IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 7, JULY 1994 1447

VI. CONCLUDING REMARKS In a related paper 161, we established necessary and sufficient

conditions for the Hurwitz stability and the Schur stability of real interval matrices, and we used these results to develop an algorithm which terminates in a finite number of steps to determine the Hurwitz and Schur stability properties of interval matrices. Using ideas which are very similar to those employed in [6], we can utilize the present results (Theorems 2a and 2b) as the basis of an algorithm to determine the controllability and observability of linear time invariant systems with interval plants. Because of space limitations, we give only an outline of this algorithm, and we state, without proof, a result for the termination of the algorithm in a finite number of steps. Further details concerning this algorithm are provided in [7] and in the related work cited above [6] (dealing with Schur and Hurwitz stability of interval matrices).

Algorithm Outline: For any given interval matrix [A” , A M ] , we first determine the controllability of system (SI) (the observability of system (SO)) for A = A I = ( 1 / 2 ) ( A m + A M ) . If ( A I , b ) is not con- trollable (if (AI, c) is not observable), then we verify if Assumption la (if Assumption lb) is satisfied for A I , A A l = ( 1 / 2 ) ( A M - A “ ) and b (and c). If Assumption l a (Assumption lb) is satisfied, then the algorithm terminates with the result that (SI) is controllable (that (SO) is observable) for all A E [A”, A M ] . Otherwise, we divide the interval [A”’, A M ] into two equal subintervals and repeat the above process for each subinterval. The algorithm continues unless system (SI) is controllable (unless system (SO) is observable) for all A in each subinterval of [A“, A M ] or until at least some matrix, say A‘ E [A”, A M ] , is found such that system (SI) is not controllable (that system (SO) is not observable) for A = A’, in the manner described above. 0

In [7], we prove the following result. Proposition: The above algorithm will terminate in a finite number

of steps if system (SI) is controllable (if system (SO) is observable) for all A E [A” , A M ] .

REFERENCES

K. Balachandran and J. P. Dauer, “Controllability of perturbed nonlinear delay systems,” IEEE Trans. Automat. Conrr., vol. 32, pp. 172-174, 1987. B. R. Barmish, “Gobal and point controllability of uncertain dynamical systems,” IEEE Trans. Automat. Contr., vol. 27, pp. 399408, 1982. M. Mansour, “Robust stability of interval matrices,” in Proc. 28th IEEE Con5 Decision and Conrr., pp. 46-51, Tampa, FL, 1989. “Control systems society index,” IEEE Tram. Automat. Conrr., vol. 37, no. 8, part II, pp. 144-147, 1992. S. S . Sastry and C. A. Desoer, ‘The robustness of controllability and observability of linear time-varying systems,” IEEE Trans. Automat. Contr., vol. 27, pp. 933-939, 1992. K. Wang, A. N. Michel, and D. Liu, “Necessary and sufficient conditions for the Hunvitz and Schur stability of interval matrices,” IEEE Trans. Automat. Contr., to appear. K. Wang and A. N. Michel, “An algorithm for the controllability and observability of a class of linear, time-invariant systems with interval plants,” submitted for publication.

A Computational Scheme for Ha Model Reduction

Davut Kavranoglu

Absfruct-In this paper, a computational scheme for the HOC, model reduction problem is introduced. The scheme is illustrated by three examples. The algorithm is based on the recent characterization of the solution to the H- model reduction problem.

I. INTRODUCTION

In control system analysis and design, it is desirable to have an accurate model with a small number of modes. In a plant with a small number of modes, less computation are required for simulation, controller design will be less involved, the resulting controller will have a lower degree, etc.

Dejinition I-H, Model Reduction Problem: Given a p x m transfer function Gn(s) E R H , with McMillan degree n, find a p x m transfer function G,(s) E RH- with McMillan degree T < n such that llGn - Grllm is minimized.

Define 70: = minG,(a)ERH, llGn - G,II,G,(s) with degree r. Remark I: In this paper, we will assume Gn(m) = 0 with no

loss of generality. Otherwise, one first computes $e approximation of (Gn(s) - Gn(m)) , say G,(s), then G,(s) = Gr(s) + Gn(m) .

