Sampled-data minimum - Springer Link

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol, 87, No. 2, pp~ 235-267, NOVEMBER 1995

Sampled-Data Minimum H -Norm Regulation of Linear Continuous-Time Systems Using Multirate-Output Controllers K. G.

ARVANITIS 2 AND

P. N.

PARASKEVOPOULOS

3

Communicated by C. T. Leondes

Abstract. This paper deals with the problem of designing multirateoutput controllers for sampled-data H~-optimal control of linear continuous-time systems. Two formulations of the problem are studied. In the first, the intersample behavior of the disturbance and the controlled output signals is not considered, whereas in the second the continuoustime nature of these signals is taken into account. It is shown that, in both cases and under appropriate conditions, it is plausible to reduce the respective initial problem to an associated discrete-time H~-optimization problem for which a fictitious static state feedback controller is to be designed. This fact has a beneficial influence on the theoretical and numerical complexity of the problem, since only one algebraic Riccati equation is to be solved here, as compared to two algebraic Riccati equations needed in known techniques conceming the H~-optimization problem with dynamic measurement feedback. Key Words. Sampled-data systems, multivariable control systems, robust control, multirate controllers, control systems design, optimal control.

1. Introduction After its original formulation (Ref. 1), the H~~ problem has drawn great attention (Refs. 2-20). Several papers treat the problem in ~The work described in this paper has been partially funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research, and Technology and by the Heracles General Cement Company of Greece. 2Director of Technical Services, State Council of Greece, Greek Ministry of Justice, Athens, Greece. 3professor, Division of Computer Science, Department of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece. 235 0022-3239/95/1100-023550%50/0 9 1995 Plenum Publishing Corporation

236

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

the frequency domain (Refs. 2-4), whereas others treat it in state space (Refs. 5-16). For an extensive overview and comparison of frequency domain and state-space techniques concerning H%optimization, see Ref. 9. These papers use either static and dynamic state or dynamic output feedback controllers to solve the problem for a variety of systems, such as linear time-invariant continuous-time systems (Refs. 2-9, 13), linear time-invariant discrete-time systems (Refs. 10-12, 14), linear time-varying systems (Ref. 15), nonlinear systems (Refi 16), and sampled-data systems (Refs. 17-20). Further interesting related extensions of H%optimization theory cover problems such as maximum entropy/H%controller synthesis (Ref. 21), mixed H2/H~-design (Ref. 22), H~/LTR-design of linear systems (Ref. 23), etc. Digital controllers containing multirate sampling mechanisms are of particular interest. The results reported in Refs. 24-28 may be grouped in two general categories. The first category (Refs. 24, 25) involves a sampling mechanism in which the output samplers operate less often than input samplers; controllers designed along this line are usually called multirateinput controllers. The second category (Refs. 26-28) involves a multirate sampling mechanism, in which the input samplers operate less often than output samplers; controllers of this type are known as multirate-output controllers. Both categories have been applied successfully to solve many important control problems such as pole assignment (Refs. 24, 26, 27), exact model matching (Ref. 25), H2-optimal control (Ref. 26), loop-transfer recovery (Ref. 27), etc. In the present paper, we deal with the problem of designing multirateoutput controllers in order to solve the sampled-data minimum H~ regulation problem for linear continuous-time systems. Two formulations of the sampled-data H ~ optimal control problem are considered. The first formulation consists in designing a multirate-output controller, with a frame period To, to achieve closed-loop stability and H~%optimal performance between the external disturbance and the controlled output, only at the sampling instants kTo; we define this problem as the sampled-data H ~ control problem. The second formulation, called the intersample H%control problem, consists in designing multirate-output controllers to achieve closedloop stability and H~~ performance by treating the external disturbance and the controlled output as continuous-time signals, thus taking into account the intersample behavior of the system. In the past, H%optimal control theory has been applied in order to design digital controllers, following two different ways to approach the problem. The first consists in the use of a discrete-time system model in order to design a discrete controller (Refs. 10-12, 14). The second consists in synthesizing the controller in continuous time and in discretizing the control law (Ref. 17). Neither of these approaches consider the intersample behavior of

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

237

the continuous-time signals involved in the control loop. In the last few years, H~176 theory has been applied also to sampled-data systems either by using an H~ in order to approximate a continuous-time controller with a discrete controller (Ref. 18) or by calculating the performance of sampled-data systems for band-limited disturbances (Ref. 19). In a recent paper (Ref. 20), the sampled-data control of linear continuous-time systems with an H~ criterion is treated and standard dynamic discrete-time output feedback controllers are designed to achieve intersample H~ The technique used in Ref. 20 takes into account the intersample behavior of the continuous-time signals and reduces the original problem to an associated discrete-time H~ problem whose solution can be obtained using standard H~ techniques; see, for example, the techniques presented in Refs. 10, 14. Generally speaking, when the state vector is not available for feedback, the H~ control problem is usually solved, in both the continuous and the discrete-time cases, by the use of dynamic measurement feedback. This approach requires the solution of two algebraic Ricatti equations, which is in general a hard task. On the basis of these Riccati equations, it is plausible to compute a dynamic controller which achieves the desired requirements; see Ref. 14 for details. Nevertheless, a complete characterization of all controllers satisfying these requirements is not yet available. The present paper gives a new perspective and a simple way to overcome these difficulties; the technique presented to treat both sampled-data and intersample H~ problems reduces, under appropriate conditions, the two original problems to two associated discrete H ~ problems, for which fictitious static state feedback controllers are to be designed. This fact has a beneficial influence on the theoretical and numerical complexity of the problem, since only one algebraic Riccati equation is to be solved here, as compared to two algebraic Riccati equations needed by the technique of Ref. 14. Moreover, a characterization of all the multirate-output controllers satisfying the design requirements is outlined. These are the basic motivations for using multirate-output controllers for sampled-data H~176

