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Proceedings o l he 3 rt Conferenceon Lbdr lon and ControlTucson.Arhona *December 1962 WP9 - 1 8 : O OState-Space Solutions to Discrete-Time and

Sampled-Data 3-12 Control Problems1T o n g w e n Ch e nD e p t . of E l e c t . & C o m p . E n gg .

Universi ty of Ca l g a r yCa l g a r y , A l b e r t a

C a n a d a T 2 N 1N4

A b s t r a c tThis paper gives a complete state-space derivation of thediscrete-time "2-optimal controller. This derivation can beextended to treat a sampled-data HZontrol problem, resulting

in a new direct solution to the sampled-data problem.

1 IntroductionA recent trend in synthesizing sampled-data systems is to use themore natural continuous-time performance measures. This broughtsolutions to several new "2-optimal sam ple d-d ata control problems(6, 4 , 131, each reducing t o an 7 2 problem in discrete time.Discrete-time ?i2 LQG) theory was developed in the 197O's, see,e.g., [2, 8, 14, 15 I , 161. A s in the continuous-time case, the discreteoptimal controller is closely related to the solutions of two Riccatiequati ons. In [9], the solution to a continuous -time "2-optimal con-trol problem was rederived using the state-space approach. This givesa clean treatme nt of the problem an d provides compact formulas forthe optim al controller. Since complete. general formulas for the dis-crete optim al controller a re not readily available in the literature, weask the question here, can a state-space treatment be accomplishedfor discrete-time H2 problems?Th e goal in this paper is twofold: to present a st ate-sp ace solutionto th e d iscre te 2 control problem and to give direct formulas for an"2-optimal sampled -data control problem with state feedback anddisturbance feedforward using the powerful lifting technique [17, 19,

Th e organization of the paper is as follows. In the n ext sectionwe collect and some preliminary results on Rjccati equations; thepresen tation follows closely tha t in [9] in continuou s time. Section 3trea ts the discrete-time "2-optimal control, f irst via stat e feedbackand disturbance feedforward and then via dynamic output feedback.Section 4 presents new direct formulas for a sampled-data 3 2 problemusing state measurement. In Section 5 we apply the optimal sampled-da ta control in Section 4 to a two-motor system and compare wi th theoptim al analog control. Most proofs ar e omitted and ca n be found in

Th e notation in this paper is quite standard: C is th e complexplane, 'D C C is the open unit disk, and 82, is the boundary of'D, namely, the unit circle. Also, 2 is the set of all integers and 22- ) s the n onnegative (negative) subset of 2. The space l z ( Z + ) ,or

simply l z consists of all square-summable sequences, perhaps vector-valued, defined on 2+. Similarly for ez(Z) nd (Z-). Th e discrete-time frequency-domain space H * ( D ) , r simply 2, is the Hardy spacedefined on V. We use R H2 for the real-rational subspace of 7 1 2 . Indiscrete time, we use A-transforms instead of z-transforms, whereX = z-l. Finally, g ( A ) stands for the transposed matrix g ( l / X ) .

3 51.

(71.

'The work was supported by t he N a t u r a l Sciences a nd Engineering ResearchCouncil of Canada.

2

B r u c e A . Fr a n c i sD e p t . of Elect . Engg .Universi ty of T o r o n t o

T o r o n t o , O n t a r i oC a n a d a M5S 1A4

Riccati EquationI t is well-known tha t Rjccati equations play an import ant role in theH2 optimization problem. The solution of a Riccati equation canbe obtained via the stable eigenspace of the associated symplecticmatri x if the stat e transition matrix of the plant is nonsingular. If

this ma trix is singular, as is the case when th e plant has a time delay,then t he symplectic matri x is not defined; but we can use the stablegeneralized eigenspace of a certain matrix pair [16].

Let A , Q , R be real n x n matrices with Q and R symmetric.Define the ordered pair of 2n x 2n matrices

A pair of matrices of this form is called a symplectic pair. (Th isdefinition is not the most general one .) N ote tha t if A is nonsingular,then H; HI is a symplectic matrix.Introduce the 2n x 2n matr ix

J := [ - I ]I OIt is easily verified that H I J H I = H 2 J H ; . Thus the generalizedeigenvalues (including those at infinity) for the matrix pair H (i.e.,those numbers X satisfying H l x = X H z z for some nonzero z r esymmetric abou t the u nit circle, i .e., X is a generalized eigenvalue iffl / X is [16].Now we assume H has no generalized eigenvalues on 8V. h e nit must have n inside and n outside. Thus the two generalizedeigenspaces X t (H ) and X , ( H ) , corresponding to generalized eigen-values inside and outside the unit circle respectively, both have di-mension n. Let us focus on the stable subspace X , ( H ) . There existn x n matrices XI and X z uch tha t

Then for some stable n x n matr ix H , ,

Some properties of the matrix XiX2 are useful.Lemma 1 Suppose H has no eigenvalues on 8 D. Then

i) X : X 2 s symmetric;ii) XiX2 2 0 i f R 2 0 andQ 2 0 .

