An LQ R weight selection approach to the discrete generalized H 2 control problem

INT. J. CONTROL, 1998, VOL. 71, NO. 1, 93± 11 An LQ R weight selection approach to the discrete generalized H 2 control problem D. A. WILSON², M. A. NEKOUI² and G. D. HALIKIAS² It is known that a generalized H 2 (GH 2 ) controller for a continuous-time system takes the form of a conventional Kalman lter together with a state feedback control law, and that the feedback gains can be determined using convex programming. In this paper the discrete-time version of the GH 2 control problem and the related convex programmes involving weight selection in a full information LQR problem are considered. 1. Introduction In a generalized H 2 (GH 2 ) control problem, the conventional H 2 norm is replaced by an operator norm (Wilson 1989, 1991, Rotea 1993, Wilson and Rubio 1995), and a stabilizing controller is sought such that the closed loop gain from time-domain input disturbances in L 2 to time-domain regulated outputs in L is either minimized or kept below a certain bound. A complete solution to the continuous-time GH 2 control problem was presented by Rotea (1993), where it was shown that the solution consists of a conventional Kalman lter together with a state feedback control law in which the gains depend on the Kalman lter. It was also shown that in the case of output feedback the GH 2 synthesis problem could be reduced to a full information/state-feedback problem involving an auxiliary system. The procedure for extending these results to the discrete-time case was given by Kaminer et al. (1993). Wilson and Rubio (1995) showed that in the case where the GH 2 norm is de ned from L 2,2 to L, (the d max problem), the solution consists of a Kalman lter with estimated feedback where the feedback gains could be determined from a weighted LQR problem. The possibility of using a convex programme for LQR weight selection for a GH 2 problem was noted by Rotea (1993), with reference to the work of Boyd and Barratt (1991). Although weight selection was not used by Rotea (1993), the convex programme associated with this method is less than or equal to n + m 2-1, where the system is of order n with m 2 control inputs which, for high-order multi-input systems, o ers a potential reduction in computational burden. The purpose of this paper is to consider the use of weight for the discrete-time GH 2 problem. Following Rotea (1993), the computations are carried out on an auxiliary system, the derivation of which is provided in section 2 for discrete-time systems. The LQR weight selection method is presented in section 3, and section 4 contains a numerical example. Received 8 August 1997. Revised 12 March 1998. ² Department of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK. e-mail: d.a.wilson@elec-eng.leeds.ac.uk. 2-7179/98$12. Ñ 1998 Taylor & Francis Ltd.

êêê þ 94 D. A. Wilson et al. 2. Discrete-time representation and the auxiliary system This section shows how an auxiliary system with the structure of a state feedback sysem can be derived from the more general output feedback representation. Consider the discrete state space system x(k + 1) = Ax(k) + B 1 w(k) + B 2 u(k) y 1 (k) = C 1 x(k) + D 1 u(k) y 2 (k) = C 2 x(k) + D 2 w(k) where x(k) Î R n, w(k) Î R m 1, u(k) Î R m 2, y 1 (k) Î R p 1 and y 2 (k) Î R p 2 are the states, disturbances, control inputs, errors and measurements, respectively. Throughout this paper we assume, without loss of generality, that C T 1 D 1 = and B T 1 D 2 =. In addition, the pairs (A,B 2 ) and (A,C 2 ) are assumed to be stabilizable and detectable, respectively. In what follows, attention is restricted to linear controllers, and we begin by determining that the Youla-parametrized (Francis 1987) closed-loop transfer function matrix between y 1 and w is given by T y1w (z) = T 11 (z) + T 12 (z)q(z) T 21 (z) (2) where T 11 (z), T 12 (z), T 21 (z) and Q(z) belong to H 2. Using the standard notation for a real rational matrix transfer function we have (Francis 1987, Gu et al. 1989) T (z) = C(zI - A) - 1 B + D = z A B T 11 = C A c - B 2 F B 1 A f B f C c - D 1 F T 12 = A c B 2 C c D 1 T 21 = A f B f C 2 D 2 with A c = A + B 2 F, A f = A + HC 2, B f = B 1 + HD 2 and C c = C 1 + D 1 F, and with F and H any matrices stabilizing A. It is also assumed that A f is devoid of eigenvalues at the origin. The H 2 norm of the system de ned in equation (2) can be found by calculating the cost matrix S(Q) = 1 T 2p j$ y1 w(z) T * y 1 w (z) dz c z (with T * y 1 w = T T y 1 w (z - 1 ) and c the unit circle). Then, the H 2 norm is de ned by i T y1 wi 2 2 = trace (S (Q)) úúú ïïü ïïý D (1) (3) (4) (5) (6)