2. Preliminaries and Problem Formulation

Consider the following continuous-time linear time-invariant system: 2(t) = Ax(t)+ Bu(t) + Dw(t), z(t) =Ex(t) + Jlu(t),

x(0) = 0,

y(t) = Cx(t) +JzU(t),

(la) (lb)

238

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

where x ( t ) ~ R " is the state, u ( t ) ~ R " is the input, w ( t ) ~ R d is the external disturbance, z(t)~ R r is the controlled output, and y ( t ) ~ R p is the measurement output. All the matrices in (1) have real entries and appropriate dimensions. To the system (1), we apply separately one of the following two multirate-output sampling mechanisms. The first multirate sampling mechanism is a generalization of the multirate-output sampling mechanism first proposed in Ref. 26. More precisely, we connect a sampler and a zeroth-order hold with period To (which can be selected as suggested in Ref. 28) to each plant input ui(t), i= 1. . . . . m, such that u(t)=u(kTo),

te[kTo, (k + 1)To),

(2a)

and we detect the ith disturbance w~(t), i= 1. . . . . d, and the ith controlled output zi(t), i= 1. . . . . r, at time kTo, such that w(t) = w(kTo),

te[kTo, (k + 1) T0),

z(kTo) = E~(kTo) + Jlu(kTo) ;

(2b) (2c)

the ith plant measured output y~(t), i= 1, 2 . . . . . p, is detected at every Tg, such that y,(~To + ~ Ti) = ci~(~To + ~ T,) + (J2)iu(lcTo), /~=0, 1. . . . . N ~ - I ,

(2d)

where we define by 4(" ) the discrete state vector obtained by sampling x(t). Note also that ci and (J2)i are the ith rows of the matrices C and J2, respectively; Ni~ Z § are the output multiplicities of the sampling, and T~ are the output sampling periods having the form T,.= To/Ni, i = 1, 2 . . . . . p. The second multirate sampling mechanism differs from the first in that the external disturbances and controlled outputs are not detected at kTo, but they are considered as continuous-time signals. Now, consider the dynamic output feedback control law of the form u[(k + 1) To] = Lu u(k To) - K~(kTo).

(3)

Controllers of this special form are well known as dynamic multirate-output controllers. Such controllers involve one of the multirate-output sampling mechanisms mentioned above, and their application to linear systems results in closed-loop systems having the two alternative configurations of Figs. 1

239

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

w(t) /

w(kTo)

z(t)

w(t)

/

z(kTo)

)

To

To PLANT

u[(k+l)T o] ~

u(kTo)~U(t)

y(t) )

I

MULTIRATE \

^

y (kTo)

SAMPLING

Fig. 1. Control strategy for sampled-data H~-control.

and 2. Note that the vector ~(kTo) is composed of the multirate sampled data of the output in the interval [kTo, (k+ 1)To) and has the form

"yl(kTo) y~[kTo+ (N~- 1)T~] ~(kro) =

:

yp(k ro) yp[kro + (Np- 1)TA In this paper, we study the problem of designing multirate-output controllers of the form (3), which solve the H~-optimal control problems, defined as follows. Sampled-Data H~ Problem. Find a multirate-output controller of the form (3), involving the first multirate sampling mechanism, which minimizes the 12-induced norm from the discretized external disturbance to the discretized controlled output and stabilizes the overall closedloop system. IntersampleH~ Problem. Find a multirate-output controller, involving the second multirate sampling mechanism, which minimizes the 1_2-induced norm for the continuous-time external disturbance to the continuous-time controlled output and stabilizes the overall closed-loop system.

240

JOTA: VOL. 87, NO. 2, N O V E M B E R 1995

z(t)

w(t)

}

PLANT [(k+ 1)To] ~ F Y ~

u(kT0) ~

y(t)

u(t)

)

, I' I'U"F--+

A

MULTIRATE SAMPLING

I Fig. 2.

K

'. y(kT0)

t

Control strategy for intersample H~-control.

3. Multirate-Output Controllers for Sampled-Data H%Control In this section, our aim is to design multirate-output controllers of the form (3) in order to achieve the following design requirement:

IlHw:(Z)II~ < Y,

(4)

for a specified ),e N+, where IIg(z)If ~o is the Ha-norm of the proper stable transfer function matrix H(z), which is defined as IIg(z) II~ = sup d[H(z)], Izl = 1

where d[H(z)] is the maximal singular value of the matrix H(z). We next investigate both the finite-time and the infinite-time horizon cases, as well as the case where disturbances are available for feedback. Concerning the sampled-data H ~ problem, we make the following assumptions: (a)

the matrix triplets (A, B, C) and (A, D, E) are stabilizable and observable;

(b)

rankIA

(c)

J~[E, J~] = [0, I];

0D]=n+d;

(5a)

JOTA: VOL. 87, NO. 2, NOVEMBER 1995 (d)

241

there is a sampling period To, such that the discrete-time system of the form

~(k+l)=(1)~(k)+Bu(k)+bw(k),

z(k)=E~(k)+Jlu(k), (5b)

is stabilizable and observable and does not have invariant zeros on the unit circle, where (I)=exp(ATo),

B=

[0

exp(A)OB d2,, D=

f)o

exp(A)OD d)c. (6)

Remark 3.1. Assumption (5a) can be formulated equivalently as follows. For the system (A, D, C), the following properties are fulfilled: (i) (ii) (iii)

d0.

