A Popov-theory-based approach to digital H control with measurement feedback for Pritchard¿Salamon systems

IMA Journal of Mathematical Control & Information (1994) 11, 277-309

A Popov-theory-based approach to digital H°° control with

measurement feedback for Pritchard-Salamon systems

*Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands

**Department of Automatic Control, Polytechnic Institute of Bucharest, 313 Splaiul Independents, 77206 Bucharest 1, Romania

[Received 17 March 1994]

In this paper, we deal with the digital output-measurement-feedback H00 control problem for Pritchard-Salamon infinite-dimensional systems with unbounded input and output operators. A discrete Popov-theory-based solution is given in terms of the solvability of Kalman-Szego-Popov-Yakubovitch systems associated with the equivalent discrete-time time-invariant system obtained by lifting the 7"-periodic continuous-time system.

1. Introduction

In [25,26], Pritchard & Salamon introduced a class of infinite-dimensional systems that allow certain unboundedness in control and observation. Specific examples of such systems are linear systems described by PDEs with point and/or boundary control and observation [9,12], and retarded systems with delays in control and observation [25]. This class has become very popular since it has been proved to be rich enough to permit one to obtain the solution to the linear quadratic control problem (see Pritchard & Salamon [26] and van Keulen [29]) as well as the extension of the classical finite-dimensional results on H°° optimal control from [15] (see van Keulen [29]). For an overview of the basic results concerning perturbation theory, exponential stability, and transfer functions of Pritchard-Salamon systems, the reader is referred to the recently announced paper of Curtain et al. [13].

The H°° control problem with state and output measurement feedback for infinite-dimensional systems has received much attention in recent years. A state-space solution to the problem has been given by van Keulen in [29] where the main two-Riccati equation result of Doyle et al. from [15] has been extended to the Pritchard-Salamon class of systems.

Among the various approaches to the H00 control problem, the Popov-theory-based solution, originally developed for discrete-time finite-dimensional systems by Ionescu & Weiss [20], has been fully generalized to infinite-dimensional systems: the continuous-time bounded-input bounded-output case by Weiss [32], and the time-dependent discrete-time case by Dragan, Halanay, & Ionescu [17].

The digital linear quadratic control problem for Pritchard-Salamon systems has been addressed in Barb & DeKoning [3] where a digital state-feedback result was given. The case of digital output-measurement feedback was treated in Barb &

277

2 7 8 F. DAN BARB. V. 10NESCU AND W. DE KONING

DeKoning [4]. In [5], a Popov-theory-based solution to the digital state-feedback Hx control of Pritchard-Salamon systems was given in terms of solvability of certain Kalman-Szego-Popov-Yakubovitch (KSPY) systems. This paper is a natural continuation of the work in [5]. We are concerned with the (7-suboptimal) digital output-measurement-feedback Hx control problem for Pritchard-Salamon

systems. Our goal is to design a discrete-time controller interconnected with the continuous-time plant via a synchronized sampler and zeroth-order holder such that the effect of the disturbance input on the controlled output is lower than a prespecified bound imposed by continuous-time design considerations. The piece-wise constant control is generated from the discretized measurement output {)'2(kT)}keN (when v2(*) can be sampled; we shall see that further on in this

paper). The difficulty of taking into account the intersample behaviour is overcome by lifting the overall time-dependent system to a discrete-time one. This is done in the spirit of the original work of Yamamoto [33], making use of the framework developed by Bamieh et al. [1, 2]. A similar lifting technique was applied by Tadmor [28] who gave a solution to the digital H00 control problem for systems with finite-dimensional state space in terms of three Riccati equations, two algebraic equa-tions, and a differential one. The Popov-theory approach we take in this paper has the advantage of allowing the main control result to be stated in a framework of Hilbert-space operators, while permitting the set of necessary and sufficient conditions to be expressed in terms of the solvability of a certain KSPY system (equivalent to a Riccati equation). This was not possible in [1], since the spaces of the lifted disturbance input and the lifted controlled output were infinite-dimensional. Within our approach, we remain in the classical framework of the solution based on the Riccati equation. There are two reasons for this. The first is that our original state space is infinite-dimensional and there is no struggle as in [1] to reduce everything to finite-dimensional spaces—the implementation issue is not one of our objectives at this stage. The second reason is that we have available a general Riccati theory for discrete-time systems on real separable Hilbert spaces that we can apply to our digital control problem.

The structure of this paper is the following. In Section 2, we formulate the problem. Some basic Pritchard-Salamon results are also given here. In Section 3, we show how to reduce the control problem to an equivalent discrete-time one. General discrete Popov-theory results are collected in Section 4, and the solution to the 7-attenuation problem for discrete-time systems is outlined in Section 5 in the case of output measurement feedback. The shortcomings of the method are discussed in Section 6. Conclusions are reached and directions of future research are indicated in Section 7.

2. Problem formulation

2.1. Pritchard-Salamon systems. Basic properties

Capital script letters will stand for real, separable Hilbert spaces. If A : X —> y, then t)(A) will denote the domain of A. We write 2(X,y) for the space of bounded operators from X to y. By L2°C(K+, A") we denote the space of locally (Bochner) square-integrable functions from K+ = [0,oo) to X, and £2(N, X) will stand for the

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 2 7 9 Hilbert space of square-summable sequences in A" (i.e. maps from N = {0,1,...} to A'). We shall say that A generates an exponentially stable semigroup of bounded linear operators { £ ( / ) } ,6 8 on X (a C° semigroup) if there exists \i > 1 and a < 0 such that

||S(0|| < / i e ° ' f o r / ^ O .

An operator A € 2(X) is called power-stable if there exist \i ^ 1 and p € (0,1) such that

\\Ak\\ < npk for keN.

Let X be a Banach space with norm || »\\x, and let W be a linear subspace of X. Assume that another norm, ||»||w> >s defined on W such that W is itself a Banach space. Consider the linear operator I :W>-* X defined by Iw = w. We call this a

continuous embedding [14] if / £ £(W. X). If W is dense in X with respect to the

norm in X (i.e. its closure is equal to X), we shall call / a dense injection. In the case when / is also continuous, we shall use the notation W<—*X.

If X is a real Hilbert space with the norm given by the inner product, we shall say that X is separable if (X, || • ||^) contains a dense subset which is countable.

Let us consider three real separable Hilbert spaces W, X, and V with continuous dense injections W<-> X^-> V. We consider a Co semigroup {£(/)}, 6R+ on V, and we shall assume that its restriction on W and X are also Co semigroups. Aw, Ax, and Av will denote the infinitesimal generators of {^W(0};eR+' {^(OJreR^' a n c' {Sv(/H.,eR+ on W, X, and V, where Av = A, and where the superscripts W and X indicuK: the corresponding restricted operators. The following definition is reported from [26,13,29].

DEFINITION 1 Let U and y be the input and output spaces.

1. An operator B e C(U, V) is called an admissible input operator for S( •) (see Weiss [30]) if there exist / > 0 and a > 0 such that

[' SV( / - T ) B « ( T ) ( 1 T 6 W (2.1)

Jo

and

[' Sv(t-T)Bu{T)dr

Jo _w (2-2)

for all «(.) € LJf^K+.W), where

2. An operator C € 2(W,y) is called an admissible output operator for S(») (see Weiss [31]) if there exist / > 0 and 0 > 0 such that

W ^ 0 I M I V (2.3) for all x G W.

2 8 0 F. DAN BARB, V. IONESCU AND W. DE KONING

3. Let B £ 2(U, V) and C £ 2(W, y) be admissible input and output operators for

S(•). The system E(S(*),B,C,D) given by

x(t) = Sv(t)xo + \'oS^-r)Bu(r)dr) ^ ^ ^ ^

y{t) = Cx(t) + Du(t) J

where x0 £ V and «(•) £ L2OC(K+]ZY), is called a Pritchard-Salamon system; this is well denned because, if (2.2) (resp. (2.3)) holds for some t > 0 and for some a > 0 (resp. (5 > 0), then it holds for any / > 0 and some a (resp. fi) depending on t (see [13]). •

Note that, if S( •) is exponentially stable on W, then a and j3 can be chosen not to depend on /, and (2.2) and (2.3) hold then for / = oo.

