The Kleinman iteration for nonstabilizable systems

We consider the computation of Hermitian nonnegative definite solutions of algebraic Riccati equations. These solutions are the limit, P = lim(i -->infinity) P-i, of a sequence of matrices obtained by solving a sequence of Lyapunov equations. The procedure parallels the well-known Kleinman technique but the stabilizability condition on the underlying linear time-invariant system is removed. The convergence of the constructed sequence {P-i }(igreater than or equal to1) is guaranteed by the minimality of P-i in the set of Hermitian nonnegative definite solutions of the Lyapunov equation in the ith iteration step.