Note that, in contrast to the continuous-time case, T y1 w (z) need not be strictly proper to obtain a nite H 2 norm. Consequently, Q(z) may have a constant term. The discrete-time H 2 control problem involves determining inf trace (S(Q)) QÎ H 2 In the discrete-time version of the GH 2 setting, obtained from its continuous-time counterpart de ned by Rotea (1993) and Wilson and Rubio (1995), w Î z 2,2, y 1 Î z,r with r = 2 or depending on the spatial norm used on the controlled sequence {y 1 (k)} where w Î z 2,2 «i wi 2,2 = å y 1 Î z,2 «i y 1 i,2 = sup - <k< w T (k)w(k) ( - ) y 1 Î z, «i y 1i, = sup - <k< The performance measure is then 1 /2 < (y T 1 (k)y 1 (k)) 1/2 < max y 1,i 1 i p 1 (k) < { (7) f r (S(Q)) = max(s (Q)) r = 2 d max (S (Q)) r = where max and d max denote maximum eigenvalue and maximum diagonal entry, respectively, of a non-negative de nite matrix. The GH 2 norm is then given by i T y1 wi = Ï ê f ê ê ê ê ê ê ê ê ê ê ê ê ê ê ê êê r (S (Q)) (8) The discrete-time GH 2 control problem involves determining inf f QÎ r (S(Q)) H 2 It is well known that the H 2 norm is both left and right unitary invariant, that is, equation (2) can be pre- and post-multiplied by a unitary matrix without altering the norm. The GH 2 norm in equation (8), however, is only right unitary invariant. This property will be used in this section to obtain a simpli ed expression for the cost matrix, and hence to derive an auxiliary system with full state accessibility. The rst step is to choose H as where Y satis es the Riccati equation and Discrete-time version of the H 2 control problem 95 H =- AYC T 2 Z 2 AYA T - Y - AYC T 2 (D 2 D T 2 + C 2 YC T 2 ) - 1 C 2 YA T + B 1 B T 1 = Z = (D 2 D T 2 + C 2 YC T 2 ) - 1 /2 Then, based on the results from Gu et al. (1989), the following factorization can be derived: T 21 = Z - 1 ZT 21 = Z - 1 T 21ci (9)

êêê 96 D. A. Wilson et al. where T 21ci = A + HC 2 B 1 + HD 2 ZC 2 ZD 2 is co-inner. Using equation (9), equation (2) can be written as T y1 w = T 11 + T 12 QZ - 1 T 21ci = T 11 + T 12 Q n T 21ci (1) where Q n = QZ - 1. Using the right unitary invariance property of the GH 2 norm, equation (1) can be post-multiplied by the unitary matrix [T * 21ci.. T 21ci^ ] without altering the norm [T 11 + T 12 Q n T 21ci][T * 21ci.. T 21ci^ ] = [T 11 T * 21ci + T 12 Q n.. T 11 T 21ci^ ] (11) The complementary part of T 21ci is chosen such that T 21ci... T 21ci^ is square and unitary, in the following way (Gu et al. 1989) where T 21ci^ = A f B f C^ D^ C^ =- D^ B T f A - T f Y + D^ (D 2 - C 2 Y Y + A - 1 f B f ) T = D^ (I + B T f A - T f Y + A - 1 f B f )D^ T = I (Y + being the pseudo-inverse of Y ). Using similarity transformations, the T 11 T * 21ci term on the right-hand side of equation (11) can be written as T 11 T * 21ci = A c AYC T 2 A - T F A - T f C T 2 C c C 1 Y = A c AYC2 T Z C c úúú Z + A- f T A - f T C2 T Z C 1 Y 7 (T 11 T * 21ci) + + (T 11 T * 21ci) - (12) where, in equation (12), T 11 T * 21ci has been decomposed into its causal and strictly anti-causal parts. Note that for an impulse response sequence {M(k)} -, the causal part will be de ned by {M(k)} and the strictly anti-causal part by {M(k)} - 1, i.e. the response at time zero is associated with the causal part. Also, again using similarity transformations,

