Mean stability for a class of discrete-time non-homogeneous positive Markov jump linear systems

Abstract

This paper investigates the mean stability and stabilisation problem for a class of discrete-time positive Markov jump linear systems where the Mode Transition Probability Matrix (MTPM) is a stochastic process instead of time invariant. By assuming the jump of system mode is governed by a low-layer Markov chain and the variation of corresponding MTPM is governed by a high-layer one, a non-homogeneous positive Markov jump linear system model with two Markov chains is then proposed. Based on this, the necessary and sufficient conditions of mean stability for this concerned model are addressed via analysing the time evolution of the first-order moment of state variables. Second, a mode-MTPM-dependent state feedback controller can then be designed according to the given stability conditions which is solvable in terms of linear programming problems. Finally, a numerical example is provided to show the effectiveness of the presented control strategy.

Keywords:

• Non-homogeneous positive Markov jump linear systems
• mean stability
• mode-MTPM-dependent state feedback controller
• linear programming

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Key Research and Development Project under Grant 2018AAA0100802, National Natural Science Foundation of China under Grant 61374073, 61503356, and the Anhui Provincial Natural Science Foundation under Grant 2008085MF198.

Notes on contributors

Jin Zhu

Jin Zhu received his BSc and PhD degree from Dept. of Automation, University of Science and Technology of China in 2001, 2006 respectively. From 2007–2009, he was with Hanyang University, South Korea as a post-doc and contract professor. He joined Dept. of Automation, University of Science and Technology of China in 2009 as an associate professor. He was a visiting scholar in University of Illinois at Urbana-Champaign from 2013–2014 and had two short visits to Carnegie Mellon University in 2016, 2017 respectively. His current research interests include the theory and application of hybrid systems and deep reinforcement learning.

Pengfei Jiang

Pengfei Jiang received his MSc degree from Dept. of Automation, University of Science and Technology of China in 2019. His research interests include Markovian jump systems.