Theory and Application of the Z-Transform Method

THEORY AND APPLICATION OF THE z-TRANSFORM METHOD I~. I. J URY,

DEPARTMENT OF ELECTRICAL ENGINEERING,

IINIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA

I{(>BERT E. KRIEGER PUBLISHING CO. IIUNTINGTON, NEW YORK

PREFACE

I luring the curly Ilfties. considerable interest and activity arose umong C'lIl!illeCrS und systems theorists in the relatively new area of discrete ,y,ICIlI theory, This activity and aroused interest were mainly motivated h)' Ihe advivcly ~Ieveloped and applied to many areas of discrete system IIII'III'Y, I hI' ,uhjec\ matter is mainly addressed to engineers and systems Ilwllli,I,; however. some of the material developed would also be of IJIII'I~",1 In applied mathematicians, Whenever it is found necessary. most III IIU' IhC'llrellls alld lemmus are proved 011 a somewhat rigorous mathe111011 h.d ha,i!>, Jlowever, to limit the text and to place morc cmphasis on IIII' .ll'l'lh'alioll!>, Ihe ri~lIr is sllllletimcs sacrificed, ('onsclluently, certain 1111'1111'11" an' tOlldled nil hricl1y without prnnl's, Cnnver~encc prnhlcms ,II" lI11t '"'I'nroll!>ly p"rs"ed, hut lIrc left til Ihe reader ror rurther study, "III 11111',1 .. I' Ih~' III1PI'll\'~'1I ll1alC'rial till' pC'rtiJlent rdi:rell~'c, arc ~'ited SIl 111.11 II ... 11';11"'1 ~'aJll'a,ily lilld Ihe ri~ol'Utis dcrivlllilllls or Ihl' thl'ory, C'h,ll'lI'l I .11"'11' ..1'" ill dl:lail thl' .:,lrallsl'lIrll1 Ihl'ury alld it-. 1IJ1l1lilka"till', ,11,,1 IIII' IIIII,," ...d ,'tl;\II .. l'ollll, I',xll'n .. ivl' Ii .. " III tlll'lIlI'lII', 01 rilles ,II" "llh"1 011'11\'1'" III 1"1"1'1111'" III ;11'l1"ailll 1IIl' n'a"I'1 With till' ,Il'taill'" II'," .. I 1111',1111'1111111 \'11

viii

PREFACE

In Chapter 2, the z-transform method is applied to the solution of linear difference or difference-differential equations. Several forms of difference equations are given with examples from network. circuit, and control theory. The problem of stability of linear discrete systems and the root distribution within the unit circle are discussed in detail in Chapter 3. This stability discussion is not yet available in any other text; it also represents some of my original work that is available only in papers. Thus the contents of this chapter could be very useful to applied mathematicians as well as to engineers and systems theorists. Further development of the z-transform theory is considered in Chapter 4, with the derivation of the convolution of the z-transform and the modified z-transform. The application of the convolution theorem. and of the two-sided z-transform to statistical study and design of discrete systems are demonstrated. The application of the convolution z-transform to the solution of certain types of nonlinear discrete systems or nonlinear difference equations is emphasized in Chapter 5. Examples of the application of this method are also described. (n Chapter 6 the periodic modes of oscillation of certain nonlinear discrete systems are emphasized and applied. The stability study of limit cycles and their identification is also included. The use of the z-transform method and its modification as applied to the approximate solution of differential equations or continuous systems is examined in Chapter 7. The methods developed are well suited for digital computer analysis. Finally, in Chapter 8 are found various examples of areas in discrete system theory. These areas include nonlinear sampled-data feedback system, discrete antenna array theory, information and filtering theory, economic systems, sequential circuits. and discrete Markov processes. The main emphasis is the application of the z-transform theory. developed in the preceding chapters, to various disciplines of discrete system theory. Extensive tables of z-transform, modified z-transforms. total square integrals, and summation of special series are in the Appendix. Furthermore, there are many problems that illustrate further the application of the z-transform theory and indicate the method of the proofs of cert:lin theorems omitted from the text. Some of the chapters have heen 11I\I~ht :It :1 ~r:ulU:ltc seminar fur a olle-lIllit. one-scmcstcr l'uursc at the Universily III' Califurnia. 'Thc pren"luisitc is all ullder,~radllatl' "(1111's,' ill lim'al' sysll'lI1 analysis. If used as a lI'xt.1 hl'lil'w thalnlllh"lI1all'rilll":1II h,'a,h"llIalc'IYl'llv,'n'" ill a tWII"Ullit, lilli' .,,'IIII,.,Il', rnlll'il' lit IIII' ,~, III I UIIIl' Ie'wl,

PREFACE

ix

The writing of this book was aided considerably by my students whose contributions and efforts are very much emphasized. To mention but a few, I would like to thank Professors M. A. Pai and S. C. Gupta, Doctors C. A. Galtieri and T. Nishimura. and Messrs. A. Chang. T. Pavlidis. and A. G. Dewey. The suggestions and valuable comments of Professor W. Kaplan have led to much improvement in the mathematics and they are greatly appreciated. Since the major part of this material grew both from my research activities and those of my students, I would like to thank the Air Force Office of Scientific Research for their generous support and interest in our group at the University of California. The patience and efforts of Mrs. L. Gilmore in typing this manuscript are gratefully appreciated and acknowledged. I wish to acknowledge the very helpful private discussions by correspondence, over many years, with my colleague Professor Va. Z. Tsypkin of the Institute of Automatics and Telemechanics in the U.S.S.R. t hope that the contents will serve the useful purpose of providing a unified and integrated transform method which can be applied extensively lind fruitfully by systems engineers and scientists for many years. /oi·hr/lory 1964