If XQ £ W, then x(») defined by (2.4a) is a continuous function on R+ with respect to the topology on W, and the output

t^y(t) = CSw(t)x0 + cl Sv(t-T)Bu(T)dT (2.5)

Jo

is also a continuous function on K+ with values in y. If x0 6 V\W, then (2.5) does not make sense directly. From (2.3), the admissibility of the output operator implies that the linear map

has a unique bounded extension

V -» U0 C(R+ ) y) : x ^ CSW( • )x

in the sense that, for every x € V, we shall use the expression CSW( • )x to denote the function in Li2C(R+,y) obtained by continuous extension to V of the operator that maps x e W into CSw(»)x in Li2C(R+,y). Thus, the output j(«) e L^iR^y) in (2.5) is interpreted as

t~y{t) = CS

(t)

+ C S

(t - r)5«(r)dr + Du(t)

Jo

and is a well defined y-valued locally L2 function.

Following [13], we assume (without loss of generality) that the Pritchard-Salamon system is smooth, i.e.

X>{AV)^>W. (2.6)

Notice that assumption is not restrictive in the sense that (2.6) is satisfied by all known examples of Pritchard-Salamon systems if W and V are chosen appropriately. We quote from [13] that (2.6) is implied by T)(AV) c W, the latter inclusion also ensuring that the resolvent sets of Aw and Av are the same. In contrast, the growth bounds u^y and u)v may not be the same (see Example 2.1 in [13]), and hence exponential stability on V (resp. W) does not imply exponential stability on W (resp. V). However, using the concept of admissible stabilizability, a satisfactory stability theory was developed (see [13,29] for details).

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 281

DEFINITION 2 Let E(S( •), B, C, D) be a Pritchard-Salamon system.

1. The pair (S( •), B) is called admissibly (boundedly) stabilizable if there exists an admissible output operator F e 2(W,U) (an operator F e 2(V,U)) such that the perturbed C° semigroup Sp(»), generated by Av + BF, is exponentially stable on W and V.

2. The pair (C, S(»)) is called admissibly (boundedly) detectable if there exists an admissible input operator H e 2(y, V) (an input operator H € 2(y, W)) such that the perturbed C° semigroup SJj(*), generated by Av + HC, is exponentially stable on W and V. •

The following result [29] gives the relationship that exists between the two concepts of stabilizability quoted above.

PROPOSITION 1 (1) The pair (S(«),B) is admissibly stabilizable if and only if is boundedly stabilizable. (2) The pair (C, £(•)) is admissibly detectable if and only if is boundedly detectable. •

We shall focus now on the notion of input-output stability.

DEFINITION 3 Let E(S( •), B, C, D) be a Pritchard-Salamon system, and let

G : D(G) C L!,0C(R+;W) -» L12oe(R+;y) defined by

(GM)(O = C\ S(t- T)BU{T) AT + Du(t). (2.7)

Jo

We call 27(S( •), B, C, D) input-output stable if 3>(G) = L^OC(1R+;W). D

The next result gives the relationship between input-output stability and exponential stability (see [29]).

PROPOSITION 2 Let £(S(*),B, C,D) be a Pritchard-Salamon system, and suppose that (S( •), B) is admissibly stabilizable and (C, S( •)) is admissibly detectable, respec-tively. Then the system is input-output stable if and only if is exponentially stable on W and V (i.e. ww < 0 and u^ < 0). •

2.2. The digital output-measurement-feedback H00 control problem for Pritchard-Salamon systems

Let us consider three Hilbert spaces with continuous dense injections W •—> X •—»V, and assume that there exist two other real separable Hilbert spaces U\ and yu and let *, efl(W,,V), B2e2{U2,V), C , e f i ( W , y , ) , C2e 2 ( W , B and Z>12 € £(^2)^1)- We consider the Pritchard-Salamon system

2 8 2 F. DAN BARB, V. IONESCU AND W. DE KONING

satisfying

*(/) = Sw(t)x0 + £ SV(l - r)[fl,M, (r) + B2U2(T)} AT

yi(l) = Clx(t)+Dllul(t) + y2(t) = C2x(t)

We shall assume that the system is smooth (i.e. (2.6) is fulfilled). In our set-up, x(t) e V is the state, ut(t) £ U\ is the disturbance input, u2(t) € U2 is the control

input, y\(t) € 3^i the controlled output, and y2(t) £ ^2 is t n e measurement output

at time t. The following assumptions are made on the Pritchard-Salamon system 1. U2 = Rm (we have m actuators).

2. y2 = Rp (we have p sensors).

3. x0 e W (the output measurement is a continuous IR^-valued function with

respect to the topology on W). 4. (S(»),B2) is admissibly stabilizable.

5. (C2, S( •)) is admissibly detectable. 6. Z>n = 0 .

7. D\2DX2 is coercive. REMARK 1

• Assumptions 1-2 are based on engineering reality, in which the number of sensors and actuators must obviously be finite.

• Assumption 3 guarantees that the output measurement can be sampled. • Assumptions 4 and 5 provide a sufficient condition for the hybrid stability of the

digital control system (see Proposition 3).

• Assumption 6 is made for simplicity. For details of how this assumption can be removed, the reader is referred to [29: § 5.4].

• Assumption 7 guarantees the well-posedness of the Pritchard—Salamon system. •

If x0 — 0, then we can express (2.8) as

where

Sv(t-T)BlUl(T)dt: o

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 2 8 3 Gu : L ^ R - R "1) — L ^

(C

//

)(0 = C| [

Let K, be another real separable Hilbert space, and consider a controller

S(K,L..M,N) of the form

&+i = K£k + t>Vk, Ct = ^ S t + # % : (2-10) where K € £ ( £ ) . L € fifR', /C), M G £(/C;E'"), yv 6 R'XI", and the initial state of the controller. £0, is given. If $0 = 0, then (2.10) can be expressed in an input-output fashion as

Qk = (GKT})k.

where

k-\

i=0

We want to stabilize the system (in a certain sense that will be defined below), and ensure that the influence of the disturbance input ii\ on the controlled output v, is smaller (in a certain sense) than a prespecified bound 7, by digital output-measurement feedback. Let us explain further what we mean by digital output-measurement

feed-back. Define first the sample and zeroth-order hold operators with period T: + )* ) - *N, C = SrC - (Co C* = C(*7").-).

A'), C(0 = (H7-O* = C* f o r / e [ * 7 \ ( * + l ) r ) , where by (£(R+,X) we have denoted the space of A"-valued piecewise continuous

functions that are bounded on compact subsets of R+, and where Xn is the space

of /f-valued sequences defined on the set of nonnegative integers. By their defini-tions, the hold operator is synchronized in time with the sampling operator, and they are well defined when X is replaced by any other space. Since the output measurement is a continuous R^-valued function (we have assumed that .v0 € W. so this condition is fulfilled) we can define its time-discretized version as

f'2 = Sy.lV We want to make sense of the feedback connection

" 2 ( 0 = Ct for kT^t<(k+l)T) { ''

284 F. DAN BARB, V. 1ONESCU AND W. DE KONING fl, B2 '12 K M L N

Fig. 1. The closed-loop configuration of digital output-measurement feedback.

Our goal is to design a controller S(K, L,M,N) with the following properties. 1 (stability). The closed-loop system E(GT, GK), obtained by the feedback inter-connection of the hybrid generalized plant GT and the controller GK is hybrid-stable. We shall explain what we mean by hybrid stability of the structure E(GT,GK) in Section 3.1.

