Indefinite Robust Linear Quadratic Optimal Regulator for Discrete-Time Uncertain Singular Markov Jump Systems

Area of Science:

• Control Theory
• Systems Engineering
• Stochastic Systems

Background:

• Singular Markov jump systems (SMJSs) present challenges in robust control due to uncertainties and system singularity.
• Designing optimal regulators for these systems requires addressing both robustness and stability under stochastic variations.

Purpose of the Study:

• To develop a robust LQ optimal regulator for discrete-time uncertain SMJSs.
• To ensure regularity, causality, and stochastic stability of the closed-loop system.
• To eliminate uncertain parameters from the closed-loop system.

Main Methods:

• Introduced a new quadratic cost function using the penalty function method.
• Transformed the indefinite robust optimal regulator problem into a positive definite problem for uncertain Markov jump systems (MJSs).
• Applied the robust least-squares method to solve the transformed problem.

Main Results:

• Established conditions for the existence and analytic form of the robust optimal regulator.
• Obtained optimal state feedback for infinite horizons.
• Demonstrated the ability to guarantee system regularity, causality, and stochastic stability.
• Successfully eliminated uncertain parameters in the closed-loop system.

Conclusions:

• The proposed method effectively solves the robust LQ optimal regulator problem for discrete-time uncertain SMJSs.
• The obtained optimal state feedback ensures desirable system properties.
• Validated through numerical and practical (DC motor) examples.