(7)

242

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

have the following forms:

H=

A

clCA~Vi)-'

cIBIN, "~ ( J 2 ) l

ctA~ 1

clBll + (J2)l

ClD1NI : ^ ClDll

A

" cp(A;P) -I

,

~u =

,

w=

(8)

c;D;N~ A

Cp2;'

c;,;,

+ (s2);

c;O~,

m

where N *= ~i= 1 xi and where .4~= exp(A Ti),

Bij =

i = 1, 2 , . . . , p,

fo-iT'exp(A2)B

(9)

dA,

i = 1, 2 . . . . . p and j = 1, 2 . . . . , Ni,

(lOa)

Dij: f -jTi exp(A2)D d2,

:o i= 1, 2 . . . . . p and j = 1, 2 , . . . , N;.

(lOb)

The proof of Lemma 3.1 is given in Appendix A (Section 6). 3.2. Realization of a Static State Feedback via Multirate-Output We next focus our attention on the problem of equivalently realizing a desired static state feedback via multirate-output controllers9 To this end, observe first that the multirate-output controller of the form (3), when applied to the system (1), has the same action as a multirate-output controller of the form Controllers.

u[(k + 1) To] = L . u(kTo) + Lw w ( k To) - K ~ ( k To),

(11)

with Lw--O. We next focus our attention on the matrix [H, | We shall give the conditions under which this matrix has full column rank. To this end, we give the following definition of observability index vector9 Definition 3.1. For an observable matrix pair (A, C), with A E ~ "• C ~ R v• C = [c~-, c~, . . . . epT] T, a set o f p integers {nl, n 2 , . . . , np} is called an observability index vector of the pair (A, C) if the following relationships

JOTA: VOL. 87, NO. 2, N O V E M B E R 1995

243

hold: P

rank[c~-, . . . ,

~. ni=n,

(AT) nl -'clx , . . . , c ,p ,m ...

(AT) ~p-lc~]T = n.

i=1

We are now able to establish the following theorem. Theorem 3.1. Let (A, C) be an observable pair, and suppose that (5a) holds. Then, the matrix [H, | has full column rank for almost every sampiing period To if the output sampling multiplicities Ni are chosen such that Ni> o-~,

i = 1 , 2 . . . . . p,

where o'i are integers which form an observability index vector for the matrix pair (M, C*), with

The proof of Theorem 3.1 is given in Appendix B (Section 7). We give now the basic idea of equivalently realizing a desired state feedback, via multirate-output controllers of the form (3). This is done in the following theorem. Theorem 3.2. Let (A, C) be an observable pair, and suppose that (5a) holds. If the output multiplicities N~, i= 1, 2 . . . . . p, are selected such that Ni> r then for almost every period To we can make the control law (11), with Lw = 0, governing the behavior of a multirate-output controller, equivalent to any static state feedback control law of the form

u(kT0) = -F~(k~Co),

(13)

by choosing properly the multirate-output controller matrices K and Lu such that K[H, | Proofi yields

= [F, 0],

Lu = K|

(14)

Multiplying the basic multirate formula from the right by K

KH~[(k + 1) To] = K~(k To) - K|

u(kTo) - K|

w(kTo).

The control law (11), with Lw==-O[or equivalently the desired control law of the form (3)] is equivalent to the static state feedback control law (13),

244

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

if the matrices K and Lu satisfy the following relationships: K[H, |

= [F, Lw] = [F, 0],

Lu = K|

Indeed, with this choice of K and L,, we obtain F~[(k + 1) To] = K#(k To) - Lu u(kTo), or equivalently, -u[(k + l) To] = K#(kTo) - L~ u(kTo), which readily yields (3). From Theorem 3. l, there is always a matrix K which satisfies (14) for any given matrix F, if we choose Ni___o-i. This completes the proof of the theorem. [] 3.3. Finite-Time Horizon. On the basis of Theorems 3.1 and 3.2, it is clear that, in order to solve the sampled-data H~-control problem defined by (4) using multirate-output controllers of the form (3), one has to refer essentially to an easier problem, i.e., the design of a fictitious static state feedback control law of the form (13), which equivalently solves the same problem for the system (5b). We next investigate the finite-time horizon case. To this end, we formulate the following associated problem. Consider the discrete-time system ( k + 1) = (I)~(k) +Bu(k) +Dw(k), ~(0) =0,

z(k)=E~(k) +J,u(k),

k = 0 . . . . . N - 1.

(15a) (15b)

Find a static state feedback controller of the form (13), such that the design requirement (4) holds, where the norm of z(k) in/2[0, N - 1] is defined by N--1

Ilzll~ = 2

zT(k)z(k)

9

k=0

Standard procedures can be applied to solve this discrete-time problem in order to parametrize the set of static state feedback controller matrices F such that the closed-loop transfer function matrix satisfies the H~176 bound (4). In particular, applying the game-theoretic approach of Ref. 12, we obtain F(k) = ( I + BTPk+ ,B)-1BTpk+l(I), where Pk is the solution of the following difference Riccati equation: p~ = ErE+ .Vp~+ , . _ .Vpk+ ,/~(I + JRTB~+ l B ) - I j~Tek + l(I) + p k 3 r ( I + Dr ^T Pk Dr) ^ -1Dr ^v Pk ,

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

245

and where

b~=r-'b. Once such a matrix F(k*) is obtained for some k = k*, the multirate-output controller matrices K and L., which in the present case depend on k, can be computed according to the following theorem. Theorem 3.3. Let F(k*) be the solution of the aforementioned associated H~176 problem for some k = k*. Define A1 ~ [H, |

A,(k*) & [F(k*), 0],

and choose N; = nil _ oi. Then, the following relation holds: rank IH' . . . ] = rank(H0, LVff/r

(16)

)d

where

n, ~ A,|

v,(k*) ~ [(Z,)T, (Z,)J . . . . . (Z,)~m],

and 0.1)7, i= 1, 2 . . . . , m, is the ith row of the matrix Al(k*). The matrices K and L. are given by 41 K(k*)=So(k*)+ Y, &Silk*), (17a) j--1

Lu(k*) = So(k*)+ ~. psSj(k*) |

(17b)

j=l

where pj, j = 1, 2 . . . . .