To the best of this author’s knowledge, the first results on the subject were reported in [ l ] and [2]. In [l], a characterization of the solution for the zeroth order H , approximation problem ( r = 0) and a computational scheme is reported. In [2], a characterization of the solution for the H , model reduction problem is reported, but no explicit computational scheme was given.

In this paper a suboptimal computational procedure for the general multivariable continuous-time rth order optimal H , model reduction problem will be developed. This procedure is based on the complete characterization of the solution to the problem reported in [2].

In Section II, the necessary background material is given. In Section 111, a computational procedure is developed for the H , model reduction problem. In Section IV three examples are given.

11. PRELIMINARIES In this section we summarize results needed in the sequel. An m x m transfer function G(s) E H , is called y-inner if

G”(s)G(s) = G(s)G“(s) = y z I . (1)

Dejinition 2: Given G(s) E R H , with a minimal realization

then ut = [X,(L,L,)]? (in decreasing order) are called the Hankel singular values (HSV) of the system G(s), where L, and Lo are defined by

AL, + L,AT + BBT = 0 and A T L , + L O A + CTC = 0.

The following is a well-known lemma (see, for example, [3]).

Manuscript received April 22, 1993; revised July 8, 1993. The author is with the Department of Systems Engineering, King Fahd

University of Petroleum and Minerals, Dhahran, Saudi Arabia. IEEE Log Number 9400372.

0018-9286/94$04.00 0 1994 IEEE

1448 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 7, JULY 1994

Lemma 1: Given an m x m, nth order transfer function G,(s,, let G,(s) be an rth order transfer function. We then have: U,+I 5 llG, - G.ll,, where u,+1 is the r + 1st HSV of GR(s).

Definition 3: A state-space realization is balanced [4] if L , = Lo = E, where C:= diag(a1, u2, u3,...,un) . The following lemma is a corollary to Theorem 6.3 in [ 3 ] .

Lemma 2: Let [XI be a minimal, balanced realization of

an m x m transfer function G ( s ) with C = diag (Cl , u I R p r ) , where U is the smallest HSV.

Partition (A, B , C ) conformably with C, as

1) See (3) at the bottom of the page. It is stable and has exactly r modes (n - r is the multiplicity of the smallest Hankel singular value), where I: is a unitary matrix satisfying L32 = -C,'U and r:= E:: - 21,. > 0.

2) G ( s ) - G,(s) is u-inner.

A. A Solution for the H , Model Reduction Problem In [2], a characterization for the H , model reduction problem has

been reported. In this paper using this characterization a computa- tional scheme is developed.

The following theorem characterizes the H , model reduction problem as a Hankel norm approximation problem through an imbed- ding process [2].

Theorem 1: Given an nth order m x m transfer function G, (s)

with a minimal realization G,(s) = [#I, any rth order

G,(s) E R H , satisfying

(4) IIG, - Grllm I: Y

for any y 2. yo, is characterized in the following steps. 1) Calculate Bo, CO such that

Amin ( P Q ) = y2, with multiplicity n - r ( 5 )

where

A ~ Q + QA + cTc + c, '~, = 0. (7)

2) Using Lemma 2, calculate H,(s) , which is the rth order Hankel approximation of

3) G,(s) = ( I 0 ) H r ( s ) ( ; ) , G,(s) is a solution for (4). Remark 2: The above result was stated, without loss of generality,

for the square case. The nonsquare case can be treated by adding an appropriate number of zeros to G ( s ) to make it a square transfer function [2 ] .

B. Computation of the Zeroth Order Approximation The zeroth order H , approximation problem is easier than the

7th order case. In this section we summarize the results reported in [ 11 on computing the zeroth order approximation. The results on the zeroth order approximation are needed in the development of the computational scheme for the rth order case.

A conceptual algorithm, which follows from Theorem I, would go V#] as follows: Given a minimal realization G ( s ) =

1) Find Bo, CO such that all of HSV of

H ( s ) = [E] are equal to 7 , i.e., PQ = 7'1.

DH = y :ii 2) Calculate a balanced realization and use Lemma 2 to obtain

such that ( H ( s ) + D H ) is y-inner. 3) Fina 0 ly DO = 1U11.

