems, solu0-486-64785-4 tions. Appendix. Index. Preface. 436pp. 5% x 8V.. STRENGTH OF MATERIAlS, J. P. Den Hartog. Full, clear treatment of basic material (tension, torsion, bending, etc.) plus advanced material on engineering 0-486-60755-0 methods, applications. 350 answered problems. 323pp. 5% x 8'h. A HISTORY OF MECHANICS, Rene Dugas. Monumental study of mechanical principles from antiquity to quantum mechanics. Contributions of ancient Greeks, Galileo, Leonardo, Kepler, Lagrange, many others. 67lpp. 5% x 8'h. 0-486-65632-2 STABILITY THEORY AND ITS APPLICATION S TO STRUCTURAL MECHANICS, Clive L. Dym. Self-contained text focuses on Koiter postbuckling analyses, with mathematical notions of stability of motion. Basing minimum energy principles for static stability upon dynamic concepts of stability of motion, it develops asymptotic buckling and postbuckling analyses from potential energy considerations, with applications to columns, plates, and arches. 1974 ed. 208pp. 5% x 8'h. 0-486-42541-X METAL FATIGUE, N. E. Frost, K.J. Marsh, and L. P. Pook. Definitive, clearly written, and well-illustrated volume addresses all aspects of the subject, from the historical development of understanding metal fatigue to vital concepts of the cyclic stress that causes a crack to grow. Includes 7 appendixes. 544pp. 5% x 8'h. 0-486-40927-9
CATALOG OF DOVER BOOKS
Mathematics FUNCTIONAL ANALYSIS (Second Corrected Edition), George Bachman and Lawrence Narici. Excellent treatment of subject geared toward students with background in linear algebra, advanced calculus, physics and engineering. Text covers introduction to inner-product spaces, normed, metric spaces, and topological spaces; complete orthonormal sets, the Hahn-Banach Theorem and its consequences, and many other related subjects. 1966 ed. 544pp. 61A. x 9\4. 0-486-40251-7 ASYMPTOTIC EXPANSIONS OF INTEGRALS, Norman Bleistein & Richard A. Handelsman. Best introduction to important field with applications in a variety of scientific disciplines. New preface. Problems. Diagrams. Tables. Bibliography. Index. 448pp. 5% X 8~. 0-486-65082-0 VECTOR AND TENSOR ANALYSIS WITH APPLICATIONS, A. I. Borisenko and I. E. Tarapov. Concise introduction. Worked-out problems, solutions, exercises. 257pp. 55,1, X 81.4. 0-486-63833-2 AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS, Earl A. Coddington. A thorough and systematic first course in elementary differential equations for undergraduates in mathematics and science, with many exercises and problems (with answers). Index. 304pp. 5% x 8~. 0-486-65942-9 FOURIER SERIES AND ORTHOGONAL FUNCTIONS, Harry F. Davis. An incisive text combining theory and practical example to introduce Fourier series, orthogonal functions and applications of the Fourier method to boundary-value problems. 570 exercises. Answers and notes. 416pp. 5% x 8~. 0-486-65973-9 COMPUTABILITY AND UNSOLVABILITY, Martin Davis. Classic graduatelevel introduction to theory of computability, usually referred to as theory of recurrent functions. New preface and appendix. 288pp. 5% x 8~. 0-486-61471-9 ASYMPTOTIC METHODS IN ANALYSIS, N. G. de Bruijn. An inexpensive, comprehensive guide to asymptotic methods-the pioneering work that teaches by explaining worked examples in detail. Index. 224pp. 5% x 8'h 0-486-64221-6 APPLIED COMPLEX VARIABLES, john W. Dettman. Step-by-step coverage of fundamentals of analytic function theory-plus lucid exposition of five important applications: Potential Theory; Ordinary Differential Equations; Fourier Transforms; Laplace Transforms; Asymptotic Expansions. 66 figures. Exercises at chapter ends. 512pp. 5% x 8~. 0-486-64670-X INTRODUCTION TO LINEAR ALGEBRA AND DIFFERENTIAL EQUATIONS, John W. Dettman. Excellent text covers complex numbers, determinants, orthonormal bases, Laplace transforms, much more. Exercises with solutions. Undergraduate level. 416pp. 5% x 8\1,. 0-486-65191-6 RIEMANN'S ZETA FUNCTION, H. M. Edwards. Superb, high-level study of landmark 1859 publication entitled "On the Number of Primes Less Than a Given Magnitude" traces developments in mathematical theory that it inspired. xiv+315pp. 5% X 8\1,. 0-486-41740-9