~ , are arbitrary parameters, with ~ = rank(II1), and

So(k*)L(rl)mj,

q,

(ro)i, i= 1 , . . . , m, are constructed

by partitioning

(18)

vl(k*)HT[1-I~I-Ii-r+

ZTE1] -1 as follows:

v,(k*)n?tn,nT

+z,~Zl]-' = [(to),, (ro)2 . . . . .

(rO)m],

(19)

where ZI is a basis for Ker(rI 0. In (17), (ti)j is the ith row of the matrices T j e ~ , x N*, j = 1, 2 . . . . . 41, which are constructed by partitioning El as follows: Z1 = [T,, T2 . . . . . Tm].

(20)

246

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

Proof.

The equation

K(k*)[H, Owl [F(k*), 01, =

or equivalently the equation

K(k*)A, = A,(k*), is equivalent to the equation /~(k*)n, = v,(k*),

(21)

where

/~(k*) = [r~, r2 . . . . .

~:,.]

and tci, i= 1, 2 . . . . . m, is the ith row of the matrix K(k*). From the Kronecker-Capelli theorem, Eq. (21) has a solution with respect to/c(k*) [hence, the equation K(k*)A~ = A~(k*) has a solution with respect to K(k*)] iff the rank condition (16) is satisfied. In the case where Ne= nil ~ O'i, the relation (16) holds. Then, the general solution of (21) for/~(k*) is given by

~(k*) = v , ( k * ) n [ [ n , n,~ + z ( z , ] - ' + ( p , , p2 . . . . .

p~,)x,.

Partitioning vI(k*)FI[[H~FI[ +Z[Z~] -~ and Z~ as in (19) and (20), respectively, and taking into account (14), we obtain (17). [] 3.4. Infinite-Time Horizon. The results for the infinite-time horizon case can be obtained from the results of Section 3.3 by formally taking the limit of the number of stages N at infinity. This leads to the following result for the fictitious state feedback controller: F = (I + ~Vp~)-l~Vpq~,

(22)

where P is the solution of the following algebraic Riccati equation;

P=E TE+r

q:

P@-e

Y

PB(I+.B Y P.B)- 1.BY P@+ Pbr(I+D~,PD~,)^T ^ -,D~,^Tp.

The multirate-output controller matrices K and Lu, which in the present case are time-invariant, can be computed by adjusting the results of Section 3.3 to the case where F does not depend on k. In the infinite-time horizon case, if the multii-ate-output controller matrix Lu (corresponding to the transition matrix of the multirate-output controller itself) is desired to have a specified form, we can design the multirate-output controllers according to the following theorem.

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

247

Theorem 3.4. Suppose that F is given by (21) and that the matrix has a specified form Define

Lu,sp.

A2 -~ [H, |

|

Lu

A2 g [F, Lu,,e, 01.

Suppose also that there exists a selection na of the output multiplicities N,-, i = 1, 2 . . . . . p, such that the following relationship holds: rank [H;] = rank(FI2),

(23)

where

n : ~ A:|

v: =~ [(z:),~, ( z : ) L . . . , (,:)~1,

and (~.2)[, i = 1, 2 , . . . , m, is the ith row of the matrix A2. Then, the matrix K is given by {2

K=S~+

p'S*,

y~ j

where p*, j = 1, 2 , . . . , 42, are arbitrary parameters, with 32 = rank(H2), and

P'< =

:

,

(24)

s,'= I i/;

LU:)'J (r*),, i= 1. . . . . m, T

T

T

are --1

V2I-I2 [I-[21-I2 -t- Z2 ~"2] T

T

constructed

by

partitioning

the

vector

as follows :

v:n: i n : n : + z :Ts : l

--1

= [(4'),, ( 4 ' ) : , . . . , (r~)ml,

(t*)j

where 22 is a basis for Ker(II2). In (24), is the ith row of the matrices T*~ R~• ~', j = 1, 2 . . . . . 42, which are constructed by partitioning Z2 as follows:

z : = [ r L r~ . . . . . r*~]. Proof.

The proof of Theorem 3.4 is similar to that of Theorem 3.3. []

Remark 3.2. In general, it seems difficult to suggest directly a choice of the output multiplicities of the sampling Ni for which (23) holds. Nevertheless, in the special case where

J2=0'

rank[; B D]

(25)

248

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

a choice of the output multiplicities of the sampling Ni such that Theorem 3.4 holds is the following: N~> o-*,

i = 1 , . . . ,p,

where o-* are integers associated with the observability index vector of the matrix pair

o

I 001}

This is due to the fact that, with this choice of Ni, the matrix [11, Ou, | has full column rank, and in this case (23) always holds. The proof of this issue can be obtained similarly to the proof of Theorem 3.1. Theorem 3.4 gives us the possibility of designing either stable dynamic or static multirate-output controllers. To make this clear, we first point out that, in many cases, the matrix L, is desired to have its eigenvalues inside the unit circle, a fact that means directly that the multirate-output controllers (3), considered as a dynamical system, is stable. We also point out that, if the desired form of Lu is Lu,se= 0, then the multirate-output controller takes on the following form:

u[(k + 1) To] = -Kf(kTo),

(26)

which represents a static multirate-output controller. 3.5. Measured Disturbances. In the case where the disturbance vector is measured, it can be used for feedback and the multirate-output controller configuration is

u[ (k + 1) To] = L,, u(k To) + Lw w(k To) - K~(k To),

(27)

where Lw #0. The overall closed-loop system is depicted in Fig. 3. In this case, it is necessary to transform Theorems 3.1 and 3.2 into the following forms. Theorem 3.5. Let (A, C) be an observable pair. Then, the matrix H has full column rank for almost every sampling period To if the output multiplicities of the sampling Ni are chosen such that

