Sliding Mode Control for Markov Jump Singularly Perturbed Systems Under Piecewise Homogeneous Stochastic...

Area of Science:

• Control Theory
• Systems Engineering
• Stochastic Systems

Background:

• Markov jump singularly perturbed systems (MJSPSs) present control challenges due to mode-dependent dynamics and external disturbances.
• Communication constraints and unmatched external disturbances further complicate controller design for these systems.
• Existing control strategies often struggle to address the interplay between system modes, controller modes, and communication protocols.

Purpose of the Study:

• To develop an adaptive sliding mode control (SMC) strategy for MJSPSs subject to unmatched external disturbances and communication constraints.
• To propose a novel three-layer nonstationary Markov model (NMM) integrating system, controller, and stochastic communication protocol (SCP) modes.
• To ensure the accessibility of the sliding surface (SS) and the stability of the closed-loop system (CLS) under the proposed control framework.

Main Methods:

• Introduction of a piecewise homogeneous Markov chain (MC) for stochastic communication protocol (SCP) scheduling, dependent on system and controller modes.
• Formulation of a three-layer nonstationary Markov model (NMM) to capture the dynamics of the system, controller, and SCP.
• Design of a nonstationary SMC law using a standard singular sliding mode surface and mode-dependent Lyapunov functions.
• Application of matrix decoupling technology to derive the specific control gain expression.

Main Results:

• The proposed adaptive SMC law guarantees the accessibility of the sliding surface (SS) and the stability of the closed-loop system (CLS).
• Sufficient stability criteria are derived using the mode-dependent Lyapunov function and piecewise homogeneous Markov model method.
• The control gain expression is explicitly obtained through matrix decoupling, facilitating practical implementation.

Conclusions:

• The developed control strategy effectively addresses the complexities of MJSPSs with communication constraints and external disturbances.
• The novel three-layer NMM provides a comprehensive framework for analyzing such systems.
• Numerical simulations validate the efficacy and correctness of the proposed adaptive SMC approach.