State-Feedback Control of Positive Switching Systems with Markovian Jumps
This chapter deals with positive linear systems in continuous-time affected by a switching signal representing a disturbance driven by a Markov chain. A state-feedback control law has to be designed in order to ensure mean stability and input–output \(\ma
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Abstract This chapter deals with positive linear systems in continuous-time affected by a switching signal representing a disturbance driven by a Markov chain. A state-feedback control law has to be designed in order to ensure mean stability and input–output L1 -induced or L1 -induced mean performance. The chapter is divided into two parts. In the first, the control action is based on the knowledge of both the state of the system and the sample path of the Markovian process (mode-dependent control). In the second, instead, only the state-variable is known (mode-independent control). In the mode-dependent case, as well as in the single-input mode-independent case, necessary and sufficient conditions for the existence of feasible feedback gains are provided based on linear programming tools, also yielding a full parametrization of feasible solutions. In the multi-input mode-independent case, sufficient conditions are worked out in terms of convex programming. Some numerical examples illustrate the theory. Keywords Positive systems • Markov Jumps • Stabilization • Input-output performance
P. Colaneri () Politecnico di Milano, DEIB, IEIIT-CNR, Milano, Italy e-mail: [email protected] P. Bolzern Politecnico di Milano, DEIB, Milano, Italy e-mail: [email protected] J.C. Geromel School of Electrical and Computer Engineering, UNICAMP, Brazil e-mail: [email protected] G.S. Deaecto School of Mechanical Engineering, UNICAMP, Brazil e-mail: [email protected] © Springer International Publishing Switzerland 2016 B. Goldengorin (ed.), Optimization and Its Applications in Control and Data Sciences, Springer Optimization and Its Applications 115, DOI 10.1007/978-3-319-42056-1_6
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1 Introduction This chapter deals with stabilization and control of (continuous-time) positive Markov jump linear systems (PMJLS). The switching signal is a Markov process associated with a given transition rate matrix. The class of positive systems in the deterministic setting has been widely studied in the past years. Relevant general textbooks are available, see [9, 11, 12], and more specific contributions dealing with Lyapunov functions and input–output norms can be found in [2, 6, 14–16]. As for the class of Markov Jump Linear Systems (MJLS), a wide corpus of results is available, see the textbooks [5, 7]. On the other hand, only a few papers on PMJLS (in continuous-time) are available up to now. To the best of the authors knowledge, the first contribution pointing out the usefulness of the linear programming (LP) approach to the study of PMJLS is [2]. More recently, in [4], various notions of stability and their relationships are studied, while results on stochastic stabilization are provided in [17]. An application to an epidemiological model can be found in [1]. A very recent survey on analysis and design of PMJLS is available in [3]. The chapter is divided into two parts. In the first, the attention is concentrated on mode-dependent state feedback laws u.t/ D K.t/ x.t/, whereas in the se
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