Ni>_ni,

i = 1 , 2 . . . . . p,

where nieZ § form the observability index vector for the matrix pair (A, C). Theorem 3.6. Let (A, C) be an observable pair. If the output multiplicities N,., i= 1, 2 . . . . . p, are selected such that N,-> ni, then for almost every

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

w(t) /

w(kT~)

w(t)

,I

249

9(t) /

,(ITe)

)

To

PLANT y(t)

ut(k+l)rol ~ . _ _ u ( k X O) ]

MULTIRATE v(kTo~ SAMPLING ]

Fig. 3. Control strategyin the case of measureddisturbances. period To we can make the control law (27) equivalent to any static state feedback control law of the form u(kTo) = - F ~ ( k T o )

by choosing the multirate-output controller matrix triplet [K, Lu, Lw] such that K H = F,

L,, = KOu,

Lw = KOw.

The proofs of Theorems 3.5 and 3.6 are similar to those of Theorems 3.1 and 3.2, respectively. On the basis of Theorems 3.5 and 3.6, it is plausible to interpret both the finite-time and the infinite-horizon cases. The respective results follow closely the results of Sections 3.3 and 3.4, with the difference that, in the case of measured disturbances, for the design of a multirate-output controller so as to achieve the design requirement (4) and simultaneously stabilize the closed-loop system, it is not necessary for Assumption (5a) to hold. We also note that, in the case where the matrix Lu is desired to have a specified form, one can design the respective multirate-output controllers according to the following corollary, which is a direct extension of Theorem 3.4.

Corollary 3.1. Suppose that F is given by (21) and that the matrix Lu has a specified form Lu,~p. Define n~ a [H, 0~1,

A~ a [F, L~,~,].

250

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

Suppose also that there exists a selection n~3 of the output multiplicities N,., i = 1, 2 . . . . . p, such that the following relationship holds: rank [rIv~] = rank(II3),

(28)

where n 3 -~ a 3 |

v3 a [ ( ; h ) L (;t3)~ . . . .

, (z3).~,],

and (~3)[, i = 1, 2 . . . . . m, is the ith row of the matrix A3. Then, the matrix K is given by r

K=8o+ j=l E ~jGj,

where pj, j = 1, 2 . . . . , ~3, are arbitrary parameters, with ~3 = rank(H3), and

_ F,'~ /, G0:/!

Cz,=

;

(29)

LO o)q (F0)i, i = 1. . . . . m, T

T

T

v3H3 (1-I31I3 + E3 E3] v3n~(n3n~

are -1

constructed

by

partitioning

the

vector

as follows : + z ~ z 3 ] - ' = [(~o),, (70)2 . . . . .

(~o)m],

where Z3 is a basis for Ker(II3). In (29), (t'i)j is the ith row of the matrices ~e~r215 j = 1, 2 . . . . . ~3, which are constructed by partitioning 23 as follows: 23 = [ ~ , , ~2 . . . .

,

~m].

Remark 3.3. In general, it is difficult to suggest directly a choice of the output multiplicities of the sampling Ni for which (28) holds. Nevertheless, in the special case where

J2=O,

rank[;

B]=n+m'oj

(30)

a choice of the output multiplicities of the sampling Ni such that Corollary 3.1 holds is the following:

N~>_m~,

i= 1. . . . . p,

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

251

where mi are integers associated with the observability index vector of the pair

([o :lI 0 } This is due to the fact that, with this choice of N,., the matrix [H, | has full column rank, and in this case (28) always holds. The proof of this issue is given similarly to the proof of Theorem 3.1 in Appendix B (Section 7).

4. Multirate-Output Controllers for Intersample H~-Control In this section, our interest is focused on the design of multirate-output controllers of the form (3), which stabilize internally the overall closedloop system and reduce the U_2[0,oo)-induced norm of the map from the continuous-time disturbances w(t) to the continuous-time controlled outputs z(t) as far as possible. In other words, defining by • such a multirate-output controller, we wish to find the set of all stabilizing ~(, such that sup(tlz~ll~2: [Iwll~2~1) < r,

(31)

for some specified y ~ •+. To design such controllers, we propose a technique which is a combination of the results of the previous sections and those presented in Ref. 20. To this end, we derive an approximate discrete-time representation of the continuous-time plant under control.

4.1. Discrete-Time Representation. Observe that, at the sampling instants, the following relationships hold: ~[(k + 1) To] = ~ ( k T o ) + Bu(kTo) +

ff0

exp[A(To- ~)]Dw(kTo + ~) dA,,

y(kTo) = C~(kTo) + J2u(k To),

(32a) (32b)

and that, for 0 < t~< To, the controlled output signal is given by z(kTo + ~ ) = E~(kTo + ~ ) + Jlu(kTo).

