Robust stability, ℋ₂ analysis and stabilisation of discrete-time Markov jump linear systems with uncertain probability matrix

Abstract

The stability and the problem of ℋ₂ guaranteed cost computation for discrete-time Markov jump linear systems (MJLS) are investigated, assuming that the transition probability matrix is not precisely known. It is generally difficult to estimate the exact transition matrix of the underlying Markov chain and the setting has a special interest for applications of MJLS. The exact matrix is assumed to belong to a polytopic domain made up by known probability matrices, and a sequence of linear matrix inequalities (LMIs) is proposed to verify the stability and to solve the ℋ₂ guaranteed cost with increasing precision. These LMI problems are connected to homogeneous polynomially parameter-dependent Lyapunov matrix of increasing degree g. The mean square stability (MSS) can be established by the method since the conditions that are sufficient, eventually turns out to also be necessary, provided that the degree g is large enough. The ℋ₂ guaranteed cost under MSS is also studied here, and an extension to cope with the problem of control design is also introduced. These conditions are only sufficient, but as the degree g increases, the conservativeness of the ℋ₂ guaranteed costs is reduced. Both mode-dependent and mode-independent control laws are addressed, and numerical examples illustrate the results.

Keywords:

• discrete-time Markov jump linear systems
• uncertain probability matrix
• robust stability
• ℋ₂ norm
• state feedback control
• LMI relaxations

Acknowledgements

This work was supported by the Brazilian agencies CAPES, CNPq and FAPESP.

Notes

Note

1. The numerical implementation of all theorems presented in the article can be downloaded from www.dt.fee.unicamp.br/~ricfow/robust.htm