2 (7 contraction). The input-output operator

T, : L2( R+, « i ) -> L2(R+,3;i) : "1 ^y\

expressed as a linear fractional transformation of the hybrid generalized plant SrG2i SrG22Hr

and the controller GK, i.e.

T\ = F(GT, GK) = GU+ GnHTGK(I - S

is well defined and is a 7 contraction of spaces, i.e.

)~ STTGG2\,

REMARK 2 Due to the sampler, Tj is not time-invariant (see [28]), and we cannot associate a transfer function with the linear system mapping u\ to yt. •

2.3. The discrete-time H°° control problem

We conclude this section by formulating the H°° control problem in discrete time. Our basic model is the infinite-dimensional discrete-time system

representing the set of equations

sx = Ax + B\ «i + B2u2

j>i = C\x + Duux + Di2u2 } ,

y2 = C2x

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 285

where s denotes the forward shift operator (.Yfc)freNt-+(**+i)freN> and where Ax = (Axk)keN etc. A G 2(X) is assumed to define a power-stable evolution,

5, e 2(UUX), B2 G 2(U2,X), C, € £ ( * , ? , ) , C2 G £(*,3>2), />„ G 2(Uuyl), and

£>i2 G £(£/2,3>i). We consider U\,Ui, 3^i» 3^ t o be the real separable Hilbert spaces of the disturbance input (i/1]Jt G Wi), control input (u2jt € i/2), output to be controlled (y\.k £ ^i)> a nd measured output (y2j! G ;y2), respectively. Here A" is a real separable Hilbert state space, with x G A". Consider for the system (2.11) the controller

u2 = Ccxc + Dcy2, (2.12)

^ belongs to the real separable Hilbert sxc = Acxc + Bcy2,

where Acxc = {Acxck)keN etc., and where

controller state-space Xc; here Ac, Bc, Cc, and Dc are operators bounded on

appropriate subspaces.

The resultant closed-loop system is given by

where again the convention ARxR = ( ^ R ^ R * ) A -£N e t c- is used, and where x A + B2DCC2 B2CC

BCC2 Ac

DnCc

The augmented state space XR = X © Xc is a real separable Hilbert space under the

inner product

(xck>xck)xK = {xk,xk)x+ (xckixck)xc

-The system (2.11) can be written in an input-output fashion as

1 2

i r

2 2

\ [

where Ttj G 2(£2(N,Uj), e2(N,yj)) for i,j = 1,2. If Tc is the input-output operator

associated with the controller, then the closed-loop input-output operator is expressed as the linear fractional transformation of the system (2.11) and the controller (2.12):

'21 r22

The 7-suboptimal H°° control problem (or 7-disturbance-attenuation problem) is to find a controller (2.12) for the system (2.11) such that

1. The closed-loop system (2.13) is power-stable (i.e. AR generates a power stable

evolution on XR). We shall refer to this as the stability requirement.

2. The closed-loop input-output operator is a 7 contraction:

2 8 6 F. DAN BARB. V. IONESCU AND W. DE KONING

3. The equivalent discrete-time H* optimal-control problem

In this section, we shall reduce the digital control problem to an equivalent discrete-time one. The basic idea is to lift a continuous-discrete-time periodic signal to a shift-invariant discrete-time one. In this way, we describe the action from the lifted input to the lifted output of a periodically time-dependent system by an operator we shall call the lifted input-output operator. Since state-space formulae are important for our develop-ments, we shall derive the expression of lifted A, B, C, and D operators of a given system. Our basic references are Bamieh & Pearson [1], Bamieh et at. [2], and Yamamoto [33], papers in which the reader can find a much more detailed treatment of this problem. We shall also investigate stability properties of the digital control system. Since the resulting closed loop is inherently hybrid in its nature, an appro-priate stability theory should be developed. However, our work is greatly simplified by the concept of hybrid stability introduced by Chen & Francis [10], which is adequate for our purposes. We generalize the main results on hybrid stability from [10] to the Pritchard-Salamon class of hybrid systems.

3.1. Hybrid stability for Pritchard-Salamon systems

The concept of hybrid stability was initially introduced for the finite-dimensional case in [10]. In [11] it has been shown that exponential stability does not imply I^-input-output stability unless a low-pass filter is used prior to the sampler. However, it has been shown that, under some pathology-excluding conditions imposed on the sampling step, a certain concept called hybrid stability is equivalent to exponential stability in finite dimensions. In this section, we shall extend the concept of hybrid stabil-ity to the Pritchard-Salamon class of systems and establish the relationship between this type of stability and input-output stability as was defined in subsection 2.1.

Consider a Pritchard-Salamon system £(S(*),B,C,D) with respect to W<—»X<—> V. Assume that .v0 € W. Suppose that the control is piecewise constant with u(t) = uk for kT ^ t < (k + \)T, and that (.xk)keN is the sequence of state

discrete values. Then the state equation becomes

.v(/) = SW(t - kT)xk + [ Sv(t - r)Buk dr. ikT

Write t - T = TJ. Then, at t = (k + 1)7, we obtain

-v*+. = Sw(T)xt + f Sv(V)Buk dV (3.1 Jo

and also

yk = Cxk+Duk. (3.2)

DEFINITION 4 Consider S(SV( • ). B, C. D) with respect to W^ X ^ V. We shall call

(<P. P. C. D) a time-discretization of E(SV( • ) . B, C, D), the latter system satisfying the

difference state equation (3.1) and the discrete output equation (3.2), if $ and F (depending on the sampling step T) are given by

f SV{T1)BAT]. • (3.3)

±

f

DIGITAL CONTROL OF PR1TCHARD-SALAMON SYSTEMS 287

Following [10], let us introduce two compatible exogenous signals v e C2(N.RP) and

»' € £2(N, Km) and add them to the closed-loop system. Clearly, we define the concept of hybrid stability on the basis of the signals

i t \ , V | . M2. »»•, v . i S

and the configuration depicted in Fig. 2.

DEFINITION 5 The digital control system is called hybrid-stable if the mapping {u\,v, >t')i-+(y1.M2) iS)

is well defined and bounded from

to

. D

REMARK 3 Notice that the equation relating (u\,v,w) to (j^-i^, iS) is / - G1 2Hr 0 0 / GK 0 SrG2 2Hr / "2 0 / 0 0 0 / "1 V

Since G2i is strictly proper (Z)21 = 0), ST-G21 is well defined and bounded, and so is

G,, 0 0 '

0 7 0 . Hence, if the exponential stability assumption on 5 ( « ) holds, SrG2 1 0 l\

then a simple sufficient condition for hybrid stability is that the operator / 0 0 step - G ST-G I •% i V T> 12H / 22H 0. T T

D

defined on L2{M.+ , i © £2(N, K'") with values in

is boundedly invertible for a given sampling

Si B2

V +A+ K

M N

w

Yl

2 8 8 F. DAN BARB, V. IONESCU AND W. DE KONING

DEFINITION 6 Let E(S( •), B, C, D) be a Pritchard-Salamon system with respect to W ' - J ^ ^ V . We shall say that it is hybrid-stabilizable if there exists a controller such that the digital control system is hybrid-stable. •

We want to characterize first the stabilizability (in a certain sense). As was argued in Section 1, a pair (5( •), B) that is stabilizable on W (resp. V) is not necessarily stabi-lizable on V (resp. VV). The next example shows that this inconvenient property of (S(»),B) is not alleviated by sampling, i.e. the power stabilizability of the time-discretized pair, (#, F), on W (resp. V) does not necessarily imply power stabili-zability on V (resp. W). Define first for every sampling step the discrete growth

constant of S(T) as

(3 = eaT/T, (3.4)

where S(T) is either SW(T) or SV(T), and let flw and /3V be their corresponding discrete growth constants. The following examples, which represent the time-discretization of the translation semigroups (see [13: Ex. 2.1]), show that /3y and f3w may in general be different (this is actually a consequence of the fact that aw and av might be different), and both situations f3v > (3W and fa < (}w might appear. Let

