Abstract
The paper deals with the problem of finite-time \(L_1\) control for positive Markovian jump systems with partly known transition rates. Firstly, by constructing a linear co-positive Lyapunov function, sufficient conditions for finite-time boundedness and \(L_1\) finite-time boundedness of the open-loop system are developed. Then, an effective method is proposed for the construction of the state feedback controller. These sufficient criteria are derived in the form of linear programming. A key point of this paper is to extend the special requirement of completely known transition rates to more general form that covers completely known and completely unknown transition rates as two special cases. Finally, two examples are given, which include a mathematical model of virus mutation treatment to illustrate the validity of the obtained results.
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References
A. Arunkumar, R. Sakthivel, K. Mathiyalagan, Ju H. Park, Robust stochastic stability of discrete-time fuzzy Markovian jump neural networks. ISA Trans. 53(4), 1006–1014 (2014)
P. Bolzern, P. Colaneri, G. Nicolao, Stochastic stability of positive Markov jump linear systems. Automatica 50(4), 1181–1187 (2014)
L. Caccetta, V.G. Rumchev, A positive linear discrete-time model of capacity planning and its controllability properties. Math. Comput. Model. 40(1–2), 217–226 (2004)
X.M. Chen, J. Lam, H.K. Lam, Positive filtering for positive Takagi–Sugeno fuzzy systems under \(l_1\) performance. Inf. Sci. 299, 32–41 (2015)
X.M. Chen, J. Lam, P. Li, Z. Shu, \(l_1\)-induced norm and controller synthesis of positive systems. Automatica 49(5), 1377–1385 (2013)
L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000)
E. Fornasini, M.E. Valcher, Stability and stabilizability criteria for discrete-time positive switched systems. IEEE Trans. Autom. Control 57(5), 1208–1221 (2012)
X.H. Ge, Q.L. Han, Distributed fault detection over sensor networks with Markovian switching topologies. Int. J. Gen. Syst. 43(3–4), 305–318 (2014)
E. Hernandez-Varga, P. Colaneri, R. Middleton, F. Blanchini, Discrete-time control for switched positive systems with application to mitigating viral escape. Int. J. Robust Nonlinear 21(10), 1093–1111 (2010)
A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003)
T. Kaczorek, Positive 1D and 2D Systems (Springer, New York, 2002)
H.Y. Li, H.J. Gao, P. Shi, X.D. Zhao, Fault-tolerant control of Markovian jump stochastic systems via the augmented sliding mode observer approach. Automatica 50(7), 1825–1834 (2014)
P. Li, J. Lam, Z. Shu, \(H_\infty \) positive filtering for positive linear discrete-time systems: an augmentation approach. IEEE Trans. Autom. Control 55(10), 2337–2342 (2010)
X. Mao, Stability of stochastic differential equations with Markovian switching. Stoch. Proc. Appl. 79(1), 45–67 (1999)
O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems. IEEE Trans. Autom. Control 52(7), 1346–1349 (2007)
K. Mathiyalagan, J.H. Park, R. Sakthivel, S.M. Anthoni, Robust mixed \(H_\infty \) and passive filtering for networked Markov jump systems with impulses. Signal Process. 101, 162–173 (2014)
K. Mathiyalagan, H.Y. Su, R. Sakthivel, Robust stochastic stability of discrete–time Markovian jump neural networks with leakage delay. Z. Naturforsch. A 69a, 70–80 (2014)
L.J. Shen, U. Buscher, Solving the serial batching problem in job shop manufacturing systems. Eur. J. Oper. Res. 221(1), 14–26 (2012)
R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: asymptotic results. IEEE/ACM Trans. Netw. 14(3), 616–629 (2006)
Y. Song, J.X. Xie, M.R. Fei, W.Y. Hou, Mean square exponential stabilization of networked control systems with Markovian packet dropouts. Trans. Inst. Meas. Control 35(1), 75–82 (2013)
M. Xiang, Z.R. Xiang, Finite-time \(L_1\) control for positive switched linear systems with time-varying delay. Commun. Nonlinear Sci. 18(11), 3158–3166 (2013)
J.F. Zhang, Z.Z. Han, H. Wu, Robust finite-time stability and stabilisation of switched positive systems. IET Control Theory Appl. 8(1), 67–75 (2014)
J.F. Zhang, Z.Z. Han, F. Zhu, Stochastic stability and stabilization of positive systems with Markovian jump parameters. Nonlinear Anal. Hybrid Syst. 12, 147–155 (2014)
X.D. Zhao, X.W. Liu, S. Yin, H.Y. Li, Improved results on stability of continuous-time switched positive linear systems. Automatica 50(2), 614–621 (2014)
S.Q. Zhu, Q.L. Han, C.H. Zhang, \(L_1\)-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: A linear programming approach. Automatica 50(8), 2098–2107 (2014)
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This work is supported by the Key Program of the National Natural Science Foundation of China under Grant No. 61433004.
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Wang, J., Qi, W. & Gao, X. Finite-Time \(L_1\) Control for Positive Markovian Jump Systems with Partly Known Transition Rates. Circuits Syst Signal Process 35, 1751–1766 (2016). https://doi.org/10.1007/s00034-015-0131-4
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DOI: https://doi.org/10.1007/s00034-015-0131-4