Fuzzy Control and Filtering for Nonlinear Singularly Perturbed Markov Jump Systems

Area of Science:

• Control Systems Engineering
• Fuzzy Logic Systems
• Stochastic Systems

Background:

• Markov jump singularly perturbed systems (MJSP) are crucial in modeling systems with abrupt changes.
• Takagi-Sugeno (T-S) fuzzy models offer a powerful framework for approximating nonlinear systems.
• Uncertainty in transition probabilities (TPs) poses significant challenges in robust control design.

Purpose of the Study:

• To design mode- and variation-dependent fuzzy static output-feedback controllers (SOFC) and filters.
• To address H∞ control and filtering problems for T-S fuzzy approximated MJSP systems.
• To ensure mean-square exponential admissibility and stability under uncertain TPs.

Main Methods:

• Descriptor representation to transform the closed-loop system into a fuzzy piecewise-homogeneous Markov jump singularly perturbed descriptor system (MJSPDS).
• Development of criteria for mean-square exponential admissibility of the fuzzy MJSPDS.
• Formulation of criteria for mean-square exponential stability of the fuzzy filtering error system.
• Matrix variable techniques for synthesizing the fuzzy SOFC and filter.

Main Results:

• A rigorous proof of mean-square exponential admissibility for the designed fuzzy MJSPDS.
• A criterion for ensuring the mean-square exponential stability of the fuzzy filtering error system.
• Explicit solutions for the fuzzy SOFC and filter are derived by setting specific matrix variables.
• Verification of the developed fuzzy control and filtering results through two practical examples.

Conclusions:

• The proposed fuzzy control and filtering methodologies are effective for T-S fuzzy approximated MJSP systems with uncertain TPs.
• The developed criteria guarantee the mean-square exponential stability and admissibility of the closed-loop and filtering error systems.
• The practical examples demonstrate the feasibility and validity of the proposed approach in real-world applications.