In general, a computational procedure to calculate the imbedding for the optimal y is not yet available, but in some special cases, optimal solutions have been obtained [5]. With the well-known definition At: = ( c 2 1 + ATA)-'AT, in the general case we have the following suboptimal result.

Theorem 2 [ I ] : Let G(s) = [XI be a minimal balanced

realization of an m x m transfer function. Define

21: =A,,, ( B T [ - ( A + AT)ltB),

7 2 : =X,,,(C[-(A + AT)ItCT). (8.2)

(8.1)

Then

7 : = (9)

Bo: =(-(A + AT)yl - B B T ) i ,

CO: =(-(A + AT)y2 - C T C ) i

is a suboptimal solution for the imbedding problem.

problem is given in the following lemma. A suboptimal solution for the zeroth order H , model reduction

Lemma 3 [ I ] : Let G(s) = [x] be a minimal balanced

realization of an m x m transfer function. Then, DO = -C(A + A T ) t B is a suboptimal solution satisfying the norm bound (IG - Doll, 5 y, where y is characterized by Theorem 2.

The following theorem relates y to the Hankel singular values. Theorem 3: [I] The suboptimal y characterized by (8)-(9) satisfies

the following 1) y 5 (10) 2) y < ut, otherwise. (11)

ut, if rank ( B ) + rank(C) = 2,

111. MAIN RESULT In this section we present a computational scheme to compute the

rth order approximation. The computational procedure is a gener- alization of the one developed for the zeroth order approximation problem reported in [I].

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 7, JULY 1994 1449

A. Computational Procedure A&Q2 + QlA12 + C,',Cz2 = 0, (27) The following is a step-by-step development of the procedure.

A&Q2 + 0 2 - 4 2 2 + C,',cz2 = 0. (28)

Let G(s) = E: z: 1 f] , dim (Al1) = T (12) Observe that the best P a , Q2 are obtained by finding the best zeroth _ _

order H , approximation of

be a minimal, balanced realization of an m x m transfer function G(s) with

Lc = Lo = = (: :2), dim(&) = T. (13)

Our strategy is as follows: we would like to find BZ1, Br2, Czl, and Cz2 such that the controllability gramian, P , and the observability gramian, Q, of

rA11 A12 I B1 B z l i

H ( s ) : = 1-1 (14)

Lczl cZ2 I o o J satisfy the condition

A,i,(PQ) = y; with multiplicity n - T. (15)

In this paper, we pursue a more restricted path, i.e., we require P and Q have the structure

with dim(P1) = dim(Q1) = T. (16)

Therefore, instead of (15) we have

PZQZ = 7'1n-r and A,(PlQ,) > yz (17)

where y is the smallest possible with the given restriction. It is clear that y 2 TO. i.e., the solution is suboptimal.

Next, we impose condition (17) and the structure of P and Q given by (16) on the development. The gramians P and Q satisfy (6) and (7). Also note that since the realization given by (12) is a balanced realization. one has

B B ~ = -AX - C A ~ , cTc = - A ~ C - CA.

Substituting (18) in (6) and (7) one gets

A P + P A ~ + B,,B: = o

A ~ Q + Q A + c ~ T c ~ = o where

From (19), we have the following equations

A i i P i + P I A T , + BziB,T1 = 0,

To calculate BZ1 and Bz2, p2 and g2 we use the results developed in [l] for the zeroth order approximation. Then, we solve for Bzl from (24) and Czl from (26) to get

Bzi = -(A12P2 + PiA?i)B,;T (30)

Cz1 = -(A?iQz + Q,A12)C,;'. (31)

Finally substitute (30) in (23) and (31) in (26). After proper manipulation one gets

ApPl+P1A~+P1AZT1CP-1A~1P1+Ai2P2CP-1P2AT2 = 0 (32)

with Ap: = A11 + A12p2CP-1A21, CP: = Bz2B,T2,

ATQQl +gl Ag+Ql AIZ'B-~ AT2Q1 +AzTlQ2'P-1Q2 Azi = 0 (33)

with Ag: = A11 + A12'B-1gzA21r 'B: = CT2Cz2.

