Singular H 2 -optimization problems are considered for the standard discrete-time control system. Two types of singularity (type I and type II) are distinguished. A detailed treatment of problems with singularity of type II, which leads to nonuniqueness of solution, is presented. New algorithms for design of optimal controllers are presented both in frequency domain and state space, which generalize standard procedures onto the case of singular H 2 -problems. A parameterization of the set of optimal controllers is given.
STATEMENT OF THE PROBLEMMany problems of digital control for discrete-time and continuous-time dynamic plants can be formulated as problems of minimizing the H 2 -norm of the transfer matrix of an equivalent standard discrete-time control system [1]. In existing standard time-domain [1-3] and frequencydomain [4][5][6][7][8][9][10][11] solutions of these problems it is assumed that some non-singularity assumptions hold, which ensure that the solution does exist and is unique. Nevertheless, there are important applied problems where these conditions are violated [1,5,12].Consider the H 2 -optimization problem for the standard discrete-time system shown in Fig. 1. Here w, z, u, and y denote vectors of disturbance, output, control and measurement, respectively. Linear discrete-time plant is described by four transfer matrices that define its operator equations 1 :(1)If all the matrices G ij (ζ) (i, j = 1, 2) are rational and causal (i.e., lim ζ→0 G i,j (ζ) = ∞), an equivalent state-space model has the formwhere x is the state vector, q is the forward shift operator, and A, B 1 , B 2 , C 1 , C 2 , D 11 , D 12 , and D 21 are constant matrices of corresponding dimensions. Feedback controller K should stabilize the plant and minimize the H 2 -norm of the closed-loop transfer matrix from input w to output zdefined by T zw (ζ) 2 = 1 2πj tr T * zw T zw dζ ζ 1/2 , 1 Hereinafter ζ denotes the backward shift operator used in modern literature on discrete control [10, 13-15]. 2 Matrix argument are omitted for brevity.