E. I.

JURY

CONTENTS

z-TRANSFORM DEfiNITION AND THEOREMS

1.1 1.2 1.3 1.4 1.5 1.6 1.7

I.K

1.'1

2

.'.1 .'.) .'. , .'..1 .'.... .',(1

.', I

Discrete Time Function and z-Transform Definitions Properties of z-i'ransforms Inverse z-Transform and Branch Points The Modified z-Transform Relationship between Laplace and z-Transforms Application to Sampled-Data Systems Mean Square Value Theorem Equivalence between Inverse Laplace and Modified z-Transforms Other Transform Methods Appendix. A Method of Determining the Coefficients of the .:-Tnlllsform Expansion .:-TRANSrORM METHOD OF SOLUTION OF LINEAR DIffERENCE EQUATIONS

LillcClr Dilference Equations with Constant Coefficients SlIlution of Difference Equations Whose Coefficients Are Pcrillllic Functions l.incClr l>illcrcncc-DiITerential Equations I )illi.'I'l'lIl·C ':'Iuatinns with Periodic Coefficients Tillll'·Var'yin~ Dillcrcncc E(llIutions '1 inl!" Varying .:-Translill'ln lind thc Systcm FUllction 1)01111'" .:-"('rallsliH·lIIaliun lind SCllutilln IIf Parlinl Dil1'erencc

2 3 9 15 20 28 29 31

36 41

45 45

48 51 57 S9 66 73

xii

CONTENTS

3

STABILITY CONSIDERATION FOR LINEAR DISCRETE SYSTEMS

79

3.1 Definition of Stability 80 3.2 "Stability Condition for Linear Time-Varying Discrete Systems 81 3.3 Tests for Stability 82 3.4 Stability Test Directly Applied in the z-Plane 84 3.5 Determinant Method 8S 3.6 Critical Stability Constraints for System Design 94 3.7 Number of Roots of a Real Polynomial Inside the Unit Circle 9S 3.8 Relationship between the Determinant Method and Hurwitz Criterion 97 3.9 Table Form 97 3.10 Division Method 109 3.1 I Aperiodicity Criterion for Linear Discrete Systems 112 3.12 Theorems Related to Stability and Number of Roots 116 Appendices l. Derivation of the Table Form of Stability 121 2. Singular Cases in Determinant and Table Forms 128 3. Summary of the Stability Criteria 136 4

4.1 4.2 4.3

5

5.1 5.2 5.3 5.4 (,

tt.! td

CONVOLUTION z-TRANSFORM

142

Complex Convolution Theorem Complex Convolution Theorem for the Modified z-Transform Applications of the Convolution Modified or z-Transform Method Appendices I. Proof of Complex Convol ution Formula 2. Derivation of Total Square Integrals Formula

142 145 147 167 168

CONVOLUTION z-TRANSFORM APPLIED TO NONLINEAR DISCRETE SYSTEMS

174

Assumptions Convolution z-Transforms of Certain Functions Method of Solution for Secon i [f(n -

I

+ m)T]z-lI,

°

,,-0

~ m ~ I

(1.93) This definition also relates to the modified z-transform of the function .I [(1/ - I m)T). By using equation (1.9), the preceding equation can be written as

+

':F(z, m)

= Z-1 ~"" [fen + m)T)z-n.

( 1.94)

n=O

I-'rom this equation by letting m

= 0, we readily deduce

ex:

z.~(z, m)lon'o = ~ f(t1T)z-II

= .~(z)

( 1.95)

n~O

Therefore the z-transform is obtained as a special case from the modified transform. Furthermore, if f(t) has no discontinuity at the sampling illstants (or jumps), the :-transformation can also be obtained as follows: .?="(z)

= .:F(z. m)l ...

(1.96)

1

Whenf(l) has a discontinuity at the sampling instants. the lime function Il'Iatl'd to this equation yields the value at the left side of the discontinuity, Ihal i~. at 1= nT , for /I = 1,2, ' .. and the value zero at 1/ = O. This '0111 Ill' readily ascertained by noting equation (1.7). SlInilar to the .::-transform we can derive several properties of the IIIlIdilil'd ::-transform, These properties and important theorems only arc ',1;111'" in thl' ftlllowin~ disl'ussions and the proofs clIn be e~,sily derived by IIH' n'adl''', Thl' steps for the derivation follows exactly as for the Ilall~fllrlll, rlII'O/',·II/.\' ,."/,/{",, 10 1/11' I~H

IIAI

VAIlII

'11

m 'u

,>I""II.~j;l/'Il/ll.~';

IIIIIIHI M

11111./111 I 111),1 II

//Ior/i/i,'"

IIIII',J

I. ",

•

'II

I~. III'

II

I) II

]8

THEORY AND APPLICATION OF THE Z-TRANSFORM

Special case, the response over the first interval, n

lim fen ,.. ... 0

+ m)T =

O~m:sa

= 0, 0 s m s 1, is

lim z.F(z, m)

(1.98)

Z:~Q)

O:S:m:S:l

FINAL VALUE

limf(n, m)T = lim (z - l).F(z, m)

(1.99)

I~l

ft-Q)

if the limit exists. REAL TRANSLATION

1 ... [f(1 - kT)] = z-Ic.F(z, m),

1",[f(1 - aT)] = z-IJF[Z, m = .F(z, m -

where

+ I - a), a],

(1.100)

k - 0, 1,2, ...

