Stability Domains

NONLINEAR SYSTEMS IN AVIATION AEROSPACE AERONAUTICS ASTRONAUTICS

STABILITY DOMAINS

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Nonlinear Systems In Aviation Aerospace Aeronautics Astronautics

A series edited by: S. Sivasundaram Embry-Riddle Aeronautical University, Daytona Beach, USA

Volume 1 Stability Domains L.Gruyitch, J-P. Richard, P. Borne and J-C.Gentina Volume 2 Advances in Dynamics and Control S. Sivasundaram Volume 3 Optimal Control of Turbulence S.S. Sritharan

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NONLINEAR SYSTEMS IN AVIATION AEROSPACE AERONAUTICS ASTRONAUTICS

STABILITY DOMAINS

L. GRUYITCH, J-P. RICHARD, P. BORNE AND J-C. GENTINA

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. © 2004 by Chapman & Hall/CRC

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Contents Preface Notations General introduction 1 Introductory comments on stability concepts

1.1 Comments on Lyapunov's stability concept 1.1.1 Lyapunov's denition of stability and the denitions of stability in the Lyapunov sense 1.1.2 Denitions of attraction 1.1.3 Denitions of asymptotic stability 1.1.4 Denitions of exponential stability 1.1.5 Denitions of absolute stability on Ni (L) 1.1.6 Denitions of attraction with nite attraction time . 1.1.7 Denitions of stability with nite attraction time . 1.1.8 Denitions of absolute stability with nite attraction time 1.2 Comments on the practical stability concept 1.2.1 Introductory comments 1.2.2 Denition of practical stability 1.2.3 Denition of practical contraction with settling time . . 1.2.4 Denition of practical stability with settling time . .

2 Stability domain concepts

2.1 Introductory comments 2.2 Domains of Lyapunov stability properties 2.2.1 The notion of domain 2.2.2 Denitions of stability domains . . 2.2.3 Denitions of attraction domains . . 2.2.4 Denitions of asymptotic stability domain 2.2.5 Denitions of exponential stability domains . . 2.2.6 Denitions of asymptotic stability domains on N( ) ( ) 2.3 Domains of practical stability properties 2.3.1 Denitions of domains of practical stability

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2.3.2 Denitions of domains of practical contraction with settling time 2.3.3 Denitions of domains of practical stability with settling time

3 Qualitative features of stability domains properties 3.1 Introductory comments 3.1.1 Denition of a motion . 3.1.2 Existence of motions . 3.1.3 Existence and uniqueness of motions . 3.1.4 Continuity of motions in initial conditions . 3.1.5 Dierentiability of motions in initial conditions . 3.2 Generalised motions 3.2.1 Motivation 3.2.2 Dini derivatives 3.2.3 Generalised motions . . . 3.2.4 Limit points and limit s 3.2.5 Limit sets, Lagrange stability, precompactness and stability domains 3.2.6 Invariance properties of sets 3.2.7 Invariance properties of limit sets . 3.3 System regimes . . . 3.3.1 Forced regimes and the free regime . . . 3.3.2 Periodic regimes 3.3.3 Stationary regimes and stationary points 3.3.4 Equilibrium regimes and equilibrium points 3.4 Invariance properties of sets of equilibrium states . 3.5 Dynamical and generalised dynamical systems . . . . . 3.5.1 Denition and properties of dynamical systems . . . 3.5.2 Denition and properties of generalised dynamical systems . . 3.6 Stability properties and invariance properties of sets 3.7 Invariance features of stability domains properties 3.8 Features of equilibrium states on boundaries of domains of stability properties

4 Foundations of the Lyapunov method

4.1 Introductory comment 4.2 Sign denite functions 4.2.1 Sign semi-denite functions . 4.2.2 Sign denite functions 4.2.3 Comparison functions 4.2.4 Positive denite functions and comparison functions . 4.2.5 Radially unbounded and radially increasing positive denite functions 4.3 Uniquely bounded sets

