Stabilizability and detectability of discrete-time time-varying systems

Stabilizability and detectability of discrete-time time-varying

systems

Citation for published version (APA):

Engwerda, J. C. (1988). Stabilizability and detectability of discrete-time time-varying systems. (Memorandum COSOR; Vol. 8816). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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Faculty of Mathematics and Computing Science

Memorandum CaSaR 88-16

Stabilizability and detectability of discrete-time time-varying

systems by

J.C. Engwerda

Eindhoven, The Netherlands

ABSTRACT

In this paper we introduce some notions of stabilizability and detectability for discrete time-varying systems. These concepts are introduced by making state-space decompositions of the system. This allows an intuitive interpretation and shows the difficulties which occur if one tries to derive general stabilizability and detectability properties.

Moreover, we show by means of a counterexample that the notions ofunifonn stabilizability and unifonn detectability, as defined by Anderson and Moore, do not imply stabilizability respectively detectability of the system.

I.Introduction

In the theory of linear time-varying difference equations the concept of "unifonn asymptotic sta-bility" plays an important role. This, since according to a theorem of Poincare-Bendixson uni-fonn asymptotic stability of the linearized system implies uniuni-fonn asymptotic stability of the non-linear system. Now, J.L. Willems proved in (1970), theorem 7.5.2, that the flow of a linear discrete time-varying system is unifonnly asymptotically stable if and only if it is exponentially stable.

Therefore, a natural question is under which conditions such a system is exponentially stabiliz-able.

Stated differently, under which conditions does there exist a control sequence in the fonn of a state feedback, such that the resulting closed-loop system becomes exponentially stable.

For time-invariant systems these conditions are well known. For time-varying systems, however, this question is more complicated, and a general theory about it is lacking.

Hager and Horowitz (1976), and Anderson and Moore (1981) took a lead with the introduction of sufficient conditions for detectability and stabilizability of discrete-time time-varying systems. Moreover, they used these tosolve some control and filtering problems. However, unfortunately the claim of Anderson and Moore that the time-varying discrete-time Kalman filter is exponen-tially stable under the conditions of unifonn stabilizability and unifonn detectability, is incorrect We provide a counterexample to this claim here. Furthennore, we believe that as well the definitions which Hager and Horowitz give as those of Anderson and Moore, do not give a clear

insight into the basic underlying structural problems.

To obtain a better insight in these problems, the state-space decomposition approach given by Ludyck in (1981) seems to be a more promising one. Therefore, we extend that analysis in this paper.

Based on two state-space decompositions we discern several types of stabilizability and detecta-bility. These concepts can be used to solve e.g. the Linear Quadratic regulator problem, as will be reported elsewhere.

Since the proofs given in Ludyck (1981) to obtain the state-space decompositions are not entirely correct, we also provide correct proofs of them.

The outline of the paper is as follows.

First, in section 2 we introduce some definitions and provide the counter example. Then, by mak-ing a decomposition of the state-space at any time into three orthogonal subspaces, we obtain in section 3 an equivalent system representation from which easily various stabilizability properties of the original system can be deduced. The decomposition originates from considering the reachability and exponential stability subspaces. In section 4 an analogous analysis is performed for detectability. Here the decomposition of the state-space into the unobservable subspace and its complement plays an important role. At last we combine the results in section 5 in which we give a state-space description based on a simultaneous decomposition of the space into the reachable and unobservable parts. The paper ends with some concluding remarks.

n.

Stabilizability

In this paper we consider a system described by the following linear discrete-time time-varying recurrence equation:

x(k

+

I)=A (k) x(k)

+

B(k) u(k); x(ko)

=

x

1;:

y (k)=C(k) x(k) .

Herex(k)e JRn isthe state ofth~ system, u(k) e JRm the applied control, andy(k) e /RP the output at timek. Moreover we assume that all matrices A ( • ), B ( • ) and C ( • ) are bounded.

We use the following notation:

Notation 0

vT(0denotes the transpose for v(i).

v[k, 1] :=(vT(k),'" ,vT(l)l . v[k,·] :=(vT(k), vT(k

+

I),···l .

