Geometric Theory for the Discrete Algebraic Riccati Equation

Abstract It is to be shown in Chapter 16 that, to solve the “linear-quadratic regulator” (or LQR) problem for a time-invariant differential system, it is necessary to solve an algebraic Riccati equation of the kind discussed in Part II of this book, equation (7.2.1) for example. The discrete analogue of the LQR problem requires the solution of a rather different algebraic equation for X: It is therefore tempting to describe this equation as a discrete algebraic Riccati equation (or DARE) and, when convenient, we shall do so. The geometric theory for the DARE turns out to be considerably more complicated than for the symmetric CARE. The identification of a main operator (whose invariant subspaces determine solutions of a DARE) is already a significant task when compared to analysis of CARE (see Proposition 12.2.2). Having done this, it is possible to obtain several conclusions for the DARE by a suitable transformation between equations of CARE and DARE types (see Lemma 12.3.2). In contrast to analysis of the CARE it is convenient to make use of techniques involving rational matrix functions at an early stage of the analysis.