${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

Abstract

For given rational matrices V a, U a, V b , U b , we find necessary and sufficient conditions for existence of a stable rational matrix Φ satisfying Φ ∞ ≤ 1, V aΦ = U a, and ΦV b = U b . A condition is the positive semidefiniteness of a matrix, denoted by R. We present a parametrization of all problem solutions. A property of the proposed algorithm is, as a first step, to reduce the problem to a minimal realization, by an orthogonal transformation. Another property is the ability to transform the problem into one with constant matrices, by another orthogonal transformation. A problem motivation is the optimal H∞ control problem of descriptor systems. We show by an example that the existing numerical H∞ control optimization algorithms, which solve the problem of obtaining stabilizing controllers such that Φ ∞ ≤ γ, where Φ is the closed loop transfer matrix, and γ is slightly greater than the optimal one, compute controllers such that the closed loop system suffers from lack of stability robustness. Our proposed algorithm doesn't require γ to be greater than the optimal one, and the closed loop system has the property of stability robustness. Consequences of the singularity of matrix R, which property generically appears for the optimal γ, are (1) that Φ(jω) = γ for all real ω and all optimal controllers, and (2) if the measured output is single, or the control input is single, then the H∞ controller is unique. A numerical example is presented to illustrate the H∞ control optimization algorithm. Introduction.Consider rational matrices (rm's) V a (s) ∈ R(s) μ×m , U a (s) ∈ R(s) μ×r , V b (s) ∈ R(s) r×ρ , and U b (s) ∈ R(s) m×ρ , where by R(s) μ×m we denote the set of rm's with real coefficients, of dimension μ × m, which are possibly improper. We state the following problem.Problem 1. Find a proper stable rm Φ(s) ∈ R(s) m×r such that Φ ∞ ≤ 1 and the following conditions are satisfied:

Almost fault‐tolerant tracking

“…Let H 2 (s)x = 0, for some s in ℜ[s] ≥ 0 and complex vector x. By algebraic manipulation with the realization (14) of H 2 , we obtain that x = Ky, where the vector y satisfies Ay = sy and  y = 0. Using iteratively the left formula in (B4), starting with i = − 1, and the condition > 0, we obtain C 1 y = 0, and y = 0 as a consequence of the detectability of (A, C 1 ).…”

“…We can restrict Y so that ‖ Y ‖ ∞ =‖ T ur ‖ ∞ is minimal. Then we have to find a proper and stable RM Y such that Two algorithms to solve the latter problem in the unknown proper stable Y are given in References and .…”

A reformulation of augmented basic interpolation problem and an application to H∞ control

“…Besides to the general bitangential interpolation problem, which can be regarded as a discrete problem, because of the interpolation points, the aBIP can be applied to the following "continuous" problems [21] (without "discrete points" in their formulations):…”

“…Introduce the matrices C 1F = C 1 − D 12 F, B 1L = B 1 − LD 21 , and B 2L = B 2 − LD 22 , and the following proper and stable rm's…”

Input Estimation Over Frequency Region in Presence of Disturbances

“…Problem 2 is elaborated in [10] and [13], by considering the FE filter realization in an observer form, and formulating a matrix inequality that includes the unknown coefficients of the observer. It can also be solved by the algorithm of [18].…”