0

~ m

a$ m~

I

$

a (1.101)

o < a < I, and zero initial conditions off

LINEARITY

where a l are constants independent of t (1.102) COMPLEX SCALE CHANGE

1 ... [e HTf(t))

= e:!:bT(m-l).F(ze±bT, m)

DIFfERENTIATION WITH RESPECT TO

1m

(1.103)

m

[1.om fen, m)TlJ = 1.om .F(z, m)

(1.J04)

MULTIPLICATION BY IT
1 •

( I. ) 27)

'/,' III :

",

II

\\T

III~"',~

dl'I\1I1!-

~ "(1

II/')·-'~ '/'( I), WI'

I'l'adily

t'~lahli .. h Ih(~

,',,"m'd ion

I I)', I "I.' I :I1lt1 1,1]7) Iwlw('rll IIII' .'·Irall .. rmm IIlId Iht, 1111'1111'1'

11,,"',1111111

-'I )

I, ",. -

""I,~).!l '1'1(,·(111 - .'I'I/It) It"lnl

(I J.lH)

22

THEORY AND APPLICATION OF THE z-TRANSFORM

By using the convolution theorem for the Laplace transform and assuming that f( t) contains no impulses and is initiaUy zero, we obtain (1.129)

where

=

F(P)

F(s)

1 T

1-

e- •

1._" = 9'[f(t)] I.~"

(1.130)

= 1 + e-T• + e- 2T• + ' , , e- nT• + ' , ,

= ~[bT(t)],

for le-T'I

1 dp + V(O+)] eT'_~

e

(1.133)

El'O/uotion of .('[F(s)] To evaluate §(z), we assume first that F(s) has two degrees in s higher denominator than numerator; thus we can use equation (1.129), The path of integration in equation (1.129) should lie in an analytic strip which does not enclose or pass through the poles of the integrand, This is assured in view of the restriction imposed on c, In effecting the line integral, we can readily enclose in a negative sense (clockwise direction) the poles of 1/1 - (' .Tc. pI in the right half of the p-plane, or alternatively we may enclose in a positive sense the left half of t Where ,r - Rr 1.\'1. rI.,. - IIh ....·I\'1I or IIh."llIlr l'''"Y!',!:!'I!"!' til' /'In, ,,1"\,111 II!'II' "., - II, " - It I' 1/'1 .

CI""h~lllulr l'llllyr'W'lIl'"

If.. , -

IIhwi~~u

%-TRANSFORM DEFINITION AND THEOREMS

23

the plane. Because of the assumed form of F(s) the integrals on both the infinite semicircles are zero. If we integrate along the left half p-plane as shown in Fig. 1.S equation (1.129) becomes I ·~(%)I.~,7'O = -. 2fT)

t

F(p)

I- T ( ' dp 1 - e "'J)

(1.134)

If f~s) has only simple poles, the integral using Cauchy's formula yields sum of the residue of the function in the closed path, that is,

I he

~(z) = rontAor ~ residue of A(p) ~ B(p) 1 - e m",

where

F(p) = F(s)/.=» =

T

IJ

I

e- , .~e7'o

~~:~

(1.135)

(1.136)

When B(s)/.~P = 0 has simple roots only, this equation becomes (1.137) wh~'rc S •• Sz, S3, ••• , s"

B'(s,,) = dB

ds

are the simple roots of B(s) = 0, and

I

(1.138)

sa ••

WIII:re /'ls) has branch points in addition to regular singularities, the can also be obtained using equation (1.129); however. here

llallsful'ln

Imaginary .f ....... ~

II

/

jt"'''Poles 01

x X

I

....... Poles 0 1 - - -

x;'-

)(

I

p-plane

x

!\.·(II)

,, ,,'MY. \

c+jfJO x

1- ,.-T/o-p)

0,0

,.'

x

"

q

'~ .......

--- ...•.... ... /

I Ie .IIIU I \

",.Ih IIllIIh'/:..IIII ... III 1111' "'1'1111111111 Ih"" ""11\1'

24

THEORY AND APPLICATION OF THE z-TRANSFORM

,

p-plane

\

.,

\

Poles of

\

F(pJ

I I I

---.l'2 .. X

x

J-

xt T x

I

x }$.k ....

c -joo

'"

I

Real

'f

JI

FIGURE 1.6 Path of integration in the right half of the p-plane.

special care is required to evaluate the line integral. An example will be presented later. The integral in equation (1.129) can also be evaluated by the contour integration which encloses only the poles of 1/1 - e-'l'II-PI in the right half of the p-plane as shown in Fig. 1.6. Thus equation (1.129) can be represented by

F*(s)

1 , = ..2'[J*(I)] = -. 217)

F(p)

] -

I - T ( ) dp e I-P

(1.139)

The integral of equation (1.139) is equivalent to the negative sum of the residue according to Cauchy's theorem,

F*(s)

=

-(sum of the residue of the integrand at the poles enclosed) (1.140)

By evaluating the residues at the infinite poles of 1/1 - r finally obtain 1 k~oo F"'(s)=- I F(s+jkw r )

T

7'CS-PI,

we

(1.141)

k--co

where (I. 142)

If we replace in this equation ,,'1'. = :, we readily ohtain the ;:-transli,rm = Z[F(.\,)J, The'two fOI'IIlS III' ( I. I.Vi an" 1.141) arl' l'lluivalent clln he relldily verified till' hpcciflc ('""mple" III' 1-'(.\,),

,~(.;:)

Z-TRANSFORM DEFINITION AND THEOREMS

25

For F(s) has one degree in s higher denominator than numerator. the integral along the infinite semicircle in the left-half plane is no longer zero. whereas in the right-half plane it is still zero. There equation (1.132) should be used. which yields on the left-half plane the same as equation (1.135). whereas for integration in the right-half plane we obtain 1·1.·10 I

k~'"

pes) = - ~ F(s T k:-..,

+ jkwr) + U(O+)

(1.143)

It should be noted that here the infinite summation is not absolutely convergent. However. if the sum is evaluated by taking pairs of terms ~'vrresponding to equal positive and negative values of the index k. the ~llIll converges to a definite value, EXAMPLE

o.