CATALOG OF DOVER BOOKS CALCULUS OF VARIATIONS WITH APPUCATIONS , George M. Ewing. Applications-oriented introduction to variational theory develops insight and promotes understanding of specialized books, research papers. Suitable for advanced undergraduate/graduate students as primary, supplementary text. 352pp. 5% x 8'h. 0-486-64856-7 COMPLEX VARIABLES, Francis J. Flanigan. Unusual approach, delaying complex algebra till harmonic functions have been analyzed from real variable view0-486-61388-7 point. Includes problems with answers. 364pp. 5% x 8'h. AN INTRODUCTIO N TO THE CALCULUS OF VARIATIONS, Charles Fox. Graduate-level text covers variations of an integral, isoperimetrical problems, least action, special relativity, approximations, more. References. 279pp. 5% x 8'h. 0-48 6-65499-0 COUNTEREXA MPLES IN ANALYSIS, Bernard R Gelbaum and John M. H. Olmsted. These counterexamples deal mostly with the part of analysis known as "real variables." The first half covers the real number system, and the second half encompasses higher dimensions. 1962 edition. xxiv+198pp. 5% x 8'h. 0-486-42875-3 CATASTROPHE THEORY FOR SCIENTISTS AND ENGINEERS, Robert Gilmore. Advanced-level treatment describes mathematics of theory grounded in the work of Poincare, R. Thorn, other mathematicians. Also important applications to problems in mathematics, physics, chemistry and engineering. 1981 edition. 1 References. 28 tables. 397 black-and-white illustrations. xvii + 666pp. 6 k x 9%. 0-486-67539-4 INTRODUCTIO N TO DIFFERENCE EQUATIONS, Samuel Goldberg. Exceptionally clear exposition of important discipline with applications to sociology, psychology, economics. Many illustrative examples; over 250 problems. 260pp. 5% x 8'h. 0-486-65084-7 NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS, Richard Hamming. Classic text stresses frequency approach in coverage of algorithms, polynomial approximation, Fourier approximation, exponential approximation, other 0-486-65241-6 topics. Revised and enlarged 2nd edition. 721pp. 5% x 8'h. INTRODUCTIO N TO NUMERICAL ANALYSIS {2nd Edition), F. B. Hildebrand. Classic, fundamental treatment covers computation, approximation, interpolation, numerical differentiation and integration, other topics. 150 new problems. 0-486-65363-3 669pp. 5% X 8'h. THREE PEARLS OF NUMBER THEORY, A. Y. Khinchin. Three compelling puzzles require proof of a basic law governing the world of numbers. Challenges concern van der Waerden's theorem, the Landau-Schnirelmann hypothesis and Mann's theorem, and a solution to Waring's problem. Solutions included. 64pp. 5'/s x 8'h. 0-486-40026-3 THE PHILOSOPHY OF MATHEMATICS: AN INTRODUCTOR Y ESSAY, Stephan KOmer. Surveys the views of Plato, Aristotle, Leibniz & Kant concerning propositions and theories of applied and pure mathematics. Introduction. Two 0-486-25048-2 aooendices. Index. 198oo. 5'k x 8'h.
CATALOG OF DOVER BOOKS TENSOR CALCULUS, J.L. Synge and A. Schild. Widely used introductory text covers spaces and tensors, basic operations in Riemannian space, non-Riemannian spaces, etc. 324pp. 5% x 8%. 0-486-63612-7 ORDINARY DIFFERENTIAL EQUATIONS, Morris Tenenbaum and Harry Pollard. Exhaustive survey of ordinary differential equations for undergraduates in mathematics, engineering, science. Thorough analysis of theorems. Diagrams. Bibliography. Index. 818pp. 5% x 8'h. 0-486-64940-7 INTEGRAL EQUATIONS, F. G. Tricomi. Authoritative, well-written treatment of extremely useful mathematical tool with wide applications. Volterra Equations, Fredholm Equations, much more. Advanced undergraduate to graduate level. Exercises. Bibliography. 238pp. 5',1, x 8'h. 0-486-64828-1 FOURIER SERIES, Georgi P. Tolstov. Translated by Richard A. Silverman. A valuable addition to the literature on the subject, moving clearly from subject to subject and theorem to theorem. 107 problems, answers. 336pp. 5% x 8'h. 0-486-63317-9 INTRODUCTION TO MATHEMATICAL THINKING, Friedrich Waisma.nn. Examinations of arithmetic, geometry, and theory of integers; rational and natural numbers; complete induction; limit and point of accumulation; remarkable curves; complex and hypercomplex numbers, more. 1959 ed. 27 figures. xii+260pp. 5% x 8'h. 0-486-63317-9 POPULAR LECfURES ON MATHEMATICAL LOGIC, Hao Wang. Noted logician's lucid treatment of historical developments, set theory, model theory, recursion theory and constructivism, proof theory, more. 3 appendixes. Bibliography. 