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4.4

4.5 4.6

4.7 4.8 4.9

4.10 4.11

4.3.1 Denition of uniquely bounded sets 4.3.2 Properties of uniquely bounded sets 4.3.3 O-uniquely bounded sets and positive denite functions 4.3.4 Denition of uniquely bounded neighbourhoods of sets 4.3.5 Properties of uniquely bounded neighbourhoods of sets Dini derivatives and the Lyapunov method 4.4.1 Fundamental lemmae on Dini derivatives 4.4.2 LaSalle principle 4.4.3 Dini derivatives, positive deniteness, positive invariance and precompactness . Stability theorems 4.5.1 Stability of a set 4.5.2 Stability of X = 0 4.5.3 Comment Asymptotic stability theorems . 4.6.1 Asymptotic stability of a set 4.6.2 Complete global asymptotic stability of sets 4.6.3 Asymptotic stability of X = 0 4.6.4 Complete global asymptotic stability of X = 0 4.6.5 Absolute stability of X = 0 of Lurie systems Exponential stability of X = 0 4.7.1 Krasovskii criterion 4.7.2 Yoshizawa criterion Stability domain estimates . . . 4.8.1 Denitions of stability domain estimates . . . 4.8.2 Estimates of the stability domain of a set 4.8.3 Estimates of the stability domain of X = 0 Asymptotic stability domain and estimates 4.9.1 Classical approach 4.9.2 Denition of asymptotic stability domain estimate . . . 4.9.3 Estimates of the asymptotic stability domain of a set . 4.9.4 Estimates of the asymptotic stability domain of X = 0 Exponential stability domain estimate . . . . 4.10.1 Denition of exponential stability domain estimate . . . 4.10.2 Estimates of the exponential stability domain of a set 4.10.3 Estimates of the exponential stability domain of X = 0 Asymptotic stability domains on N( ) ( ) 4.11.1 Denition of estimate of the asymptotic stability domain on N( ) ( ) 4.11.2 Algebraic approach 4.11.3 Frequency domain approach

5 Novel development of the Lyapunov method 5.1 Introductory comment 5.2 Systems with dierentiable motions . 5.2.1 Smoothness property 5.2.2 Two-stage approach

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5.2.3 Approach via O-uniquely bounded sets 5.2.4 General one-shot approach 5.2.5 Exponential stability 5.3 Systems with continuous motions (generalised motions) 5.3.1 Smoothness property 5.3.2 Approach via O-uniquely bounded sets 5.3.3 General one-shot approach 5.4 Conclusion

6 Foundations of practical stability domains

6.1 Introductory comment 6.2 System aggregation function and sets . . 6.2.1 System description and sets . 6.2.2 Denition of estimates of practical stability domains of systems . . 6.2.3 System aggregation function extrema and sets . . . 6.3 Estimate of the system practical stability domain . 6.4 Estimate of the domain of practical stability with settling time s 6.5 Conclusion

7 Comparison systems and vector norm-based Lyapunov functions 7.1 Introductory comments and denitions 7.1.1 Presentation 7.1.2 Comparison systems . . . 7.1.3 Dierential inequalities, overvaluing systems . 7.1.4 ;M matrices . . 7.2 Vector norm-based comparison systems 7.2.1 Denition and aim of vector norms . 7.2.2 A rst statement 7.2.3 Computation of overvaluing systems . 7.2.4 Overvaluation lemma 7.3 Vector norms and Lyapunov stability criteria 7.3.1 Stability of equilibrium points 7.3.2 Stability of sets 7.3.3 Examples . 7.4 Vector norms and practical stability criteria with domains estimation . . . 7.5 Conclusions

References

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Preface \At the same time it becomes clear why some stability investigations should not be taken too seriously. One needs to know the size of the region of asymptotic stability." J.P. LaSalle and S. Lefschetz1 The kind of persistency of dynamical behaviour is important not only for various sciences like mathematics, mechanics, uid mechanics, thermodynamics, electricity, electronics, chemistry, control, econometrics, biology, medicine, and not only for engineering and science, but also for individuals, social life and state development. An appropriate persistency is the essential sense of the corresponding stability property. Relativeness of the notion of persistence resulted in a number of stability concepts among which was that of practical stability probably initiated by Chetaev in the thirties of the twentieth century. The above cited comment by LaSalle and Lefschetz, who were probably those who attracted the attention and interest of English speaking scientists to the Lyapunov stability theory, explains well the need to study stability domains in general and stability regions in particular. This book is aimed at what we consider basic from the theory of stability domains for various direct eective applications and/or for further research. In doing so we express freely and openly our views on stability theory in the framework of the Lyapunov stability concept and practical stability concept. Authors