å å T 11 T 21ci^ * = A- f T A - T f C 1 Y C^ T Z The transfer function matrix in equation (11) can now be decomposed as follows: N 1 L N 2 ( ) ) ) ) ) ) ) *, (T 11 T 21ci) * ) ) ) ) ) ) ) & ( ) ) ) ) ) ) ) ) ) ) ) - + (T 11T 21ci) * ) ) ) ) )) *, ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )) &. ( ) ) ) ) ) *, - + T 21 Q n T11 T 21ci^ * ) ) ) ) )) & [ ] Denote the operation of taking the discrete inverse Fourier transform by I - 1, then {N(k)} - 7 I - 1 [ (T 11 T * 21ci) - + (T 11 T * 21ci) + + T 21 Q n.. T 11 T * 21ci^ ] = [{N 1 (k)} - 1 + {L (k)}. {N 2 (k)} - ] The cost matrix is then - 1 S (Q n ) = å N(k)N T (k) = - å N 1 (k)n1 T (k) + - å - N 2 (k)n T 2 (k) + å L (k) L T (k) The rst two summations on the right-hand side can be determined using discrete Lyapunov equations: - 1 - N 1 (k)n T 1 (k) = C 1 YP 1 YC T 1 - C 1 YC T 2 Z 2 C 2 YC T 1 where P 1 satis es the discrete Lyapunov equation Similarly, - P 1 = A T f P 1 A f + C T 2 Z 2 C 2 N 2 (k)n T 2 (k) = C 1 YP 2 YC T 1 where P 2 satis es the discrete Lyapunov equation P 2 = A T f P 2 A f + C^ T C^ It can then be shown that C 1 Y (P 1 + P 2 ) YC1 T = C 1 YC1 T and hence S(Q n ) = å N(k)N T (k) = - å L (k) L T (k) + C 1 YC1 T - C 1 YC2 T Z 2 C 2 YC1 T The transfer matrix function analysis associated with the causal sequence {L (k)} (T 11 T * 21ci) + + T 12 Q n = A c AYC T 2 Z C c C 1 YC T 2 Z + A c B 2 C c D 1 is Q n (13) This is the sum of two transfer functions, and can be written in terms of the following sets of state-space equations: and Discrete-time version of the H 2 control problem 97 x 1 (k + 1) = (A + B 2 F)x 1 (k) + AYC T 2 Zr(k) z 1 (k) = (C 1 + D 1 F)x 1 (k) + C 1 YC T 2 Zr(k)

å å þ 98 D. A. Wilson et al. x 2 (k + 1) = (A + B 2 F)x 2 (k) + B 2 Q n (x )r(k) z 2 (k) = (C 1 + D 1 F)x 2 (k) + D 1 Q n (x )r(k) where x denotes the forward shift operator. Now let x a = x 1 + x 2, y 1a = z 1 + z 2, then the system with impulse response {L (k)} has the state-space realization x a (k + 1) = Ax a (k) + B a r(k) + B 2 u a (k) y 1a (k) = C 1 x a (k) + D 1 u a (k) + C 1 YC2 T Zr(k) y 2a (k) = x a(k) [ r(k) ] where B a = AYC T 2 Z and u a (k) = Fx a (k) + Q n (x )r(k). This is a state-space model of a full information plant and will be referred to as the auxiliary system. For a given F, with A + B 2 F stable, the closed-loop impulse response sequence between {r(k)} and {y 1a (k)} will be denoted by {L (Q n; k)} to display the explicit dependence on Q n (x ). Then, it follows that the cost matrix is given by S(Q n ) = å k= ïïïïïü ïïïïïý (14) L (Q n ; k) L T (Q n ; k) + C 1 YC T 1 - C 1 YC T 2 Z 2 C 2 YC T 1 (15) This new form of the cost matrix explains how the output feedback problem is related to a state feedback problem through the constant term S c 7 C 1 YC T 1 - C 1 YC T 2 Z 2 C 2 YC T 1 This will be used in the next section to develop the weight selection method. 3. Weight selection method The continuous-time version of the weight selection method for the GH 2 problem was considered by Wilson and Rubio (1995) and Nekoui and Wilson (1996). This problem involves determining inf d QÎ max (S(Q)) = inf H 2 QÎ H 2 max s i (Q) 1 i p 1 where s i (Q), i = 1... p 1 are the diagonal entries of S(Q). Using a result from Medanic and Andjelic (1971), this problem is equivalent to determining inf max QÎ H 2 iî X p 1 i=1 is i (Q) p where X = { i : i, å i i=1 i p = 1}. Since å 1 i=1 is i (Q) is convex in Q and concave in i, and the sets H 2 and X are convex, the minimax theorem (Balakrishnan 1981) implies that where K solving inf max QÎ H 2 iî X p 1 i=1 is i (Q) = sup iî X inf QÎ H 2å p 1 i=1 is i (Q) = sup iî X inf trace QÎ H 2 [ K S(Q) ] = diag ( i). Hence, the d max version of the GH 2 control problem consists of

