Generalized Discrete-Time Riccati Theory

Abstract

A Riccati-like equation, termed the generalized (discrete-time) algebraic Riccati equation, which incorporates as special cases both the standard and the constrained discrete-time algebraic Riccati equations, is introduced and investigated under the weakest possible assumptions imposed on the initial data. A complete characterization of the conditions under which such an equation of general form has a stabilizing solution is presented in terms of the so-called proper deflating subspace of the extended Hamiltonian pencil. An evaluation of an associated quadratic index along constrained stable trajectories is given in terms of the stabilizing solution to the generalized Riccati equation. Possible applications of the developed theory range from nonstandard spectral and inner-outer factorizations to $H^2 $ and $H^\infty $ control in singular cases. The results exposed in the present paper are the discrete-time counterpart of those stated in the authors’ previous paper concerning the generalized (singular) continuous-time Riccati theory. The results could be also seen as an extension to singular cases of the usual discrete-time algebraic Riccati equation theory (of indefinite sign).