BUMPLESS SWITCHING CONTROLLERS. William A. Wolovich and Alan B. Arehart 1. December 27, Abstract

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1 BUMPLESS SWITCHING CONTROLLERS William A. Wolovich and Alan B. Arehart 1 December 7, 1995 Abstract This paper outlines the design of bumpless switching controllers that can be used to stabilize MIMO plants which are both switching stabilizable and switching detectable. A primary feature of such controllers is that their order is no greater than that of the plant. 1 Introduction Consider a linear, time-invariant MIMO dynamical system or plant dened by the state-space representation: _x(t) = Ax(t) + Bu(t); y(t) = Cx(t); (1:1) with A < nn, B < nm and C < pn, which is characterized by the open-loop transfer matrix T yu (s) = C(sI? A)?1 B Such plants will be called switching systems if one or more of the elements or parameters which comprise the switching matrices A, B and C can asynchronously switch from one xed value to another. The (nite) set of all possible xed plants, or fa; B; Cg triples which belong to a particular switching system, will be called its switching group. In particular, if A i (for i = 1; ; : : :; ), B j (for j = 1; ; : : :; ) and C k (for k = 1; ; : : :; ) denote all possible xed values of A, B and C in a particular switching system, then the ( ) triples fa i ; B j ; C k g will dene its switching group. To illustrate the preceding, a sudden failure of (say) only input u q (t) could be modeled by a switching system where all of the q-th column elements of B instantaneously switch to. In such a case, two xed plants fa 1 ; B 1 ; C 1 g and fa 1 ; B ; C 1 g would dene its switching group. More generally, if a switching system were dened by the condition that any (but only one) of its m > 1 inputs could fail, its switching group would consist of exactly m + 1 xed plants, with both A = A 1 and C = C 1 xed, and B = B j, for j = 1; ; : : :; m + 1 = Laboratory for Engineering Man/Machine Systems, Box D, Brown University, Providence, RI 1

2 A switching system will be called bumpless if its output y(t) remains a continuous function of time when u(t) is continuous, despite all possible parameter switches. Clearly, a switching system will be bumpless if the output matrix C remains xed, and only the parameters of A and B (which appear before the state integrators) can switch values. A switching system will be called switching controllable if all members of its switching group are (completely state) controllable. Switching observability, switching stabilizability and switching detectability are dened in an analogous manner. It is of interest to note that xed controllers sometimes can be found that stabilize more than one member of a switching group. Such would be the case, for example, if a single xed controller were to simultaneously stabilize two or more xed plants[1][], all of which belong to the switching group. Morse and Pait have done extensive work on switching controllers, which are switching systems used as feedback controllers [5] [4] [6] [7]. In [6], they discuss the design of bumpless switching controllers for MIMO plants, noting a need to reduce the order of such controllers. This paper will address this issue. n-th Order Non-Bumpless Switching Controllers If a switching system is both switching stabilizable and switching detectable, then it is well known that ( ) and ( ) switching gain matrices, namely G ij and K ik, can be determined which arbitrarily assign all of the controllable eigenvalues of A i + B j G ij and all of the observable eigenvalues of A i + K ik C k for all switching group members. In such cases, both A i +B j G ij and A i +K ik C k can be stabilized in the sense that all of their eigenvalues will lie in the half-plane Re(s) <. Therefore, a switching observer, as dened by the state-space representation: _~x(t) = (A i + K ik C k )~x(t) + B j u(t)? K ik y(t); u(t) = G ij ~x(t) + r(t); with r(t) an external reference input, would then imply a (generally non-bumpless) switching compensator, as dened by the transfer matrix T uy (s) =?G ij (si? A i? B j G ij? K ik C k )?1 K ik ; (:1) and a corresponding family of closed-loop systems dened by the transfer matrices T yr (s) = C k (si? A i? B j G ij )?1 B j jsi? A i? K ik C k j jsi? A i? K ik C k j = C k(si? A i? B j G ij )?1 B j

