Regulation of Markov Jump Linear Systems Subject to Polytopic Uncertainties

• Published in: IEEE transactions on automatic control Vol. 67; no. 11; pp. 6279 - 6286
• Main Authors: Bueno, Jose Nuno A. D., Marcos, Lucas B., Rocha, Kaio D. T., Terra, Marco H.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Apexes; Convergence; Convexity; Cost function; Discrete time systems; Linear systems; Markov jump systems; Markov processes; Numerical stability; Optimization; Penalty function; polytopic uncertainties; Power system stability; Regulation; Robust control; Uncertainty
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3181567

When discrete-time Markov jump linear systems are prone to the damaging effects of polytopic uncertainties, it is necessary to address all the vertices of each Markov mode in order to properly design robust controllers. To this end, we propose a robust recursive linear-quadratic regulator for this class of systems. We define a quadratic min-max optimization problem by combining least-squares and penalty functions in a unified framework. We design a one-step cost function to encompass the entire set of vertices of each mode altogether, while maintaining its quadratic structure and the convexity of the problem. The solution is then obtained recursively and does not require numerical optimization packages. We establish conditions for convergence and stability by extending the matrix structure of the recursive solution. In addition, we provide numerical and real-world application examples to validate our method and to emphasize recursiveness and diminished computational effort.