1mAdenotes the image of the mapping defined by matrixA;KerAdenOtes its kernel.

A (k

+

i, k):=A (k

+

i-I)

* ... *

A (k), ifi ~ I, andA (k, k) :=I . S[k,k-N] :=[B(k)IA(k+l,k)B(k-I)1 ... IA(k+l,k+I-N)B(k-N)] W[N, N+i]:= [CT(N)I {C(N+I)A(N+I,N)}TI ... I (C(N+OA(N+i, N)}T]

OP,fi :=zero matrix withprows andqcolumns.

x(k, ko, xo, u) is the state of the system at time kresulting from the initial state Xo at time k o

when the inputu [ko, k -I]is applied.

y(k, ko,xo, u) :=C(k)x(k, ko, xo, u) .

o

Ifin

1;,

C(k) equals the identity matrix at any time k (i.e. we have full state observations), the subscript y is dropped.

We startOUfanalysis by giving fonnal definitions of several notions of stability and stabilizabil-ity. In these definitions we use the concept of exponentia! convergence of a sequence u[ko,' ].

This is defined as follows. We say thatu(· ) converges exponentially fast to zero if there exist positive constants

a

andM such that IIu (k)II

<

Me~k-ko)forallk

>

ko.

Definition 1

The initial statexof the system

l:y

is said tobe stable atko ifk __lim x(k, ko, x,0)

=

O.

exponentially stable atko if there exist positive constantsaandM such that

IIx(k, ko,x, O)II~Me--a(k-ka)lIxllforanyk

>

ko.

stabilizable atkoif there exists a control sequenceu [ko,'

l,

with the property thatu (. )4 0, such that limk __x(k, ko, x, u)=O.

exponentially stabilizable atko if there exists a control sequenceu [ko,'

l,

with the property that

u ( • )

converges exponentially fast to zero, and positive constants

a

andM such that

IIx(k, k o, x, u)1I~Me-a(k-ka)IIxII for anyk

>

ko.

The system

l:y

is called stable (respectively exponentially stable, stabilizable, exponentially sta-bilizable) at_{k o}if any initial state of

l:y

possesses the corresponding property atko.

As announced in the introduction, we give in this section a counterexample for a result obtained by Anderson and Moore in (1981).

To that end we firstintroduce their concepts of uniform stabilizability. and uniform detectability and quote their corollary 5.4.

Definition 2

l:y

is uniformly stabilizable if there exist integers

s,

t~0 and constants d, b with O~ d

<

1,0

<

b

<

00,such that whenever

IIA(k+I, k+l-t)vll~dllvll

for some v, k,then

l:y

is uniformly detectable if there exist integers s, t~0 and constants d, b with O~d

<

1,0

<

b

<

00,such that whenever

IIA(k+I,k+ I-t)vII~dllv II for some v,k,then

"Corollary 3" (Corollary 5.4 in Anderson and Moore (1981» If

l:y

is uniformly detectable, there exists a bounded

x(k+1)

=

(A (k)-K(k)C (k»x(k)is exponentially stable.

o

sequence K (k) such that

o

This corollary is an immediate consequence of theorem 5.3 in the above mentioned paper.

Inthis theorem it is claimed that if the Eystem is uniformly stabilizable and uniformly detectable, then the Kalman filter is exponentially stable (under the usual system noise assumptions).

Now, consider the following example:

Counterexample 4

Let

[001

[0

l]

A(2k)= 4 OJ; C(2k)=(0 1); A(2k+1)= 0

b ;

C(2k+1)=(O 0), k=O, 1,2,··· Then, withs=t=O, d=t andb=t,we see that

1:,

is uniformly detectable.

However, if we consider the initial state x

= [

~]

at time zero, we see that for

any

sequence

K(· ), x(k+ l)=(A (k)-K(k) C(k»x(k), x(O)=xisnot stable. Which contradicts corollary 3.

0

A direct implication of this example is that the above mentioned theorem 5.3 is incorrect too. By dualizing this example, Le. take A ( • ) =AT( • ) andB (. )

=

CT( • ). we see that the uniform sta-bilizability condition is not sufficient eitherto conclude that there exists a state feedback such that the closed-loop system becomes (exponentially) stable.

