A GAIN MATRIX DECOMPOSITION AND SOME OF ITS APPLICATIONS

Any real square matrix M can be written as M = U (I + L)S, where U is a matrix of 0's, I's and - I's having exactly one nonzero element in each row and column, L is a strictly lower triangular matrix, and S is a {symmetric}, positive-semidefinite matrix. The aim of this paper is to demonstrate the utility of this easily derived fact. This is done in two ways. First, the decomposition is used to develop an identifier-based solution to a simplified multivariable adaptive stabilization problem solved previously using nonidentifier-based methods. Second, it is briefly explained how to use the decomposition together with. hysteresis switching and a certain ''lifted'' discrete-time system representation, to obtain an excitation-free, identifier-based, adaptive stabilizer for the entire class of n-dimensional, siso, controllable, observable, discrete-time linear process models. This is accomplished by exploiting a new method of discrete-time parameter adjustment called ''pseudo-continuous'' tuning.