Abstract
This paper is concerned with robust stochastic stabilization for positive Markov jump systems with actuator saturation. The considered systems contain interval and polytopic uncertainties, respectively. First, a stochastic co-positive Lyapunov functional is constructed for the systems. By virtue of the presented Lyapunov functional, a new controller design approach is addressed using matrix decomposition technique. Under the designed controller, robust stochastic stabilization of the systems with interval and polytopic uncertainties is achieved, respectively. Furthermore, an effective method for estimating the attraction domain is established by solving an optimization problem. An implemental algorithm is provided based on linear programming to solve the corresponding conditions. Finally, two numerical examples are provided to illustrate the reduced conservatism and effectiveness of the proposed design.
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References
M. Ait Rami, J. Shamma, Hybrid positive systems subject to Markovian switching, in Proceedings of the 3rd IFAC Conference on Analysis and Design of Hybrid Systems (Zaragoza, Spain, 2009), pp. 138–143
A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (SIAM, Philadelphia, 1994)
E.K. Boukas, A. Haurie, Manufacturing flow control and preventing maintenance: a stochastic control approach. IEEE Trans. Autom. Control 35(9), 1024–1031 (1990)
W. Chen, J. Xu, Z. Guan, Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays. IEEE Trans. Autom. Control 48(12), 2270–2277 (2003)
X. Chen, J. Lam, P. Li, Positive filtering for continuous-time positive systems under \(L_{1}\) performance. Int. J. Control 87(9), 1906–1913 (2014)
Y. Chen, W. Zheng, \(L_{2}\)-\(L_{\infty }\) filtering for stochastic Markovian jump delay systems with nonlinear perturbations. Signal Process. 109, 154–164 (2015)
J.G. Da Silva, S. Tarbouriech, Anti-windup design with guaranteed regions of stability for discrete-time linear systems. Syst. Control Lett. 55(3), 184–192 (2006)
D.P. De Farias, J.C. Geromel, J.B. Do Val, O.L.V. Costa, Output feedback control of Markov jump linear systems in continuous-time. IEEE Trans. Autom. Control 45(5), 944–949 (2000)
P. De Leenheer, D. Aeyels, Stabilization of positive linear systems. Syst. Control Lett. 44(4), 259–271 (2001)
H. Dong, H. Wang, H. Gao, Fault detection for Markovian jump systems with sensor saturations and randomly varying nonlinearities. IEEE Trans. Circuits Syst. I. Regul. Pap. 59(10), 2354–2362 (2012)
V. Dragan, I.G. Ivanov, Sufficient conditions for Nash equilibrium point in the linear quadratic game for Markov jump positive systems. IET Control Theory Appl. 11(15), 2658–2667 (2017)
Y. Ebihara, D. Peaucelle, D. Arzelier, LMI approach to linear positive system analysis and synthesis. Syst. Control Lett. 63(63), 50–56 (2014)
L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000)
T. Hu, Z. Lin, Control Systems with Actuator Saturation: Analysis and Design (Birkhäuser, Boston, 2001)
T. Hu, Z. Lin, B.M. Chen, An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica 38(2), 351–359 (2002)
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, Practical stability of positive fractional discrete-time linear systems. Bull. Pol. Acad. Tech. Sci. 56(4), 313–317 (2008)
T. Kaczorek, Positive 1D and 2D Systems (Springer, London, 2002)
V. Kapila, A.G. Sparks, H. Pan, Control of systems with actuator saturation non-linearities: an LMI approach. Int. J. Control 74(6), 586–599 (2001)
F. Knorn, O. Mason, R. Shorten, On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica 45(8), 1943–1947 (2009)
T. Li, Y. Zhang, Fault detection and diagnosis for stochastic systems via output PDFs. J. Frankl. Inst. 348(6), 1140–1152 (2011)
H. Liu, E.K. Boukas, F. Sun, D.W. Ho, Controller design for Markov jumping systems subject to actuator saturation. Automatica 42(3), 459–465 (2006)