3 Switching Conversion Matrices If all ( ) of the (m n) switching gain matrices G ij are of full rank m, then for any xed (m n) gain matrix F, also of full rank m, ( ) non-unique, but nonsingular (n n) switching conversion matrices P ij can be determined such that G ij = F P ij (:1) One way to determine such P ij is to rst determine any m linearly independent columns of G ij and F, which will be dened as G mij and F m, respectively. The permutation matrices P Gij and P F could then be used to reorder the columns of G ij and F so that G ij P Gij = [ G mij G rij ] and F P F = [ F m F r ] A subsequent choice of F?1 m M ij = G mij F m?1 [G rij? F r ] I (:) would then imply that G ij P Gij = F P F M ij, or that G ij = F P ij for P ij = P F M ij P?1 Gij (:) Note that if F = [I ] = [F m F r ], then P F = I. Moreover, if the rst m columns of all G ij = [G mij G rij ] are linearly independent, then P Gij = I as well. In such cases, (.) and (.) will imply that Gmij G P ij = M ij = rij (:4) I 4 n-th Order Bumpless Switching Controllers We rst observe that F P (si? A)?1 B = F (si? P AP?1 )?1 P B Therefore, when a switching permutation matrix P ij is determined such that (.1) holds, the switching compensator dened by (.1) also can be dened by T uy (s) =?F (si? P ij [A i + B j F P ij + K ik C k ]P?1 ij )?1 P ij K ik ; (4:1) Nonsingular matrices which have a single nonzero entry of 1 in each row and column.

4 and implemented in the \standard" observer conguration with T ur (s) = F (si? P ij [A i + B j F P ij + K ik C k ]P?1 ij P )?1 ij B j + I This \replacement" of G ij by F P ij, with F xed, converts a generally non-bumpless compensator dened by (.1) into the bumpless compensator dened by ( 4.1). An Example of Integrity To illustrate the preceding, consider a two-input state-space system dened by the matrix pair? 1 A = 4?1 1 5 and B = [ b 1 b ] = ; 1? which is (completely state) controllable using either input alone. We assume that either of the two inputs can fail suddenly with a resultant, instantaneous switch of B = B 1 to either B = [ b ] = 4 5 or B = [ b 1 ] = ; 1 4 the (= ) values of B j (with A = A 1 xed) which dene the switching group in this case. We next determine (see the Appendix) that if the switching gain matrix?1?1 G 11 = = [ G?17 8? m11 G r11 ] switches to G 1 = G 1 =?8 8?11 = [ G? 11?6 m1 G r1 ] = [ G m1 G r1 ] ; if either input fails, then the eigenvalues of A 1 +B 1 G 11, A 1 +B G 1 and A 1 +B G 1 will remain at?1 and?1 j in the complex plane. Moreover, G m11 and G m1 = G m1 have full rank in this case. Therefore, if we choose (a xed) 1 F = = [ F 1 m F r ] = [ I ] 4

5 then (.4) will imply that in the event of a failure of either input, a switch of to P 11 = M 11 =?1?1 G r11 = 4?17 8? 5 I 1 Gm11?8 8?11 P 1 = P 1 = M 1 = M 1 = 4? 11?6 5 ; 1 and a corresponding switch of B 1 to either B or B will insure input integrity[]; i.e. a closed-loop system which remains stable despite the failure of either input. 5 General Compensators We next show that any two strictly proper loop compensators of minimal dynamic order n 1 and n, which have the same number m of (linearly independent) outputs always can be switched in a bumpless manner. In particular, suppose T uy1 (s) = C 1 (si? A 1 )?1 B 1 and T uy (s) = C (si? A )?1 B dene the transfer matrices of the compensators in terms of their state-space representations. If we assume for convenience that n 1 n, then n 1? n = ~n uncon- def trollable and unobservable (stable) modes can be added to the second compensator by appropriately increasing the dimensions of A, B and C. In particular, ~n zero columns (rows) can be added to to C (B ) and a stable (~n ~n) lower right \block" can be added to A to dene C ~ ( B ~ ) and A ~, respectively, which would imply a state-space system of order n 1 with transfer matrix T uy (s). A nonsingular matrix P could then be determined, as in (.), so that C ~ = C 1 P, thereby implying that T uy (s) = ~ C (si? ~ A )?1 ~ B = C 1 (si? P ~ A P?1 )?1 P ~ B Since the output matrix of both compensators would be the xed matrix C 1, either compensator could be switched to the other while maintaining a continuous output. APPENDIX By switching A 1 and B 1 to P ~ AP?1 and P ~ B, respectively, or vice-versa. 5