(iiven F(s) = 1'( I - /3)SII 1",2R, ~ > f/ > O. and noninteger, we obtain ..... (.:) ~ Z[F(s)] as follows: \Ising the contour integration of equation (1.129). we can write for

,. . I .F(z)

= Z[F(s)] =

1 iC+iOC 'rJI 1I-lr(1 R) -. e p -.1:- I' dp 27TJ c-;-,; I - Z e

The integrand has a branch cut (for

~

(1.144)

not integer) in the left half of

lIu' plane. Using Cauchy's formula the integral of equation (1.144) is "lllIivalcnt 10 integration around the branch cut as shown in Fig. 1.7. lit-lin' (1.145)

c +joo

fl-plane

J' u -om

Rf'1I1

\

\ f'" 1,'·-'· \

,, ,

111,1 'itl I I

"-

..... ,

I',"h III

IIIh'I~IIII""1 ,111111111111

hllllll h 1111

26

THEORY AND APPLICATION OF THE z-TRANSFORM

Setting p = xe/'ff on r I' and p around r 0 vanishes, we have

g&"(z)

r(1 - ~}[iO e-«zxfJ-1e;flrr dx 2'TTj "" 1 - z-le- z

==

= xe -/.. on r 2' and

r(l - {J) sin {J'TTi"" e-«"xfJ - 1

0 1 - z-le- Z

'TT

= 1-

r(fJ)

i"" 0

noting that the integral

e-«Zx/J-1e-1fJrr' ] +1"" dx 0 1 - z-le- z

dx

e-a"xfJ - 1 dx

(1.146)

1 - z-l e-.,

i"

We note from integral tables that

-If'" fJ-l e Xl dx = r(p)lI>(z-l, o 1 - z- e-'"

p, (X),

{J

> 0, (X > 0

(l.I47)

This equation is also defined for {J integer. Therefore g&"(z)

= lI>(z-\ p, (X),

for (X

>0

and

P> 0

(and also integer) (1.148) It is known that the definition of lI>(z-1, {J. (X) is lI>(z-x,

p, (X) =

i

n=O (n

1

+ (X)fJ

z-n

(1.149)

which is the definition of the z-transform oif(t) = 1/(1 a > 0, and T = 1. This result is expected because

+

f(t) = g.>-l[F(s)] = g.>-l[r(l - P)S,s-le,,·] = 1/(t

(X)II, for

P> 0,

+ (X),s

for all p's different from I, 2. 3. Althougp (z - I, ~. Ct) can be represented only in a summation form, however. if Z-1 = I, the function (1, P. (X) = '(P, (X) is tabulated for certain (X and po This function is referred to as generalized Riemann-zeta function. Therefore the z-transform does not exist in a closed form; 0

0

0

0

00

however. the infinite summation I I/(n ,,=0 ically for certain -x's and p's.

+ (Xt

can be calculated numer-

£L'aluation of ~(z. m) ~ Zm[F(s)]

The relationship between the Laplace transform and the modified ztransform can be readily obtained similarly to the relationship of the z-transform. From equations (I. 93) and (1.123), we can write o~(z. m)11

fT.

A P(s. m) = .:1'1/*(/. m>l '-j

:/'1/( t

'f"

III /') ,~ -,.(1)

I. 0

.-~

m ..:: (I.I 'til)

27

Z-TRANSFORM DEFINITION AND THEOREMS

This equation can also be written as S;(z, m)I"ae7" = !l'[f(t - T =e- Ts

+ mT) ~T(t -

.2"[/(t+mn8 T (t)].

T)] Osm 0 (1.183) c(t) = ds 21Tj c-Ico (1 - e-T·)(s + a) The integrand in this equation has simple poles at $ = -a and s = ± j(27rklT), k = 0, I, 2, . . .. Evaluating the integral by the sum of the residues, we obtain e-at 1 2 GO c(t) = ++ Tk-I -I I - ~T aT X a cos (21TkIT)t + (27rk/T) sin (27rk/T)t (1.184)

+ (27rk/T)2

a2

The steady-state part of this equation is c(t)•• = ,

..!... + ~ i aT

a cos (27rk/T)t

TIt=l

a2

+ (2'1rk/T) sin (27rkIT)r + (27rk/T)2

(1.185) If the substitution t = (n - I + m)T is inserted in this equation. the expression is 1 2 ~ a cos 27rkm + (27rk/T) sin 27rkm () ( 1.186) ct.,=-+-k aT

TIt=l

a2

+ (27rklT)2

On the other hand, the modified z-trunsform of F( ...) is readily obtained as .'F(z, m)

= ":",(F(.~)J

'"

I

I::

I

~",(' ,~

II

/I

) '"

:: ,.,'

t -"",.