Since the major reason for introducing the uniform stabilizability and detectability condition is to have a criterion from which exponential stabilizability respectively exponential detectability of the system can be concluded, we concentrate ourselves in the rest of this paper on finding such criteria.

IlL Sufficient conditions for exponential stabilizability

Inthis section we derive sufficient conditions for exponential stabilizability of 1:.

To that end we first consider the exponential stability subspace at ko, denoted byX;(A (. ,ko».

This subspace consists of all initial states at ko which are exponentially stable. Similarly, we define the stability (or modal) subspace atko,denoted byX-(A (. , ko».

ThatX;(A (. ,k

_o

»

is indeed a linear subspace is easily verified.

The next lemma tells us, moreover,thatthe exponential stability subspace isA(k)-invariant. This property is used later on in this sectiontomake an appropriate state-space decomposition.

LemmaS

A(k)X;(A(· ,k»cX;(A(· ,k+l).

Proof:

Consider 1: at timek. Let xbean exponentially stable state.

Then, we have by definition that for some positive constantsa andM

IIx(t+1,k, x, 0) IIS; Me-a(t+l-k)IIxII

at any timet~k. But, sincex(t+1,k, x,0)=x(t+1,k+1,A (k)x, 0) we obtain immediately that

A(k)xE X;(A (. , k+1». Which proves the lemma.

0

Another property that plays an important role in our analysis concerns the reachability subspace. Its definition reads as follows.

Definition 6

The statexis said tobereachable atkofrom zero if there exists a control sequence u[N, ko-1]

with- 0 0<N <kosuchthatx(ko, N, 0,u)=X.

The subspace consisting of all reachable states from zero atk

_o

is called the reachability subspace atko,and denoted by_Rko' Its dimension is denoted byTko.

The reachability property we are interested in is stated in the next lemma. A formal proof canbe found in Engwerda (1987).

Lemma 7

Consider timeko,and defineA(k)=O andB(k)=Ofork <ko.

A(k)Rk+ImB(k)=Rk+1 •

TIlls lemma tells us in particular that the reachability subspace is alsoA(k)-invariant. Now consider the following state-space decomposition:

XI_(k)=Rkn ~(A( •• k» ;

X2(k)~XI(k)=Rk ;

X3(k)~X2(k)~X

I(k)=

JRrI •

o

where XI •X2and X3are chosen orthogonal.

With respect to a basis adapted to this state-space decomposition the next important corollary holds (see lemmas 5 and 7).

Corollary 8

There exists an orthogonal state transfonnationT'(. )which does not affect the boundedness

pro-pertyof the system parameters such that with x (k)

=

T'(k) x'(k).1: is described by

[

X'I(k+l)] [A'l1(k) A'12(k) A't3(k)]

lX'I(k)]

[B'l(k)] 1:'1: x'2(k+l) 0 A'22(k) A'23(k) x'2(k)

+

B'2(k) u(k)

x'3(k+l) 0 0 A'33(k) x'3(k) B'3(k)

where

1:'I :X'I(k

+

l)=A'11(k)x'I(k)+B'1(k) u (k) is exponentially stable at any timek~k

_{o .}

1:'2 :x'2(k

+

1) =A'2,2(k)x'2(k)+B'2(k) u(k) is reachable at any timek~k

_o•

and is not exponentially stable.

Inthe remainder of this section we derive sufficient conditions in tenns of the transfonned system for exponential stabilizability of 1:.

From corollary 8 we have immediately the following result:

Lemma 9

Consider the transfonned system (l).

If1: is exponentially stabilizable atko.then1:'2 has tobeexponentially stabilizable atko and

r

3

Infact we can conclude much more from thestate-s~acedecomposition.

We see namely that all disturbances entering the system!/2at any time k~ko in a specific way

I

(namely viaImA'23(k)A'33(k, ko» can be controlle4 exponentially fast to zero. We investigate this phenomenon in more detail now!

I I

I

Definition10 i

i

Consider 1:d:x(k+l)=A(k)x(k)+B(k)u(k)+G(k~d(k),where d(· ) is a known disturbance

I

andx(ko)=x.

