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602 I EEETRANSACTIONS ON AUTOMATICOhTROL, VOL. AC-30.O. 6. J U N E 1985

Then by using

(5)

and

(9)he

equality (8) yields

n n

n n

= C C A , [ ~ ( ~ + I + ~ ,+ 1 + t ) - A I + ( i + 1 + k ,

j + t )

i = O

J = Q

- A 2 $ ( i + k , j + l + t ) - A o ( i + k ,

j + t ) ] .

10)

Then, for

k

> 0 or f > 0, we ought to consider the following cases:

follows from Theorem

1 ;

to zero according to the second property of

stm

(2-DGM);

equation and Theorem

1

yield

i ) fo r i

+

k

> 0

and

j + t >

0 the right side of

(10)

is

equal

to zero, it

ii)

f o r i

+ k + 1 >

O o r j

+

f

+

1 > Otherightsideof l0) isequal

iii

for

i

+

k -I- 1

=

0

and j

f > 0

the right side of the above

A@(O,

j +

1+f)-A19 0,

j + t ) l

=A,-[A1 (O,+t)-A,+(O, j + t ) l = O ;

and

zero analogously to

iii).

iv) for i +

k

> O a n d j

+ +

1 = Otherightsideof 10)isequalto

Thus, we get

a i i @ ( i + k , +t =o ,

for /c>Oor

t > o or

k = t = ~

n n

=o j = o

which is equivalent to

( 6 ) .

For completing the proof one should note that

for

k = 1

and

t

=

- n the

above

equality

may be IWWitten in the form

i = O

Next use k =

1

and f =

-n

+ 1, etc. E

Remarks:

1) Based on the proof one can note for

k

> 0 and

0 6

< n, hat is

i = O j = O

The

similar formula for 0

0 is obvious.

well-known Cayley-Hamilton theorem

2) For k = t = 0 Theorem 3 may be written as a generalization of the

n

slm(2-DGM)

fulfills

ce(Z-DGM), i.e.,

uo+ i, j ) = 0.

n n

r=o

1-0

However, taking into account the presented definitions and proof of the

theorem we can state a more general result.

Theorem

4 : A state-transition matrix of linear time-invariant digital

system always satisfies the characteristic equation of a system.

V . CONCLUDING

EMARKS

Th e general state-space model of 2-D linear digital system

is

considered

in this note. However,

the

results may be easily applied to any linear

digital system, e.g.,

N-D.

Moreover, we must point out that

the

concepts

of state-transition matrix, etc., apply

to

continuous systems, too.

By using the proposed defmition of state-transition matrix of a causal

system it is easy to find

a

recursive formula for state-transition matrix

calculation as

shown

in Theorem 1. Then, having a state-transition matrix

one can simply get a general r e s p o n s e formula for a system

wth

any

boundary conditions analogously to Theorem 2.

It

is rather evident, e.g.,

[4], [ 5 ] ,

that

the

state-transition matrix is

essential for structural properties of a system, for instance, stability,

controllability, observability. Then Theorem 3 and characteristic function

of the model definition can be useful when the state-transition matrices are

calculated.

Finally, it should

be

noted that the presented theorems

are

valid for

matrices

A i

and

Bi,

i = 0,

1, 2,

over any field, not only real

as

it

was

assumed in Section II.

ACKNOWLEDGMENT

The author w ishes to express

his

grateful thanks to

Prof.

T. Kaczorek

from the Technical University of Warsaw for stimulating discussions on

two-dimensional systems theory.

REFERENCES

S.

A m i , Systemes

lineaires

homogenes a dwx indices,

Rapporr Laboria,

E.

Fornasini and G. Marchesini. State-spaceealization

thwry of two-

vol.

31, Sept. 1973.

dimensional filters,

IEEE

Trans. Automat.

Conrr.,

vol. AC-21, no. 4, pp.

484-492, Aug.1976.

E.

Fornasini and

G .

Marchesini.Doubly-indexeddynamical ystems:State-

spacemodels and structural pmpenies , Math.

Syst. Theory,

vol. 12, pp. 59-72,

1978.

E.

Fornasini and G. Marchesini, A critical review

of

recent

results

on 2-D

systems

theory

(Preprints) Proc.

8th IFAC Congress,

1981, vol.

II,

pp. 147-

153.

S.-Y.

Kung,

B.C. Levy,M. Mod, and T. Kailath, New

results

in 2-D systems

thwry-Pan

U:

2-D state-space models-realization and the notions f controllabil-

ity, observability and minimality,

ROC.

EEE, vol. 65, pp.945-961, une

1977.

W. Marszalek, Two dimensional state-space discrete models

for

hyperbolic

partial

differential

equations. Ap p l . M a th. Model.,

vol.