I

-- ',,.' l , .. t ..

..

lI<m ao. !lased on this definition. the use of Dirac (delta) function in the defini111111 the =-lransform could be bypassed and thus retain a certain rigor III Ihl' mathematics of this transform. It may be noted that the LaplaceSllc·lc.ics integral reduces to the one-sided Laplace transform when ~(I) = I.

nr

71,(, xeneraliud Laplace transformS IIII'. IIIdhnd, which has been introduced by W. M. Stone for the opera111111,11 ,,"lution of difference equutions. is defi!led as follows:

. \ or: 1-'(11) /.I/·WIJo: =f(·~) = l . "

/I

\\llh III,· l'IIIHlilinn limf{.\')

C'"

"

(1.210)

S"II

n.

The symhol H dennle" a shiflin~

.'1

"111"1,11111 1lt-lilll'lI hy IIIl' rl'ialiclll HI. I II FI·"(I.), lIy 1I"IIII~ IIII'. 11:I1I .. 1'11I1tI. Stiliit' ha" llt-wIIlP"cl all c'Xll'lI,iw 1i .. 1 III' 11'\11',1111111 l'ail .. (I"(/,) 1(,\,)1. whidl ,1It'l'I""dy u'lah'clll' IIIC' . 110111'01'''1111 111111 ..... ",n'pl 1111 1\ lill'lIlI ' - c,"'··,

38

THEORY AND APPLICATION OF THE %-TRANSFORM

;y=Ix r--

4

I

3

2

o FIGURE 1.14 A jump function.

The jump function method? This method was introduced by Gardner and Barnes and is similar to the Laplace-Stieltjes integral discussed earlier. The transform in this case is deli ned as yes)

6

(00

== 9"[Iy(x)] == Jo

(1.211)

Iy(x)e-·:e dx

where Iy(x) denotes the jump function of y(x). If y(x) function of y(:c) is as shown in Fig. 1.14 and gives Iy(:c)

==

== :c,

the jump

00

I:c

== Irp(:c -

(1.212)

r)

r-l

where p(:c) denotes a unit pulse of duration one, that is, p(:c) = u(:c) - u(:c - I)

(1.213)

where u(:c) is the unit step function. In general, the jump function Iy(:c) can be written as

00 Iy(:c) = !y(r)p(:I: - r). r-O

where p(:I: - r) £, u(:c - r) - u(:c - r - J) (1.214)

Dirichlet series transform8 This transform has been proposed by T. Fort and has the following definition: f(.~)

.£

'r,

/)[,,(1)) ... ~ m "tI(I),

,·-0

fur

III

>

I,!I

> I,

I -

intcoller

(1.21~)

z-TRANSFORM DEFINITION AND THEOREMS

39

It is of interest to note that this transform reduces to the z-transform when m = e, 1 = nT, that is,

..,

/(s)

=I

a(nT)z-II £ ~(z)I,_.'"

.. -0

= J[a(nT)]

(1.216)

This change of notation could also be performed without influencing the application of the method.

Discrele Laplace Iransjorm 9 •41 The discrete Laplace transformation, which is denoted by the operator symbol ~, is a transformation applied to the sequence (fll}' that is, 00

(1.217) ·~{fn} ~ ~ i"e-"' == F*(s) ,,=0 This transformation can also be described as a Laplace transformation applied to a certain pseudofunctionl*(t) described as 01)

1*(t)

== /(1) 2 ~(t - nT)

( 1.218)

n=9

I"hcrcfore the!!) transformation may be considered as a special case of the I apllicc transformation. The discrete Laplace transformation has been applied extensively to discrete system theory by Tsypkin and his colleagues in the U.S.S.R. More recently. Doctsch U has presented an illuminating exposition of this 1II1'Ihud and its relationship with both Laplace transform and the z1IIIIlsfunn methods.

1.ollren/-Cauchy·s transjorm lO II .... lransfurm is again related to the z-transform or the generating 'ullrlinn method and is defined as (1.219)

S"!/~ "P,II',lIm· II •44

:\ 11,.","1

III",ralll" Oil sl"llIl'lIl'l'S is 111l~ shift opcralor H, dclillCll hy Ihe FII,. Thl'lI II". 1':11" I /'711,. 111 gl'lIl'l'I1l,

,,'1,1111'11" II" I

I,. "", . - II""

(I.n!))

40 THEORY AND APPLICATION OF THE Z-TRANSFORM If u(t) is a function of the continuous variable t and if T is a given constant, we define the shift operator E here by the formula Eu(t)

= U(I + T)

(1.221)

or, more generally, ERu(t} = u(t

+ nT)

(I.222)

In applications to difference equations, the shift operator plays the same role as the differentiation operator D in differential equations. B. M. Brown has applied this method extensively to the analysis and design of discrete systems.

,- Transform method!3 This transform, which is applicable to sampled-data systems and to certain difference equations, is defined as

ya, e) = ~~Y ..+a}

'--I. , =e •-

co

=

I

y,,+&(l

+ ,)-R,

n-O

t,

0

~

e~ 1

n

= 0, I, 2, . . .

(1.223)

The inverse integral transform is given by y,,+&

=~ .1 Ya, e)(1 + ,)"-1 d" 2'1TJ Jpole8 or Ule Intesrand

o s: e s:

I,

n

= 0, I, i, ...

