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 \(\mathcal{L}_{\infty }\)-induced or \(\mathcal{L}_{1}\)-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.

[1]  Anders Rantzer,et al.  Scalable control of positive systems , 2012, Eur. J. Control.

[2]  Jeff S. Shamma,et al.  Hybrid positive systems subject to Markovian switching , 2009, ADHS.

[3]  Zhengzhi Han,et al.  Stochastic stability and stabilization of positive systems with Markovian jump parameters , 2014 .

[4]  Patrizio Colaneri,et al.  Positive Markov Jump Linear Systems , 2015, Found. Trends Syst. Control..

[5]  Corentin Briat,et al.  Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1‐gain and L∞‐gain characterization , 2012, ArXiv.

[6]  M. Fragoso,et al.  Continuous-Time Markov Jump Linear Systems , 2012 .

[7]  S. Rinaldi,et al.  Positive Linear Systems: Theory and Applications , 2000 .

[8]  Anders Rantzer,et al.  Distributed control of positive systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[9]  T. Kaczorek.  Positive 1D and 2D Systems , 2001 .

[10]  Takashi Tanaka,et al.  The Bounded Real Lemma for Internally Positive Systems and H-Infinity Structured Static State Feedback , 2011, IEEE Transactions on Automatic Control.

[11]  Dimitri Peaucelle,et al.  Optimal L1-controller synthesis for positive systems and its robustness properties , 2012, 2012 American Control Conference (ACC).

[12]  Patrizio Colaneri,et al.  Minimax control of Markov jump linear systems , 2016 .

[13]  M. A. Rami,et al.  Stability Criteria for SIS Epidemiological Models under Switching Policies , 2013, 1306.0135.

[14]  Victor M. Preciado,et al.  Optimal Design of Switched Networks of Positive Linear Systems via Geometric Programming , 2015, IEEE Transactions on Control of Network Systems.

[15]  W. Haddad,et al.  Nonnegative and Compartmental Dynamical Systems , 2010 .

[16]  Patrizio Colaneri,et al.  Stochastic stability of Positive Markov Jump Linear Systems , 2014, Autom..