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 of0's, l's and l's having exactly one nonzero element in each row and column, L is a strictly lower triangular matrix, and S is a {symmetric}, positivesemidefinite matrix. The aim O f 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 withhysteresis 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 discretetime parameter adjustment called "pseudo-continuous" tuning.