(32c)

252

JOTA: VOL 87, NO. 2, NOVEMBER 1995

Relation (32c) may further be written as

z(kTo + 6 ) = E exp(A ~ ) ~(kTo) +

fo

E exp[A(8 -

~)lDw(kTo + ~) d~

+{fo~Eexp[A(~-;O]Bd~+J,}u(kTo). Let the linear operators 0 1 : R " --, a_~[O, ~ ) , 02: L~[O, ~ ) --, ~", 03:

--,

04: •" --* 0_~[0, m) be defined as (Ref. 20)

(01 x)( 5 ) = E exp(A ~ )x, O2v =

exp[A(To - ~,)]Dv(X) d~, ~0

({~3v)(~) =

E exp[A(~ - ~,)]Dv(~) d~,,

(04u)(8)={fo~ Also, let 5~ be the shift operator defined as

($~x)(t) = x ( t +

a),

and let the sequence of operators {Oztkr0,~k+l)T01}, k=O, 1 , . . . , be defined as the sequence of orthogonal projections from B_2[O,~) onto n_2[kTo, (k+ 1)T0), k = 0 , 1. . . . . According to the results reported in Ref. 20, the external disturbance and controlled output signals can be uniquely decomposed as

W= ~ P[kTo,(k+I)To]W, k=0

2= ~ P[kTo,(k+l)To]Z. k=0

JOTA: VOL. 87, NO. 2, NOVEMBER1995

253

Furthermore, the operators O~ and Oz have the following corresponding representations in terms of their Schmidt pair decompositions: ql

q2

o, = E ajCjoT,

o2= E

j=l

joj(ej,. );

j=l

here, q~ and q2 are the rank of O~ and 0 2 ; aj and ]~j a r e the singular values of O1 and O2; 7/j~ gO"and Cj~ n_~[0, To) are orthonormal right and left singular vectors of the operator O1; 0j~ ~" and ej~ kza[0, To) are orthonormal left and right singular vectors of the operator 02. The singular values and the right singular vectors of the operator O1 are given by the following matrix eigenvalue problem:

R(A,E,

To)rlj=a~rlj,

j = l , 2 . . . . . ql,

where

R(A, E, To)=foT~exp(AV~)EVE exp(As d2. The left singular vectors of O1 are given by

rj(5)=a~lEexp(Af)rb,

j = l , 2 . . . . ,q,.

The singular values and the left singular vectors of the operator by the following matrix eigenvalue problem:

W(A, D, T0)0j = fl} 0j,

0 2 are

given

j = 1, 2 . . . . . q2,

where

W(A, D, To) =

foT~exp(A,l)DD v exp(AVX) d&.

The right singular vectors of 02 are given by ej(5) = fl)-I Dr exp[AV(T0 - 8)]Oj,

j= 1, 2 , . . . , q2.

Now, let ~ / b e the closed subspace of n_2[0, ~ ) , which is defined as

W={w~Ild[0, ~):w = ~ k=0

where q2

A=V Ej, j=l

P[,~ro,(~+,)ro]wandSkroP[kro,(k+,)ro]W~l~},

254

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

and where Vq2_-1 Ej denotes the linear span of the set {0j}. Also, let 7/be the closed subspace of L2[0, o0) defined as 7/= {zell_~[0, oo): z= ~ lPtkro,(k+1)rolz and $kroPtkro,(k+ i)rolZ~ 1/3}, k=0

where B is the finite-dimensional subspace of 0_2(0, To), defined as ql

q2

B= V ~:+o3 V ,j+ o,~ m j=l

j=l

Next, we focus our attention to the case where wsW and z~Z. Denoting by qw and q~. the dimensions of A and B, one can construct orthonormal bases { Vtj} and {Zj} for A and B, following the procedure proposed in Ref. 20, so as to expand uniquely a signal w s W and a signal z~Y_ as follows: q2 ~, qz w= S(-kro) ~ w:k~:, * z= ~(-kro) Y~ Zjk * Zj ," (33) k=o

:=~

~=o

:=l

here, the sequences {w*} and {z~'}, with w~ - ( w ~ , w*~ . . . . , w * ~ ) w ~ Rq.,, : = (z*~, ~*~ . . . . .

z * : ) z ~ Rq:,

are in l qwand I q--. Note that l~ denotes the Hilbert space of sequences r * = ( r o * , r ~ . . . . ),

r * ~ R ~,

with inner product defined as (p*, v*)= ~ (p*)Tr*,

with p*, r*el~.

k=0

disturbances w in the set W, the sampled-data system described by (32) has a discrete-time representation of the following form: For

~[(k+l)To]=~(kTo)+Bu(kTo)+Rw~,

~(0) =0,

(34a)

z* = Q~(kTo) + Wu(kTo) + Nw~,,

(34b)

y(kTo) = C~(kTo) + J2u(kTo),

(34c)

where the matrices R ~ n•

Q~lt~qz• ~

R = [•101,

p202,...,

Pq2Oq2],

Q = [air/l,

0~2q2, . . 9 ,

Otqlqqi, 0 , . . . ,

qzxm, N E ~ qz•

0],

are defined as

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

255

qJ = [Xj, O3 ~'k], N = [Zj, O4ek], where e, is the kth unit vector in ~m. 4.2. Basic Multirate Formula in the Intersample Case. In what follows, a formula analogous to that given in (7), for the case where w~W, will be outlined. To this end, we start by observing that

Yl (k To + 12Ti) = c, exp(A12 T~)~(k To)

f F

+ ~ce Jo

+ci

}

exp(A,~)B d~,+ (J2)i u(kTo) exp[A (12T,-- ~)]Dw(kTo + ,~) d~,,

(35)

where cl and (J2),, i = 1, 2 . . . . . p, are the ith rows of the matrices C and J2, respectively. Furthermore, relation (32a) yields

~(kTo)=exp(-ATo){~[(k + 1)To]- for~

d~,u(kTo)

- for~

(36)

in deriving (36), use was made of (6). Introducing (36) into (35) yields y; (k To + 12Ti) = ce exp [A (12T~- To)]~ [(k + 1) To]

+fc,ffT'-T~ -c,- exp[A (12T~- To)]

+c~

f?

