On state covariance bounds for linear stochastic uncertain systems

For the case of robustness of closed-loop linear systems in the face of uncertainty in the system parameters, it is shown that a Riccati equation approach can be pursued in the discrete-time case. The extension requires the analysis of a suitable symplectic pencil. It turns out that, depending on the scalar parameter β which appears in a H_∞-type Riccati equation, the picture is more involved than in continuous-time. For some values of β the stabilizing solution exists but does not yield a covariance bound. The second issue addressed is the problem of computing the value of β that minimizes the trace of the covariance bound. It is proven that such an optimization problem is convex and therefore amenable to iterative methods of solution