Optimal discrete systems with prescribed eigenvalues

In this article, the problem of the numerical computation of the stabilising solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H _∞ control problem for a class of stochastic systems affected by state-dependent and control-dependent white noise and subjected to Markovian jumping. The stabilising solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilising solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The proposed algorithm extends to this general framework the method proposed in Lanzon, Feng, Anderson, and Rotkowitz (Lanzon, A., Feng, Y., Anderson, B.D.O., and Rotkowitz, M. (2008), ‘Computing the Positive Stabilizing Solution to Algebraic Riccati Equations with an Indefinite Quadratic Term Viaa Recursive Method,’ IEEE Transactions on Automatic Control, 53, pp. 2280–2291). In the proof of the convergence of the proposed algorithm different concepts associated the generalised Lyapunov operators as stability, stabilisability and detectability are widely involved. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.