W = L2(R+,R), V = { / e U0 C( R+ )R ) : e - ' / € W }> (3.5a, b) with ||/||v = ||e */(*)llw- Consider the translation semigroup S(») defined by

[S(t)f\{x)=f{x + tl

which is a C° semigroup on W<-> V. Let T > 0 be arbitrary chosen. It is straight-forward to prove that ||S(/)||e(VV) = 1 and l|S(Olle(V) = e2' (s e e again t1 3 : E x- 2 1J ) which implies aw = 0 and av = 2 and hence /3W = \/T and /Jv = e2T/T. Hence,

0V > (3W. Let W and V be given by (3.5) and consider the semigroup S( •) defined by

(x-t)

Following [13], we have that S(») is strongly continuous on W"—>V with ||S(r)||£(w) = 1 and ||S(0llc(V) = l/e2'> implying that aw = 0 and av = -2. Then, by (3.4) we get fiw - \/T and pv = \/Te2T. Hence, /?v < 0W. We give now the discrete version of Definition 2.

DEFINITION 7 A pair (#, F) is called power-stabilizable with respect to VV <—• V if there exists Fx e fi(W, W) such that <P + TF, is power-stable on VV and V. •

DEFINITION 8 A pair (C, $) is called power-detectable with respect to VV <-• V if there exists # , G 2{y, V) and i/2 € 2(y, VV) such that # + CHX and ^ + CH2 are both power-stable on VV and V. •

The following central result is reported from [6].

LEMMA 1 Let X denote either VV or V, and let E{SV{ •), B, C, D) be a smooth Pritch-ard-Salamon system with respect to W<-> V. Then the following statements hold.

1. If Sv(>) is exponentially stable on Vand VV, thentf> = SV(T) is power-stable on V and VV.

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 2 8 9

2. f ₎

3. Assume that (Sv( •), B) is admissibly stabilizable. Then there exists a sufficiently small sampling period To such that, for every T < To, the time-discretized pair ($, F) associated with (Sv( •), B) is power-stabilizable on V and W.

4. Let (SV(»),B) be admissibly stabilizable and let (<?,r) be the time-discretized pair associated with (Sv( •),£). If an admissible feedback operator makes the perturbed Co semigroup s£(«), generated by Av + BF, to be stable both on W and V, it ensures also the power stability of the perturbed time-discretized pair <2> + FF on both spaces. •

COROLLARY 1 Let S(S('),B, C,D) be a Pritchard-Salamon system such that

(C,S(»)) is admissibly detectable. Then there exists a sufficiently small sampling

period To > 0 such that, for every sampling step T smaller that To, the pair (C,<P) is power-detectable with respect to W ^ V . D

The following result is the generalization to Pritchard-Salamon systems of [10: Thm 1]. Since the extension is rather trivial, we shall only sketch the proof.

PROPOSITION 3 The Pritchard-Salamon system

is hybrid-stabilizable if (<2>, F2,C2), is power-stabilizable and power-detectable with respect to W <-» V.

Proof. Let U(K, L,M,N)bea stabilizing compensator for SJ-G^HJ-, an£I let Mi = 0-Let y2 = ST-_V2, and let u2 be the compensator output. The operator

is bounded from ^2(N,Km) ©^(N.R') to e2{N,Rp) ©^(N,Km). The power stabilizability and power detectability of (<P, F2, C2) implies that the operator

is bounded from ^2(N, Rm) © £2{N, Rp) to £2{N, W) and to £2{N, V) respectively. Let X

denote either W or V. Since

*(/) = S

(t - kT)x

+ f S

(t - T)B

dru

,

ikT

and if W| = max,6[Orj ||Svv(/)||w and u2 = ^ \\Sv(T)B2\\^dT, then it follows immediately that

IWOIU < ^ I I I - M R * + W2||M2^IIR" for kT ^ t < (k+l)T,

which implies that (v, w)>->x is a bounded operator from ^2(N,K/') ©£2(N,Rm) to L2(R+, X). It is sufficient to show that the operator

2 9 0 F. DAN BARB. V. 1ONESCU AND W. DE KONING

equation can be re-written at the sampling instants as

x

= Sx

+ r, u

+ r

u

, (3.6)

where uik stands for ii\{kT + •) f (0, 7"), with

W, f , H , . ^ [ S

(T

-Jo

(T

A : E!" -* W. A M , k = \ Sv( T - T)B2 dr u2 k

Jo

One can check immediately that M, I-> M, is bounded from L,(R+.£/|) to^,(N. L,(0, T; U\)). Indeed, writing the condition for Bx to be admissible on the time interval R+, we obtain that T, € £(L,(R+.W|), W). From the power stability of the solution to the state equation under the control u2 = GKy2, we get that it; "-*(J',!«2) 's bounded from £,(N.L,(0,r;W|)) to ^(N.R'eR"1). Finally, the boundedness of M, ^ v, follows from standard results. •

COROLLARY 2 Let

be a smooth exponentially stabilizable Pritchard-Salamon system. Then there exists a sufficiently small sampling step To such that, for every T < To, the Pritchard-Salamon system is hybrid-stabilizable. •

3.2. Lifting continuous-time systems

In this section, we give the main results on lifting a continuous-time periodic system to a discrete-time time-invariant one. The idea is to rearrange the original periodic system in such a way that its periodicity is reflected by shift invariance in the new set-up. We need first to define the lifting operator. Let Z be a Banach space, and let L2(K+,Z) be the space of square integrable Z-valued functions and £2(^>Z) the space of square summable Z-valued sequences. Notice that L2(K+,Z) and £2(N,Z) are Hilbert spaces with respect to the norms induced by the inner products. Let T > 0 be a fixed positive constant. Let

ft:L2oc(R+JZ)->S(N,L2(0,r;Z))I fK = C, (3.7) where S(N, Z) is the space of Z-valued sequences C, = (OOtei^ a r |d where

It is easy to show (see [1,2,33] for details of the proof) that Q is a linear bijective isometry from L ^ R + . Z ) to S(N,L2(0, T\Z)). Notice also that, if C(«) is a piece-wise constant function, say

C(0 = Ct f o r / € [ * r , ( A : + l ) r ) ,

C-DIGITAL CONTROL OK PRITCHARD-SALAMON SYSTEMS 2 9 1 Let us focus on state-space formulae for lifted systems. Consider the continuous-time system S(S( •). B, C. D) on a real separable Hilbert space X given by

(3.8a, b) where S(») is a strongly continuous semigroup generated by A. Let U and y be the real separable Hilbert spaces of the inputs and outputs respectively. The state equation can be reformulated as

J

for kT ^ / < (k +1)7". For 77 = kT + r, equation (3.8a) becomes

x{kT + t) = S(t)x(kT) + f S(t-T)Bu(kT + T)dT. Jo

At the sampling instants, the state equation (3.8a) becomes

xk+l == S{T)xk +\ S(T- T)Buk{T)dT. Jo Let A.X^X, A=S(T), [T (3-9) B:L2(0,T:U)^X. Buk=\ S(J - T)Buk{r)dT. Jo

where uk = u(kT + . ) f ( 0 , 7 " ) . F r o m (3.8b), we finally obtain

fa = Cxk + Duk = CS( • )xk + c\ S( • - T)Buk(r) dr + Duk. Jo Define now C :X^L2(0,T,y), Cxk = CS(.)xk,

r

r

()

D :L2(0,T;U) - L2(0, T-,y), Duk = C\ S{- - r)Buk(r)dT + Duk. Jo

Then the original system admits at the sampling instants the following representation

sx = Ax + Bit, y = Cx + Du,

where s is the forward shift operator, A, B, C, D are defined via (3.9)—(3.10), and where x € XN, u e S(N, L2(0, T;U)), and y 6 S(N,L2(0, T-,y)) respectively; here again the convention Ax = (Axk)k€fi etc. is used.

