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.

KEYWORDS:

• Discrete-time systems
• mixed stochastic and deterministic disturbances
• unknown covariances
• performance measure γ₀
• H_∞/γ₀ filtering
• H_∞/γ₀ control

Acknowledgements

This work was supported in part by the Russian Foundation for Basic Research under Grants 13-01-00603 and 14-01-00266. The paper is prepared within the terms of the research project 3021 financed by the Ministry of Education and Science of the Russian Federation within the basic part of the Research Assignment.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Russian Foundation for Basic Research [grantnumber 13-01-00603], [grant number 14-01-00266].