If (32) and (33) have symmetric positive (semi) definite solutions, the aim is achieved, i.e., the imbedding is complete and the achieved level is y, which is computed for solving the zeroth order approxima- tion of G 2 ( s ) defined by (29). If (32) and (33) do not have symmetric positive (semi) definite solutions then we have to increase the y until a solution is possible.

Computational Algorithm Given G(s) in (12) with a balanced realization: Step I) Construct the imbedding for the zeroth order approxima-

tion of Gz(s) = r&], i.e.,

Step 2 ) Solve (32) and (33) when possible, if not possible in- crease y and go to Step 1.

Step3) Compute Bzl and Czl from (30) and (31). Then the imbedded system in (14) is obtained.

Step 4 ) Using Lemma 2 compute the rth order approximation of H ( s ) , which is Hr(s).

Step 5 ) Calculate G,(s) via the compression G,(s) = (1 O)Hr(s) (i).

B. A New Norm Bound For Balanced Truncation Approximation

known [3]. In balanced truncation approximation the following result is well

Theorem 4: Consider the system G(s) given by (12), and its balanced trun-

cation approximation G, (s) = [*I, then one has IIG -

Corollary 1: If U,+I = 1 . . = U", then IIG - Grllm I 2ur+1. The following theorem gives a new H , norm bound for the

Grllm 5 2 Z = = P + l , o , + u ~ + l

balanced truncation approximation.

1450 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 7, JULY 1994

Theorem 5: G ( s ) E R H , be given as in (12), and its balanced truncation

= I%], then one has (IG - G.11, 5 27, where y

is defined by the compitational algorithm given in Section 111-A. Proof: Consider the imbedded system R ( s ) given by (14). After

the imbedding, the gramians of H ( s ) will have the property Pz = ~ I L - ~ , Q 2 = y21n-r (see (16)). Notice that H ( s ) can be balanced by a similar transformation of the form T = diag (TI, t l In-r), tl being a scalar. Truncate the balanced realization of H ( s ) and apply the compression as the one given in Step 5 of the computational algorithm given in Section 111-A. Finally the result follows from Corollary 1.

IV. EXAMPLES In this section we study three examples, one analytical and two

numerical.

A. A Class of Systems-Analytical Solution

Consider the class given by G(s) = [#I, where A =

AT < 0, BBT = CTC = I . To compute the solution we apply the steps in the computational

algorithm. Without loss of generality assume the A matrix is di- agonalized; Otherwise one finds an eigenvalue decomposition of A = UAUT such that the

A: = diag (-XI, -Az,. . . , --Az, . . , -A,)

with A, > 0 and IX, I 5 IX,+, I and applies a similarity transformation T = U to get the balanced realization of G(s) . Therefore, we assume A = A, BBT = CTC = I . It follows that the balanced gramian C is given as C = -;A-’.

Partition the realization of G ( s ) as All = A I , A22 = A z ,

Note that in balanced coordinates Ai2 = 0, Azi = 0.

BIB; = 0, BZB; = CTC2 = -2A2Cz

CTCz = 0, BIB; = CTCl = -2A1C1,

where

z z = - + A T 1 with dim(A1) = T , dim(&) = n - T .

Step I ) From Theorem 2 we get

1 y = 71 = yz = gr+l = - , Bx2 = Cx2 = (-2Azy - I)’”.

2Xr+1

Step 2 ) Since A12 = 0 and A21 = 0, we have A p = AQ = A l . Then (32) and (33) become

A P 1 + P I A = 0

from this equation we conclude PI = a, = 0, and finally PI = 621 = C l .

Step 3) Since A12 = 0 and A21 = 0 from (40) and (41) we have Bxl = C,1 = 0. And the imbedding is completed.

Calculation of the solution from Step 4 and Step 5 is straightfor- ward.

Remark 3: The proposed H , approximation scheme gives solu- tions that are optimal in both Hankel norm and H , norm sense. This class of systems was also studied by Mustafa [6], where it is shown that the balanced truncation scheme is Hankel norm optimal (of course, not necessarily optimal in the H , norm sense since the Hankel norm is independent of the “D” term). It is easily checked that for balanced truncation, the 1 1 . 11, norm of the error is 20,+1.