REMARK 4 1. Notice that the lifting operator (3.7) is defined on locally square-integrable 2-valued functions and its values are sequences of square-square-integrable Z-valued functions on [0,7"]. However, if we restrict to square-integrable Z-Z-valued functions then the lifted signal C is a square-summable sequence of square-integrable Z-valued functions on [0, T\.

2. Assume that, in (3.8a), the operator A is the infinitesimal generator of a stable Co semigroup on X. Then, for any u 6 L2(K+! U), the state is also square-integrable, i.e.

2 9 2 F. DAN BARB. V. IONESCU AND W. DE KONING

.Y 6 L2(R+, X), which implies y e L2(R+, y). Let w and y be the lifted image of u and y. It is routine to show that u e £2(N, L2(0, T;U)) and y 6 E2(N, L2(0, T; y)).

3. If we assume stability, then the input-output operator (2.7) of the linear system (3.8a) and (3.8b) is a well defined and bounded map from L2(R+ ,U) to L2(R+,y). It is well known (see [2]) that the lifted image of G is then

G = QGQ-{ € 2(<?2(N; L2(0, T;U)), <?2(N; L2(0, T; y))), and furthermore ||G|| = ||G||. D

3.3. Lifting the hybrid Pritchard-Salomon system

The theory developed in subsection 3.2 is now applied to lifting the hybrid Pritchard-Salamon system. Consider the smooth Pritchard-Pritchard-Salamon system

with respect to W1—»X<-^ V and satisfying (2.8). Assume that there exist two other real separable Hilbert spaces H] and y{ such that Bx e 2(UUV), B2 € 2{R"',V), Cx € 2{V,y\), Dn e 2(Rm,yl). As in [3,4] we shall restrict to the case when the system is smooth. Let us apply the lifting technique developed in subsection 3.2 under the assumption that u2 is obtained in a piecewise fashion from the discrete values x(kT) of the state, say u2{kT + T) = u2k (0 *S T < T). The state equation written at the sampling instants becomes

xk+i = Sxk + r , wu + r2uu (3.ii)

with

4-.V-+V, S = S

(T),

f, :L2(0,TjWO-W, f,filit4 f Sv(T-T)BlulJc(T)dT,

Jo

A : M

-» W, f

«

^ f 5

( T -

dr.

Jo

The equation for the controlled output is now

with

), (3.12a)

(3.12b) = C, f Sv(.-T)B2u2.kdT + Di2uu. (3.12c)

Jo

DIGITAL CONTROL OF PR1TCHARD-SALAMON SYSTEMS 293

system:

v,.* = C].xk (3.13a,b)

(3.13c) where ulJc, u2k, \'\.k, and v2Jt are respective klh elements of the sequences

u, G L2(0, T- u2 G (R1")"1, v2 6

REMARK 5 We have assumed u2 to be constant on [AT, (k + 1)7"), for k e N, so the

lifted image of w2( •) is the same as its sample sequence w2, and hence ii2 = u2 6 (R'")N. It follows that the expression of f2 simplifies to A = Jor SV(T)B2 dr. •

COROLLARY 3 The operators in (3.13a, b) satisfy the boundedness conditions

C,€ fi(W,L

(0,r;y,)),

Proof. The result for <? is immediate from the definition. By Proposition 3, r , G fi(L2(0,r;W,),W)), and the result A e fi(R'",W) is obtained by applying Lemma 1 to the pair (S(»),B2). This yields the results for Cu Du, and Dn

immediately from their definitions. •

REMARK 6 After lifting, the compensator's input and output spaces remain

unaltered, i.e. W and R"1. The lifting operator affects only the spaces where the new disturbance input u( and the new controlled output V| belong. They are now L2(0,r;Wi) and L2{0, T-,y{) respectively. Any finite-dimensionality of U\ and/or

yx is not preserved in the new problem, since the new spaces are

infinite-dimensional. •

The digital control system, being hybrid in its nature, becomes discrete-time after lifting. The resultant closed-loop system is then

where • + r2NC2 A M LC2 K . = [Ct+Di2NC2, DUM], DR = DU Ba =

r,

o

It is rather straightforward that, if XR = W © K, stands for the augmented state space

294 F. DAN BARB. V. 10NESCU AND W. DE KONING

satisfied:

A

e2(x

)

B

e2(L

(o,TM

).,x

),

C

Gfi(^

,L

(O,r;y,), D

e2(L

(0,T-M

).L

(0,T-y

)).

The following proposition establishes the equivalence between the hybrid stability of Pritchard-Salamon systems and the power stability of its lifted counterpart.

PROPOSITION 4 AR = r2NC2 F2M

LCi K is power-stable on XR = W © fC if and only if the Pritchard-Salamon system E\ £ ( • ) , \BX, B2},\ ' , " U ) is

hybrid-stable. V L Q J L 0 0 \) Proof, ("if) Suppose that AR is power-stable on XR. Then GK is a stabilizing

compensator for the lifted hybrid generalized plant

T L^S7-G2i^"' tfS7-G22H7-.fr1 J [Sj-Gz,/?"1 SrG22HrJ

(see [2] for details about lifting SrG, GHr, and SrG Hr operators). It follows that GK stabilizes S7-G22H7-. Then, by Proposition 3, we get that the Pritchard-Salamon

system

[ ^ ] , [° ^ j ) is hybrid-stable.

('only if) Let the Pritchard-Salamon system E[S('), [Bx, B2],\ ' , ' | )

( ) ( u 2 fi d

)

be hybrid-stable. It follows that the map (u\,v,w)i->(y\,u2,y2) is well defined and

bounded from

to

which implies that (ii\,v, w)>-» (y\,u2,y2) is well defined and bounded from

*

(N,L

(0, r;W,))©^(

to

This follows from the fact that the lifting operator is an isometry of spaces. We conclude that the discrete-time system (3.13a), closed under output feedback via the controller (2.10), is input-output-stable. But we know that, for a system that is power-stabilizable and power-detectable, input-output stability and power stability are equivalent in the sense that the transfer function is holomorphic and bounded outside the closed unit disc. The reader is referred to [23]. We conclude that the closed-loop system is internally stable, i.e. AR is power-stable on XR. •

REMARK 7 Notice that power stability of the closed-loop system on XR = W © K.

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 2 9 5

an open question whether simultaneous stability of the closed-loop system on both spaces ALR and V © K. is equivalent to hybrid stability. •

We return now to the 7-attenuation condition we have imposed for our digital control system. Let us notice first that the following holds.

PROPOSITION 5 Let

be the input-output operator mapping Wi to y\, and let f, :*2(N1L2(0,7';W1))-»*2(N,L2(0 be the input-output operator mapping ul to i>\. Then

Proof. Notice that the input-output operator mapping «, to y>\ can be expressed as a linear fractional transformation of the hybrid generalized plant, and the controller

G\\ G|2Hr 1 , OK

JJCJ^I o(j22ri7-J

and the input-output operator mapping i/| to V| can be expressed as a linear fractional transformation of the lifted hybrid generalized plant and the controller

\ [ SrG2 lr ? ' ST-G^HT-J )

Exploiting the fact that the lifting operator preserves the norms, we get II fill = ||^(G|i +Gl 2HrA:(/-SrG2 2H7-/f)-1S7-G2 1)/?-|||

= DC,, +Gl2HTK(I -STG22HTK)-lSTG2l\\ = \\Tt\\. D

We can conclude this section with the following result.