6 Consider any controllable system dened by ( 1.1) with B = [ b 1 b : : : b m ] of full (column) rank m < n. If we lexicographically reorder the rst n linearly independent columns of the controllability matrix C = [B AB : : : A n?1 B], as in [9], we will dene the (n n) nonsingular matrix C nn = [b 1 Ab 1 : : : A d 1?1 b 1 b Ab : : :A d?1 b : : :A dm?1 b m ]; the m controllability indices d i, and the (m) integers k = P k d 1 k, so that 1 = d 1, = d 1 + d, etc., with m = d 1 + d + : : : d m = n. If the (m) row vectors q k are then dened as the k rows of C nn,?1 def ^B m = 6 4 q 1 A d 1?1 B q A d?1 B. q m A dm?1 B will be a nonsingular, upper right triangular matrix, with all diagonal elements equal to 1, so that j ^Bm j = 1. If we choose m arbitrary stable monic polynomials i (s), each of degree d i, and dene the state feedback gain matrix G =? ^B?1 m 6 4 q 1 1 (A) q (A). q m m (A) it then follows[8] that the (n) eigenvalues of A + BG will correspond to the roots of (s) def = Q m 1 i(s). It might be noted that in the single-input (m = 1 and B = b) case, a single vector q would be dened as the last row of C?1. For any stable monic polynomial (s) of degree n, a choice of g =?q(a) 4 would then imply that the (n) eigenvalues of A + bg will correspond to the roots of (s) ; References [1] A. B. Arehart and W. A. Wolovich, \A Nonlinear Programming Procedure and a Necessary Condition for the Simultaneous Stabilization of or More Linear Systems," Proc. IEEE Conf. on Decision and Contr., Which is known as Ackermann's formula[1]. 6

7 [] Blondel, Vincent, Simultaneous Stabilization of Linear Systems, Springer-Verlag Lecture Notes in Control and Information Sciences 191, [] Davison, Edward J., \The Design of Controllers for the Multivariable Robust Servomechanism Problem Using Parameter Optimization Methods," IEEE TAC, Vol. 6 (1), February, [4] A. S. Morse, \Towards a Unied Theory of Parameter Adaptive Control{Part : Certainty Equivalence and Implicit Tuning," IEEE Trans. Automat. Contr., vol. 7, pp. 15{9, Jan [5] A. S. Morse, \Control Using Logic-Based Switching," Proc. European Contr. Conf., [6] A. S. Morse and F. M. Pait, \MIMO Design Models and Internal Regulators for Cyclicly Switched Parameter-Adaptive Control Systems," IEEE Trans. Automat. Contr., vol. 9, pp. 189{1818, Sept [7] F. M. Pait and A. S. Morse, \A Cyclic Switching Strategy for Parameter- Adaptive Control," IEEE Trans. Automat. Contr., vol. 9, pp. 117{118, Jun [8] J. Wang and Y. Juang, \A New Approach for Computing the State Feedback Gains of Multivariable Systems," IEEE Trans. Automat. Contr., vol. 4, pp. 18{186, Oct [9] W. A. Wolovich, Linear Multivariable Systems. Springer, [1] W. A. Wolovich, Automatic Control Systems: Basic Analysis and Design. Saunders,