If ~

exp[A(To - ~,)]Dw(kTo + X) d~

exp[A (12Ti - ~,)lDw(kro + ~) dL

(37)

Relation (37) may further be written as

Yi (k Zo + 12Zi) = Ci (.~N/-/.t)-I ~[(k + 11To] + {c, Bi,N,-,,, + ( J2l,}u(k To) .~._Ci (

~. -(AiANi--lt ) --1 R+Guri}Wk,

(38)

256

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

in deriving (38), use was made of (33a). In (38), .~,-is given by (9), B;,N,-, is given by (10a) f o r j = N i - p , and G~r,~R n• has the following form: GuT, = [(GuT)I, (Gut,)2 .... , (GuT,)qj.

Note that (G~T,):= W(A, D, uT,)(~I'~)N'-uflf ~ 0:,

where W(A, D, p T~)~ ~"• W(A, D, p T~) =

j = I, 2 ..... qw,

is the positive-semidefinite matrix

f?

exp(A~)DD r exp(AV,~) d~

and

G0=G~rJ:0=0. Next, expressing Eqs. (38), for i= 1, 2 , . . . , p and p = 0, 1. . . . . N i - 1, in vector-matrix form, we obtain the following relationship: ~( k To) = n~[ (k + 1)T0] + |

k To) + |

w~,

(39)

or equivalently, H~[(k + 1) To] = r

- | u(kTo) - 0 " w*,

k=0, 1 , . . . ,

(40)

where the matrices H ~ N*• and O u ~ N*• have the same forms as in relation (8) and the matrix | :g ~ RN * X qw has the following form: O* = - H R + Gw.

The matrix Gw is given by

F~ 1

(G~)/= cG i T~.

L(a),J

,

i = 1. . . . . p.

Lc,

Relation (40) is the basic multirate formula sought for the output sampiing mechanism in the case where weW. In order to design multirateoutput controllers of the form (3) such that relation (31) is satisfied, we generalize the procedure presented in Section 3. To this end, instead of the problem of designing multirate-output controllers of the form (3), consider again the equivalent problem of designing multirate-output controllers of

J O T A : VOL. 87, NO. 2, N O V E M B E R 1995

257

the form

u[(k+l)To]=Luu(kTo)+L*w'~-K~(kTo),

with L* =0.

(41)

Now, let HK,w~z~be the discrete transfer function matrix from w~ to z~' of the closed-loop system (34). Its Hoo-norm is given by (Ref. 20)

lln~,wZ.-Z [I -- sup(llz~li2: IIw~ll2-< 1, {Uk} given by (3)), or equivalently, * * =sup( IIz~ll2: Ilwll2---1, w~W). IIH ~,w,:kll

An approximate solution to the problem of designing stabilizing multirate-output controllers of the form (3) so as to achieve the performance bound (31) can now be found by considering the problem of designing stabilizing multirate-output controllers of the form (41), such that IIHK,wZ:: II< ~*.

(42)

Remark 4.1. From the above analysis, it is clear that the only approximation involved, when problem (31) is replaced by problem (42), is due to the restriction of the compact integral operator (}3 to the subspace A. Note however that, according to the results in Ref. 20, one can arbitrarily ameliorate the accuracy of the approximations (33) and (38), by substituting the subspaces A and B with the subspaces A O"i. To this end, dropping appropriate rows of the matrix E[H, | exp{MTo}, we obtain the following square matrix: 'c*

~=

c* exp{MTl - I} a'-I :

c* c* exp{MTp- I} ap-' The matrix f~ has the same rank as the matrix

c* (c,/T, o'1--1 ) exp{MT,-I} ~ f2*=

:

(c*/T;p-') exp(MTp - I} a'-' As the frame sampling period tends to zero, the matrix f~* tends to the following matrix: "c* c~ M a l - 1

~=

:

r C~ M ap - 1

and the determinant of fl* tends to the determinant of matrix ~, which takes a nonzero value, due to the assumption that the o'i formulate an observability index vector of the pair (12). This means that the determinant of the matrix (55) takes also a non-zero value for sufficiently small values of To. Due to the fact that the determinant of the matrix (56) is an analytic function of To, we conclude that this determinant takes nonzero values for

266

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

almost all sampling periods To. Consequently, the matrix [H, | column rank for almost all sampling periods To.

has full []

