An Algorithm for Optimizing Singular Values of Feedback Linear Multivariable Systems

Abstract

We present an algorithm for finding an x ∈ R n to (1a) minimize f x Δ – – max σ G f x,w : w ∈ Ω f (1b) s . t. g x Δ – – max σ G g x, w − b x,w : w ∈ Ω g ≤ 0 (1c) g j x ≤ 0 for j ∈ J, where σ [•] denotes the maximum singular value, G f and G g are m x m complex valued transfer matrix functions locally analytic in x, b and g j are continuously differentiable, Ω f and Ω g are finite sets of frequencies, and J is finite. Such problems arise in the design of feedback linear multivariable control systems, where x is a compensator parameter and G f and G g are the sensitivity and complementary sensitivity functions. When (lb) is absent, one may equivalently minimize f ˆ x = max h fwu x : u ∈ U, w ∈ Ω f subject to (lc), where h fwu x = < u ∈ G f x , w x G f x , w u > and U = u ∈ C m : u = 1 . At the k-th iteration, our algorithm replaces f around iterate x k by f ˆ x = max h fwu x k + < ∇ h fwu x k , x-x k > : u ∈ U, w ∈ Ω f and finds a search direction d k d by solving approximately (2) min d max f ˆ k x k + d − f x k , g j x k + ∇ x k , d > ∈ J + 1 2 d 2 . Then an Armijo-like search from x k along d k produces x k+1 . Instead of (2), we solve its semi-infinite QP formulation (3a) minimize v+ 1 2 d 2 over all v,d ∈ R 1+n satisfying (3b) h fuw x k − f x k + < ∇ h fuw x k , d > ≤ v, u ∈ U, w ∈ Ω f , (3c) g j x k + ∇ g j x k , d > ≤ v, j ∈ J by a column generation technique from [1] that requires eigenvalue analyses for evaluating max h fuw x k + ∇ h fuw x k , d > : u ∈ U for trial d. An efficient dual method is described for solving the resulting finite sequence of QP subproblems with only n+2 constraints (3b,c). Also ways of using only sections of U in (3b) to reduce computational effort are given. To treat (lb), we include constraints similar to (3b) in (3). The algorithm is both readily implementable and globally convergent to stationary points of problem (l). We describe an implementation of the method and give test results for several problems of the design of compensators for multivariable systems. We also discuss ways of handling the case when an original formulation of (1) involves infinite Ω f and Ω g . The algorithm presented is more widely applicable than the one in [3], and exploits.the structure of (1) differently than does the method of [2].