Now assume fu r ther tha t X I s nonsingular, i.e., the two subspaces

are complementary. Set X := X 2 X ; . Then

CH3229-2/92/0000-1111 1OO 1992 IEEE 1111


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Note tha t the n x n matrix X is uniquely determined by the pair H( though X1 and X2 are not) , that is, H X is a function. We shalldenote this function by R ic and write X = Ri c ( H ) .To recap, R ic is a function R2nxZn Rnxn h a t m a p s H t oX , w h er e X is defined by equation (2). The domain of Ric, de-

noted dom Ric, consists of ll symplectic pairs If with tw o proper-ties, namely, H has no generalized eigenvalues on dV and the twosubspaces - -are complementary.Some properties of X are given next.Lemma 2 Suppose H E dom Ric and X = Ric(H) . Then

(i) X is symmetric;(i i ) X satisfies the algebraic Riccati equation

A X ( I + RX )- A - X Q = 0;(i i i) I RX)-A is stable.

Lemm a 2 is quite s tan dar d, see, e.g., [16, 121. Th e following resultgives verifiable conditions under which H belongs to dom Ric.Theorem 1 Suppose H has the form

with (A, B ) stabilizable and (C,A) having no unobservable modes ondV. Then H E d o m Ri c a n d R i c ( H ) 2 0.

Proof Th e proof that H has no generalized eigenvalues on the unitcircle follows from [16]. Now we will show that the two subspaces

are complementary. Bring in matrices XI , Xz, Hi and write 1) astwo equations below ( R = EB,Q = CC):AX1 = XlHi BBX2H;- CCXI X2 = AX2Hi.

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We want t o show tha t Xi is nonsingular, i.e., KerX1 = 0. First, i t isclaimed t ha t K er X1 is Hi-invariant. To prove this, let z E Ker XI .Pre-multiply (3) by zlH, XJ and pos t-mu ltiply by z o ge tz H ~ X ; X I H ~ Z z H ~ X ; E B X ~ H ; Z 0.

Note tha t since X;X1 2 0 (Lemma l ) ,both te rms on the left are 2 0.T h u s B X z H p = 0. Now post-multiply (3) by z o get X1H ;z = 0,i .e. , Hiz E Ker X1. This proves the claim.Now t o prove tha t X1 is nonsingular, suppose on the contrarytha t Ker X I # 0. Then HilKer X1 has an eigenvalue, p , and a corre-sponding eigenvector, z:

H , z = p z , ( 5 )IpI < 1, 0 I E KerX1.

Post-multiply (4) by z nd use (5):(PA - 1 ) X z ~ 0.

If p = 0 then Xzz = 0. Otherwise, since EX zz = 0 from BXzHiz =0 and ( 5 ) , we have

Then stabilizability implies Xzz = 0 as well. Bu t if X lz = 0 andXzx = 0, t h e n z = 0 a contradiction. This concludes the proof ofcomplementarity.Now set X := Ric(H) . By Lemma 1 ( R = EB,Q = C C, X l =

QEDThis theorem has various forms in the literature; for example, in

[14] imilar results were given when t he ma trix A is nonsingular andin [16] n indirect proof was given that X1 is nonsingular. Our proofhere is along the lines of a continuous-time proof in [lo].

I,Xz = X ) , x 2 0 .

3 Discrete-Time CaseThis section rederives in a state-space approach t he perhaps-knownresults for a disc rete- time 2-optimal control problem.We begin with the standard setup:

................... ............................ .........

1 1 ; i v............. .............

