Robust stabilisation of discrete-time time-varying linear systems with Markovian switching and nonlinear parametric uncertainties

Abstract

In this paper, the problem of the robustness of the stability of a discrete-time linear stochastic system is addressed. The nominal plant is described by a discrete-time time-varying linear system subject to random jumping according with a non-homogeneous Markov chain with a finite number of states. The class of admissible uncertainties consists of multiplicative white noise type perturbations with unknown intensity. It is assumed that the intensity of white noise type perturbations is modelled by unknown nonlinear functions subject to linear growth conditions. The class of admissible controls consists of stabilising state feedback control laws. We show that the best robustness performance is achieved by the stability provided by a state feedback design based on the stabilising solution of a suitable discrete-time Riccati-type equation.

Keywords:

• robust stability and robust stabilisation
• nonlinear parametric uncertainties
• discrete-time stochastic systems
• non-homogeneous Markov chains
• discrete-time Riccati-type equations

Additional information

Funding

This work was supported by CNCS Romania [grant number 145/2011].

Notes on contributors

Vasile Dragan

Vasile Dragan was born on 2 February 1950, in Valea Bolvasnita, Caras-Severin, Romania. From 1965 to 1969, he studied at ‘Traian Lalescu’ high school in Mehadia.He received his undergraduate degree in mathematics in 1974, and his Ph.D. degree from the Faculty of Mathematics, University of Bucharest, in 1979. He is now a senior researcher in the Department of Differential Equations, Institute of Mathematics of the Romania Academy. His research interests are in qualitative theory of differential equations, mainly in singular perturbation techniques, stability theory and asymptotic expansions. His current research interests also include control theory, robust control for deterministic and stochastic systems, and for finite-dimensional and infinite-dimensional systems, both in the time-invariant and time-varying cases. He is the author or coauthor of six books and over 170 scientific papers dedicated to various topics on differential equations and control theory. In 1994, the Romanian Academy awarded him with the ‘Gheorghe Lazar’ prize for the papers published in the control field. He is a member of the American Mathematical Society. Since 2005, he has been the associate editor of the International Journal of Innovational Computing and Information and Control and of the ICIC-Express Letters.