H2-Filtering for discrete-time hidden Markov jump systems

ABSTRACT

In this paper, we consider the ${\mathcal{H}}_2$ filtering problem for a discrete-time Markov jump linear systemin which the Markov parameter is not available. In the spirit of active fault-tolerant control systems, it is supposed that there is a discrete-time hidden Markov model $(\theta \left(k\right),\widehat{\theta }\left(k\right))$ in which the observable part $\widehat{\theta }\left(k\right)$ represents the information coming from a detector and available to the filter while the hidden part θ(k) of the process represents the dynamics of the real system. Several models found in the literature are encompassed by this framework, like the complete observation case, the clustering information case, the mode-independent case, and Markov models in fault-tolerant control. We start by analysing an auxiliary filtering problem in which it is assumed that both $(\theta \left(k\right),\widehat{\theta }\left(k\right))$ are available to the filter, so that necessary and sufficient conditions in terms of LMI (linear matrix inequalities) for the existence of the optimal filter can be obtained. In the sequel, we consider the realistic case in which only $\widehat{\theta }\left(k\right)$ is available for the design of the filter. A sufficient condition based on an LMI optimisation problem to design a guaranteed ${\mathcal{H}}_2$ cost filter that depends only on $\widehat{\theta }\left(k\right)$ is presented. The results are strengthened for two particular cases, named the cluster case and the Bernoulli case. The paper is concluded with some numerical examples to illustrate the obtained results.

KEYWORDS:

• Markov jump linear systems
• fault-tolerant control
• filtering
• discrete-time systems
• linear matrix inequalities

Acknowledgements

The authors would like to thank the Associate Editor and the reviewers for the very helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported in part by the São Paulo Research Foundation (FAPESP), Grant No. 2015/09912-8 for the first author, and the Brazilian National Research Council - CNPq, Grant No. 304091/2014-6, and FAPESP/BG Brasil through the Research Center for Gas Innovation, FAPESP Grant No. 2014/50279-4 for the second author.