On exponential stability of linear networked control systems
Miad Moarref
Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, H3G 2W1, Canada
Search for more papers by this authorCorresponding Author
Luis Rodrigues
Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, H3G 2W1, Canada
Correspondence to: Luis Rodrigues, Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, H3G 2W1, Canada
E-mail: [email protected]
Search for more papers by this authorMiad Moarref
Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, H3G 2W1, Canada
Search for more papers by this authorCorresponding Author
Luis Rodrigues
Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, H3G 2W1, Canada
Correspondence to: Luis Rodrigues, Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, H3G 2W1, Canada
E-mail: [email protected]
Search for more papers by this authorSUMMARY
This paper addresses exponential stability of linear networked control systems. More specifically, the paper considers a continuous-time linear plant in feedback with a linear sampled-data controller with an unknown time varying sampling rate, the possibility of data packet dropout, and an uncertain time varying delay. The main contribution of this paper is the derivation of new sufficient stability conditions for linear networked control systems taking into account all of these factors. The stability conditions are based on a modified Lyapunov–Krasovskii functional. The stability results are also applied to the case where limited information on the delay bounds is available. The case of linear sampled-data systems is studied as a corollary of the networked control case. Furthermore, the paper also formulates the problem of finding a lower bound on the maximum network-induced delay that preserves exponential stability as a convex optimization program in terms of linear matrix inequalities. This problem can be solved efficiently from both practical and theoretical points of view. Finally, as a comparison, we show that the stability conditions proposed in this paper compare favorably with the ones available in the open literature for different benchmark problems. Copyright © 2012 John Wiley & Sons, Ltd.
REFERENCES
- 1 Daniel JP. Fly-by-wireless Airbus end-user viewpoint. Fly-by-Wireless Workshop, Grapevine, TX, 2007; 1–16.
- 2 Zahmati AS, Fernando X, Kojori H. Emerging wireless applications in aerospace: benefits, challenges, and existing methods. Proceedings of the 4th CANEUS Fly-by-Wireless Workshop (FBW11), Montreal, QC, 2011; 87–90.
- 3 Ost M, Pichavant C. Update on wireless avionics intra-communications (WAIC). ICAO WG-F Meeting, Dakar, Senegal, 2011; 1–13.
- 4 Zahmati AS, Fernando X, Kojori H. Transmission delay in wireless sensing, command and control applications for aircraft. Proceedings of the 4th CANEUS Fly-by-Wireless Workshop (FBW11), Montreal, QC, 2011; 91–94.
- 5 van de Wouw N, Naghshtabrizi P, Cloosterman M, Hespanha JP. Tracking control for sampled-data systems with uncertain time-varying sampling intervals and delays. International Journal of Robust and Nonlinear Control 2010; 20: 387–411.
- 6 Zhou Q, Shi P. A new approach to network-based H ∞ control for stochastic systems. International Journal of Robust and Nonlinear Control 2012; 22: 1036–1059.
- 7 Yang H, Xia Y, Shi P. Stabilization of networked control systems with nonuniform random sampling periods. International Journal of Robust and Nonlinear Control 2011; 21: 501–526.
- 8 Hale JK, Lunel SMV. Introduction to Functional Differential Equations. Springer: New York, 1993.
- 9
Gu K,
Kharitonov VL,
Chen J. Stability of Time-delay Systems. Birkhauser: Basel, 2003.
10.1007/978-1-4612-0039-0 Google Scholar
- 10
Boukas EK,
Liu ZK. Deterministic ans Stochastic Time Delay Systems. Birkhauser: Basel, 2002.
10.1007/978-1-4612-0077-2 Google Scholar
- 11 Yamamoto Y. New approach to sampled-data control systems—a function space method. Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, HI, 1990; 1882–1887.
- 12 Bamieh B, Pearson JB, Francis B, Tannenbaum A. A lifting technique for linear periodic systems with applications to sampled-data control. Systems & Control Letters 1991; 17(2): 79–88.
- 13 Yamamoto Y. A function space approach to sampled data control systems and tracking problems. IEEE Transactions on Automatic Control 1994; 39(4): 703–713.
- 14
Chen T,
Francis B. Optimal Sampled-data Control Systems, Communication and Control Engineering Series, Springer-Verlag: Berlin, 1995.
