Optimal discrete-time H∞/γ0 filtering and control under unknown covariances

ABSTRACT New stochastic γ0 and mixed H∞/γ0 filtering and control problems for discrete-time systems under completely unknown covariances are introduced and solved. The performance measure γ0 is the worst-case steady-state averaged variance of the error signal in response to the stationary Gaussian white zero-mean disturbance with unknown covariance and identity variance. The performance measure H∞/γ0 is the worst-case power norm of the error signal in response to two input disturbances in different channels, one of which is the deterministic signal with a bounded energy and the other is the stationary Gaussian white zero-mean signal with a bounded variance provided the weighting sum of disturbance powers equals one. In this framework, it is possible to consider at the same time both deterministic and stochastic disturbances highlighting their mutual effects. Our main results provide the complete characterisations of the above performance measures in terms of linear matrix inequalities and therefore both the γ0 and H∞/γ0 optimal filters and controllers can be computed by convex programming. H∞/γ0 optimal solution is shown to be actually a trade-off between optimal solutions to the H∞ and γ0 problems for the corresponding channels.