A Newton‐like approximation algorithm for the steady‐state solution of the riccati equation for time‐varying systems

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.