(We have used dotted lines for discrete signals and will reserve con-tinuous lines for continuous signals.) The inpu t w is standard whitenoise ero mean, unit covariance matrix. Th e problem is to designa K that stabilizes G and minimizes the root-mean-square value ofC ; i t can be shown that this is equivalent to minimizing the norm onHz of the transfer matrix from w t o (.3.1 State Feedback and Disturbance FeedforwardFirst we allow the controller t o have full information. In this case, aswe will see he optimal controller is a cons tant sta te feedback with adisturbance feedforward. With the exogenous input being some pulsefunction, say, w = wo6d WO is a constant vector and 61 he discreteunit pulse), we can even think of U as unconstrained. Th e preciseproblem is as follows:

Given the system equationsG : ((k 1) = A((k) Elw(k) U(k ) , W = WO6d

C(k) = Clt (k ) D l l W( k ) + D l Z V ( L )with the assumptions

(i) (A, Bz) is stabilizable;(ii) D;,D12 = I and D12 is nonsquare;

(iii) th e mat rix

has full rank V X E dV.Solve the optimization problem

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Note that for ease of presentation we initially allow U to be in e,,,t h e eztended space for ez; however, the optimal U , to be seen la te r ,will actually lie in e . Assumptions ( i ) and (iii) are mild restr ictionsand (ii) basically means th at th e system has more outpu ts to be con-trolled than control inputs and the control weighting is nonsingular.If DizD12 is nonsingular but not identity, we can normalize it bydefining the new U t o b e ( D ~ z D 1 ~ ) l / z ~ .he th ree assumpt ions to -gether guara ntee that the symplectic pair below belongs t o dom R i c .Define

Theorem 2 The unique optimal control is uOpt= FE t F1w More-over,

min I lCl l z = IIJCwollz.In co ntrast w ith the full- information continuous-time case wherethe o p t imal con tro l i s a constant state feedback, the discrete-timeoptimal co ntrol law involves a disturbance feedforward term, an d thisis true even when D l l = 0.

3.2 Output FeedbackNow we study th e Hz-optimal control problem posed at th e star t ofSection 3, where the measured ou tpu t II, does not have full infor-mat ion an d therefore dyna mic feedback is necessary. All discussionper ta ins t o the s tandard d iscre te-t ime se tup . Let Tcw deno te theclosed-loop syst em from o (. We say a causal, finite-dimensional,linear time-invariant controller Ii is admissible if it achieves internalstability. Our goal is to f ind an admissible Ii to minimize Il ,llz.Consider plants of the form

i X ) = [VjI DIIC2 DZITh e following assum ptions are m ade:(i) (A, Bz) s stabilizable and (Cz, A ) is detectable;(ii) D:2D12 = I with D12 nonsquare and D zl D ; , = I with Dzlnonsquare;

(iii) t he matricesA - X Bz A - X B[ CI D I Z ] [ Cz D ~ I

have full rank VX E aV;The first parts of assumptions (i)-(iii) were seen in Section 3 1 T h esecond part s of assumptions ( i)-( iii) are dua l to their f irst parts: To-gether they gu aran tee tha t the symplec tic pa i r J introduced belowbelongs to om R i c . In assumption (ii) i t is essential that the twomatrices D{,D,z and D z l D i , be just nonsingular; for they ca n benormalized via changing coordinates in U and :

Theorem 3 The unique optimal controller is

Moreover, 9. l k w l l = l l i c l l ; + Ilgrll;-All the proofs ar e contained in [7]. Th e first term in the minimumcost, Iljclli, s associated with optimal control with s tate feedback anddisturbance feedforward and th e second, Ilprll; with opt imal filtering.

4 Sampled-Data CaseThe formulas in the preceding section have direct application insam pled-d ata control problems. We will look at the case when thecontrol signal is the out put of a D /A device, but is otherwise uncon-strained. Then the optimal control law is a sampled sta te feedbackwith a suitable disturbanc e feedforward.

Consider the sampled-data se tup

Here the continuous-time system G is described by the s tate equationsi ( t ) = A z ( t )t B i w ( t ) B z ~ ( t )z ( t ) = Ciz(t) D i i w ( t ) D 1 2 4 t ) .

The con trol inpu t U is obtained through a zero-order hold HO withsampling period h , processing a discrete signal v , the control se-quence. Thus U and U satisfy

u ( t ) = v(k), k h 5 t < ( k 1)h.The exogenous input w is assumed t o be f ixed and affects the systemonly through the f irst sampling period. So w has support in [ O , h ) .For example, w could be the impulse w ( t ) = w06(t - o ) , where WO isa constant vector and 0 5 to < h.Our sampled-data problem is

Define the norm being on [ O , w . We shall assume tha t G h a s zero initial1113


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s tate .Now we use the lifting technique in [ 5 ] to set the problem in th elifted space. Following the no tation in [ 5 ] , let & denote any finite-dimensional Euclidean space (i ts dimension will be irrelevant) and Kdenote & [ O , h ) . The sequence space & ( 2 + , K ) , or simply & ( K ) , isdefined to be

m

eZ(K) := 11 : $k E K , ll$k11' < m}.k=OT h e n o rm fo r $k is the one on K and the norm for & ( K ) s given by

The l i ft ing operator W , mapping [O, 00) to l z ( K ) , s defined by= wy $ k ( t ) = Y ( t k h ) , 0 5 t < h.