T H E O R E M 1 Let E(S(»), [BU B2],\ ' L l 2 J be a smooth P r i t c h a r d -Salamon system with respect to W"-+ <¥<-» V satisfying

X{l) - S{t)WX0 + f' SV(t - T)[BiUl(T) + B2U2{T)]dT, X0

Jo

yi(t) = ClX(t) + Dl2u2(t), y2(t) = C2x(t). Then a discrete-time controller S(K, L,M,N) satisfying

K£k + LVk, Ck = M£,k + Nr]k,

W,

solves the digital output-measurement-feedback H°° control problem if it is a solution to the discrete-time output-measurement-feedback H°° control problem for the lifted

2 9 6 F. DAN BARB, V. IONESCU AND W. DE KONING discrete-time system satisfying where «,,* ^ [ Sv(r-T)fi,«l i i t(T)dr, T2^ f 5V( r - r ) f i2d r , Jo Jo , *0 6 W, ,k - C\xk + DUUIJC

0,

= C, [ S

('-T)B

dr

Jo

Proof. Consider the disturbance-attenuation problem for the lifted system (3.14).

Since £{K, L, M, N) is its solution, we obtain the following results.

1. E(K,L,M,N) provides power stabilizability and power detectability for

{&,r2,C2) with respect t o W ^ V . By Proposition 3, £(K,L,M,N) then provides hybrid stability for the original Pritchard-Salamon system.

2. The input-output operator from ut to y{ is a 7-contracting map from e2(N,L2(O,T;U1)) to £2(N, L2(0, r;3>,)). It follows by Proposition 5 that the input-output operator mapping ux to yt is also a 7-contracting map from L2(R+)W,)toL2(R+;;y,).

We conclude that £(K,L,M,N) is then also a solution to the digital output-measurement-feedback control problem. •

Theorem 1 is the cornerstone result which facilitates our application of the discrete Popov-theory-based solution of the discrete-time H°° control problem, to solve the digital H°° control problem for Pritchard-Salamon systems.

4. General discrete Popov theory

Originally developed by Ionescu & Weiss [20] for finite-dimensional discrete-time systems, the main results on discrete Popov theory have been completely generalized to time-dependent discrete-time systems on Hilbert space in [16]. Among the various results of Popov theory, the one giving the link between a quadratic cost functional and the solution to the so-called Kalman-Szego-Popov-Yakubovitch (KSPY) system, strongly related to Riccati-equation theory, is (in our opinion) the most relevant. The central result proved in [19,16] replaced the Popov positivity condition with a more general one, expressed in terms of the invertibility of a certain operator.

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 2 9 7

An immediate application, probably the most important one, was to express the solution to the discrete-time H00 (7-suboptimal) control problem. This section is devoted to collecting the main results of discrete Popov theory developed in [20,16], which provide a basis for deriving the solution to the output-measurement-feedback discrete-time H°° (7-suboptimal) control problem.

Our formal model is described by the difference state equation

**+i = Axk + Buk (k G N), A-O = & (4.1)

where A € 2(X) and B e Q(U, X). We shall assume that A is power-stable on X and consequently, for every given (£, M) € X x £2(N,W), that equation (4.1) has a unique solution JC?1" € £2(N,X) given by

Xs-'" = $x + Lu, (4.2)

where £ : X -> £2(N, X) and L : e2{N,U) -* £2{N, X) are defined as

k-\ ($x)k = Akx, {Lu)k = Y^ Ak~'~lBuk, (Lu)0 = 0, 1=0 for k € N. Let Q L * R

with Q = Q* € £ ( * ) , L € fi(W, AT), and R = R* e fi(W), and call (^, B,iT) a />o/?ov //•/p/e on (A"^) (in this definition, A is not necessarily power-stable). Associate with ( 1. B, n) two objects, as follows.

1. The KSPY system

{

L + A*XB=H, (4.3) Q + A*XA- X = HG~lH*,

is considered as a set of simultaneous equations in X — X*, G, and H. We shall write

FGM±-G-]H*. (4.4)

2. Given an operator V on Z, we denote by V_ the operator on Zn defined by

Then the self-adjoint operator R : e2{N;U) -+ ^2(N;W) is defined by

R = R + LtL + LtL + L*QL. (4.5)

It is easy to check that (4.3) is equivalent to the discrete-time algebraic Riccati equation (DTARE)

A*XA -X - (A*XB + L){R + B*XB)~\L* + B*XA) + 0 = 0. (4.6) A triplet (X,G,H) is called a stabilizing solution to EKSpy (4.3) if G~l is bounded

and A + BFGH defines a power-stable evolution on X for Fc H defined via (4.4). Clearly, in this case, X is also a stabilizing solution to (4.6) and, as is well known, is unique.

2 9 8 F. DAN BARB, V. IONESCU AND W. DE KONING The main result of discrete Popov theory can be stated as follows.

THEOREM 2 Let (A, B, 77) be a Popov triplet, with A power-stable. Then the follow-ing statements are equivalent.

1. 7*~' is well defined and bounded.

2. The KSPY system (4.3) (or equivalent DTARE (4.6)) has a stabilizing solution.

Proof. See [16]. •

For more comments on the general framework of Theorem 2, the reader is referred to [16,17].

We have assumed that A is power-stable on X. Here we treat the situation when A does not necessarily satisfy this assumption. Let (A,B,II) be a Popov triple on

(X,U) with A not necessarily defining a power-stable evolution on X, and suppose

that (A,B) is power-stabilizable. Let Fe 2(X,U). We shall call (AF,BF,nF) the

F-equivalent of {A, B, 77) if .

[Q + LF + F*L* + FtRF L + F*R

AF = A + BF, BF = B, nF=\*

L L* + RF R

(4.7)

PROPOSITION 6 Let (A,B,IJ) be a discrete Popov triple, and F e 2(X,U) an arbitrary feedback operator. The stabilizing solution (X,G,H) to the KSPY system associated with 77 exists if and only if the stabilizing solution to the KSPY system associated with IJF exists. Moreover, the solution to the latter is (X, G,H + F*G). Proof. See [17]. •

Notice that X is also a solution to the discrete-time Riccati equation (4.6) for

(AF,BF,IIF). Hence, in order to remove the power-stability assumption made on A we proceed as follows.

Step 1. Choose F such that AF = A + BF is power-stable, and consider the F-equivalent Popov triple (AF, BF, I7F).

Step 2. Prove via Theorem 2 the existence of the solution to the KSPY system

associated with (AF, BF, IJF).

We conclude with two necessary results for the developments of the next section. PROPOSITION 7 Assume that (X,G,H) is a stabilizing solution to (4.3), where A is assumed to be power-stable. Then R given by (4.5) admits the factorization

R = S*GS.. (4.8)

where 5 : £2(N,W) -• £2(N,U) is the input-output operator associated with the system

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 2 9 9 Proof. See [16].

(A more general result is proved in [20]). The idea is to show that the associated quadratic forms on t2(H:U) are the same for the r.h.s. and for the l.h.s. in (4.8). •

An immediate consequence is the following.

COROLLARY 4 Assume that the K.SPY system associated with (A,B,II) has a stabilizing solution (X.G.H), and let (AF.BF,IIF) be its F-equivalent, where

F = FG H is given by (4.4). Then (4.8) becomes

GF = G = RF..

where GF and RF are associated with (AF. BF, 77F). •

5. Disturbance-attenuation problem for discrete-time systems

The 7-disturbance-attenuation problem for discrete-time systems has recently received a lot of attention. Starting with the leading work of Stoorvogel [27], Iglesias [19], and Ionescu & Weiss [21], where a two-Riccati-equation set of formulae has been written for the solution to the finite-dimensional H°° control problem, a complete generalization of the latter results, based on the extension of the Popov function theory developed in [20], has been obtained for time-dependent systems on Hilbert spaces by Halanay & Ionescu [17] and by Halanay et at. [16], respec-tively. However, the theory in [17,16] cannot be applied to solve the digital Hx (7-suboptimal) control problem for the Pritchard-Salamon class of infinite-dimensional systems. The motivation of this is the strong-epicity assumption made on the D2i term {D2\D*2i ^ vl). In our case, the 7-attenuation problem we have obtained

for the lifted hybrid Pritchard-Salamon system was essentially singular, having the £>2| t e r rn equal to zero (recall that this fact was a direct consequence of the fact that the sampler operator is not well defined over L2 spaces).

