Numerical solution of Lyapunov equations related to Markov jump linear systems
Provides a practical comparison of solution methods for a specific class of equations in control theory, but the results are incremental and confirm the superiority of existing fixed-point approaches.
The paper compares numerical methods for solving Lyapunov-like equations in Markov jump linear systems, finding that fixed-point iterations outperform optimization-based methods (steepest descent, conjugate gradient, trust-region) in large-scale problems, with application to networked control systems.
We suggest and compare different methods for the numerical solution of Lyapunov like equations with application to control of Markovian jump linear systems. First, we consider fixed point iterations and associated Krylov subspace formulations. Second, we reformulate the equation as an optimization problem and consider steepest descent, conjugate gradient, and a trust-region method. Numerical experiments illustrate that for large-scale problems the trust-region method is more effective than the steepest descent and the conjugate gradient methods. The fixed-point approach, however, is superior to the optimization methods. As an application we consider a networked control system, where the Markov jumps are induced by the wireless communication protocol.