10.1007/978-1-4471-3037-6 Google Scholar
- 15 Hu LS, Lam J, Cao YY, Shao HH. A linear matrix inequality (LMI) approach to robust H2 sampled-data control for linear uncertain systems. IEEE Transactions on Systems, Man, and Cybernetics 2003; 33(1): 149–155.
- 16 Naghshtabrizi P, Hespanha JP, Teel AR. Exponential stability of impulsive systems with application to uncertain sampled-data systems. Systems & Control Letters 2008; 57(5): 378–385.
- 17 Naghshtabrizi P, Hespanha JP, Teel AR. Stability of delay impulsive systems with application to networked control systems. Transactions of the Institute of Measurement and Control 2010; 32(5): 511–528.
- 18 Fridman E, Seuret A, Richard JP. Robust sampled-data stabilization of linear systems: an input delay approach. Automatica 2004; 40(8): 1441–1446.
- 19 Gao H, Chen T, Lam J. A new delay system approach to network-based control. Automatica 2008; 44(1): 39–52.
- 20 Fridman E. A refined input delay approach to sampled-data control. Automatica 2010; 46(2): 421–427.
- 21 Zhu X, Wang Y. New stability and stabilization criteria for sampled data systems. Proceedings of the 8th IEEE International Conference on Control and Automation, Vol. 1, Xiamen, China, 2010; 1829–1834.
- 22 Liu K, Fridman E. Networked-based stabilization via discontinuous Lyapunov functionals. International Journal of Robust and Nonlinear Control 2012; 22: 420–436.
- 23 Mirkin L. Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Transactions on Automatic Control 2007; 52(6): 1109–1112.
- 24 Krasovskii N. Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay. Stanford University Press: Stanford, CA, 1963.
- 25 Hespanha JP, Naghshtabrizi P, Xu Y. A survey of recent results in networked control systems. Proceedings of the IEEE 2007; 95(1): 138–162.
- 26 Strum JF. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 1999; 11-12: 625–653. (Available from: http://sedumi.ie.lehigh.edu/) [accessed on December 1, 2012].
- 27 Löfberg J. YALMIP: A toolbox for modeling and optimization in MATLAB. Proceedings of the CACSD Conference, Taipei, Taiwan, 2004; 284–289. (Available from: http://users.isy.liu.se/johanl/yalmip/) [accessed on December 1, 2012].
- 28 Liberzon D. On stabilization of linear systems with limited information. IEEE Transactions on Automatic Control 2003; 48(2): 304–307.
- 29 Shao H. New delay-dependent stability criteria for systems with interval delay. Automatica 2009; 45(3): 744–749.
- 30 Sun J, Liu G, Chen J, Rees D. Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica 2010; 46(2): 466–470.
- 31 Dong H, Wang Z, Gao H. Robust H ∞ filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts. IEEE Transactions on Signal Processing 2010; 58(4): 1957–1966.
- 32 Dong H, Wang Z, Gao H. Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts. IEEE Transactions on Signal Processing 2012; 60(6): 3164–3173.
- 33 Cloosterman M, van de Wouw N, Heemels M, Nijmeijer H. Robust stability of networked control systems with time-varying network-induced delays. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, 2006; 4980–4985.
- 34 Nilsson J. Real-time control systems with delays. Ph.D. Thesis, Lund Institute of Technology, 1998.
- 35 Zhang W, Branicky MS, Phillips SM. Stability of networked control systems. IEEE Control Systems Magazine 2001; 21(1): 84–99.
- 36 Fridman E, Orlov Y. Exponential stability of linear distributed parameter systems with time-varying delays. Automatica 2009; 45(1): 194–201.
- 37 Hoffman K, Kunze R. Linear Algebra, (2nd edn), Prentice Hall: Englewood Cliffs, NJ, 1971.
- 38
Boyd S,
El Ghaoui L,
Feron E,
Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. SIAM: Philadelphia, 1994.
10.1137/1.9781611970777 Google Scholar
- 39 Khalil HK. Nonlinear Systems. Prentice Hall: New Jersey, 2002.
- 40 Park P, Ko JW. Stability and robust stability for systems with a time-varying delay. Automatica 2007; 43(10): 1855–1858.