1 Stability

by Lyapunov's Direct Method

© 2004 by Chapman & Hall/CRC

, Academic Press, New York, 1961, p. 121.

Notations Capital script Roman letters denote sets and spaces, capital block Roman letters designate matrices, lower case script Roman letters are used for scalars, lower case block Roman letters represent vectors, most Greek letters denote scalars. Latin

A A A A a

Ba Ba

+

B B b b

C C (k)(S ) C0(I0  X0 ) Ct(I0  X0 ) C c

D D(A) Da

a nonempty subset of t]".

Denition 3.8 A function X : I0  X0  I ! X is a generalised solution (a generalised motion) of the system (3.4) through X0 at t = 0 if and only if X0 2 X0 and X satisfy (i){(iv): (i) X is dened in X , X(t X0 i) 2 X , if and only if t 2 I0, I0 = (l;
l+ ), for every i 2 I , (ii) for every i 2 I and each X0 2 X0, X is (a) continuous in t 2 I0, (b) dierentiable almost everywhere in t 2 I0. At t 2 M0 , M0 has the zero measure, M0  I0 , at which X is not dierentiable in time t the following holds (c) X has both Dl X(t X0  i) and Dr X(t X0  i) if t is an interior point of I0, t 2 ]l;
l+ , (d) X has Dr X(t X0  i) if l; 2 I0, I0 = l;
l+ ) and t = l; , (e) X has Dl X(t X0  i) if l+ 2 I0, I0 = (l;
l+ ] and t = l+ , (iii) for every i 2 I (a) X identically satises (3.4) almost everywhere on I0,

d X(t X  i) = fX(t X  i)
i(t)]
8 t 2 I
0 0 a 0 dt

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(b) if t 2 M0 is an internal point of I0 then Dl X(t X0 i) = fX(t;  X0  i)
i(t;)]
t = t
Dr X(t X0  i) = fX(t+  X0  i)
i(t+)]
t = t
(c) if l; 2 I0 and t = l; then

Dr X(t X  i) = fX(t  X  i)
i(t )]
t = t = l;
(d) if l 2 I and t = l then Dl X(t X  i) = fX(t;  X  i)
i(t;)]
t = t = l
+

0

+

0

+

+

0

0

+

0

and (iv) X fulls the initial condition,

X(0 X0 i)  X0 : The motivation for introducing the notion of generalised motions will be illustrated by reconsidering the systems x_ = sgn x (Example 3.2) and x_ = ;sgn x (Example 3.3).

Example 3.6 Generalised motions of the rst-order system dx = sgn x dt

are determined by 8= 0
t 2 ] ; 1
;jx0j
) > > x0 2 > = x 0 + t sgn x0
t 2 ;jx0j
1
>

> > > does not exist
t 2 > > > does not exist
t 2 ] ; 1
0
) > x = 0
if (sgn 0 6= 0) 2 f;1
+1g
<
t 2 0
1
x(t x ) > = t sgn x > 9 > does not exist
t 2 ] ; 1
;jx j
) > > > > if sgn x = sgn 0
> > > = x + t sgn x
t 2  ;j x j
1 
= (x 6= 0) 2 < > > and > ) > > does not exist
t 2 ] ; 1
;j x j ]
> > sgn 0 6= 0: > > : = x + t sgn x
t 2 ] ;jx j
1
if sgn x 6= sgn 0
>  0

0

0

0

0

0

0

0

0

0

0

0

0

0

They are depicted in Fig. 3.7. Now, we can compare motions with generalised motions as follows:

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0

x(t;x0) x(t; x0*)

sgn 0 ∈]0,1]

x0*>0 -|x0*|

x(t;0)

sgn 0=0

0

x(t;0)

sgn 0=0

x(t;0)≡0

t

^

-|x0|

^x ", which is the consequence of the assumption that  > 0. Hence,  = 0, or X 1 ( X0 ) = X 2 ( X0 ). This further contradicts the existence of  2 < at which the (generalised) motions X 1 and X 2 dier. Therefore, X 1(t X0 ) = X 2 (t X0) = X(t X0 ) for all t 2 < and every X0 2 < X(t X ) = > 0
t2<
X = 0
> 9 > (jX j ; t) sign X
t 2 ] ; 1
jX j ; 1] = > > > : exp (;t+jX j;1) sign X
t 2 jX j ; 1
+1 
jX j 2 1
+1: 0

0

0

0

0

0

0

0

0

0

0

0

The motions are dened and continuous in (t
X0 ) 2 <  0 for all (X 6= 0) 2 N . 2) v(x) = sin jxj is positive denite on ];
. It is not positive denite on ;
] because v(x) = 0 for jxj = . ( 0
jxj  1 , is globally positive denite with respect to the 3) v(x) = jxj ; 1
jxj 1 set A = fx : x 2 0 out of the origin. ( 0
jxj  1
4) v(x) = is positive denite with respect (jxj ; 1)(4 ; jxj)
jxj 1
to the set A = fx : x 2 < ( ) = >   1  : j j 1 + cos 
2 < : +

It is not strictly monotonously increasing on 0
 for any  > 0. Notice that ( ) 2 C( ;1

is an approximation of Dc , which may not be necessarily contained in Dc (that ^ = 0). happens if there is X^ 2 S2 such that vm (X) © 2004 by Chapman & Hall/CRC

Example 4.18 198] The following system is known as the state{space representation of the Van der Pol equation,

x_ 1 = x2
x_ 2 = x1 ; x2 + x21 x2: Then, for m = 4, e4 = 0:12295
+ r3 (X) + r4 (X) v4 (X) = r2 (X) 1 + q1(X) + q2(X)
r2 (X) = 32 x21 ; x1x2 + x22
r3 = 0
r4 = ;0:3186x41 + 0:7124x31x2 ; 0:1459x21x22 + 0:1409x1x32 ; 0:03769x42
q1 = 0
q2 = ;0:2362x21 + 0:31747x1x2 ; 0:1091x22


 = 5:4413
S1 = fX : v4 (X) <  g  Dc : The strict asymptotic stability domain Dc and its approximation S1 are shown in Fig. 4.1.

Estimate of the domain of asymptotic stability with nite attraction time Denition 4.20 (a) The state X = 0 of the system (4.1) has the domain of asymptotic stability with nite attraction time a = a (X ), which is denoted by D
, if and only if

0

1) it has the asymptotic stability domain D, 2) there is nite time a = a (X0 ) such that X(t X0 ) = 0 for all t 2  a
1 provided only that X0 2 D
, 3) D
is a neighbourhood of X = 0. (b) A set S , S 0:

(5.97)

The function q (5.96) obeys 1) and 2) of Denition 5.4. The solution v of (5.2) for such a choice of the function q is

" 1 ;1 # 10 v(X) = ln 10 ; X T HX
H = = H T > 0: (5.98) ;1 4 The function v is continuously dierentiable on S , which shows that the function q obeys 3) and 4) of Denition 5.1. Moreover, the function v is positive denite on © 2004 by Chapman & Hall/CRC

S and v(X) ! +1 as X ! @ S , X 2 S . All conditions of Theorem 5.10 through Theorem 5.13 corresponding to q 2 Q (S
f) have been veried. Any of them shows 1

that X = 0 of the system (5.95) is asymptotically stable. They clarify that the set N = S is the asymptotic stability domain of X = 0 of the system,

D = fX : X ; 2X X + 4X < 10g: 2 1

1

2

2 2

5.2.5 Exponential stability What follows is based on 85].

Introductory comment

In this section the following problem is addressed and solved.