1981 edition. ix + 283pp. 5% x 8'h. 0-486-67632-3 CALCULUS OF VARIATIONS, Robert Weinstock. Basic introduction covering isoperimetric problems, theory of elasticity, quantum mechanics, electrostatics, etc. Exercises throughout. 326pp. 53k x 8'h. 0-486-63069-2 THE CONTINUUM: A CRITICAL EXAMINATION OF THE FOUNDATION OF ANALYSIS, Hermann Weyl. Classic of 20th-century foundational research deals with the conceptual problem posed by the continuum. 156pp. 5% x 8'h. 0-486-67982-9 CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLUTIONS, A. M. Yaglom and I. M. Yaglom. Over 170 challenging problems on probability theory, combinatorial analysis, points and lines, topology, convex polygons, many other topics. Solutions. Total of 445pp. 5',1, x 8'h. Two-vol. set. Vol. I: 0-486-65536-9 Vol. II: 0-486-65537-7 INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS WITH APPUCATIONS, E. C. Zachmanoglou and Dale W. Thoe. Essentials of partial differential equations applied to common problems in engineering and the physical sciences. Problems and answers. 416pp. 5% x 8'h. 0-486-65251-3 THE THEORY OF GROUPS, Hans]. Zassenhaus. Well-written graduate-level text acquaints reader with group-theoretic methods and demonstrates their usefulness in mathematics. Axioms, the calculus of complexes, homomorphic mapping, p-group theory, more. 276pp. 5% x 8'h. 0-486-40922-8
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CATALOG OF DOVER BOOKS
History of Math '·
THE WORKS OF ARCHIMEDES, Archimedes (T. L. Heath, ed.). Topics include the famous problems of the ratio of the areas of a cylinder and an inscribed sphere; the measurement of a circle; the properties of conoids, spheroids, and spirals; and the quadrature of the parabola. Informative introduction. c!xxxvi+326pp. 5% x 8'h. 0-486-42084-1 A SHORT ACCOUNT OF THE HISTORY OF MATHEMATICS, W. W. Rouse Ball. One of clearest, most authoritative surveys from the Egyptians and Phoenicians through 19th-century figures such as Grassman, Galois, Riemann. Fourth edition. 0-486-20630-0 522pp. 5% X 8'h. THE HISTORY OF THE CALCULUS AND ITS CONCEPTUAL DEVELOPMENT, Carl B. Boyer. Origins in antiquity, medieval contributions, work of Newton, Leibniz, rigorous formulation. Treatment is verbal. 346pp. 5% x 8'h. 0-48660509-4
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THE HISTORICAL ROOTS OF ELEMENTARY MATHEMATICS, Lucas N.H. Bunt, Phillip S.Jones, andjack D. Bedient. Fundamental underpinnings of modem arithmetic, algebra, geometry and number systems derived from ancient civiliza0-486-25563-8 tions. 320pp. 5% x 8'h. A HISTORY OF MATHEMATICAL NOTATIONS, Florian Cajori. This classic study notes the first appearance of a mathematical symbol and its origin, the competition it encountered, its spread among writers in different countries, its rise to popularity, its eventual decline or ultimate survival. Original 1929 two-volume edi0-486-67766-4 tion presented here in One volume. xxviii+820pp. 53,1, X 8'&. GAMES, GODS & GAMBUNG: A HISTORY OF PROBABIUTY AND STATISTICAL IDEAS, F. N. David. Episodes from the lives of Galileo, Fermat, Pascal, and others illustrate this fascinating account of the roots of mathematics. Features thought-provoking references to classics, archaeology, biography, poetry. 0-486-40023-9 1962 edition. 304pp. 5% x 8'h. (Available in U.S. only.) OF MEN AND NUMBERS: THE STORY OF THE GREAT MATHEMATICIANS, Jane Muir. Fascinating accounts of the lives and accomplishments of history's greatest mathematical minds-Pythagoras, Descartes, Euler, Pascal, Cantor, many more. Anecdotal, illuminating. 30 diagrams. Bibliography. 0-486-28973-7 256pp. 5% X 8'h. HISTORY OF MATHEMATICS, David E. Smith. Nontechnical survey from ancient Greece and Orient to late 19th century; evolution of arithmetic, geometry, trigonometry, calculating devices, algebra, the calculus. 362 illustrations. 1,355pp. Vol. I: 0-486-20429-4 Vol. II: 0-486-20430-8 5% x 8'h. Two-vol. set. A CONCISE HISTORY OF MATHEMATICS, Dirk]. Struik. The best brief history of mathematics. Stresses origins and covers every major figure from ancient l'I.T~~- '~;'~·• •~
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