Abstract
This chapter presents results on the stability of reset control systems with finite-dimensional base systems. The stability problem is addressed from different, complementary points of view: (i) internal or Lyapunov stability, (ii) external or input–output stability with passivity analysis, and (iii) stability by the describing function method. Internal stability techniques are subdivided into techniques giving rise to stability conditions that do not depend directly on the reset instants (reset-times independent) or, alternatively, are reset-times dependent. The first case is obtained directly using continuous time Lyapunov functions (that gives rise to the so-called H
β
condition), while the second case (reset-times dependent) requires a discretization at the after-reset values and a subsequent discrete-time Lyapunov analysis. Then, the input–output 
stability is studied, and a number or results are presented in connection with passivity and dissipativity properties of reset feedback loops. Finally, the standard describing function tool is used for approximately predicting the appearance or absence of oscillations and as a good practical tool to evaluate the phase lead obtained by reset compensation. The material is based on several published works (Baños et al. in European Control Conference, Kos, Greece, 2007; Baños et al. in Proc. IEEE International Symposium on Industrial Electronics, Spain, 2007; Baños et al. in 3rd IFAC Conference on Analysis and Design of Hybrid Systems, Zaragoza, Spain, 2009; Baños et al. in Nonlinear Anal. Hybrid Syst., 2010, doi:10.1016/j.nahs.2010.07.004; Baños et al. in IEEE Trans. Autom. Control 56(1):217–223, 2011; Carrasco et al. in 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, Florida, USA, 2008; Carrasco et al. in Syst. Control Lett. 59(1):18–24, 2010).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aangenent, W.H.T.M., Witvoet, G., Heemels, W.P.M.H., van de Molengraft, M.J.G., Steinbuch, M.: An LMI-based
gain performance analysis for reset control systems. In: Proceedings of the American Control Conference (2008)
Bainov, D.D., Simeonov, P.S.: Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood, Chichester (1989)
Baños, A., Carrasco, J., Barreiro, A.: Reset times-dependent stability of reset control system. In: European Control Conference, Kos, Greece (2007)
Baños, A., Carrasco, A., Barreiro, A.: Reset times dependent stability of reset systems with unstable base system. In: Proc. IEEE International Symposium on Industrial Electronics, Spain (2007)
Baños, A., Dormido, S., Barreiro, A.: Stability analysis of reset control systems with reset band. In: 3rd IFAC Conference on Analysis and Design of Hybrid Systems, Zaragoza, Spain (2009)
Baños, A., Dormido, S., Barreiro, A.: Limit cycles analysis in reset systems with reset band. Nonlinear Anal. Hybrid Syst. (2010). doi:10.1016/j.nahs.2010.07.004
Baños, A.: Carrasco, J., Barreiro, A.: Reset times-dependent stability of reset control systems. IEEE Trans. Autom. Control 56(1), 217–223 (2011)
Bemporad, A., Bianchini, G., Brogi, F.: Passivity analysis and passification of discrete-time hybrid systems. IEEE Trans. Autom. Control 53(4), 1004–1009 (2008)
Beker, O.: Analysis of reset control systems. Ph.D. Thesis, University of Massachusetts, Amherst (2001)
Beker, O., Hollot, C.V., Chait, Y., Han, H.: Fundamental properties of reset control systems. Automatica 40, 905–915 (2004)
Camlibel, M.K., Heemels, W.P.M.H., Schumacher, J.M.: On linear passive complementary systems. Eur. J. Control 8(3), 220–237 (2002)
Carrasco, J., Baños, A., van der Schaft, A.: A passivity approach to reset control of nonlinear systems. In: 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, Florida, USA (2008)
Carrasco, J., Baños, A., van der Schaft, A.: A passivity-based approach to reset control systems stability. Syst. Control Lett. 59(1), 18–24 (2010)
Bergen, A.R., Franks, R.L.: Justification of the describing function method. SIAM J. Control 9, 568–589 (1971)
Fernández, A., Barreiro, A., Baños, A., Carrasco, J.: Reset control for passive bilateral teleoperation. IEEE Trans. Ind. Electron. (2010). doi:10.1109/TIE.2010.2077610
Guo, Y., Wang, Y., Xie, L., Zheng, J.: Stability analysis and design of reset systems: theory and an application. Automatica 45(2), 492–497 (2009)
Haddad, W.M., Chellaboina, V., Nersesov, S.G.: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, Princeton (2006)
Horowitz, I.M., Rosenbaum, P.: Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. Int. J. Control 24(6), 977–1001 (1975)
Khalil, H.K.: Nonlinear Systems, 2nd edn. Prentice Hall, New York (2002)
Krishman, K.R., Horowitz, I.M.: Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances. Int. J. Control 19(4), 689–706 (1974)
Lozano, R., Brogliato, B., Egeland, O., Maschke, B.: Dissipative Systems Analysis and Control. Springer, London (2000)
Mees, A., Bergen, A.: Describing functions revisited. IEEE Trans. Autom. Control 20(4), 473–478 (1975)
Michel, A., Hu, B.: Towards a stability theory of general hybrid dynamical systems. Automatica 35, 371–384 (1999)
Nešić, D., Zaccarian, L., Teel, A.R.: Stability properties of reset systems. Automatica 44(8), 2019–2026 (2008)
Pogromski, A., Jirstrand, M., Spangeus, P.: On stability and passivity of a class of hybrid systems. In: Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, Florida, USA (1998)
Rockafellar, R.T., Wets, J.-B.: Variational Analysis. Series of Comprehensive Studies in Mathematics, vol. 317. Springer, Berlin (1998)
Rugh, W.J.: Linear System Theory, 2nd edn. Prentice Hall, New York (1996)
Sanders, S.R.: On limit cycles and the describing function method in periodically switched circuits. IEEE Trans. Circuits Syst. I 20(4), 473–478 (1975)
Van der Schaft, A.:
-Gain and Passivity Techniques in Nonlinear Control. Springer, Berlin (2000)
Vidal, A., Baños, A.: QFT-based design of PI + CI compensators: application in process control. In: IEEE Mediterranean Conference on Control and Automation, pp. 806–811 (2008)
Vidal, A., Baños, A.: Reset compensation for temperature control: experimental application on heat exchangers. Chem. Eng. J. 159(1–3), 170–181 (2010)
Vidyasagar, M.: Nonlinear Systems Stability. Prentice-Hall, London (1993)
Willems, J.C.: Dissipative dynamical systems—Part I: General theory. Arch. Ration. Mech. Anal. 45, 321–351 (1972)
Wright, R.A., Kravaris, C.: On-line identification and nonlinear control of an industrial pH process. J. Process Control 11, 361–374 (2001)
Zefran, M., Bullo, F., Stein, M.: A notion of passivity for hybrid systems. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, USA (2001)
gain performance analysis for reset control systems. In: Proceedings of the American Control Conference (2008)
-Gain and Passivity Techniques in Nonlinear Control. Springer, Berlin (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag London Limited
About this chapter
Cite this chapter
Baños, A., Barreiro, A. (2012). Stability of Reset Control Systems. In: Reset Control Systems. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-4471-2250-0_3
Download citation
DOI: https://doi.org/10.1007/978-1-4471-2250-0_3
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2216-6
Online ISBN: 978-1-4471-2250-0
eBook Packages: EngineeringEngineering (R0)