B. Computation1 Example I In this section we calculate the H , and the Hankel norm approx-

imations for the ninth order stable nuclear reactor model reported in [7]. The Hankel singular values are: 150.9439, 17.874, 0.6334, 0.0647, 0.0042, 0.00067, 0.00042, 0.000032, 0.000025.

The results are summarized in Table I where the H , errors for reduced models orders one through seven are given for both the H , and the Hankel norm approximation schemes.

The lower bounds in the tables follow from the well-known Nehari’s theorem. The upper bound for the H , norm approximation error in the Hankel norm approximation with the “D” term is equal to the sum of the tail of the Hankel singular values [3].

As seen from Table I, both the H , norm approximation and the Hankel norm approximation schemes give almost optimal solutions, i.e., the lower bound, which is not necessarily achievable, is almost attained. This is confirmed with the previously well-known observa- tion that if the Hankel singular values are well separated, then the Hankel norm approximation scheme with the “D” term will give good results.

C. Computational Example II

erated “A” matrix, given by Now consider the following stable system, with a randomly gen-

(where 17 is 7 x 7 identity matrix) (34)

with the equation found at the bottom of the page. The Hankel singular values are: 122.0241, 47.9925, 46.1334,

40.1786, 35.8851, 27.0061, 11.4352. The results are summarized in Table I1 where the H , errors for

reduced models orders one through five are given for both the H , and the Hankel norm approximation schemes.

As shown in Table 11, the H , norm approximation scheme generates results that are essentially optimal since the estimated upper bounds and the actual computed errors are almost equal to the lower bounds. For the case of the Hankel norm approximation scheme, the estimated upper bounds are very loose in all cases, and except for the fifth order, the actual approximation errors are far from the optimal

1.9554 0.2521 0.6293 0.3893 0.4103 0.2531 0.9317 0.5561 1.4885 0.1267 0.2033 0,1312 0.1351 0.6516 0.1482 0.4640 1.6513 0.0284 0.8856 0.7832 0.2152 0.9833 0.9611 0.6216 1.9017 0.0922 0.4553 0.6796 0.4088 0.1260 0.8031 0.4265 1.1622 0.3495 0.9089 0.1418 0.1998 0.2478 0.1420 0.0711 1.4523 0.2501 0.5649 0.3192 0.4764 0.9475 0.3653 0.8089 1.8609