8.

pp.11-14,Feb.

1984.

R. R. Roesser, A discrete state-spacemodel for linear image processing,

IEEE

S.G . Tzafestas and T. G. Fimenides. Exact model-matching control of t h r e e

Trans. Automat. Contr., vol. AC-20, pp. 1-10. Feb. 1975.

dimensional systems using state and output feedback, In?. J .

Sysr.

Sci., V O ~ .13,

~ ~~

pp.1171-1187. 1982.

Asymptotic Recovery

for

Discrete-Time Systems

J .

M. MACIEJOWSKI

Abstract-An asymptotic recovery design procedure is proposed for

square, discrete-time, linear, time-invariant multivariable systems, which

allows a state-feedback design tobe approximately recovered b y a

dynamic outpnt feedback scheme. Both the

case

of negligible processing

lime (compared to the sampling interval) and of significant processieg

time are discussed. In the former case, it is possible to

obtain

perfect

Paper

recommended by

Pas t

Associate Editor, D.

P.

LoozeThis work was

supported

by

Manuscript received June 23, 1983; revised October 8, 1984 and October 26, 1984.

the Science and Engineering Research Council.

The

author is with heDepartmentof Engineering, Cambridge University, Cam-

bridge, England.

0018-9286l85/0600-062 01.00 0 1985 IEEE


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TRANSACTIONS ON AUTOMATIC

CONTROL.

VOL. AC-30.

NO.

6. J U N E 1985

603

if the plant is minimum -phase and has he smallest possible

of

zeros at infinity. In other cases good recovery

is

frequently

ble. New conditions are found which ensure tha t the return-ratio

good

robustness properties.

I . INTRODUCTION

been developed by Kwakernaak

121, [3] for continuous-time, minimum-phase

s. These are design techniques which allow the excellent robust-

was the

be attained with a

two pairs of matrices, namely a pair of cost-weighting

the asymptotic recovery

values according to an automatic procedure. Further-

the designer can obtain considerable guidance

of

the pair of matrices to be designed, in such

to approach the spe cification more closely.

This

results in a

re for discrete-time systems, in spite of the act that discrete-time

the attractive features which are

in

the continuous-time case-for example, stability margins and

two cases. In the first case he processing time

is negligible when compared to

uk

can

be d o w e d to depend on the output observations up

h he second case, he processing time is comparable to the

nd u k can

be

allowed to depend only on

y k -

I

with a consequent impairment of performance.

e

shall

find that, unlike the continuous-time case, it is possible to

II.

DESIGNPROCEDURE

be controlled is modeled as

xk+l=AXk+BUk; Yk=CXk 1)

u and y are rn-dimensional input and output vectors, x is the n-

state

vector

n

2 rn),and

A , B , C

re constant matrices. As

is stabilizable and detectable.

noise covariance matrices, W an d

V ,

are

used

to obtain a steady-

in

[4],

this takes the following form:

~ k - c / k = A ~ k r / k - I + B U k - K ; C F k / k - I - Y k ) 2)

j k / k - 1= c f k l k - I 3)

~ k , k = ~ k l k - l - K ; ~ k / k - l - Y k ) 4)

K;=K ;

5 )

K ; = P c ( c P c + V)-1 (6)

is the positive semidefinite solution of the Riccati equation

P = A P A - A P C ( C P C +) - l C P A W. 7)

k/k-l

yk

-

-E

Fig.

1. The

structure of

the discrete-timeobserver.

~

lic

-

Fig.

1. The

structure of

the discrete-timeobserver

A block-diagram representation of these equations is shown in Fig. I , in a

form which emphasizes that KY is a feedback matrix, while K$ is a

feedforward matrix.

The dynamics of the plant are augmented if necessary, and the W and V

matrices are adjusted until the frequency response characteristics of the

Kalman filter are those which the designer would like to obtain at the

output of the compensated plant. By frequency response characteristics

we mean the behavior of indicators such as the characteristic loci or the

singular values of the fdters open-loop return ratio

+ ( z ) = C ( Z Z - A ) - K Y

a nd o r its closed-loop transfer function

W Z )

= + Z ) [ Z + q z ) ]

.

9)

Next a cheap optimal state feedback controller is synthesized for the

augmented plant, with state weighting matrix Q = CC and control

weighting matrix R = 0. This requires one to solve equations dual to ( 5 ) .

6), 7)

to obtain the state feedback m atrix Kc (see

[5]

for details) which is

used to generate the control according to

uk = K& k. ( loa)

If it is impractical to use yk for the estimation of &, we replace (lo a) by

uk= -K&?k/k- l .

(lob)

Shaked has recently found closed-form expressions for Kc for the cheap

control problem

[6].