(1.224)

The advantage of this transform is that the Laplace transform is readily obtainable as a limiting process, that is,

Yes)

= .9'[y(t)] = r"" y(l)e-

Jo

sf

dt = lim TY(', 0)

(1.225)

7''''0

Mikusinski's operational calculus42.43 Mikusinski's operational calculus is the most general method that encompasses both the continuous and the discrete theory. The z-transform method can be obtained as a special case of a series of the translation operator defined by Mikusinski. In this operator the numerical cocmcients are the values of the function at the sampling instants and the exponents are 0, T, 2T, ... , nT. . . .. We can define for the functiun [j(t)) the operator

,"

f· = llcl'J:alllnn

I'rcs~,

Ncw Yllrk. V.. 1. K, 1')5').

II 1'.1'."'111. Iialln·hm·. "lh'lallll/l,hip 1ll'IWC\'1I MiJ..u~ill\J..i·s 0p"rauIIlIal ('"I,'ullls alld lilt' 1\1."hll.·.1 11.",,1 .."11." Ma''''1 IIfS.-i,·tIl·,· I'I"i",·I. ''''1'1. .. I' I .I .• tllllVl't',ily Clf t ,lhlllllll;l. "'·It.,·I,·y ..hllll· 1'1,.. (111111"1 ',up"lvhlllll .. llh,· :lIl1hlll). +I

I It

lit "" II. II 1\1. /I,.. 1111111"'/11,,1,...,1 1"" I. t h I I

\\ \ til".

n"·,,, I' ,./1 ",,',1/ ,\"·,It,·"". I.. hll Wlh-y .111.1 S....•••

44

THEORY AND APPLICATION OF THE %-TRANSFORM

4S. Volgin, L. N., Elements of the Theory of Control Machines, Soviet Series, Moscow, 1962. 46. Kuo, B. C., Analysis and Synthesis of Sampled-Dala Control Systems, PrenticeHall, Englewood Cliffs, New Jersey, 1963. 47. Tsypkin, Y. Z., Theory of Linear Sampled-DQtQ Systems, State Press for Physics and Mathematical Literature, Moscow, 1963 (in Russian). 48. Tamburelli, 0., Alternate;:..Transform, Alta Frequenza, Vol. 30. No. lQ, October 1961, pp.74S-7S0.

2 z-TRANSFORM METHOD OF SOLUTrON OF LINEAR DIFFERENCE EQUATIONS

The field of linear difference equations has been studied extensively by many mathematicians in the last century or more. Several excellent textbooks! 8 have been written on this subject that can be studied by the interested reader. Several methods exist for solution of difference equations such as the classical method, the matrix method, the recurrence method, and the transform methods. In this chapter we shall apply the :-transform method (or widely known among mathematicians as the !tenerating function method) for the solution of only certain types of linear difference equations. The formulation of the difference equation can be expressed in several forms such as the backward and forward dilTerence method or the translational form. The theorems developed in the preceding chapter will be readily applied to any of the several forms used in the mathematical formulation of these equations. Several examples l'nlln network and system theory will be presented and solved, using the ,lilTerence equations as the basis of description. Although this chapter is necessarily a brief study of difference equations, ",'vera I types of linear difference or difference-differential equations, usually encountered in physical. economic. and physiological systems, will he studied. This study will be based primarily (although other IIll'thmlli exist) Oil the ::-trunsl'orm or the modified :-lransform theory.

2.1 I.incur dilTcrcllcc c«.Juuti()ns with (·I1I1!.icll'"

Ilw

,,'OI1Slillll

rC1l1nwill~

,." ",.1'" ~

,

II

c()cllicicnts2H .:1I

,IiITcrcncc Cllllillil1l1.

,., v

I' ...... ,."

'.11" t.

(1.1 )

46

nn·:ORY AND APPLICATION OF THE Z-TRANSFORM

lind h, are constants, Y .. is the forcing function and is known for all villues of n ~ O. It is desired to find z". z" and Y,. are both functions of thr pnmmeter 1= nT. Assuming T = I in all the discussions of this dUlpter. I = n. where n is an integer equal to or greater than zero. Tilking the z-transform of equation (2.1). we obtain

tI,

(2.2) wIH~I'l' .'1'(:) ~ j[znJ. qv(z) ~ 1[Y,.], and zl' Ys are the initial conditions. SCllvill~

equation (2.2) for

:"(z) =

'W(z)

i 1-0

~(z),

we obtain

biZ; + f QiZi(I1Z/Z-/) ;-0

1-0

f b•.zi(!yr- /) ;=0

1-0

p

IQ;Zi

i-O

(2.3)

The inverse z-transformation zn A 1-1[.~(z)J can be accomplished by Cauchy's integral formula, the power series expansion, the partial II IIl·tiCln expnnsion, or the available table of inverse z-transforms. It is "l'plln~III frum the foregoing that initial conditions are readily inserted IIl1hlllllllicnlly ut the start, thus avoiding evaluating the integration conIIIIIlIls liS in the clnssical solution of difference equations. This fact with Iht' Wit' III' lahles of z-transform pairs indicates that this method of solution I... " ,'(,l'llIin ndv~\Otages over the other methods. Thl" l"xtellsinn of the z-transform to solution of simultaneous linear ,hllrlrlll't' "'JlUltions with constant coefficients can be performed in a 11111111111 fllshilln ns for the preceding case. ""illf~

II.IIISTR.ATIVE EXAMPLE·'

Irll"il'ed tn find the current in the nth loop i" for the ladder network IIhuWII ill H~. 2,1. Assume all the resistances except the load R£ have Ihr slime villue It II

III

lie II Ilet I I

Iluhl....... IWill It.

