Advertisement

Stability Analysis and Stabilization—Polytopic Representation Approach

  • Sophie TarbouriechEmail author
  • Germain Garcia
  • João Manoel Gomes da SilvaJr.
  • Isabelle Queinnec
Chapter
  • 1.6k Downloads

Abstract

Considering polytopic differential inclusions to model the saturation effects on the closed-loop system dynamics, this chapter addresses the stability analysis and stabilization of systems presenting control saturation. First, conditions for the regional asymptotic stability of the closed-loop system are derived. In particular, it is deeply discussed how to obtain estimates of the basin of attraction from the conditions. Similarly, conditions to ensure the external stability of the system with saturating inputs are stated considering that the system is subject to the action of amplitude or energy bounded exogenous signals. Secondly, the problem of designing a control law taking explicitly into account the possibility of actuator saturation is addressed. Although the results are mainly focused on the state feedback control laws, results regarding the design of dynamic output feedback control are briefly presented. The extension of the approach to cope with model uncertainties is also briefly discussed. Finally, the discrete-time counterpart of the results are presented. In this case, some particularities regarding the determination of polyhedral regions of stability are considered.

References

  1. 1.
    Alamo, T., Cepeda, A., Limon, D.: Improved computation of ellipsoidal invariant sets for saturated control systems. In: Conference on Decision and Control, Sevilla, Spain, December, pp. 6216–6221 (2005) Google Scholar
  2. 2.
    Alamo, T., Cepeda, A., Limon, D., Camacho, E.F.: Estimation of the domain of attraction for saturated discrete-time systems. Int. J. Syst. Sci. 37(8), 575–583 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 16.
    Bateman, A., Lin, Z.: An analysis and design method for linear systems under nested saturation. Syst. Control Lett. 48(1), 41–52 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 24.
    Benzaouia, A., Akhrif, O., Saydy, L.: Stability and control synthesis of switched systems subject to actuator saturation. In: American Control Conference, Boston, USA, pp. 5818–5823 (2004) Google Scholar
  5. 26.
    Bernussou, J., Peres, P.L.D., Geromel, J.C.: A linear programming oriented procedure for quadratic stabilization of uncertain systems. Syst. Control Lett. 13, 65–72 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 33.
    Blanchini, F.: Feedback control for linear time-invariant systems with state and control bounds in the presence of disturbances. IEEE Trans. Autom. Control 35(11), 1231–1234 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 34.
    Blanchini, F.: Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions. IEEE Trans. Autom. Control 39(2), 428–433 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 36.
    Blanchini, F.: Set invariance in control. Automatica 35(11), 1747–1767 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 37.
    Blanchini, F., Miani, S.: Best transient estimate for linear discrete-time uncertain systems. In: European Control Conference, Rome, Italy, pp. 1010–1015 (1995) Google Scholar
  10. 45.
    Boyd, S., El Ghaoui, L., Féron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1994) zbMATHGoogle Scholar
  11. 52.
    Burgat, C., Tarbouriech, S., Klaï, M.: Continuous-time saturated state feedback regulators: theory and design. Int. J. Syst. Sci. 25(2), 315–336 (1994) zbMATHCrossRefGoogle Scholar
  12. 54.
    Cao, Y.Y., Lin, Z.: Min-max MPC algorithm for LPV systems subject to input saturation. IEE Proc. Control Theory Appl., 152–266 (2005) Google Scholar
  13. 55.
    Cao, Y.Y., Lin, Z., Hu, T.: Stability analysis of linear time-delay systems subject to input saturation. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49, 233–240 (2002) MathSciNetCrossRefGoogle Scholar