K inf iî [- inf trace X QÎ H 2 [ K S(Q) ]] (16) This de nes a weight selection problem which can be solved using convex programming (Boyd and Barratt 1991, pp. 335± 337). To avoid a singular LQR problem, the search will be restricted to over strictly positive weights. Using the auxiliary system, the cost matrix is as de ned in equation (15), and equation (16) can be replaced by inf iî X { } - inf trace QÎ H 2 å K L (Q n : k) L T (Q n; k) + trace (K S c ) [ k= ] = inf - trace (K S iî c ) - inf trace X QÎ H 2 å 1/2 L (Q n : k) L T (Q n ; k)k 1 /2 [ k= ] (17) where L (Q n; k) denotes the closed-loop impulse response sequence of the auxiliary system between r and y 1a de ned in equation (14). The second term in equation (17) corresponds to a conventional H 2 problem for the weighted full information plant given by x a (k + 1) = Ax a (k) + B a r(k) + B 2 u a (k) 1 y 1a (k) = K /2 C 1 x a (k) + K y 2a (k) = x a(k) [ r(k) ] 1 /2 D 1 u a (k) + K 1/2 C 1 YC T 2 Zr(k) The solution to this full information problem is given by (Zhou et al. 1996) u a (k) = Fx a (k) + Gr(k) (Note that, unlike the continuous-time case, the full information solution to the discrete-time case involves feedback of the disturbance. Also, the form of the solution is precisely that required by equation (14).) The feedback matrices F and G depend on K, and are obtained from the weighted full information discrete LQR problem F(K ) =- (D T 1 K D 1 + B T 2 XB 2 ) - 1 B T 2 XA G(K ) =- (D T 1 K D 1 + B T 2 XB 2 ) - 1 B T 2 XB a = F(K ) YC T 2 Z where X (which also depends on K equation Note that C T K Discrete-time version of the H 2 control problem 99 ) is the solution to the discrete algebraic Riccati A T XA - X- A T XB 2 (D T 1 K D 1 + B T 2 XB 2 ) - 1 B T 2 XA + C T 1 K C 1 = D 1 = has been assumed, although this condition can easily be relaxed (Zhou et al. 1996). It follows that Q n (x ) = G(K ) and that Q(x ) = Q n (x )Z = F(K ) YC2 T Z 2. The form of the controller for the d max problem is therefore a standard observer-based controller with Q(x ) contributing additional dynamics: x c (k + 1) = (A + HC 2 )x c (k) + B 2 u(k) - Hy 2 (k) u(k) = F(K )x c (k) + F(K ) YC T 2 Z 2 (y 2 (k) - C 2 x c (k)) } (18)

