Polynomial approach to H∞ control problem with additional constraints

Abstract EN

The H∞control problem for linear stationary systems was studied in detail in a great number of articles beginning from the work by G.

Zames [1].

Nowadays different approaches to its solution were proposed.

They include spectral methods [2], state-space solution [3], polynomial approaches [4, 5].

All these methods are based on the main procedure which solves one the following equivalent problems: Nehari problem in the spectral method, J-spectral factorization in polynomial approaches and Riccati equation for the state-space solution.

We show in this article that only standard spectral factorization is needed for to solve the standard problem of minimization of nonnegative cost function in the case of full information.

The polynomial approach proposed below is based on a polynomial analogue of the Riccati equation.

This approach allows one to fulfil all operations with initial polynomial matrices only, without their preliminary transformations into state-space form or different factorizations.

The problem is solved also for the general case in which the quadratic form in the cost function is not assumed to be nonnegative.

In the first part of this paper the conditional problem of H°°-minimization is considered.

It is reduced to H°° control problems with parameters but without additional restrictions.

The method of this reduction is based on so-called S-procedure which was previously used in the absolute stability theory [6].