  14. 62.
    Castelan, E.B., Tarbouriech, S., Gomes da Silva Jr., J.M., Queinnec, I.: \(\mathcal{L}_{2}\)-stabilization of continuous-time systems with saturating actuators. Int. J. Robust Nonlinear Control 16, 935–944 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 64.
    Chen, C.T.: Linear System Theory and Design. Holt, Rinehart & Winston, New York (1984) Google Scholar
  16. 68.
    Chilali, M., Gahinet, P.: H design with pole placement constraints: an LMI approach. IEEE Trans. Autom. Control 41(3) (1996) Google Scholar
  17. 84.
    Dolphus, R.M., Schmitendorf, W.E.: Stability analysis for a class of linear controllers under control constraints. In: Conference on Decision and Control, Brighton, UK, pp. 77–80 (1991) Google Scholar
  18. 89.
    Fang, H., Lin, Z., Hu, T.: Analysis of linear systems in the presence of actuator saturation and \(\mathcal{L}_{2}\)-disturbances. Automatica 40(7), 1229–1238 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 98.
    Fridman, E., Pila, A., Shaked, U.: Regional stabilization and H control of time-delay systems with saturating actuators. Int. J. Robust Nonlinear Control 13, 885–907 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 99.
    Fuller, A.T.: In the large stability of relay and saturated control systems with actuator saturation. Int. J. Control 10(4), 457–480 (1969) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 112.
    Garcia, G., Tarbouriech, S.: Stabilization with eigenvalues placement of a norm bounded uncertain system by bounded inputs. Int. J. Robust Nonlinear Control 9, 599–615 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 126.
    Gilbert, E.G., Tan, K.T.: Linear systems with state and control constraints: the theory and application of maximal output admissible sets. IEEE Trans. Autom. Control 36(9), 1008–1020 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 129.
    Goh, K.C., Safonov, M.G., Ly, J.H.: Robust synthesis via bilinear matrix inequalities. Int. J. Robust Nonlinear Control 6(9/10), 1079–1095 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 130.
    Gomes da Silva Jr., J.M.: Sur la stabilité locale de systèmes linéaires avec saturation des commandes. PhD thesis, Université Paul Sabatier, Toulouse, France, October 1997. Rapport LAAS No. 97383 Google Scholar
  25. 133.
    Gomes da Silva Jr., J.M., Tarbouriech, S.: Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach. In: American Control Conference, Philadelphia, USA, pp. 92–96 (1998) Google Scholar
  26. 134.
    Gomes da Silva Jr., J.M., Tarbouriech, S.: Stability regions for linear systems with saturating controls. In: European Control Conference, Karlsruhe, Germany (1999). F924 Google Scholar
  27. 137.
    Gomes da Silva Jr., J.M., Tarbouriech, S.: Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach. IEEE Trans. Autom. Control 46(1), 119–124 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 141.
    Gomes da Silva Jr., J.M., Fischman, A., Tarbouriech, S., Dion, J.M., Dugard, L.: Synthesis of state feedback for linear systems subject to control saturation by an LMI-based approach. In: IFAC Symposium on Robust Control Design (ROCOND), Budapest, Hungary, pp. 229–234 (1997) Google Scholar
  29. 144.
    Gomes da Silva Jr., J.M., Tarbouriech, S., Garcia, G.: Local stabilization of linear systems under amplitude and rate saturating actuators. IEEE Trans. Autom. Control 48(5), 842–847 (2003) MathSciNetCrossRefGoogle Scholar
  30. 146.
    Gomes da Silva Jr., J.M., Lescher, F., Eckhard, D.: Design of time-varying controllers for discrete-time linear systems with input saturation. IET Control Theory Appl. 1, 155–162 (2007) MathSciNetCrossRefGoogle Scholar
  31. 172.
    Henrion, D., Tarbouriech, S.: LMI relaxations for robust stability of linear systems with saturating controls. Automatica 35, 1599–1604 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 175.
