H₂-control and the separation principle for discrete-time jump systems with the Markov chain in a general state space

ABSTRACT

This paper deals with the H₂ optimal control problem of discrete-time Markov jump linear systems (MJLS) considering the case in which the Markov chain takes values in a general Borel space $\mathcal{S}$. It is assumed that the controller has access only to an output variable and to the jump parameter. The goal, in this case, is to design a dynamic Markov jump controller such that the H₂-norm of the closed-loop system is minimised. It is shown that the H₂-norm can be written as the sum of two H₂-norms, such that one of them does not depend on the control, and the other one is obtained from the optimal filter for an infinite-horizon filtering problem. This result can be seen as a separation principle for MJLS with Markov chain in a Borel space $\mathcal{S}$ considering the infinite time horizon case.

KEYWORDS:

• Markov processes
• stochastic systems
• optimal filtering
• optimal control

Acknowledgements

The authors are grateful to anonymous referees for their suggestions which have greatly improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the Brazilian National Research Council – CNPq [grant number 304091/2014-6]; the São Paulo Research Foundation – FAPESP/BG Brasil, through the Research Center for Gas Innovation [FAPESP grant number 2014/50279-4]; project INCT [grant numbers CNPq-465755/2014-3, CNPq-380610/2017-5, FAPESP-2014/50851-0].

Notes on contributors

Danilo Zucolli Figueiredo

Danilo Zucolli Figueiredo holds a bachelor's, a master's, and a Ph.D. in Electrical Engineering from the University of São Paulo, Brazil. Currently, he is a postdoctoral researcher at the Polytechnic School of the University of São Paulo. His research focuses on jump systems (stability conditions, state estimation, optimal control, and its applications).

Oswaldo Luiz do Valle Costa

Oswaldo Luiz do Valle Costa was born in 1959 in Rio de Janeiro, RJ, Brazil. He obtained his BSc and MSc degrees both in Electrical Engineering from the Catholic University of Rio de Janeiro, Brazil, in 1981 and 1983 respectively, and the PhD degree in Electrical Engineering from Imperial College of Science and Technology in London in 1987. He is presently Professor in the Control Group of the Department of Telecommunications and Control Engineering of the Polytechnic School of the University of São Paulo, Brazil. His research interests include stochastic control, optimal control, and jump systems.