$L_1$-Stochastic Stability and $L_1$-Gain Performance of Positive Markov Jump Linear Systems With Time-Delays: Necessary and Sufficient Conditions

2017 ◽  
Vol 62 (7) ◽  
pp. 3634-3639 ◽  
Author(s):  
Shuqian Zhu ◽  
Qing-Long Han ◽  
Chenghui Zhang
Keyword(s):  
Linear Systems ◽  
Stochastic Stability ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Markov Jump ◽  
Jump Linear Systems ◽  
Markov Jump Linear Systems ◽  
Necessary And Sufficient

Stochastic stability of positive Markov jump linear systems with completely/partially known transition rates

2018 ◽  
Vol 40 (9) ◽  
pp. 2724-2731 ◽  
Author(s):  
Ying Chen ◽  
Yuming Bo ◽  
Baozhu Du
Keyword(s):  
Linear Programming ◽  
Linear Systems ◽  
Stochastic Stability ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Markov Jump ◽  
Jump Linear Systems ◽  
Markov Jump Linear Systems ◽  
Necessary And Sufficient

This paper addresses the stochastic stability problem of positive Markov jump linear systems (PMJLSs). A necessary and sufficient condition of stochastic stability for PMJLSs is given which can be checked by solving linear programming feasibility problems due to the positivity property. A common co-positive Lyapunov function is constructed to solve the problem, and the equivalence among stochastic stability, 1-moment stability and exponential mean stability is proved afterwards. Considering the uncertain transition rates situation, PMJLSs with partially known transition rate matrix are investigated, and a necessary and sufficient condition for robust stochastic stability is proposed in linear programming form. Numerical examples are presented to show the effectiveness of the proposed results.

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