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39. NO. 7, JULY 1994 1451

TABLE I

H A ” L NORM MODEL REDUCnON SCHEMES H, NORM ERROR~OMPARISON OF H, AND

Degree of Approximant Estimated upper 18.3149 0.6695 0.0681 0.0050 0.0010 O.ooo4 0.00006 bound-H,A Estimated upper 18.5776 0.7035 0.0701 0.0054 0.0011 0.0005 0.00006 bound-HA Calculated 18.0538 0.6397 0.0671 0.0047 0.0010 O.ooo4 O.ooOo5

~~~-~~ 17.9865 0.6375 0.0649 0.0043 0.0007 O.ooo4 O.ooOo5 Lower bound 17.8740 0.6334 0.0647 0.0042 0.0007 O.ooo4 O.ooOo3 where H,A stands for H, approximation and HA stands for Hankel approximation.

TABLE II H, NORM ERROR~OMPARISON OF H, AND

HANKEL NORM MODEL REDUCTION SCHEMES

1 2 3 4 5

48.4821 46.6046 40.3982 36.1560 27.1572 Estimated upper bound-H, A

208.6309 160.6384 114.5050 74.3265 38.4414 Estimated upper bound-HA

48.2333 46.4131 40.2446 36.0034 27.0345 Calculated norm-H, A Calculated nom-HA 80.2037 71.9559 56.6557 42.9025 27.0372 Lower bound 47.9925 46.1334 40.1786 35.8851 27.0061

Degree of Approximant

achievable errors. This is in contrast with the example given in Sec- tion IV-B. Note that the Hankel singular values are not well separated.

The above computations were done by the Matlab m-files written by the author for the H, approximation. The Hankel norm approx- imations with the “D” term are computed using the “p-TOOLS” toolbox available with MATLAB.

ACKNOWLEDGMENT

The author acknowledges King Fahd University of Petroleum and Minerals for its support.

REFERENCES

D. Kavranoglu, “Zeroth order H, norm approximation of multivariable systems,” Numerical Functional Analysis and Optim., vol. 14, no. 1, 2,

D. Kavranoglu and M. Bettayeb, “Characterization of the solution to the optimal H, model reduction problem,” Syst. Conrr. ktr., vol. 20,

K. Glover, “All optimal Hankel-norm approximations of linear multi- variable systems and their L,-error bounds,” Inr. J. Cont., vol. 39, pp.

B. C. Moore, “Principal component analysis in linear systems: Con- trollability, observability, and model reduction,” ZEEE Trans. Automat. Conrr., vol. 26, pp. 17-31, 1981. D. Kavranoglu, “H, norm approximation of systems by constant matrices and related results,” in Proc. CDC 92, Tucson, AZ, Dec. 1992,

D. Mustafa, “A class of systems for which balanced truncation is Hankel-norm optimal,” in Proc. 1991 CDC, Brighton, England, Dec.

X. Fu and L. Fu, “Optimal control system design of a nuclear reactor by generalized spectral factorization,” Int. J. Contr., vol. 47, pp. 1479-1487, 1988.

pp. 89-101, 1993.

pp. 99-107, 1993.

11 15-1 193, 1984.

pp. 3271-3275.

1991, pp. 1943-1948.

Control of Asynchronous Sampled Data Systems

Petros Voulgaris

Abstnrct-This paper is concerned with the problem of controller design in the case of asynchronous sampled data systems. Optimal LQG controllers are obtained for the class of two-rate systems where all the control inputs are held at a rate which is asynchronously related to the rate with which all of the outputs are sampled. Furthermore for this class, a parameterization of all stabilizing controllers is provided.

I. INTRODUCTION Control design of sampled data systems has received a lot of

attention lately. Several recent results have led to the development of optimal and robust design methodologies (X’, X w , Z1) that directly deal with the hybrid (continuous and discrete) nature of the system (e.g., [2]-[4], [8], [12]-[15], [17]). In these developments the sample and hold elements were assumed to operate at the same rates. Similar developments in the case of synchronous multirate sampling have also recently occurred in [9], [18], [19] where lifting techniques are used to convert the problem to a single rate. In certain applications of feedback control, due to implementation constraints, the sampling and hold elements may operate asynchronously, i.e., their rates may not be integer multiples of a common time period. For example, systems with distributed, multiple processors with possible communication delays are typical situations where asynchronous operation may arise. Or, when independent clocks dictate the sampling time of the various samplers in use (without a central clock), then these samplers may not stay perfectly synchronized. In these cases the feedback system is no longer periodically varying, and therefore the controller design becomes significantly harder. For such systems there are only a few results available. In particular, we refer to the work of [16] where a sufficient condition for stability of an asynchronous loop is given.

In this paper we investigate a particular class of asynchronous sampled data systems where all the control inputs are held at a rate h which, however, is asynchronously related to the rate s with which all of the outputs are sampled; that is, s / h is an irrational number. For this class of systems we consider the linear quadratic Gaussian (LQG) problem and provide its solution based on dynamic programming arguments. The optimal controller is the combination of a state feedback and a Kalman filter as in the standard LQG problem. The distinction here is that the filter should provide estimates of the state at times which do not coincide with the measurement instants. Furthermore by considering observer based controllers with structure analogous to the optimal steady state LQG controller, we obtain a coprime factorization of the asynchronous system which also provides a characterization of all stabilizing controllers.

II. O m M a LQG CONTROL In this section we consider the LQG problem for sampled data

systems where the control inputs are held at a different rate at which the outputs are sampled and these rates are not integer multiples of a common time period.

Manuscript received April 2, 1993; revised June 30, 1993. The author is with the Coordinated Science Laboratory, University at

IEEE Log Number 9401660. Illinois at Urbana, Champaign, Urbana, IL 61801 USA.

0018-9286/94$04.00 0 1994 IEEE