Solutions of the Continuous and Discrete Time Algebraic Riccati Equations: A Review

Abstract

This review is concerned with two algebraic Riccati equations. The first is a quadratic matrix equation for an unknown n {\texttimes} n matrix X of the form 2.1$$XDX + XA + A*X - C = 0,$$where A, D, C are n {\texttimes} n complex matrices with C and D hermitian. Further hypotheses are imposed as required, although Section 2.3 contains some discussion of more general non-symmetric quadratic equations. The second equation has the fractional form 2.2$$X = A*XA + Q - (C + B*XA)*{(R + B*XB)^{ - 1}}(C + B*XA),$$where R and Q are hermitian m {\texttimes} m and n {\texttimes} n matrices, respectively, and A, B, C are complex matrices with respective sizes n {\texttimes} n, n {\texttimes} m, and m {\texttimes} n. The two equations are frequently referred to as the ``continuous'' and ``discrete'' Riccati equations, respectively, because they arise in physical optimal control problems in which the time is treated as a continuous variable, or a discrete variable.