Z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

47

The equation for the loop with current in+ \ (2.4)

Equation 2.4 is true for any n except -I and k - I (the beginning and end loops). Equation (2.4) with end conditions is sufficient to describe the network. Applying the z-transformation to equation (2.4) and dividing by R, we obtain ,F(z) - 3:f(z) + 3z;0 + z2.1(Z} - Z2;0 - zi\ = 0 (2.5) Solving for ..f(z), we have ...f(z) = z(zio - 3;0 + i 1) Z2 3z + I

(2.6)

We can eliminate one of the initial conditions by writing the equation for the first loop of Fig. 2.1.

2Rio - Ri\ = V

(2.7)

'I. = 2''0 --RV

(2.8)

or

I'Iws .1'(.:::) becomes

fez) = z{z - [J + (V/Rio»}io z2-3z+1 h'ol11

(2.9)*

tables of inverse z-transform (Table I) we readily obtain ; .. as

1"lIows: i" = ,:y-l[..I'(Z)] = io[COSh won

I

l'US 1 /"11

nlll hI'

J (2.10)

3 2

(2.1 J)

J"5.,

(2.12)

= -

!oillh /"n :: • IIII'. 1'11",1111111

+ I - ~Rio) sinh won

2

III", Wlilh'n inn

1'111'111 ~lIilllhll'

II II

I

fill IlIhll'

,'16, 11'/ Rill) I \.

I I

rnllil'~ ;IS

48 THEORY AND APPLICATION OF THE Z-TRANSFORM

The value of io can be found by substituting equation (2.1O) into the equation for the end loop and solving for i o• The current in can also be obtained from the power series expansion of (2.9) in powers of Z-I.

2.2 Solution of difference equations whose coefficients are periodic functions ll8 ,31 In the solution oftQis type of difference equation the modified z-transform iii readily applied. Consider the following type of difference equation: p

(/

= i-O ! h(t)xnH + i-O ~ gi(t)Yn+i (n+p) s: t s: (n+P+ll

(2.13)

x{t)

I'ere /;(1) and gi(l) are arbitrary periodic functions of the independent vllriahle I with periods equal to unity. UcclIuse of the periodicity of/;(I) and gi(t), we can replace these functions hy 1;(111) and gi(m) , respectively, where m is a parameter which varies frolll zero to unity. With this change of variable equation (2.13) becomes p

x(n

(/

+ p + m) = i-O ~ };(m)xn+i + ~ gi(m)Yn+i i-O

(2.14)

I'wm Ihe definition of the modified z-transform, we have

1 + m)) ~ ,q-(z, m),

","[x(" -

.1mfx(n

+ k + m)] £

(2.15)

Zk+l[,q-(Z, m) - Ix(j + m - 1)z-/J, i-O

o~ m ~

1 (2.16)

'I'hl' lIIotlificd z-tran!.form of the difference equation in (2.14) gives

Z"'I[.,y·(Z, m) - Ix(j .

+m -

I)Z-IJ

I' 0

= i~.r.(m)Zfr(Z) - XXSZ-1J I

i R.(m):tr·:q(Z) - }::lIIZ '1,

nllih

,n.,

():5:m~1

(2.17)

I-~

• "

;n:,

IIl1tl III) lilt' Ullknown in Ihis ('tllllilion. lIt1wc"vl'r, :1'(z) ht' ,11'11'11111111',1 hy noling Ihlll :V'(:) - :n·,III)I", ,. Thlls,sllhslillllin~~ ", - I III rllllllllOIl p.1 II gavt''I Il'lIllily :n'). Tht' n"lIlliu/', ""pn" .. itlll

1'1111

1.111

Ill'

1111111,1.11111'11 III ()

Ill, whit-II

ellll

Ihl'lI ht' '1IIlvI'cI 1'111

.n . , m). Thill

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

49

would finally give ,q"(z, m) =

C?!I(Z)[ 11+1

f g1(1)zi f h(m)z' + I z,,+1 _ f h(l)z'

Q

.-0

z

]

gi(m)zi

.-0

i-O

t~o

x

[~oh(m)zi

%

ziz-i

+ i~glm)zi ~ y;Z-i]

(2.18)

The time function is obtained by the inverse modified z-transform t

= (n - J + m),

0!5:

m

!5: J (2.19)

ILLUSTRATIVE EXAMPLE

hnd ;r(t) which is a solution of the following equation. x(t)

=

:1:"+1(2.582 - 1.582e-1 - t) - x,,(J - e-I )

+ Yl/+l(e- I + t -

1)

+ 0.418Yn(1

- e-') (2.20)

Wc CCln construct the coefficients of X/t and ?Ik to be periodic functions pcriod one by replacing t in these functions by m. This gives the I .. llowing clJmuion: Wllh

x( n + J + m) = xn+ 1(2.582 - 1.582e- m

-

m)

+ ?I" I(£' + III - 1) + 0.418!1,,(1 - (' ./11), I

1\

+-

xn(l - e- m )

III

0 !5: m !5: 1

(2.2L)

111'11' III" ~'lldlkil'lIl!1 ur ."" II' ."". /1" .1_ !/" Clrc pcriodic. IIII' IIlllllilkd :-lransJ'1lI'1II ul'CtluHtion 12.21) givcs l.}I:f'(1..m)

Ix(rn 'l)+x(mh Il}

(2.~K2

1.5.'2c'

1·"'( l) xIII . Ie,. "' I '" I

1I.oIIH II

(I

1) 1.1

III

m)l,

c' "').'1"( l) .y( 1.)

