Solutions to Discrete-time Riccati Equations

Abstract

The discrete-time algebraic Riccati equation (DARE) has been investigated extensively in the literature (see, for example [9,68,72,101,105,123]). Here, most of the work was based on the discrete-time algebraic Riccati equation appearing in a linear quadratic control problem (hereafter we will refer to such a DARE as the H 2-DARE). Recently, the problem of H ∞, control and that of differential games for discrete-time systems, have been studied by a number of researchers including [4,63,78]. This work gives rise to a different kind of algebraic Riccati equation (hereafter we call it an H ∞-DARE). Analyzing and solving such an H ∞-DARE are very difficult primarily because of an indefinite nonlinear term and because we cannot a priori guarantee the existence of solutions. In this chapter, we recall the results of Chen et al. [38] on non-recursive methods for solving general DAREs, as well as H 2-DAREs and H ∞-DARES. In particular, we will cast the problem of solving a given H ∞-DARE to the problem of solving an auxiliary continuous-time algebraic Riccati equation associated with the continuous-time H ∞ control problem (H ∞-CARE) for which the well known non-recursive solving methods are available. The advantages of this approach are: it reduces the computation involved in the recursive algorithms while giving much more accurate solutions, and it readily provides the properties of the general H ∞-DARE. More importantly, the results given in this chapter build an interconnection between the discrete-time and continuous-time H ∞ optimization problems.