D.G. Luenberger, Introduction to Dynamic Systems: Theory, Models, and Applications (Wiley, New York, 1979)
S. Ma, E.K. Boukas, A singular system approach to robust sliding mode control for uncertain Markov jump systems. Automatica 45(11), 2707–2713 (2009)
S. Ma, C. Zhang, \(H_\infty \) control for discrete-time singular Markov jump systems subject to actuator saturation. J. Frankl. Inst. 349(3), 1011–1029 (2012)
W. Qi, X. Gao, State feedback controller design for singular positive Markovian jump systems with partly known transition rates. Appl. Math. Lett. 46, 111–116 (2015)
W. Qi, Y. Kao, X. Gao, Further results on finite-time stabilisation for stochastic Markovian jump systems with time-varying delay. Int. J. Syst. Sci. 48(14), 2967–2975 (2017)
W. Qi, J.H. Park, C. Jun, Y. Kao, Robust stabilization for nonlinear time-delay semi-Markovian jump systems via sliding mode control. IET Control Theory Appl. 11(10), 1504–1513 (2017)
W. Qi, X. Gao, Y. Kao, L. Lian, J. Wang, Stabilization for positive Markovian jump systems with actuator saturation. Circuits Syst. Signal Process. 36(1), 374–388 (2017)
T. Sathyan, T. Kirubarajan, Markov-jump-system-based secure chaotic communication. IEEE Trans. Circuits Syst I. Regul. Pap. 53(7), 1597–1609 (2006)
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)
S. Tarbouriech, J.G. Da Silva, Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs. IEEE Trans. Autom. Control 45(1), 105–111 (2000)
J. Wang, J. Zhao, Stabilisation of switched positive systems with actuator saturation. IET Control Theory Appl. 10(6), 717–723 (2016)
M. Xiang, Z. Xiang, Stability, \(L_{1}\)-gain and control synthesis for positive switched systems with time-varying delay. Nonlinear Anal. Hybrid Syst. 9, 9–17 (2013)
J. Xiong, J. Lam, H. Gao, D.W. Ho, On robust stabilization of Markovian jump systems with uncertain switching probabilities. Automatica 41(5), 897–903 (2005)
J. Zhang, Z. Deng, Y. Wang, Robust stability and stabilization of positive interval systems subject to actuator saturation. Asian J. Control 16(5), 1553–1560 (2014)
J. Zhang, Z. Han, F. Zhu, Stochastic stability and stabilization of positive systems with Markovian jump parameters. Nonlinear Anal. Hybrid Syst. 12, 147–155 (2014)
J. Zhang, X. Zhao, R. Zhang, An improved approach to controller design of positive systems using controller gain decomposition. J. Frankl. Inst. 354(3), 1356–1373 (2017)
S. Zhu, Q. Han, C. Zhang, Investigating the effects of time-delays on stochastic stability and designing \(l_{1}\)-gain controllers for positive discrete-time Markov jump linear systems with time-delay. Inf. Sci. 355, 265–281 (2016)
S. Zhu, Q. Han, C. Zhang, \(L_{1}\)-stochastic stability and \(L_{1}\)-gain performance of positive Markov jump linear systems with time-delays: necessary and sufficient conditions. IEEE Trans. Autom. Control 62(7), 3634–3639 (2017)
S. Zhu, Q. Han, C. 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)
Acknowledgements
The authors would like to express the most sincere gratitude to the anonymous reviewers for their valuable suggestions and comments to improve the quality of this paper. This work is supported by the National Natural Science Foundation of China under Grants 61503107, 61473107, and U1509205, the Zhejiang Provincial Natural Science Foundation of China under Grants LR16F030003, and the Cross-Discipline Innovation Team Building Project of Hangzhou Dianzi University.
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Li, S., Zhang, J., Chen, Y. et al. Robust Stochastic Stabilization for Positive Markov Jump Systems with Actuator Saturation. Circuits Syst Signal Process 38, 625–642 (2019). https://doi.org/10.1007/s00034-018-0878-5
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DOI: https://doi.org/10.1007/s00034-018-0878-5