References

I. ZAMES,G., Feedback and Optimal Sensitivity: Model Reference Transformat&ns, Multiplicative Seminorms, and Approximate Inverses, IEEE Transactions on Automatic Control, Vol. 26, pp. 301-320, 1981. 2. CHU, C., DOYLE, J. C., and LEE, E. B., The General Distance Problem in Ho~Optimal Control Theory, International Journal of Control, Vol. 44, pp. 565-596, 1986. 3. FRANCIS, B. A., and DOYLE, J. C., Linear Control Theory with an H~-Optimality Criterion, SIAM Journal on Control and Optimization, Vol. 25, pp. 815-844, 1987. 4. KIMURA, H., Conjugation, Interpolation, and Model Matching in H ~ International Journal of Control, Vol. 49, pp. 269-307, 1989. 5. PETERSEN, I. R., Disturbance Attenuation and Hoo-Optimization: A Design Method Based on the Algebraic Riccati Equation, IEEE Transactions on Automatic Control, Vol. 32, pp. 427-429, 1987. 6. Znou, K., and KHARGONEKAR,P. P., An Algebraic Riccati Approach to H~Optimization, Systems and Control Letters, Vol. 1I, pp. 85-91, 1988. 7. DOYLE, J. C., and GLOVER, K., State Space Formulas for all Stabilizing Controllers that Satisfy an Ho~-Norm Bound and Relations to Risk Sensitivity, Systems and Control Letters, Vol. 11, pp. 167-172, 1988. 8. BERNSTEIN,D. S., and HADDAD, W. M., LQG Control with an Hoo-Performance Bound: A Riccati Equation Approach, IEEE Transactions on Automatic Control, Vol. 34, pp. 293-305, 1989. 9. DOYLE, J. C., GLOVER, K., KHARGONEKAR,P. P., and FRANCIS, B. A., State Space Solutions to Standard He and Ho~-Control Problems, IEEE Transactions on Automatic Control, Vol. 34, pp. 831-847, 1989. 10. Gu, D. W., TSAI, M. C., O'YouNG, S. D., and POSTELTHWAITE,I., State Space Formulas for Discrete Time H~-Optimization, International Journal of Control, Vol. 49, pp. 1683-1723, 1989. 11. BASAR,T., A Dynamic Games Approach to Controller Design : Disturbance Rejection in Discrete Time, Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1, pp. 407-414, 1989. 12. YEASH, I., and SHAKED, U., Minimum Hoo-Norm Regulation of Linear DiscreteTime Systems and Its Relation to Linear-Quadratic Discrete Games, IEEE Transactions on Automatic Control, Vol. 35, pp. 1061-1064, 1990. 13. SCHERER,C., Hoo-Control by State Feedback and Fast Algorithms for the Computation of Optimal Ho~-Norms, IEEE Transactions on Automatic Control, Vol. 35, pp. 1090-1099, 1990. 14. STOORVOGEL,A. A., The Discrete Time Hoo-Control Problem with Measurement Feedback, SIAM Journal of Control and Optimization, Vol. 30, pp. 182-202, 1992.

JOTA: VOL. 87, NO. 2, NOVEMBER 1995

267

15. LIMEBEER,D. J. N., ANDERSON, B. D. O., KHARGONEKAR,P. P., and GREEN, M., A Game-Theoretic Approach to H~-Control of Time- Varying Systems, SIAM Journal on Control and Optimization, Vol. 30, pp. 262-283, 1992. 16. VAN DER SCHAFT,A. J., Le-Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H~-Control, IEEE Transactions on Automatic Control, Vol. 37, pp. 770-784, 1992. 17. SAFONOV, M. G., CHIANG, R. Y., and FLASHNER, H., H~-Robust Control Synthesisfor a Large Space Structure, Proceedings of the 1988 American Control Conference, Atlanta, Georgia, Vol. 3, pp. 2038-2045, 1988. 18. KELLER, J. P., and ANDERSON, B. D. O., A New Approach to the Discretization of Continuous-Time Controllers, Proceedings of the 1990 American Control Conference, San Diego, California, Vol. 2, pp. 1127-1132, 1990. 19. LEUNG, G. M. n., PERRY, T. P., and FRANCIS, B. A., Performance Analysis of Sampled-Data Control Systems, Automatica, Vol. 27, pp. 699-704, 1991. 20. TOIVONEN, H. T., Sampled-Data Control of Continuous-Time Systems with an H~-Optimality Criterion, Automatica, Vol. 28, pp. 45-54, 1992. 21. GLOVER, K., and MUSTAFA, D., Derivation of the Maximum Entropy Ho~Controller and a State Space Formulafor Its Entropy, International Journal of Control, Vol. 50, pp. 899-916, 1989. 22. KHARGONEKAR,P. P., and ROTEA, M. A., Mixed H2/H~-Control: A Convex Optimization Approach, IEEE Transactions on Automatic Control, Vol. 36, pp. 824-837, 1991. 23. SAEKI,M., H~ with Specified Degree of Recovery, Automatica, Vol. 28, pp. 509-517, 1992. 24. ARAKI, M., and HAGIWARA, T., Pole Assignment by Multirate Sampled-Data Output Feedback, International Journal of Control, Vol. 44, pp. 1661-1673, 1986. 25. ARVANITIS, K. G., and PARASKEVOPOULOS,P. N., Exact Model Matching of Linear Systems Using Multirate Digital Controllers, Proceedings of the 2nd European Control Conference, Groningen, The Netherlands, Vol. 3, pp. 1648-1652, 1993. 26. HAGIWARA, Z., and ARAKI, M., Design of a Stable State Feedback Controller Based on the Multirate Sampling of the Plant Output, IEEE Transactions on Automatic Control, Vol. 33, pp. 812-819, 1988. 27. HAGIWARA, T., FUJIMURA, T., and ARAKI, M., Generalized Multirate-Output Controllers, International Journal of Control, Vol. 52, pp. 597-612, 1990. 28. ER, M. J., and ANDERSON, B. D. O., Practical Issues in Multirate-Output Controllers, International Journal of Control, Vol. 53, pp. 1005-1020, 1991. 29. DAVISON, E. J., and WANG, S. H., Properties and Calculation of Transmission Zeros of Linear Multivariable Systems, Automatica, Vol. 10, pp. 643-658, 1974. 30. McFARLANE, A. G. J., and KARCANIAS, N., Poles and Zeros of Linear Multivariable Systems: A Survey of the Algebraic, Geometric, and Complex-Variable Theory, International Journal of Control, Vol. 24, pp. 33-74, 1976. 31. GANTMACHER,F. R., The Theory of Matrices, Chelsea, New York, New York, 1959.