Then1:_d iscalled exponentially disturbancestabiliz~ble atko if for any initial state atko and for any disturbance there exists a control sequence u

[1'0, • ]

converging exponentially fast to zero,

I

such thatx(k, ko,x, u, d) converges exponentially rast to zero. Herex(k, ko,x, u, d) is defined

similartox (k, ko, x, u).

0

From the above considerations we have the following theorem.

Ii

Theorem

11

'

1:is exponentially stabilizable atk

_o

iffthefollowin~two conditions are satisfied atk

_o:

i) !/3is exponentially stable atko.

ii) !/2d:x'2(k

+

l)=A'22(k)x'2(k)+B'2(k) u (k) +W23(k)A'33(k, ko)d is exponentially

distur-I

bance stabilizable atko. I

Proof:

I

But, since i'l :x'l (k+1)=A'11(k) X'l (k)is

exponen~allY

stable at any time

k~

ko,and matrixB

is bounded, this implies that the first state

compone~t

ofx'(k, ko,x', u)converges also

exponen-I

tially fastto zero. As thethird state component ofxf(k, ko,x', u)converges exponentially fast to zero irrespective of what uis, it is clear now, that

~th

u=U,we have found an appropriate con-trol sequence which stabilizes 1:' exponentially fast. "

0

That both the conditions are necessary was argued

~

lemma 9 and the ensueing remark..

I

That they are also sufficient is seen as follows. que to assumption ii) we know that for any

x'3(ko) there exists a control sequence

u(· ),

whic~ converges exponentially fast to zero, such that the second state component ofx'(k, ko, x', u)iconverges exponentially fast to zero. Here

'. ( ' , ')T

X

.=

Xl, X2,X3 .

So, the main problem left to be solved istogive concilitions under which!/2dis exponentially

dis-I

turbance stabilizable.

We provide sufficient conditions. Therefore, we

in~duce

the concept of periodic smooth con-trollability, as defined by Engwerdain (1987).

RO~y

spoken, a system is called periodically

smoothly controllable if there exists a finite time lriod such that whenever such a time period has passed, the system has been at least once Jntrollable during that period. Fonnally its definition reads as follows (for the definition of S

seq

notation 0).

Definition12

:I:is called periodically smoothly controllable atk

_o

lif there exist constants e,k₁ andN such that

for all k>O there exists an integer k 2(k) in the inferval (ko+(k-l)*k1,ko+k*k 1) for which

T I D

S_{[k2 -N, k 2]}S [k2 -N, k2]~d. !

\

Note that without loss of generality we can take N=2*k 1 in this definition since, whenever

N

<

2*k1owe have that S[k 2-N, k 2]ST[k2 -N,

kJ]~

e/ implies that the same inequality holds

forS[k2 -2*k_{lok2] ST[k 2-2*klok 2].}

Theorem 13

Consider "£:2dfrom theorem 11. I

Let:I:'2_{be periodically smoothly controllable at k o}~d"£:3 exponentially stable at ko.

I

Then, "£:2dis exponentially disturbance stabilizable

ail

ko.

Proof:

First of all we note that dueto the exponential stab~lity assumption on :I:'3' A'23(k)A'33(k, ko)d converges exponentially fast to zero.

Now consider the time interval(k

_o,

k1). I

I

Let e(k2(l» denote the sum of all disturbances ientering "£:2d during this time period, Le.

k2(1)

e(k2(l»

=

L

A'22(k2(l), i)A'23(i)A'33(i, ko)d.

i=/c Consider the input

u [ko, k 2(1)-N -1] =0, and

u[k2(1)-N, k2(l)]=-s'I(S'2S'I)-1(e(~2(1»+X'2),

where S'2 :=S [k2(1), k 2(1)-N] W.f.t. "£:2 and X'2 iSfe initial state of"£:2d' With this input, x'2(k 2(1)

+

1) becomes zero. I

I

We show now by induction that it is possible to ~gu1atex'2(k2(k)+1) to zero fOf any k. Let thereforet be any integer greater than one. !