The purpose of this section is to give necessary and sufficient conditions for the existence of the solution to the 7-disturbance attenuation problem under the mildest possible assumptions on the initial discrete-time data. We also set the basis for writing the solution to the digital Hx (7-suboptimal) control problem for the

Pritchard-Salamon class of systems in terms of two Riccati-equation formulae for the Hx (7-suboptimal) control problem associated with the discrete-time system

obtained by lifting the hybrid Pritchard-Salamon system.

REMARK 8 (1) If a state-space feedback law is applied, i.e. if u2 is replaced by

u2 + F2x, then the system (2.11) is updated by keeping all the coefficients the same

except A and C\, which become A + B2F2 and C\ + DnF2, respectively. (2) If a

state-space injection is applied, then the system (2.11) is updated by keeping all the coefficients the same except A which becomes A + K2C2. •

Before we state and prove the necessary solvability conditions, we introduce the following definition.

300 F. DAN BARB. V. IONESCU AND W. DE KONING

DEFINITION 9 We say that the infinite-dimensional discrete-time system system

(2.11) is:

(1) right-coercive if (i) there exists F2 such that the infinite-dimensional

discrete-time system (2.11) updated with A <— A + B2F2 and C\ <— C\ + DnF2 is

power-stable and (ii) there exists 6]2 > 0 such that

("i, 7".2«>>* > M h l h2 for U | G £2(N,W,); (5.1) (2) left-coercive if (i) there exists K2 such that the infinite-dimensional discrete-time

system system (2.11) updated with A <— A + K2C2 is power-stable and (ii) there exists

62i > 0 such that

("2, T2]u2)y] > 62l\\u2\\22 for u2 e 12{H,U2), (5-2)

where Tv_ : £2(N,U\) -> ^ W ^ ) and r2, : ^2(N;W2)-• ^2(N,^i) are the input-output operators of the linear systems mapping u\ to y2 and w2 to _V|, respectively.

5.1 Necessary solvability conditions

In this section, we give the set of necessary conditions for the solvability of the 7-attenuation problem in terms of associated KSPY systems. The minimum set of assumptions we make on the initial data is:

(1) {A.B2) is power-stabilizable;

(2) (C2.A) is power-detectable;

(3) The infinite-dimensional discrete-time system (2.11) is right-coercive and left-coercive.

Let us define the Popov triplets {A,B2,n2)= [A,B2,\ R₂₂ = \A,B2, CtC, Ct DUC, DUD, (A,B,n)= [A,B, where B=[BUB2), \D*n~\ 2 and = C?[Du,Dl2],

ii

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 301 where C = C, C2 = Q2, = Bl[D*u,0], * 1 2 * 2 2 7?|, 0 0 0

Introduce also J =

-i

o l .

r-ij,

o

o J

a n d y =

[ o i*

The main result giving a necessary set of conditions for the solvability of the 7-attenuation problem is the next theorem, which depends on the following two lemmas for its proof.

LEMMA 2 Consider the Popov triplets Q L2

J * D

(A,B,n)= \A,B, Q L

L* R = \A[BUB2), \ R\

. L2

with A power-stable, and let 7} : £2(N,W, ®U2)

R\ R22.

Assume that there exists 6 > 0 such that

^ ^

Then the KSPY systems

V*JV = R + B*XB and ( WJV = L + A*XB WJW = O + A*XA-X (5.3) W2*V2 = L2 + A*X2B2 W* W2 = Q2 + A*X2A - X2

have stabilizing solutions. Proof. See [16]. D

LEMMA 3 Let Tu : £2{N,Uj) —» £2{N, 3^) be the linear bounded operator mapping Uj

to y{. Then the operator 7^ admits a representation of the form 7? = ^ 1 1 ^ 1 2

3 0 2 F. DAN BARB, V. IONESCU AND W. DE KONING

Proof. See [16]. •

THEOREM 3 If there exists a 7-attenuator (2.12) for (2.11), then the KSPY systems V*_ V2 = R22 + B\X2B2 \ ( v; V2 = C2 Y2C2

W; V2 = L2+ A*X2B2 \, \ W2* V2 = A Y2a

W2 W2 = Q2 + A*X2A -X2) [ W2 W2 = Q2 + A Y2A* - Y2

W*JV = L + A*XB I, i W*JV = L + AYC* -O + AYA*- Y have stabilizing solutions whose respective first components satisfy

X 2* X2 2s 0, Y > Y2 > 0,

with V=\ " I and V =

. Vv.\ [ 0 V22

Proof. Notice first that the problem of 7-disturbance attenuation is compatible with the discrete Popov function theory, i.e. Theorem 2 is applicable to finding the solution. As has been shown in [16], the first statement of Theorem 2 allows a signature for 7^. Let U = U\ xU2 and suppose that R has the form

Then R~l is well defined and bounded if

K22 sg> U, ~"11 T ^12^22 "12 - ^ ^

where ' » 0 ' means that the operator is coercive. This situation corresponds to the typical game situation

(

\

\W

max min 7 U «, € u, 11, e u, \ ' I u2 J /

encountered in [7]. We have assumed that the infinite dimensional system is right-coercive and left-right-coercive. This implies that Ti2 and T2] are coercive operators. But

the operator R22 written for (A,B,n) is exactly Tf2Tn, and its counterpart for

(A*,C*.n*) is T2\T2l, respectively, which is obtained by applying Lemma 3. Now

apply Lemma 2 to obtain the existence of the solutions to the KSPY systems (5.3) associated with the Popov triplet (A,B,II) and do the same for their counterparts associated with (A*. C*, 77*), respectively, and the proof is complete. •

5.2. Sufficient solvability conditions

In this section, we give a set of sufficient conditions for the existence of a solution to the 7-attenuation problem. Consider the equivalent system

S.Y = A0X + B0,M, + ^02^2!

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 303

where

Wx - V2V W2,

where P^ ^ I ' | is partitioned conformally with U = UX @U2, and consider also

the Popov triplet

10* L* (A* r* TT*\ - I A* C* \^° where 2o Qo — Lo = Boi [ ' - v2T -7 Vv, 0 0 ^ 022 0 0 • _J _ Introduce also J = \ ^

THEOREM 4 If there exists a 7-attenuator (2.12) for the system (2.11), then the KSPY system

has a stabilizing solution corresponding to the triple (Fo, Vo, Wo) with Yo ^ 0 and

^oll Von 0 ^o22. Proo/. See [16]. •

The following theorem is our main result on sufficient conditions for the existence of a solution to 7-attenuation problem.

