A Newton‐like approximation algorithm for the steady‐state solution of the riccati equation for time‐varying systems
Erol Emre, Gareth J. Knowles · Optimal Control Applications and Methods · 1987
Abstract An approximation technique is developed for the steady‐state solution of the time‐varying matrix Riccati equation. We show how the Newton‐type algorithm of Kleinman, developed for computing the steady solution to the algebraic Riccati equation for time‐invariant systems, can be extended for time‐varying linear systems. The time‐varying case is considerably more involved than the time‐invariant one. Consider a linear time‐varying system x(t) = F(t)x(t) + G(t)u(t). If (F, G) is uniformly completely controllable, we show how one can construct a recursive sequence of matrix functions (using linear techniques) which converge to the steady‐state solution of the associated time‐varying matrix Riccati equation (a non‐linear object). At each successive state, the next approximation is in terms of the steady‐state solution to a linear Lyapunov differential equation (which is the extension of the algebraic Lyapunov equations used by Kleinman) for which an explicit expression exists. This provides an approximation technique for obtaining infinite‐time, linear‐quadratic, optimal controllers and steady‐state Kalman—Bucy filters for time‐varying systems using purely linear techniques. Thus, we provide new types of suboptimal stabilizing feedback laws for linear time‐varying systems.