Consider the interval (k o+(t-2)

*

klo~o+t

*

k1).1be sum of all exogenous influences entering

the reachable subsystem via matrix A'(. ) from k2(t~1)+ 1 until k 2(t) on is then

I

I

k₂(1)

e(k2(t»:=

L

_A'22(k2(t),i)A'23(i)X'3(i) .

i=k2(I-l)+1

Since by induction hypothesisx'2(k 2(t)

+

1) is zero, application of the input

U_{[k 2(t-l)+}1,_{k 2(t)-N}-1]=0,and

u[k2(t)-N, k z(t)]=-S'r(S'2S

'I}-

le(k2(t» ,

yields thatx'2(k2(t)+1)=0.Here S'2:=S [k2(t), k2(t)-N]w.r.t.I:'2. This completes the induction argument

Moreover we observe that IIu(k)IISMII e(k 2(t»11 for some constantM,sinceS'2S'r~£1andS'2

is bounded. ThereforeIlx'z(k)IISM'1I e(k2(t»11 for allk E (k2(t-l), kz(t».Due to the

exponen-tial convergence ofe(k2(t» to zero whent tends to infinity, we conclude that bothx'2(k) and

u

(k)converge exponentially fasttozero whenktendstoinfinity. This completes the proof.

D

Corollary 14

1:is exponentially stabilizable atko if i) I:'3is exponentially stable atk

_o

IV. Sufficient conditions for exponential detectability.

We now give a necessary and sufficient condition for exponential detectability of

Iy.

To that end we first introduce the notion of obselVability.

Definition15

The initial state x of the system

k,

is said to be unobselVable at_{k o}ifY (k. ko.x. 0)=0 for any k~ko·

The set of all unobselVable statesatk

_o

is denoted by U_to and calledtheunobselVable subspace.

Iy

is saidtobeobselVable atk0if

x

= 0 is the only unobselVable state of

k,

atko.

0

Remark:

Note that

Iy

is observable atkoiffUk_o=O!

Analogoustothe lemmas 4 and 5 we have thatUk isA(k)-invariant. This is the content of lemma 16.

Lemma 16

Proof:

LetXl bean element ofA (k)Uk'

By definition there exists then aXo such that i) Xl=A(ko)xo, and

ii) y(k, ko,xo. O)=Ofor allk~_{k o}

o

The rest of the proof follows now from the obselVation thaty(k._ko+1.XloO)=y(k. ko.xo. 0).

0

We define now the notion of (exponential) deteetability.

Definition 17

The initial stateXof

Iy

is said tobedetectable atko if there exists a finite integerN

>

0 such that x(ko)moduloX-(A( • • ko»is detetmined from any_{y[ko• ko+N}-~] andu [ko. ko+N-2].

17

is called exponentially detectable at_{k o}ifinthe above definition of detectabilityX-(A (. ,ko»

is replaced byX;(A ( • ,k

_o

».

0

Next, consider the state-space decomposition: Xl(k)=K;(k)

n

Uk ,

X 2(k)E9X1(k)=Uk ,

X 3(k) E9X2(k)E9Xl(k)=lRll ,

whereXl,X2and X3are chosen orthogonal.

With respectto a basis adapted to this decomposition the following analogue of corollary 8 holds.

Corollary 17

There exists an orthogonal state-space transformationx(· )=T"(· )x"(· ) which does not affect the boundedness property of the system parameters such that

17

is described by the recurrence equation:

y(k)=(0 0 G"3(k»x"(k) ,

where

1:"1:X"l (k

+

1)=A"l1(k)X"l (k), is exponentially stable at any timek2: k o .

o

is observable at any timek2: k o .

With the notation from the previous corollary we have:

Theorem

18

17

is exponentially detectable atkoiff.

1:"2: x"2(k

+

l)=A"22(k) x"z(k)is exponentially stable atko.

Proof:

"~lfConsider the transformed system1:",.

Since

17

is exponentially detectable, the inclusion_{X z cX;(A (. ,ko»}must hold. Consequently, 1:"2has to be exponentially stable atko.

