A novel optimization approach to robust deadbeat control is proposed. The design problem is cast into a convex programming task in which a special measure of closed-loop eigenvalue sensitivity is minimized. Advantages of the proposed method include: (1) global optimality is guaranteed when the solution set is nonempty; (2) constraints on the feedback gain can be catered naturally; (3) minimum-gain deadbeat control design can be readily treated.published_or_final_versio

Lam, J

Tam, PHK

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Title LMI solution to gain-constrained robust deadbeat poleassignmentAuthor(s) Tam, PHK; Lam, JCitation American Control Conference, Philadelphia, USA, 24-26 June1998, v. 1, p. 97-98Issued Date 1998URL http://hdl.handle.net/10722/46642Rights©1998 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising orpromotional purposes or for creating new collective works forresale or redistribution to servers or lists, or to reuse anycopyrighted component of this work in other works must beobtained from the IEEE.Proceedings of the American Control Conference Philadelphia, Pennsylvania * June 1998 LMI Solution to Gain-constrained Robust Deadbeat Pole Assignment Hei Ka Tam James Lam* Department of Mechanical Engineering University of Hong Kong Pokfulam Road, HONG KONG Abstract A novel optimization approach to robust deadbeat control is proposed. The design problem is cast into a convex programming task in which a special measure of closed-loop eigenvalue sensitivity is minimized. Ad- vantages of the proposed method include: (I) Global optimality is gualranteed when the solution set is non- empty; (2) Constraints on the feedback gain can be catered naturally; (3) Minimum-gain deadbeat con- trol design can he readily treated. 1. Introduction Consider a linear time-invariant MIMO discrete time system xt+i = .Axt + But , t O,1,2 , .  . . (1) where A E Itnx7’,  B E Rnxq, xt E RnX1, ut E Rqxl. Assume B is of full column rank while (A,  B )  is reach- able with reachability indices IC = IC1 L k2 L . . . 1 kq. By applying a time-invariant state-feedback law U t  = Kxt (2) (3) there results the closed-loop system given by xt+i = ( A  + BK)xt The reachability of (A ,  B )  implies that the spectrum of the closed-loop state matrix M Li A + BK can be assigned to any arbitrary set of self-conjugate complex numbers by proper choice of the feedback gain matrix K E Rqxn. It is often desirable to assign the set of closed-loop poles at the origin of the complex plane. In such case, the closed-loop system exhibits dead- beat characteristics, with which zero-input response of the system dies down to zero in finite time steps. Furthermore, in order for a system under Manimum- tame Deadbeat Control (MTDC), that is, the regula- tion time to drive every initial state of the closed-loop system to the origin is made shortest, M is made sim- ilar to the block-diagonal matrix J = diag(JI,Jz,. . . , Jq)  (4) where Jp is a Jordan block of dimension kp (for kp = 0, Jp  does not appear in J ) .  * Supported by HKU CRCG Grant 0-7803-4530-4/913 $10.00 0 1998 AACC 97 By observing that M is nilpotent with order of nilpotence equals k ( Mk = 0 ) and generally has a nontrivial Jordan structure, it is especially suscepti- ble to spectral variation as a result of perturbation. As the feedback gain to achieve pole assignment for multi-input systems is in general non-unique, it is par- ticularly meaningful for the non-uniqueness to be ex- ploited in searching for a more robust eigenstructure. 2. Affine Parametrization of MTDC Feedback Gain The idea of explicitly parametrizing the class of dead- beat regulators proposed by h a h a s h i  et al. [2] is borrowed. Initially, (A ,B)  is transformed via S E j p x n  into ’ the Luenberger canonical form yt+i = SAS-lyt + SBut , Y t  = Szt (5) where SAS-’=J+EA , S B = E B  (6)  in which E = blockdiag(el,e2,. . . ,eq) E Rnxq ei = (0. .  . o llT E R~~~~ a E Rqxn and B E W Q x q  are determined uniquely by (A,  B).  