A derivation of the Glover-Doyle algorithms for general H∞ control problems

We show that the Glover-Doyle algorithm can be formulated simply by using the (J, J’)-lossless factorization method and chain scattering matrix description. This algorithm was first stated by Glover and Doyle in 1988. Because the corresponding diagonal block of the (J,J’)lossless matrix in the general 4-block H” control problem of the Glover-Doyle algorithm is not square, a new type of chain scattering matrix description is developed. With this description in hand, we obtain two types of state-space solution, which are similar to each other. Thus a similarity transformation between these solutions in the 4-block H” control problem can also be obtained. The main idea of the solution is illustrated by means of block diagrams. 1. Introducdon Since Zames (1981) proposed the concept of sensitivity minimization in the H” domain, many researchers have made valuable contributions to the study of the H” domain. Transparent controllers for the standard 4-block H” problem were not obtained until Glover and Doyle (1988,1989) developed their well-known dual GD algorithms. After Glover and Doyle (1989), Green et al. (1990) and Kimura (1991a) offered alternative developments using a J-spectral factorization, a characteristic of a (J, J’)-lossless matrix. These methods are all based on the model-matching problem. Green (1992) combined an analytic system with J-lossless factorization to solve the H” control problem, which gradually yielded a problem in the form of the model-matching problem. Using (J, /‘)-lossless factorization and a chain-scattering matrix description, Kimura (1991b) and Ball et al. (1991) gave a fictitious signal method for solving the 4-block case of the problem. Furthermore, Kondo and Hara (1990) and Tsai and Tsai (1993) obtained results similar to those of Green (1992). However, in these papers the (1,l) block or the (2,2) block of the (J, J’)-lossless matrix is required to be square or to need additional fictitious signals. Consequently, the results obtained by using the (J, J’)-lossless factorization method to solve the H” control problem were not the same as those obtained by the Glover-Doyle algorithms. In this paper we combine a normalized coprime factorization of the plant and (J,J’)-factorization of one of the coprime factors, together with an alternative type of chain matrix description to recover precisely the results of Glover and Doyle (1988) (by using a left-coprime factorization) and Glover and Doyle (1989) (by using a right-copime factorization). Despite the specific features of the two cases, the transfer functions for the resulting compensators turn out to be the *Received 24 March 1994; revised 23 February 1995; received in final form 4 September 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. Tempo under the direction of Editor Ruth F. Curtain. Corresponding author Professor Ching-Cheng Teng. Fax +886 (035) 715998. IIF,(P, K)II= < Y, Y E R+. For simplicity and without loss of generality of the derivations in subsequent sections, we let IIF,(P,, K)llx < 1 instead of IIF,(P, K)112 < y, i.e. F,(P,, K) =;F,(P, K) =;P,, ++P,,K(I -PzZK)-‘P2,. Figure 1 shows a general set-up for linear fractional transformation (LFT). t Institute of Control Engineering, National Chiao-Tung The assumption of the standard 4-b&k H” control University, Hsinchu, Taiwan. problem are as follows. same. We also obtain an explicit state-space similarity between the realizations for the two compensators thus obtained. In Section 2 we briefly state the standard H” control problem. The (J, J’)-lossless, conjugate (/, J’)-lossless and conjugate (J, J’)-expansive matrices are also discussed. In Section 3 we develop alternative chain-scattering matrix descriptions, and discuss their chain properties. In Section 4 the relationship between the H^ control problem and the chain scattering matrix description is stated. The main results and the solution are presented in Section 5. 2. Notation and preliminaries Throughout this paper R denotes the real numbers, RL” denotes the set of proper real rational function matrices with no pole on the jw axis, and RH” denotes the RL” subspace with no poles in the right half-plane. Furthermore, F?H” denotes the units of RH’ (i.e. if @ E SH” then CD E RH” and a-’ E RH”) and BH”:= {@ E RH” 1 [[@[lx < l}, yBH' := {@ E RH” 1 ll@llX < y}. dom (Ric) denotes the set of Hamiltonian matrices with no pure imaginary eigenvalues, and Ric (H) is the unique solution of the corresponding ARE of the Hamiltonian matrix H. G-(s) denotes GT( -s) and G*(s) A B denotes CT@). As usual, the packed form c D is eauivalent to C(sl A)-‘B + D. [-+I i.1. The stan‘dard i-block H’ control problem. In the standard H” framework, the transfer functions from [:lta[:l are denoted by where z(t) E UPI, y(r) E W, w(t) E WI, and u(r) E R”‘2 are the error, observation, disturbance and control input respectively. The suboptimal H” control problem is then modeled so as to choose a controller K, connecting the observation vector y to u, such that K internally stabilizes the closed-loop system. Furthermore, the closed-loop transfer function, denoted by F,(P, K) ~4 P,, + P,,K(I &K)-‘P2,, satisfies the H” norm bound