"<="From corollary 17 we know that !:-"1 is exponentially stable at any time k~ko. Due to the assumption thatr!'2is exponentially stable atk

_o

we have that the following system

[

X"l(k+l)] _ [A"l1(k) A"12(k)] [X"l(k)]

x"2(k+l) - 0 A"22(k) x"2(k) is also exponentially stable atko .

SO,X1EeX2cX;(A(· ,ko».

Since !:-"3 is observable atko,it is cle2I now that

r!'y.

and therefore

1:,

is exponentially

v.

Sufficient conditions for simultaneously exponentially stabilizable and detectable sys-tems.

In the previous two sections we gave necessary and sufficient conditions for exponentially stabil-izable and exponentially detectable systems, respectively. In the present section we combine these results.

We derive now sufficient conditionsto conclude that the system

r,

is both exponentially stabiliz-able and exponentially detectstabiliz-able.

To that end we make again a state-space decomposition Consider

Xt(k)=R_{k (')Uk ;}

X 2(k)EBX1(k)=Rk ;

X3(k)EBX2(k)E9X l(k)=/Rn ,

whereXI,X2and X3are chosen again orthogonal. Then, analogoustothe corollaries 8 and 17 we have:

Corollary 19

There exists an orthogonal state-space transfonnation

x ( • )

=

T '( • )

x '( • )

which does not affect the boundedness property of the system parameters such that

r,

is described by the recurrence equation

y(k)=(O C'1(k)C'2(k))x'(k) ,

where

1:'1 :x'1(k+1)=A'11(k)x'1(k)+ B'1(k)u '(k) is reachableatany timek~k o

is both reachable and observable at any timek~ko ;

Theorem 20

With the notation as in corollary 19 we have that

1:y

is both exponentially stabilizable and exponentially detectable atkoif the following three conditionsaresatisfied:

i) 1:'1 is exponentially stable for allk~ko;

ii) 1:'2 is exponentially disturbance stabilizable and observable atk

_o;

iii) 1:'3 is exponentiallysta~leatko.

Proof:

That1:'yis eXJX>nentially stabilizable at k_ounder these conditions is proved similarly to the proof

of theorem 11.

To prove exponential detectability of1:~ atk

_o,

we note that the conditions i) andiii)imply that Xl _{€9X3 cX;(A '(. ,ko}

».

Using this and the property that any state from X2 canbe observed, we have that consequently any element of the factor space /Rn modulo_{X;(A '(. ,ko}

»

canbeobserved. So,1:~ is

exponen-tially detectable atk

_o.

0

Note that conditionii) inthis theorem is also a necessary condition, and that, moreover, exponen-tial stability of 1:'1 and 1:'3 at ko are also necessary requirements.

VI. Concluding remarks

In this paper we showed that exponential stabilizability and exponential detectability properties of a system can be analyzed like in the time-invariant case by using appropriate state~space decompositions.

On the one hand this is duetoour choice of the definitions of stabilizability and detectability. In our definition of stabilizability we required, namely, that additional to the property that the closed-loop system must be stable after a well-chosen input has been applied, the input itself must be stable too.

On the other hand this is duetoour chOIce of the state-space decompositions. Theyareallchosen in such a way that the convergence properties of the transformed and original system remain the same.

A direct consequence of this last mentioned prerequisite was that, when we analyzed systems which are both stabilizab1e and detectable, we did not choose the state-space decomposition which seems at a first glance to be the most appropriate one for analyzing these systems.

Taking in regard the several attempts which have been taken in the past to analyze stabilizability and detectability aspects of time-varying systems and the relative ease by which results are obtained when using this analysis, it seems worth while to deepen this analysis in the future.

References:

Anderson B.D.O. and Moore lB., 1980, Coping with singular transition matrices in estimation andcontrol stability theory; International Journal of Control, pp.571-586.

Anderson B.D.O. and Moore J.B., 1981, Deteetability and stabilizability of time-varying discrete-time linear systems; Siam Journal Control and Optimization V01.19, no.I, pp.2Q-32.

Engwerda J.C., 1987-a, Control aspects of linear discrete time-varying systems; Memorandum Cosar 87-21, T.ll.E., to appearin: International Journal of Control.

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