An explicit state-space solution to the one-block super-optimal distance problem

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent repository link AbstractAn explicit state-space approach is presented for solving the super-optimal Nehari-extension problem. The approach is based on the all-pass dilation technique developed in [JL93] which offers considerable advantages compared to traditional methods relying on a diagonalisation procedure via a Schmidt pair of the Hankel operator associated with the problem. As a result, all derivations presented in this work rely only on simple linear-algebraic arguments. Further, when the simple structure of the one-block problem is taken into account, this approach leads to a detailed and complete state-space analysis which clearly illustrates the structure of the optimal solution and allows for the removal of all technical assumptions (minimality, multiplicity of largest Hankel singular value, positive-definiteness of the solutions of certain Riccati equations) made in previous work [LHG89], [HLG93]. The advantages of the approach are illustrated with a numerical example. Finally, the paper presents a short survey of super-optimization, the various techniques developed for its solution and some of its applications in the area of modern robust control.Keywords: super-optimal Nehari-extension problems, Hankel operator, all-pass dilations, H ∞ -optimal control, maximally robust stabilization. NotationHere we define the main notation used in the paper. Additional notation is introduced in subsequent sections as needed. All systems considered in this paper are assumed linear, time invariant and finite dimensional. Let R p×m (s) denote the space of proper p × m rational matrix functions in s with real coefficients. Associated with P ∈ R p×m (s) of McMillan degree n is a state-space realization:the para-hermitian conjugate of P . Throughout the paper we distinguish transfer matrices by making use of bold lettering which shall imply the s dependence. Let P be partitioned in 2 × 2 sub-blocks P ij , i = {1, 2}, j = {1, 2}. Then a state space realization of P can be written as:is a state-space realization of P ij . A lower linear fractional transformation of P and K is defined as for a compatible partitioning of P with K and provided that the indicated inverse exists.The spaces RL 2 consist of all real-rational matrix functions G(s) which are square-integrable on the imaginary axis, i.e. whose L 2 norm:is finite. This coincides with the space of all strictly proper real-rational matrix functions which are analytic on the imaginary axis. Similarly, RH 2 (RH ⊥ 2 ) denotes the spaces of all strictly proper real-rational transfer matrix functions which are analytic in closed right-half complex plane (closed left-half complex plane), respectively. We let ∥·∥ 2 stand simultaneously for the L 2 -norm, the H 2 -norm or the H ⊥ 2 -norm (for G belonging to the appropriate space). RH 2 and RH ⊥ 2 are subspaces of RL 2 and we define P + and P − to be the orthogonal projections from RL 2 to RH 2 and ...