We denote the l ifted signal Wy by g.Now we lift the system in th e preceding figure to geti = G [ E ]

G = WG [ Wilo].Here the l i f t ed sys tem C satisfies the d iscrete-t ime equations [5 ] [sincew h as s u p p o r t i n [O, h), 3 s a pulse sequence in (K)]

[(k -I-) = A d((k) -k Bi B zdv (k) , 3k = coad(k) ( 6 )f k = e l ( (k ) b l l c k b 1 ' 2u (k ) , (7)

where [(k) := z kh) and the operators a re g iven by

The sys tem G can be regarded as a l inear t ime-invariant systemin discrete t ime, with ik an d t i r k being infinite-dimensional (func tionsin K). Since the l ift ing operator is norm-preserving, the equivalentdiscrete Hz problem is

m;tn I lE l lzsubject to equat ions (6-7), the norm being on 2 (K ) .

This problem looks almost l ike the one we studied in Section 3,the difference being that now we are treating operators instead ofmatrices. So the derivation for th e optima l control in Section 3 carriesover except for a few changes such asusing operator adjoints, denotedby *, ins tead of transposes.In view of the a ssumptions in Section 3, we assume here that

(i) (A d ,BZd) is stabil izable;(ii) t h e m a t r i x d12 is invertible;( i ii ) the mat r ix operator

is injective VX E aV.To write down th e formulas, we need to normalize d1~irst. SOdefine the mat r ix

Q = (b:2b12)-1'2 ( 8 )and vneW= -'U to get the normalized equat ions

((k 1 ) = A d ( ( k ) k 816 BzdQvnew(k)i k = C, k) b 1 1 f i k d ~ ~ Q e w ( k ) .

We can now give the optim al control . Define

(9)

whole is a m at r ix (operator on &). Th e formulas for Rzcl anbe derived easily. Similarly, F , though involving operators, isalso a matrix. However, the feedforward gain F1 is an operatormapping K t o b ; i ts action on a fixed an be determined apriori.Th e opt imal cont rol can be w ri t t en as

Th e opt imal s tate feedback ( F z ( k h ) ] s independent of the ex-ogenous input and can be realized by sampling z ( t ) a t t h es am e r a t e as the hold operator . In part icular, if the rate ofthe D/A device is chosen, one does not gain any advantage bysampling z(1) faster, or even by measuring z ( t ) continuously.Assu mption (i) is satisfied if ( A ,B z ) s stabil izable in continuoustime a nd if the sampling is non-pathological in a certain sense,see, e.g., [Ill.

4. It is not hard to show th at assumption (i ii) is satisfied ifd - Bzd

is injective VX E OV, hich can be checked easily since it is ama trix expression.

5 ExampleTh e theory of the preceding section is now applied to a simple setupconsist ing of two mo tors controlled by one PC. The block d iagramfor the system is as follows:

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IW

- 10 0 -10 0

c,= 0 0 0 " 1 , D l , = \ i - ,, o o 0 0

0 0D 1 2 = \ 1

0 0.1

I IU 2

Shown are two identical motors, with shaft angles 81 and 82. T h eleft-hand motor is forced by an external torque W . Th e controller,A', inputs t he two shaft positions and their velocities, and out putstwo voltages, u1 and u2 to the motors. The goal i s tha t the systemshould act like a telerobot: When a human applies a torque w , t h e"mwte r" ( left-hand) motor should turn appropriately and the "slaven(right-h and) motor should follow it.Th e s ta te vec to r is taken to he

I = [e, i , 8 2 e 2 yFor certain values of the physical parameters, the st ate matrices are

0 1 0 0A = [ 2r1 1

0 -24.51

0, 1 > 0.1and the result is shown in Figure 1 (0, solid, Bz dash, in degreesversus time in seconds).gain F one must com pute the matr icesTurning t o the optima l sampled-data control, for the state-feedback

Figure 1: Optimal analog controller

Th e disturbance-feedforward gain is an operato r K E , but sincew ( t ) here is co nstant over [0, h ) , he action of Fl is to multiply by amatrix, denoted, say, 4 his matrix is computed as follows:

These matrices, F and F1, were computed for h = 0.1 (quite large,for illustration) and the resulting s ampl ed-data system was simulatedwith the same v ( t ) s above. T he responses are shown in Figure 2:

