Stabilization and Optimization of Discrete-Time Markovian Jump Linear Systems via Mode Feedback Control

This article is devoted to a specific kind of discrete-time switched linear systems, where the switching signal is governed by a Markov chain, i.e., Markovian jump linear systems. For such systems, a mode feedback control mechanism is adopted to adjust the mode transition probability matrix, which i...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2022-12, Vol.67 (12), p.6505-6520
Main Authors: Zhu, Jin, Xia, Kai, Ling, Qiang, Chen, Wei, Dullerud, Geir E.
Format: Article
Language:English
Subjects:
Algorithms
Control systems
Costs
Discrete time systems
Discrete-time Markovian jump linear systems (MJLSs)
Feedback control
Linear systems
Markov chains
mode feedback control
Numerical stability
Optimization
Performance analysis
Performance indices
quadratic performance index
Stability criteria
stochastic stability
Switches
Switching
Transition probabilities
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Summary:This article is devoted to a specific kind of discrete-time switched linear systems, where the switching signal is governed by a Markov chain, i.e., Markovian jump linear systems. For such systems, a mode feedback control mechanism is adopted to adjust the mode transition probability matrix, which is referred to as the switching law design, and the optimal mode feedback controller is sought to minimize a quadratic performance index containing both the system state and the mode feedback control input. First, the admissible set of the mode feedback control is investigated, and a sufficient condition is derived to ensure the desired stochastic stability. Second, under the assumption that the concerned system is mode-cost sequential, the original performance index is approximated into a new tractable one and a suboptimal mode feedback controller is then derived within the admissible set. Third, we consider a general situation where no specific requirements are imposed on the Markovian jump linear systems' dynamics, derive the optimal mode feedback controller via a value-iteration-based algorithm, and prove the convergence of the obtained optimal controller. Finally, simulation results are provided to illustrate the validity of the proposed mode feedback control mechanism.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2021.3131547