,. "''''''11.1

)'11

I

c.'.J ,,)

50 THEORY AND APPLICATION OF THE Z-TRANSFORM 1.6 1.4

t

1.2 1.0

0.8

-=wO. 6 0.4

ra_

0.2

°0~~--~--~~~~--~--~~~~9

FIGURE 2.2 Particular solution for 1/.. = a unit step.

To obtain ~(%). we replace m = 1 in equation (2.22) to obtain

~(%)

= (0.368% + 0.264)tzV(z) + 222:0 + %(2:1 Zl -

21

+ 0.632

Zl -

21

2:0 - 0.368yo)

+ 0.632 (2.23)

If we assume y,. = unit step (Yo = I) and 2:0 = 0, we readily obtain from equation (2.21), by letting n = -1, m I, the value 2:1 = 0.368. Therefore ~(z) becomes

=

o'!'(%) =

(0.36821

+ 0.264)% Z + 0.632)

(2.24)

(z - 1)(21· -

Substituting (2.24) in equation (2.22) and noting that 2:(m - I) = O. we obtain %2[~(Z, m) - x(m)z-I] = (2.582 - l.S82e- m - m)~(z) - (1 - e-m)~'(z)

+ (1

- e- m)-%%-

%2 + (e- m + m - 1 ) - 21 - 1 - (e- m + m - 1)%

1

(2.25)

The initial value 2:(m) can be obtained from equation (2.21) by letting n = -I to give x(m)losms 1

= yo(e- m + m -

1) = e-'"

+m

- I

(2.26)

Substituting equation (2.24) and (2.26) in equation (2.25) an~l. expandinl in powers of Z·". we ohtain the particulnr Rulution .t·pC z. m) "~ f()lIuw!l: z:"p(:' III) .... (I'

III

·1 III"

I):

II

-I- (O..\fIH I O.t,]2m)..:

-f (!.f"2

I

O.M1,. Ill).. • -I· . ..

(7..27)

z-TRANSFORM SOLUTION OF LINEAR DIFFERENCE EQUATIONS

51

The coefficient of Z-k represents x,,(t) for k ~ t < k + I. The final value of X,,(I) which is known to exist is obtained from applying the final value theorem to !r(z, m) to give lim x(n - t n"'''X)

+ m) =

lim (z - l)~,,(z, m) = 1

(2.270)

z-I

The particular solution is shown in Fig. 2.2.

2.3 Linear difference-differential equations5 ,36 In many applications of control systems a pure delay (or transportation lag) often appears in the controlled process. The mathematical description of such systems is carried through the use of mixed difference-differential equations. In the following discussion only linear constant coefficient equations involving one-time delay are considered. Extension to other cases can be performed using the same or other techniques. The modified z-transform of the continuous functions C(/) can be written in the following form:

z'Z,m

...:.P--:-...,....,.... e-U:/dm = __ 1 - ,,-Ie-Ill.

Expansion of. (., .J(

0 -,

)

_

n -

8(%-1, n) A(z-t, n)

(2.149)

is called the Jrozen transfer Junclion. Equation (2.148) is very useful in obtaining an approximate solution for slowly varying discrete systems. In some cases the frozen transfer function will give a good indication of the response behavior. Such !oituations may arise in sampled-data systems where the sampling rate l'IHlIlges very slowly. Another application of this technique is in the analysis II\' automatic gain control where the pulse-frequency modulation varies according to a sinusoidal law. Such a system will be studied as an illustnttive example. Furthermore, if .11 o{z, n) has its poles inside the unit circle for al\ n, all the iterative solutions HI> H 2 • ••• will 'have their poles inside the unit I'Irde and vice versa. ILLUSTRATIVE EXAMPLE

The following example was studied by Tratakowskii l3 using a similar approach. A more detailed analysis of this system will be discussed and n:lt't conditions on stability as well as the response approximation IlIt· ... t·nted. The system under study is referred to as "pulse feedback with \',11 iahle pulse repetition frequency." This system is shown in Fig. 2.5 ,11111 its etjuivalent feedback system in Fig. 2.6. The automatic amplifiI al ion control in u receiver of pulse signals of constant amplitude when an

...... (h'III,1)

/I

c y L-_r-----'--,---o

r-

I II iI/IO ."

I

\OllIIl'h'

fur II,,· v;II'iahlc pulse I'eetlback system.

(Output)

70

T-Y

THEORY AND APPLICATION OF THE z-TRANSFORM

T

FIGURE 2.6

PFM

H;i, I

Block diagram of the variable pulse feedback system.

RC circuit is used for filtering in the AVe leads to the study of such a system as shown in Fig. 2.5. The pulse-frequency modulator (PFM) yields the following equation In

< t < In+1 (2.150)

where In is the time of sampling which is determined by In - 1._ 1 = T(l

+ Msin On)

(2.151)

where T is the average sampling period. When 10 = 0, h is a positive constant with the following constraint: h ~ T(I

-IMD

(2.152)

The relation between y and :r for Fig. 2.6 is given by the following differential equation:

y + fly = kx

(2.153)

The solution of equation (2.153) yields y(t)

+ ke-p'l'X(T)ePr dT '0 = Itc and '0 = 'n-I, since X(/) = 0, for '

= y(to)e-P(Ho)

In particular when t

In' equation (2.154) yields

y(t",}

= y(tn_l)e-fJ(I.-tn-l) + ke- fJt •

f'.-1

'·-I+h

(2.154) 1+ h