êê Ï êê úú ê êê úú úú êê ú úú êê úú 1 D. A. Wilson et al. Remark 1: The conventional discrete-time H2 control problem, as detailed by Zhou et al. (1996), can be recovered by setting equal to the identity matrix in equation (18). Remark 2: The control in equation (18) can be written in the form u(k) = F(K ) (x c (k) + YC T 2 Z 2 (y 2 (k) - C 2 x c (k))) = F(K )x (k) where x c (k) can be interpreted as the optimal (Kalman lter) estimate of x(k) based on the measurements up to stage (k - 1), and x (k) as the optimal estimate of x(k) based on measurements up to stage k. Remark 3: At the outset, Q(z) could have been restricted to be strictly proper, and in this case the optimal solution would be Q(z) =. The optimal control would then be u(k) = F(K )xc(k), and would not involve the measurement update term. This solution is clearly more appropriate for real-time implementation. In fact one could take the view that the use of the measurement update in equation (18) implies a non-causal controller. It should be noted, however, that the matrix F(K ) will be di erent in the strictly proper case as compared with that for the non-strictly proper case. 4. Example The following example involves a discretized version of the system taken from Rotea (1993) using a sampling time of.1 s. An ellipsoid algorithm (Boyd and Barratt 1991) with an exit tolerance of 1-9 was used. A = 1.1 1.94.948 C 1 =, B 1 =.19.387 1428.6-142.9 The optimal cost matrix is given by 25 1-3, B 2 =, D 1 = 6.667 C 2 = [1-1 ], D 2 = [.1] S = with the corresponding optimal control law.9994. -.1222..9994 -.3681 -.1222 -.3681.9994 u(k) =- 231x c1 (k) - 1948x c2 (k) - 191x c3 (k) - 8.66y 2 (k).48.9516 1-3 Note that the diagonal entries of the optimal cost matrix are all equal, and that the optimal cost is ê.9994 ê ê ê ê ê ê ê ê ê ê ê êê. If the controller is constrained to be strictly proper, the optimal cost matrix becomes

ê ú Ï S = with the corresponding optimal control law 1.115. -.1239. 1.115 -.3698 -.1239 -.3698 1.115 u(k) = 323x c1 (k) - 1951x c2 (k) - 192x c3 (k) Note that for this example the cost has increased only slightly to strictly proper controller is used. 5. Conclusions Discrete-time version of the H 2 control problem 11 ê ê ê ê ê ê ê ê ê ê ê ê êê 1.115 when a A discrete-time version of the weight selection method for the d max version of the generalized H 2 control problem has been presented. The form of the solution is similar to that in continuous-time, and, as in the conventional H 2 problem, one has the choice of using either a proper controller, or a strictly proper controller, which in general will incur some loss in performance. It is not clear at present whether the max version of the GH 2 problem can be solved using some form of weight selection. The technique of Kaminer et al. (1993) therefore remains the only option for solving this version of the problem. References Balakrishnan, A. V., 1981, Applied Functional Analysis (2nd edn) (New York: Springer- Verlag). Boyd, S. P., and Barratt, C. H., 1991, L inear Controller Design: L imits of Performance (Englewood Cli s, NJ: Prentice-Hall). Francis, B. A., 1987, A Course in H Control Theory (New York: Springer-Verlag). Gu, D. W., Tsai, M. C., O Young, S. D., and Postlethwaite, I., 1989, State-space formulae for discrete-time optimisation. International Journal of Control, 49, 1683± 1723. Kaminer, I., Khargonekar, P. P., and Rotea, M. A., 1993, Mixed H 2 /H control for discrete-time systems via convex optimization. Automatica, 29, 57± 7. Medanic, J., and Andjelic, M., 1971. On a class of di erential games without saddle point solutions. Journal of Optimization Theory and Applications, 8, 413± 43. Nekoui, M. A., and Wilson, D. A., 1996, LQR weight selection in generalized H 2 control for continuous and discrete systems using convex programming. Proceedings 13th IFAC World Congress, C, San Francisco, CA, pp. 121± 126. Rotea, M. A., 1993, The generalized H 2 control problem. Automatica, 29, 373± 385. Wilson, D. A., 1989, Convolution and Hankel operator norms for linear systems. IEEE Transactions on Automatic Control, 34, 94± 97. Wilson, D. A., 1991, Extended optimality properties of the linear quadratic regulator and stationary Kalman lter. IEEE Transactions on Automatic Control, 36, 583± 585. Wilson, D. A., and Rubio, J. E., 1995, Computation of generalized H 2 optimal controllers. International Journal of Control, 61, 99± 112. Zhou, K., Doyle, J. C., and Glover, K., 1996, Robust and Optimal Control (Englewood Ci s: Prentice-Hall).