Optimal boundary interpolation and one-block optimal control problem with invariant zeros on the imaginary axis and infinity

Abstract

We formulate a matrix interpolation problem with existing interpolation points on the imaginary axis and infinity and existing equal left and right interpolation points, using the concept of parametrisation of stabilising controllers. Then, we solve the problem of obtaining all its solutions. If interpolation points at infinity are absent, we show that the introduced problem is equivalent to the existing one. We apply this result to solve the problem of optimal interpolation with existent interpolation points on the imaginary axis and infinity. We show by an example that the solution of optimal interpolation is directly applicable to the one-block H ∞ optimal control with existent invariant zeros on the imaginary axis and infinity. It is seen from the example that not only the transfer matrix of the closed-loop system is constrained on the extended imaginary axis, but also its derivatives.

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

“…Indeed, with γ = 1, the Pick matrix for the reduced interpolation problem, obtained by avoiding the interpolation conditions on the imaginary axis, is positive definite. The same argument for the existence of a minimum of the interpolation problem has been used in Theorem 1.6 of [2] and Section 5 of [22]. In the opposite case, i.e.…”

“…Define also the positive numbers [22]), when the minimum exists and equals β. With γ = β and U 1 = U /γ, form an interpolation problem with matrices A and C corresponding to the points not on the imaginary axis, and those conditions with points on the imaginary axis such that the corresponding minimum of the single-condition-problem equals β.…”

“…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…”

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

“…Let x be a complex vector such that x(E 21S1 + E 22 ) = 0. We obtain xE 22 Further we have to prove thatS is stable contractive if S is stable contractive. It suffices to prove thatS 1 is stable contractive.…”

“…then by (4.30), jω must be a point such that G22 (jω)[ 0 N1 ] = 1. Therefore, by G * 22 G 22 = I − G * 12 G 12 , jω must be a point such that the rm G 12 [ 0 N1] has no full column rank at jω, i.e., matrix pencil (4.3) has no full column rank at jω.…”