Problem. What are the necessary and sucient conditions for an exact algorithmic

one-shot construction of a Lyapunov function v for the system (5.1) with the specic smoothness property such that it and its total time derivative v_ along system motions prove the following estimate:

kx(t x )k  kx k exp (;t)
for every (t
x ) 2 <  B
0

0

0

+

(5.99)

for some numbers  2 1
+1 and  2 ]0
+1, and for obeying (5.99),

0 <  ;1$


(5.100)

where $ > 0 satises B S .

Problem solution

In this framework the system will be assumed to possess the specic smoothness property (Denition 5.1). The complete solution to the problem reads as follows:

Theorem 5.14 Let the system (5.1) possess the specic smoothness property and

$ > 0 obey B S . In order for system solutions to obey (5.98) for some numbers  2 A = (aij )ij > :

 a : : : a  9 > n   >  =   . . . . : a > 0
: : :
> 0 :   :::n . .   > >  an : : : ann   11

=1

1

11

1

iv) an M -matrix, (Metzlerian), or A 2 M, if and only if both A 2 Z and A 2 P. v) the opposite of an M -matrix, or an ;M -matrix, or A 2 ;M, if and only if ;A 2 M.

A characteristic of ;M-matrices is to have nonnegative o-diagonal elements, and a negative diagonal.

Theorem 7.3 48] Let M be a constant n  n matrix with non-negative elements (M 2 ;Z). Then the proposition \M is a ;M matrix" is equivalent to any of the following propositions:

1) Any eigenvalue of M has a negative real part. 2) Any real eigenvalue of M is negative. 3) M veries the Koteliansky conditions (that is, ;M 2 P):

 m m < 0
 m

11

11

21

m   > 0
m  12 22

  m  m  m

11 21 31

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m12 m13   m22 m23  < 0
: : :  m32 m33 

   m : : : m k   ..  > 0 : : : (;1)n det M > 0: (7.24) (;1)k  ... .    mk : : : mkk  11

1

1

4) There exists a constant vector X 0 such that MX < 0. 5) There exists a constant vector X > 0 such that MX < 0. 6) There exists a diagonal matrix $ with positive diagonal such that N = M$ is a matrix with dominant negative principal diagonal, i.e.:

nii +

X j 6=i

jnij j < 0 8i = 1 : : :n (with here, jnij j = nij ):

7) M ;1 exists and all its coecients are non-positive, i.e.: ;M ;1 0 8) If N 2 Z and N  M , then N ;1 exists. 9) For any vector X 0, X 6= 0 there exists an index i such that if Y = MX , xiyi < 0. 10) If d(A) denotes the diagonal of A, then for each diagonal matrix R such that R ;d(A), the inverse R;1 exists and (R;1 (A ; d(A))) < 1, where ( ) is the spectral radius (i.e. the maximum of the moduli of all eigenvalues). 11) There exists a permutation matrix P such that P MP T = T1 T2 , T1 lowertriangular matrix, T2 upper-triangular, T1 and T2 2 Z and have strictly negative diagonal.

rrr

Further properties of ;M matrices 1) If M 2 ;M, then there exists a negative real eigenvalue (M) of M, called the importance value of M, such that the real part of any eigenvalue of M is at most (M). 2) If M 2 ;M, there exists a nonnegative eigenvector u(M) 0 associated to

(M) and called the importance vector of M. 3) If M 2 ;M and M is irreducible, then u(M) > 0, and in Theorem 7.3 one can choose X = u(M) (in proposition 5) and $ = diag(u(M)) (in proposition 6).

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Dynamic systems dened by ;M matrices

We consider here the linear system dened as in (7.22) by dX = MX + q
M(n  n) is a ; M matrix
q and X 2 0 and an n-vector c such that c q, the sets:



9 Ba = X 2



+ + n n = Ba = X 2 (7.27)

> A = BXe = X 2 >  A+ = B +Xe = X 2 q, which proves K = 6 . Invariance of B a : Let a = ( : : :n)T > q, b = ;M a = ( : : :n )T q, and X(t) = (x : : :xn )T 2 @ B a . Then there is an index i such that at time t, jxij = i and jxj j  j for j =6 i. Since M 2 ;Z, we have 1

1

1

1

Dt jxij  +

n X i=1

mij jxj j + qi 

n X i=1

mij j + qi = i + qi  0


and jxij cannot increase at time t. Then, any trajectory of (7.25) crossing @ Ba enters B a or lies on @ B a . This proves the positive invariance of B a .