Note that in the discrete-time case it is quite possible

to set R =

0,

whereas in the continuous-time case this would lead to the

use of infinite gains.

Finally, a feedback compensator is synthesized as the series connection

of rhe

Kalman

filter and the optimal state-feedback controller in the usual

way.

From 2)- 4) and (loa) it follows that the resulting (filtering)

compensator is defined by

1; I = ( A

-

B K N K ;c)


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604 IEEE

TRANSACTIONSNUTOMATIC

CONTROL,

VOL. AC-30,

NO. 6,

JUNE

1985

If one

or

both ofhese conditions fail to hold, (16) often holds

approximately over a useful frequency range.

m SE OF H f ( Z )WITH det (CB) 0 AND G Z ) MINIMUM-PHASE

First, consider the case in which processing time is negligible, so that

the filtering version of the compensator can be used. Suppose that det

(CB) 0 and that G ( z )has no finite zeros outside the unit circle (we shall

call such G(z)minimum-phase, even though this term should really be

reserved for discrete-time systems with neither finite nor infinite zeros

outside the unit circle). Let

S(A,

B , C denote the plant model

( l ) ,

and let

Kc be the state feedback matrix obtained as the optimal solution to the

problem

minimize J = y&. 15)

k = O

Lemma (Shaked

[6]): If det

(CB)

0, then

K , = ( C B ) - l C A . 16)

(This is the simplest case of Shaked s much m ore general results. It is

easily verified that C C is the solution of the Riccati equation for this

problem, from which (16) follows.)

It

is

easy to show that

det

(CB)#O

iff W = range

( B ) B

ker C ) 17)

so that

II= B(CB) IC 18)

is

the projector onto range

( B )

along ker 0 . herefore,

range ( A-B K J = range ( [ Z - m A ) C ker

(C).

19)

Now H&) can be manipulated into the form

H~(Z)=ZK,[ZZ-(Z-K~~(A-BK,)]-~K

20)

= z K , ( z I - A + B K , ) - ' K $ 21)

in view of (19).

0, then

Theorem: If G ( z )has

no

(finite) zeros in ( z : zI > 11 and det (CB)

A z) = G(z)HJ(z) (z)

=0

22)

Proof:

A(Z)=C(ZZ--A)-~[ZBK~(ZZ-A+BK,)-'K -K~] 23)

=C(ZI-A)-~{Z~IA[ZI-(Z-T~)A]-'-A}K 24)

=C(tI-A)-'{zII[zZ-A(Z-II)-'-I)AK 25)

= C ( Z I - A ) - ' [ ( A - Z I ) ( Z - ~ ] [ Z Z - A ( Z - ~ ) ] - ' A K ~26)

=

- C ( I - T T ) [ Z Z - A ( Z - I T ) ] - ~ A K $ 27)

= O

since C(Z-rr)=O.

This shows that in this case we obtain perfect recovery. Note that, as in

the continuous-time case, the recovery does not depend on any properties

of K$ Any (stable) observer with the given structure can be recovered,

providing that KT = A K f

J

I V.

USE OF H f ( Z )

WITH G(Z)

NONMINIMUM-PHASE

It is known

[5]

hat (with

Q

=

C C

and

R = 0),

the eigenvalues of A

i) those zeros

of

C ( z ) which

lie

in { z : l z l

I } ;

iii) and the remainder at the origin.

It is also known [7l that the condition det (CB) 0 ensures that G ( z )has

BK,

are located at:

the

maximum

possible number n - m ) of finite zeros andhe

minimum possible number ( m ) of infinite zeros. The perfect recovery

obtained in Section

IU

s only possible because the nonzero po les of H f ( z )

cancel the n -

m

finite zeros of G ( z ) ,and the

m

origin poles of HAz).

Cancel the m origin zeros introduced by the factor z in (21).

Th e mechanism by which recovery

is

achieved is thus essentially the

same as in the continuous-time case: the compensator cancels the plant

zeros and possibly some

of

the stable poles, and inserts the observer s

zeros. C learly, this will fail if the plant has zero s outside the unit circle,

since

the

compensator

H J z )

guarantees internal stability. This

is

potentially a more serious limitation for disciete-time than for continuous-

time systems, since the standard sampling process is

known

to introduce

zeros, some of which usually lie outside the unit circle.

How ever, in the following paragraphs w e shall show that H&) always

cancels those zeros of G ( z ) which lie inside the unit circle. The

importance of this is that the zeros introduced by sam pling usually lie near

the negative real axi s [SI, and thus any zeros which remain uncanceled

will usually lie near the negative real axis (unless the original continuous-

time plant has zeros in the right half-plane [SI . This raises the possibility

that

G ( z ) H f ( z )

iffers significantly from

+ z)

only at high frequencies

0.5

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Digital Control System-1