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Figure 2: Optimal sampled-data controllerTh e response of the sampled-data sys tem i s comparable to t h a t o f t h eanalog syste m, except for the DC gains. [The same weights (i.e., C1and D12) are used for both th e analog and sampled-data cont ro ller ,but in general weights that are good for the analog controller willnot necessari ly be good for the sampled-data controller, and viceversa.] However, the analog F does not even stabil ize the sampled-da ta sys tem for th i s l arge h . The point is, therefore, that when h isgiven and is appreciably large, the optima l sampled-d ata controller ismuch superior to the discretized optima l analog controller. Finally,for in teres t th e sampled-data sys tem wi th only s tate feedback andnot d i s turbance feedforward , that i s ,

k = Ok 1,opt(k) = { : z ( k h ) ,

was simulated and the responses are shown in Figure 3:

Figure 3: Opt imal s tate feedbackNot surprisingly, the response is very poor: The slave motor does notbegin to move unti l the s tar t of the second sampling period, by whicht ime the t racking error is very large.Acknowledgement Th e authors wish to thank P. P. Khargonekarand P. A. Iglesias for helpful discussions.

References[I] B. D. . Anderson and J . B . Moore, Optimal Filtering, Prentice-Hall, nglewood Cliffs, NJ, 1979.[2] M. Atha ns, Th e role and use of the stochastic l inear-quadratic-Gaussian problem in control system design, IEEE Tmns. Au-

tomat. Control, vol. 16, pp. 529-552, 1971.[3] B. Bamieh and J . B. Pearson, A general framework for l inearperiodic systems with application t o H sample d-data control ,

IEEE Tmns. Automat. Control, vol. 37, pp. 418-435, 1992.[4] B . Bamieh and J. B. Pearson, The Hz roblem for sampled-datasystems, Tech. Report No. 9104, Dept. of Elect . and Comp.

Engg., Rice Univ., 1991.[5 ] B. Bamieh, J. B. Pearson, B. A. Francis, and A. Tannenbaum,A lifting technique for l inear periodic systems with applications

to sam pled-data cont ro l , Systems and Control Letters, vol. 17,pp. 79-88, 1991.[6] T . C h en an d B. A. Francis, Hz-optimal sampled -data control ,

ZEEE Tmns. Automat. Control, vol. 36, No. 4 , pp. 387-397,1991.(71 T. Chen and B. A. Francis, S tate-spa ce solutions to discrete-

t ime and sampled-data Hz control problems, Tech. Report No.9205, Dept. of Elect. Engg., Univ. of Toronto, Revised August1992.

[8] P. Dorato and A.H. Levis, Optimal l inear regulators: thediscrete-t ime case, IEEE Tmns. Automat. Control, vol. 16, pp.613-620,1971.

[9] J. C. Doyle, K. Glover, P. P. Khargonekar, and B . A. Francis,S tate-space solu t ions to s tandar d Hz and 3-1 problems, ZEEETmns. Automat. Control, vol. 34, No. 8 , pp. 831-847, 1989.

[lo] B. A. Francis, A Course in H Control Theory, Springer-Verlag,New York, 1987.l l] B. A . Francis and T. T. Georgiou, Stabil i ty theory for l inear

t ime-invariant plants with periodic digital controllers, IEEET M ~ s . utomat. Control, vol. 33, No. 9 , pp. 820-832, 1988.[12] P. A. Iglesias and K. Glover, S tate space approach t o discrete

t ime 31, control , Tech. Report , Dept. of Engg. , CambridgeUniv. , 1990.[13] P. P. Khargonekar an d N Sivashankar, HZ ptimal control forsampled-data systems, Pm. CDC, 1991.

[14] V. KuEera, The discrete Riccati equation of optimal control ,hybernetica, vol. 8, No. 5, pp. 430-447, 1972.[15] B. P. Molinari , The stabil izing solution of the discrete algebraic

Riccati equation, IEEE h n s . Automat. Control, vol. 20, No.3, pp. 396-399,1975.

[16] T. Pappas , A. J. Laub, and N. R. Sandell, Jr., On the numericalsolution of the discrete-t ime algebraic Riccati equation, IEEETmns. Automat. Control, vol. 25, No. 4 , pp. 631-641, 1980.

[17] H . T. Toivonen, Sampled-data control of continuous-time sys-t ems wi th an 1-1, optimality cri terion, Automatica, vol. 28, No.1, pp. 45-54, 1992.[18] H. K. Wimmer, Normal forms of symplectic pencils and thediscrete-t ime algebraic Riccati equation , Linear Algebm and itsApplications, vol. 147, pp. 411-440, 1991.[19] Y. Yamamoto, A new approach to sampled-data control sys-tems function space approach method, P m . CDC, 990.

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