THEOREM 5 Let Assumptions 1, 2, and 3 hold, and assume that the KSPY systems

R + B*XB \

L + A*XB \ and

Q + Q*XA -X]

have stabilizing solutions. Then the 7-disturbance-attenuation problem admits a

vjv;

Wojv; = L0 + A0 YOCO

304 F. DAN BARB, V. IONESCU AND W. DE KONING

solution, and a compensator which simultaneously achieves stability and 7-attenuation is given by (2.10) for

A — A — R ^n~L\C , A- (I A- K W .-, V~\C -A (5 4a ^ •^c — -^o uo 2 * ^ o l 2 l ^ o l ' \* ' " o l / o l 2 o22^o2J: v a/

Bc = Ko2 + {Bo2D-}2 + Ko]) Von Kol2, (5.4b)

Proo/. See [17]. •

5.5. The output-measurement-feedback solution to the digital control problem

The Popov-theory-based output-measurement-feedback solution to the digital H00

control problem can be now written in an explicit way. Assume that (Av, B2) is

exponentially stabilizable. It follows by Lemma 1 that the discretized pair (<£, F2) is

power-stabilizable for a sufficiently small sampling step. Hence, by Corollary 3, the original Pritchard-Salamon system is hybrid-stabilizable. Consider

c,

0 0

which is the system obtained by lifting the hybrid Pritchard-Salamon system

c

o o

(5.5)

(5.6) If <P is not power-stable, then choose an arbitrary power-stabilizing feedback, and obtain the F-equivalent of the lifted system which is power stable. According to

Theorem 1, finding a output-measurement-feedback solution to the digital H00

control problem is equivalent to finding an output-measurement-feedback solution to the discrete-time Hx control problem for the power-stable lifted system. Check

if (5.3) is right-coercive and left-coercive. If this is the case, then the K.SPY systems, written for the lifted system as

W2* V2 = L2+ £*X2f2

w* w

= Q

+ S*x

4 - x

V*JV = R + f*Xf

have stabilizing solutions such that X > X2 > 0, where W2 W2 = Q2 + &Y2<P* - Y2 V*JV = R+CYC* W*JV = L + SYC* *

-

c,

c

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 305

with V =

J = 0

0 Ir and J =

partitioned conformally with

. Then the stabilizing and 7-contracting compensator we are looking for is given by (5.4) updated with data provided by (5.5) respectively.

6. Shortcomings of the method

In this section we focus on the drawbacks of the Popov-theory-based solution to the H°° control problem as developed in this paper. Recall that, in Section 5.1, the minimum set of assumptions made on the discrete-time system was: (1) (A,B2) is

power-stabilizable; (2) (C2,A) is power-detectable; and (3) the infinite-dimensional

discrete-time system (2.11) is right-coercive and left-coercive. We would like to give a set of assumptions that, when made on the original data, i.e. on the Pritchard-Salamon system

\c

Hc

2

r

would guarantee that (1), (2), and (3) hold for the lifted system

c

Dl2

0

Notice that admissible or bounded stabilizability and admissible or bounded detect-ability of (5(»), B2, C2) and a sufficiently small chosen sampling step, T > 0, ensures

that (S,F2,C2) is power-stabilizable and power-detectable with respect to W^> V.

This is a natural constraint which is always imposed in digital control. Let us focus now on the third item regarding left and right coercivity of the discrete-time system

E(A,[BUB2],

0 0

PROPOSITION 8 Assume that (1) D\2Dn is coercive; (2) {n{2C\,A - B2D\2C\) is

power-detectable, where D\2 = (D\2Dn)~xD\2 and nn = I - DnD\2. Then

E[A,[BUB2],

o

Di2

0 is right-coercive.

Proof. The assumptions made in Proposition 8 are sufficient conditions for the existence of a positive semidefinite stabilizing solution to the DTARE with

Q = C;CU L = C;D12, R = D\2D12.

306 F. DAN BARB. V. IONESCU AND W. DE KONING

Then the DTARE can be rewritten as

A*XA - X + Q = 0, (6.1)

where A = A + B2F2 and Q - Q + F2L + L*F2RF2. A simple computation

performed on (6.1) gives

R + B*XB=f;2fn, (6.2)

where fu = C,(s/ - A)~l B2 + Dl2 and C]=C]+Dl2F2. Since R = D*l2Dl2 » 0

and X ^ 0, it follows from (6.2) that t\2fn » 0, which completes the proof. •

Let us see now what assumptions one should make on the original Pritchard-Salamon system such that (1) and (2) from Proposition 8 hold for the lifted system

C-, 0 0 Recall that r , « , . , ^ [ Sv(T-T)BlulJl(T)dT, r24 f SV{T-T)B2AT, Jo Jo ±Cl \' Jo Sv(,-T)BluLk(T)dT, = « - l

J>-

"2,A-From the expression of Du, one can easily notice that, if D]2DI2 » 0 for the

Pritchard-Salamon system, then D*\2D\2 » 0 for its lifted counterpart. Some

straightforward computation shows that, if (C|,S(«)) is considered admissibly or boundedly detectable, then {fl\2C\,<& — r2b\2C\) is power-detectable, where

D\2 = {D*nb\2)~xb\2 and fln = / - DnD\2 is power-detectable. Unfortunately

Proposition 8 does not apply for the dual problem, since D2X = 0 and hence

D2\D2\ is no longer coercive. The best we can prove is the following.

PROPOSITION 9 Let S[ S(>),[BUB2]A l\ , \ " l 2| ) be such that (S(»), # , ) is exactly controllable on [0, T] and C2C2 » 0. Then the lifted system

s[$,[f

r

],

c,

0 0

is left-coercive.

DIGITAL CONTROL OF PRITCHARD-SALAMON SYSTEMS 3 0 7

controllability gramian

f 2?,Z?rS*(r)dT (6.3)

V = f

Jo

is coercive [14: Thm 4.1.7]. But PT - t\f\, and hence f,/1,* » 0. Let us prove that this implies left coercivity of the lifted system. Indeed, let K2 be an output-stabilizing

injection. Then the updated system is now power-stable, and a realization for 77, = r , ( / - s r ) -1s C | i s

A- = <2>*s.v + Cjfj, r\ = £*, s.v (6.4a, b) with fi€e2(N:W), xee2(N:V), and r\ 6 £2(N;L2(0, T\U\)). Exploiting the coercivity of the controllability gramian (6.3) as well as the fact that C% C2 » 0, we get

UrfSA-lb^HSA-lb, ||C5/X||2 > S2IMI2 (6-5) for some &x > 0 and d)2 > 0. Hence from (6.4a) we obtain, by exploiting (6.5),

= \\x - #*s*||i ^ 2||.v||i

from which we obtain ^IHli ^ INli for S3 = (5,<52/\/2(l + ||^||2)l / 2. Thus ll^iMlb ^ ^311 A*! 12, which shows that 7y2r,2 » 0. D

REMARK 9 1. In general, it is hard to prove that a system is exactly controllable; indeed, large classes of partial differential and delay systems are not exactly trollable but only approximately controllable. Thus, checking the exact con-trollability of (S(»),B\) is not a sensible way of checking whether T2*\T2\ is

coercive. In particular, when U\ has finite rank, then (5(»),fl|) will not be exactly controllable.

2. Testing for the coercivity of F\T\ entails an investigation of the exact con-trollability of (S(»),Bi); this problem does not arise when using the coercivity of

C2C2 as a test of that of T2lT2]. By an adequate choice of the measurement

sensors, right coercivity can be achieved for large classes of systems.

3. If we restrict to the finite-dimensional case, then the exact controllability of (5(»),B|) collapses to a trivial coercivity of TiTi*—which i.e. the coercivity of PT

is generic—which (by a suitable choice of the design configuration) can also be achieved.

7. Conclusions and future research directions

In this paper, a discrete Popov-function-based solution to the digital H°°

7-suboptimal control of a Pritchard-Salamon system has been given. The set of necessary and sufficient conditions for the solvability of the problem has been expressed in terms of bounded invertibility of a certain Toeplitz-type operator

3 0 8 F. DAN BARB. V. IONESCU AND W. DE KONING

associated with the system, as well as in terms of the solvability of two Kalman-Szego-Popov-Yakubovitch systems. In our opinion, a natural continuation of this research would cover

• obtaining a finite-dimensional controller from the digital infinite-dimensional one obtained by the discrete Popov-function approach

• a case study on a PDE model which matches the framework of the Pritchard-Salamon class

• the relationship between the discrete-time controller obtained by the discrete Popov-theory approach and the one obtained by discretizing the continuous-time controller derived by van Keulen in [29]

• finding a reliable way of expressing the coercivity of T^ T2l in terms of the original data of the Pritchard-Salamon system.

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