An Improved Jump Model for Two-Dimensional Markov Jump Roesser Systems and Its H_∞ Control

Area of Science:

• Control Theory
• Systems Engineering
• Stochastic Systems

Background:

• Conventional 2-D Markov jump systems (MJSs) often struggle with modeling complex, abrupt changes due to single Markov chain limitations.
• Mode ambiguity is a common issue in existing 2-D MJS models, hindering accurate real-world application.

Purpose of the Study:

• To propose an improved jump model for Roesser-type 2-D MJSs utilizing two independent Markov chains.
• To develop a dual-mode-dependent state feedback control law for stabilizing the enhanced 2-D MJS.
• To establish sufficient criteria for asymptotic mean square stability and H∞ disturbance attenuation.

Main Methods:

• A novel 2-D jump model employing two independent Markov chains for horizontal and vertical dynamics.
• Dual-mode-dependent Lyapunov functional technique to enhance feasibility of stability criteria.
• A nonconservative separation principle to derive equivalent conditions, including linear matrix inequalities (LMIs).

Main Results:

• The proposed 2-D jump model offers superior modeling capabilities and avoids mode ambiguity.
• A sufficient criterion for asymptotic mean square stability and H∞ disturbance attenuation is derived.
• Equivalent conditions, including LMIs, are developed for control law design, validated by a Darboux equation example.

Conclusions:

• The novel 2-D jump model and control strategy effectively stabilize 2-D MJSs with abrupt parameter changes.
• The use of dual independent Markov chains significantly improves modeling accuracy and avoids mode ambiguity.
• The LMI-based convex optimization algorithm provides a practical method for designing robust controllers for these systems.