Invariance of B +a : Suppose that at time t, X(t) 2 @B +a , and X(t) 62 @Ba . this means that there is an index i such that at time t, xi (t) = 0, and 0  xj (t)  j . Then

dxi(t) = X m x + q 0
ij j i dt j 6=i

and X(t) cannot go out of B +a . This, together with positive invariance of B a , proves the positive invariance of B+a . Proof of (iii): This can be found in 195], in the case q = 0, and is easy to generalise for q 0 by setting Y = X + M ;1 q, Y (t
Y0 ) = exp (Mt) Y0 , and K comes down to

fa 2 0g. Proof of (iv):

Domains D (): Let the positive (semi)denite function v dened by v (X) = u(M T )T jY j. Then Dt v (X)  u(M T )T M jY j = (M)v (X)  0, and v (X) exponentially tends to zero as t ! +1. Domains D1 (): Let I = fi 2 f1
2 : : :ng
ui(M) =6 0g and v1(X) = max u;i jyij i2I = u;i0 jyi0 j. 1

+

1

1

1

1

1

1

1

Dt v1 (X)  u;i0 +

Dt v1 (X)  +

1

X

X mi0 j jyj j uuj  u;i01 v1 (X) mi0 j uj j

j 2I j 2I ; 1 ui0 v1 (X) (M)ui0 = (M)v1 (X):

Then v1 (X) exponentially tends to zero as t ! +1. © 2004 by Chapman & Hall/CRC

Example 7.5 Let the linear system be dened by " ! " ! dX = ;2 1 X + 1 : dt 1 1 ;1

(7.32)

This yields (see Fig. 7.3): "2! Xe =
K = fa = (a1
a2 )T : 1  1 + a1  a2  2a1 ; 1g
3

A = fX = (x
x )T : 0  x  2
0  x  3g
+

For a =

"5! 7

1

2

1

2

2 K
B+a = fX = (x
x )T : 0  x  5
0  x  7g: p T T T 1

2

1

2

An importance vectorp of M = M is u = u(M) = u(M ) = (2
1+ 5) , associated with (M) = ;3 +2 5 .

p D () = fX = (2 + y
3 + y )T : 2jy j + (1 + 5)jy j  g
p D1 () = fX = (2 + y
3 + y )T : jy j  2
jy j  (1 + 5)g: 1

1

2

1

2

1

1

2

2

Several trajectories of (7.32) are given on Fig. 7.3, which shows invariant domains K
A
A+
Ba

B+a
Du ().

Relation between quasi-increasing functions and ;M matrices

The stability theory literatures suggest a tight connection between Wa,zewski's conditions and ;M matrices. However, until 163] (see also 162], 164]), this link always appeared in an implicit manner, and the following Theorem 7.5 aims at clarifying the existing implications. @f (X) Let us suppose that f(X) has a bounded Jacobian matrix fX (X) = @X for X 2 S (connected subset of > < dX dt = g(X
d)
> > : dp = h(X
p)
dt

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(7.35a) (7.35b)

is a local overvaluing system of

8 > > < dX dt = g(X
d)
> > : Dt p(X) = DX p]T h(X
d): +

+

(7.36a) (7.36b)

Then (7.35b) will be used as a comparison system of (7.35a), since its solution veries Z(t)  p(X(t)), provided that initial conditions are compatible: Z(t0 ) = p(X(t0 )). However, the stability study of (7.35b) has to be carried out without solving (7.35a), and this will motivate the use of decoupled overvaluing systems (see Denition 7.5) that are such that h depends only on Z:

8 > > < dX dt = g(X
d), > > : dZ dt = h(Z).

