Abstract
This paper introduces a problem of positive \(L_{1}\) controller design for positive piecewise homogeneous Markovian jump system. The difference with the existing achievements is that the considered transition rates of positive Markovian jump system is time-varying. This time-varying nature is finite piecewise homogeneous. The controller is designed by intentionally introducing a proper delay, this method considers the information about the current and delayed state. The obtained closed-loop system is positive piecewise homogeneous Markovian jump system with time-delay. Firstly, by means of choosing a linear co-positive Lyapunov function, stochastic stability and \(L_{1}\) performance are analyzed for positive piecewise homogeneous Markovian jump system with time-delay. Then, based on the obtained achievements, positive \(L_{1}\) controller is designed for positive piecewise homogeneouss Markovian jump system. Finally, a numerical example is considered to illustrate the effectiveness of theoretical results.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bolzern, P., Colaneri, P., Nicolao, G.: Stochastic stability of positive Markov jump linear systems. Automatica 50(4), 1181–1187 (2014)
Chen, X., Lam, J., Li, P., et al.: \(l_1\)-induced norm and controller synthesis of positive systems. Automatica 49(5), 1377–1385 (2013)
Ding, Y., Liu, H.: Stability analysis of continuous-time Markovian jump time-delay systems with time-varying transition rates. J. Franklin Inst. 353(11), 2418–2430 (2016)
Du, B., Lam, J., Shu, Z., et al.: On reachable sets for positive linear systems under constrained exogenous inputs. Automatica 74, 230–237 (2016)
Ebihara, Y., Peaucelle, D., Arzelier, D.: LMI approach to linear positive system analysis and synthesis. Syst. Control Lett. 63(63), 50–56 (2014)
Eden, J., Tan, Y., Lau, D., et al.: On the positive output controllability of linear time invariant systems. Automatica 71, 202–209 (2016)
Faraji-Niri, M., Jahed-Motlagh, M., Barkhordari-Yazdi, M.: Stochastic stability and stabilization of a class of piecewise-homogeneous Markov jump linear systems with mixed uncertainties. Int. J. Robust Nonlinear Control 27, 894–914 (2017)
Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York (2000)
Kaczorek, T.: Positive 1D and 2D Systems. Spring, London (2002)
Li, J., Zhang, Q., Yan, X., et al.: Integral sliding mode control for Markovian jump T-S fuzzy descriptor systems based on the super-twisting algorithm. IET Control Theory Appl. 11(8), 1134–1143 (2017)
Rami M., Tadeo F.: Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits Syst. II, Exp. Briefs 54(2), 151–155 (2007)
Shen, J., Lam, J.: \(l_\infty \)/\(L_\infty \)-gain analysis for positive linear systems with unbounded time-varying delays. IEEE Trans. Autom. Control 60(3), 857–862 (2015)
Wang, G., Zhang, Q., Yang, C.: Stabilization of singular Markovian jump systems with time-varying switchings. Inf. Sci. 297, 254–270 (2015)
Wu, Z., Ju, H.P., Su, H., et al.: Stochastic stability analysis of piecewise homogeneous Markovian jump neural networks with mixed time-delays. J. Franklin Inst. 349(6), 2136–2150 (2012)
Zhang, J., Han, Z., Zhu, F.: Stochastic stability and stabilization of positive systems with Markovian jump parameters. Nonlinear Anal. Hybrid Syst. 12(1), 147–155 (2014)
Zhu, S., Han, Q., Zhang, C.: \(L_1\)-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)
Zhang, L., Boukas, E.K.: Stability and stabilization of Markovian jump linear systems with partly unknown transition probability. Automatica 45(2), 463–468 (2009)
Zhang, Y., He, Y., Wu, M., et al.: Stabilization for Markovian jump systems with partial informationon transition probability based on free-connection weighting matrices. Automatica 47, 79–84 (2011)
Zhang, L.: \(H_\infty \) estimation for discrete-time piecewise homogeneous Markov jump linear systems. Automatica 45(11), 2570–2576 (2009)
Zhao, F., Zhang, Q., Wang, G.: \(H_\infty \) filtering for piecewise homogeneous Markovian jump nonlinear systems. Int. J. Syst. Sci. 47(13), 3258–3271 (2016)
Zhu, S., Han, Q., Zhang, C.: \({l_1}\)-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)
Zhang, J., Lin, Y., Feng, G.: Analysis and synthesis of memory-based fuzzy sliding mode controllers. IEEE Trans. Cybern. 45(12), 2880–2889 (2015)
Zhang, B., Han, Q., Zhang, X., et al.: Sliding mode control with mixed current and delayed states for offshore steel jacket platforms. IEEE Trans. Control Syst. Technol. 22(5), 1769–1783 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Zhang, D., Zhang, Q. (2019). Positive \(L_{1}\) Controller Design for Positive Piecewise Homogeneous Markovian Jump Systems. In: Lam, J., Chen, Y., Liu, X., Zhao, X., Zhang, J. (eds) Positive Systems . POSTA 2018. Lecture Notes in Control and Information Sciences, vol 480. Springer, Cham. https://doi.org/10.1007/978-3-030-04327-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-04327-8_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04326-1
Online ISBN: 978-3-030-04327-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)