## DESIGN OF LINEAR MULTIVARIABLE CONTROLLERS

##### Author

CHANG, BOR-CHIN

##### Date

1984##### Degree

Doctor of Philosophy

##### Abstract

In this thesis, a recently developed computational method for the design of linear multivariable controllers is presented. For a given linear time-invariant multi-input-multi-output system, our objective is to find a realizable controller such that: (1) the closed-loop system is internally stable, (2) tracking/regulation takes place, and (3) the system response due to noise/disturbance is minimized subject to some stability-margin and control-input constraints, where the noise/disturbance is unknown but belongs to a prescribed set. The notion (2) with (1) is the so-called regulator problem with internal stability (RPIS). The notion (3) is formulated as the H('(INFIN))-norm optimization problem which was initiated by Zames in 1981. The solution to RPIS will guarantee a good steady-state response and the solution to H('(INFIN))-norm optimization problem will give an optimal transient response. By Cheng and Pearson's approach, we will find a quite general set of controllers which solve RPIS. Furthermore, among this set of controllers, we will find one such that the H('(INFIN))-norm optimization problem is solved.
All the computational algorithms for constructing the controllers are well developed and therefore the design can be implemented on the computers. These algorithms are used for solving the following computational problems: the construction of the unimodular matrices in the generalized Bezout identity, the solution of the polynomial equations A(s)X(s) + B(s)Y(s) = C(s) and A(s)X(s) + Y(s)B(s) = C(s), the inner-outer factorization of a rational matrix, and the solution to a generalized interpolation problem.
Our main tool for solving the first two computational problems in the previous paragraph is a modification of the Kung-Kailath algorithm. This modification allows us to solve several important polynomial equations arising in multivariable control theory.
The problem of the matrix inner-outer factorization arises in H('(INFIN))-norm optimization problem. This factorization is accomplished by using some simple transformations, polynomial matrix manipulations, and Bauer's algorithm.
The H('(INFIN))-norm optimization problem can be transformed to a generalized interpolation problem. This interpolation problem is then divided into two parts: first, computation of the minimal H('(INFIN))-norm and second, the construction of an optimal transfer function matrix. The first part of the problem is simplified to a generalized eigen-value problem by using Sarason's theory and the second part of the problem can be solved by the matrix Nevanlinna algorithm.

##### Keyword

Electronics; Electrical engineering