(7.37a) (7.37b)

This could have been obtained by means of the general vector Lyapunov's functions instead of V.N., however the main advantage of V.N. is to give direct and systematic computations.

7.2.2 A rst statement

Lemma 7.1 a) Consider the system (7.1), and let a R.V.N. p(X) and a constant positive vector c 2 : dZ = h(X
Z) dt

(7.40a) (7.40b)

has a unique continuous solution for d 2 Sd . Then a sucient condition for system (7.40) to be a (Xc  Zc
Sd)-local overvaluing system of

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8 > dX > < dt = g(X
d)
> > : Dt p(X) = Dx p]T g(X
d) +

(7.41a) (7.41b)

+

is that Zc is Xc -positively invariant for (7.40b), which means

Z0 = p(X0 ) 2 Zc ) Z(t) 2 Zc 8t t0
8X 2 Xc:

(7.42)

b) If system (7.40) is decoupled, that is, if (7.39) can be written as:

Dt p(X)  h(p(X))
8X 2 Xc


(7.43)

+

with h a locally quasi-increasing function on Zc , then the above condition is that Zc is positively invariant for (7.44):

dZ = h(Z): dt

(7.44)

rrr

Proof. a) The proof refers to Denition 7.4, with notations:

0 B X(t t
X  d) X(t t
X  d) = B B @

1 9 > > CC > CA 2 S = Xc  Zc  > > p(X(t t
X  d)) > = (7.45) > 0 1 > > > X(t t
X  d) C B > C > OS(t t
OS  d) = B 2 S
@ A >  Z(t t
X
Z  d) 0

0

0

+

0

0

0

0

0

0

0

0

0

0

Z0 = p(X0 ): (7.46) i) Solution X(t t0
X0  d) exists and is continuous by hypothesis. ii) Since (7.39) holds with h(X
Z) locally quasi increasing w.r.t. Z, as long as X remains in Xc and Z in Zc , similarly to Theorem 7.2 we have Z0 = p(X0 ) ) Z(t t0
X0
Z0 d) p(X(t t0
X0 d))
(7.47) © 2004 by Chapman & Hall/CRC

and for any t t0 OS0 = X0 ) OS(t t0
OS0  d) X(t t0
X0  d)

(7.48)

supposed that OS and X remain in S and d 2 Sd . iii) Let us suppose that (7.42) holds, and consider Z0 = p(X0 ) < c. Then 0  Z(t t0
X0
p(X0) d) < c
8t t0
8d 2 Sd :

(7.49)

At time t = t0 , X0 is in Xc, therefore (7.47) holds until time t1 when p(X(t1 )) = c. However, such a time t1 does not exist because of (7.49). So, (7.47) remains true for any t t0. This proves 0  p(X(t t0
X0 d))  Z(t t0
X0
p(X0) d)


(7.50)

and Sd -positive invariance of Xc for system (7.1). b) The decoupled case appears as a particular case of (a).

7.2.3 Computation of overvaluing systems General formula Let the system (7.1) be decomposed into dX = A(X
d)X + b(X
d)
dt A(X
d) and v(X
d) bounded for bounded X, and any d 2 Sd :

9 =  (7.51)

This decomposition is obviously non-unique: as we shall see afterwards, it is preferable to work it out in such a way that dX = A(X
d)X dt

(7.52)

has its equilibrium X = 0 stable, or attractive, or both. Algorithms will also be proposed in order to optimize this decomposition. Then dene for any i
j 2 f1
2
: : :
kg the following notation: © 2004 by Chapman & Hall/CRC

9 > > Y yi = Pi Y
> > > bi(X
d) = Pi b(X
d)
> > > > i (X) = sup Dxi pi (xi)]bi (X
d)
> > d2Sd > > > i = sup i (X)
> x2S > > > Aij (X
d) = Pi A(X
d)pj
> > T Dyi pi (yi )] Aij (X
d)yj > mij (X
Y
d) =
> = pj (yj ) > > ij (X) = sup mij (X
Y
d)
> Y > d d > > > ij = sup ij (X)
> X 2S > > > M(X) = ij (X)]ij 2f :::kg
> > > > q(X) = i(X)]i2f :::kg
> > > > M = ij ]ij 2f :::kg
> > > >  q = i]i2f :::kg: 2 > > > > > > = > > > > > > > 

