An algebra for the analysis of linear stationary and non-stationary multivariable systems using the time-domain infinite matrices†

Abstract

A brief survey is given for the method of impulse analysis of scalar systems to show the distinct advantages of time-domain infinite matrices (t.d.i.m.) methods. To tackle the problem of describing and manipulating multivariable systems by these techniques. an extension of the algebra of lower semi-infinite matrices to the multivariable case is developed. Of special importance in this respect are now recursion formulae for inverting stationary and non-stationary impulse response matrices of any dimensions. It is shown that for stationary matrices of dimensions n× n × ∞. the adjoint matrix has to be evaluated at the first sampling instant only. In terms of this evaluation the inverse matrix is generated at any sampling instant by carrying out simple algebraic operations (i.e. addition and multiplication). For non-stationary matrices similar conditions arc derived. The formulae of this paper are useful for the solution of a variety of problems in functional analysis.

A numerical example is given to illustrate the use of the algebra developed in this paper. The results are compared with those obtained by using the variation of parameters and Laplace methods.

Notes

†Communicated by the Author.