The transformation (6)  results in the closed- loop state matrix being transformed into S(A + BK)S- l= J + E R  where From Funahashi et al. [2] , it is not difficult to observe that K can be expressed as K = A + BKS-l E Rqxn (7) 4 T-1 k- where a ( s )  = ICp in which GT,,+,(,) denotes a matrix in Rqxn whose elements equal zero except the (r, w + ~ ( s ) )  element equal to one. It follows from (7) and (8) that N K(w)  = KO + C20iKi  , w = (201..  .  ON)^ (9) i=l where KO = -(B)-'As Kt = ( ' ) - lGT,V+O(S)s > w Z  = YT,'U+O(S)(lO) with the indices r ,  s,  U ,  and i being related by T-1 j-1 i = CC(kl - k j )  + (V + ~ ( s ) )  - sk, (11) j=2  1=1 N = nq - C;=,(2p - l)kp is the minimum number of independent free parameters wi in parametrizing K.  3. Measure of Robustness Recently, Lam et al. [3] have shown that the spectral norm of M serves as a special measure of eigenvalue sensitivity for deadbeat control systems. In the fol- lowing, llMll2 and llMllmax denotes respectively the spectral norm and the maximum norm of M ( maxi- mum absolute value of the entries in M ). Theorem 1 Suppose M is subjected to a perturba- tion Q where X is an eigenvalue of M + Q, then we have: (I) and (11) 1x1 5 (1 + ~ ~ l l M k - l l l ~ l l Q l l ~  5 (1 + ~ ~ l l ~ l l ~ l l ~ l l ~  (13) where E = O(lX1) + 0 as llAll2 + 0. 0 It turns out that variations on the perturbed closed- loop eigenvalues would be small if I IMI 12 is minimized. 4. Robust Pole Assignment The Gain-constrained Robust Deadbeat Pole Assign- ment (GCRDPA) problem is now formulated as find- ing a state feedback gain K of constrained size to as- sign the set of closed-loop poles into the origin of the z-plane and at the same time minimize the sensitivity of the closed-loop zero eigenvalues. The problem is cast into a constrained minimization task: minimize 4(w) = IIA+BK(w)ll2 (14) w € m Q  where and Since $(w) together with P ,  Q, and P n Q are all convex in w, (14) can be readily solved by state-of- the-art convex optimization routines. Moreover, (14) P i5 {w I II~K(w)112 < Q} Q 5% {w I IIK(w)llmax < P )  (15) (16) can be recast into a Semidefinite Programming (SDP) task in which a functional z is minimized subject to a set of Linear Matrix Inequality (LMI) constraints [l] : minimize z (17) W subject to and -pC < h +  Fw < PC where F = ( vec(K1) v e c ( ~ 2 )  . . . vec(KN) ) E ~ ( q ~ ~ ) ~ ~ ,  h = vec(K0) E R(qxn)xl ,  < = ( 1 . .  . l)T E R(qxn)xl ,  and vec( .) denotes the lexicographical ordering of the elements of a matrix. Notice that (20) is a pair of polyhedron constraints which can be represented as (20) diag(P6 - h - Fw) > 0 diag(pC + h + Fw) > 0 (21) (22) and It turns out that if (18), (19), and (20) are feasible ( i.e. P n Q # 0 ), the solution produced by (17) will be globally optimal. Remark 1 for  a! and /3 to achieve deadbeat pole assignment can be computed using the following SDP settings. I n  (1 5)  and (1 6), the greatest lower bound To determine inf, a: minimize W a! (23) subject to To determine i d w  p: minimize W p (25) subject to and Notice that (23) and (25) by themselves serve as com- putational procedures for solving the Minimum-gain Deadbeat Pole Assignment (MGDPA) problem. 0 diag(PC - h - Fw) > 0 diag(p6 + h + Fw) > 0 (26) (27) References [l] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakr- ishnan. Linear Matrix Inequalities in System and Control Theory, volume 15. SIAM Studies in Ap- plied Mathematics, 1994. [2] Y .  Funahashi and M. Yamada. Explicit parame- terization of state deadbeat controllers. I.E.E. E. Bans. Automatic Control, 37(10):1584-1588, 1992. [3] J. Lam, H. K. Tso, and N. K. Tsing. Robust dead- beat regulation. Int. J .  Control, 67587-602, 1997. 98 

LMI solution to gain-constrained robust deadbeat pole assignment

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LMI solution to gain-constrained robust deadbeat pole assignment

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