(7.67)

for pi(Xi ) = kXi k2, 8i = 1
: : :
k. The expression (7.65) together with (7.63) is not easy to compute if A strongly depends on X or d, since eigenvalues are to be calculated. However, one can use the following equations, which are less precise but easier to compute:

2 3 X ii(X) = sup max 64ass + 12 jasr + ars j75 d2Sd s2Ii r Ii r=s  X X 1

9 > > > > > > > = ij (X) = 2 sup max jasr j] + max jars j] > d2Sd s2Ii r2Ii > > > "X ! = > > > ki(X) = sup bi :  d2Sd s2I 2 6

(7.68)

1 2

2

i

This expression of ij was given in 55] or 87].

7.2.4 Overvaluation lemma

The above formulas and properties simplify and make explicit the expression of Lemma 7.1, leading to the following Lemma 7.2.

Lemma 7.2 Let the system (7.1) (7.51) have a unique continuous solution X(t t
X  d). Let a R.V.N. p(X) and a constant positive vector c 2 > < dt = g(X
d) = A(X
d)X + b(X
d) > > : dZ = M(X)Z + q(X) = h(X
Z) dt

© 2004 by Chapman & Hall/CRC

(7.69a) (7.69b)

to be an (Xc  Zc
Sd )-local overvaluing system of (7.70):

8 dX > > < dt = g(X
d) > > : Dt p(X) = DX p]T g(X
d) +

+

(7.70a) (7.70b)

is that Zc is Xc -positively invariant for (7.69b) (then X is considered as a disturbance), which means:

Z0 = p(Xo ) 2 Zc ) Z(t) 2 Zc 8t t0
8X 2 Xc :

(7.71)

b) A sucient condition for the decoupled system (7.72):

8 dX > > < dt = g(X
d) > > : dZ = MZ + q dt

(7.72a) (7.72b)

to be (Xc Zc
Sd )-local overvaluing system of(7.70) is that M is asymptotically stable and c belongs to the simplicial cone K dened by (7.29) this, i.e.: 1) M veries the Koteliansky conditions (Theorem 7.3, proposition 3) 2) ;M c q (7.73) c) Dening Ze = ;M ;1 q, ((7:72b)
Zc
Ze) is a comparison system of ((7:1)
Xc
XZe ) with regard to the asymptotic stability property with the estimate of asymptotic stability domain, i.e.: \Ze is asymptotically stable for (7.72b), with Zc an estimate of its asymptotic stability domain 0 such that M(X) + "Ik is a ;M -matrix 3) the non constant elements of M(X) are grouped into one unique column. Let

(M(X1 )) = sup f (M(X))g  ;" < 0 the upper (with regard to X 2 S ) X 2S importance value of M(X), obtained for X = X1 (possibly not unique). Then, as soon as disturbances d remain in Sd : i) X = 0 is exponentially stable for system (7.1) (with rate = ") and any set D1() included in S : D1() = fX 2 2:

(7.110)

Under this hypothesis, Theorem 7.9 (iii) ensures that the set A: " 1 !) ( 2 2 3=2 2( + ) (7.111) A = X 2 < =   B C C =2 D1 () = >X 2 < : jX j < B  S () = 2 @ A> 6 :  

(7.127)

3

=

that are depicted on Fig. 7.11. Obviously, D1 ()  D1 () is also an estimate.

Example 7.12 (Theorem 7.8) Consider the equation (with f and f piecewise 1

continuous functions)

dX = dt

" ;1:5 f (X) ! 1

2

f2 (X)

X:

2

(7.128)

Using p(X) = jX j and Theorem 7.8 shows that X = 0 is globally exponentially stable (with exp (;t) convergence) if det

" ;0:5 jf (X)j ! 1

2

1 + f2 (X)

that is, f2 (X) < ;1 ; 4jf1(X)j. © 2004 by Chapman & Hall/CRC

> 0
8X 2