    Henrion, D., Tarbouriech, S., Garcia, G.: Output feedback robust stabilization of uncertain linear systems with saturating controls: an LMI approach. IEEE Trans. Autom. Control 44(11), 2230–2237 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 187.
    Hu, T., Lin, Z.: On enlarging the basin of attraction for linear systems under saturated linear feedback. Syst. Control Lett. 40(1), 59–69 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 188.
    Hu, T., Lin, Z.: Control Systems with Actuator Saturation: Analysis and Design. Birkhäuser, Boston (2001) zbMATHCrossRefGoogle Scholar
  35. 190.
    Hu, T., Lin, Z.: Controlled invariance of ellipsoids: linear vs. nonlinear feedback. Syst. Control Lett. 53(3–4), 203–210 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 192.
    Hu, T., Lin, Z., Chen, B.M.: Analysis and design for discrete-time linear systems subject to actuator saturation. Syst. Control Lett. 45(2), 97–112 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 193.
    Hu, T., Lin, Z., Chen, B.M.: An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica 38, 351–359 (2002) zbMATHCrossRefGoogle Scholar
  38. 199.
    Huang, H., Li, D., Lin, Z., Xi, Y.: An improved robust model predictive control design in the presence of actuator saturation. Automatica 47, 861–864 (2011) zbMATHCrossRefGoogle Scholar
  39. 214.
    Kerrigan, E.C.: Robust constraint satisfaction: Invariant sets and predictive control. PhD thesis, University of Cambridge, Cambridge (2000) Google Scholar
  40. 215.
    Khalil, H.K.: Nonlinear Systems. MacMillan, London (1992) zbMATHGoogle Scholar
  41. 218.
    Kim, J., Bien, Z.: Robust stability of uncertain linear systems with saturating actuators. IEEE Trans. Autom. Control 39(1), 202–207 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 219.
    Kiyama, T., Iwasaki, T.: On the use of multi-loop circle for saturating control synthesis. Syst. Control Lett. 41, 105–114 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  43. 220.
    Klaï, M.: Stabilisation des systèmes linéaires continus contraints sur la commande par retour d’état et sortie saturés. PhD thesis, Université Paul Sabatier, Toulouse, France (September 1994). Rapport LAAS No. 94323 Google Scholar
  44. 231.
    Labit, Y., Peaucelle, D., Henrion, D.: Sedumi interface 1.02: a tool for solving LMI problems with sedumi. In: Proceedings of the CACSD Conference, Glasgow, Scotland (2002) Google Scholar
  45. 238.
    Limon, D., Gomes da Silva Jr., J.M., Alamo, T., Camacho, E.F.: Improved MPC design based on saturating control laws. Eur. J. Control 11, 112–122 (2005) MathSciNetCrossRefGoogle Scholar
  46. 254.
    Ma, C.C.H.: Instability of linear unstable system with inputs limits. ASME J. Dyn. Syst. Meas. Control 113, 742–744 (1991) CrossRefGoogle Scholar
  47. 273.
    Nguyen, T., Jabbari, F.: Output feedback controllers for disturbance attenuation with actuator amplitude and rate saturation. Automatica 36(9), 1339–1346 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  48. 280.
    Paim, C., Tarbouriech, S., Gomes da Silva Jr., J.M., Castelan, E.B.: Control design for linear systems with saturating actuators and L 2-bounded disturbances. In: Conference on Decision and Control, Las Vegas, USA (2002) Google Scholar
  49. 288.
    Petersen, I.R.: A stabilization algorithm for a class of uncertain linear systems. Syst. Control Lett. 8, 351–356 (1987) zbMATHCrossRefGoogle Scholar
  50. 309.
    Saberi, A., Lin, Z., Teel, A.: Control of linear systems with saturating actuators. IEEE Trans. Autom. Control 41(3), 368–377 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  51. 314.
    Scherer, C., Gahinet, P., Chilali, M.: Multiobjective output feedback control via LMI optimization. IEEE Trans. Autom. Control 42(7), 896–911 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 337.
    Sussmann, H.J., Yang, Y.: On the stability of multiple integrators by means of bounded controls. In: Conference on Decision and Control, Brighton, UK, pp. 70–72 (1991) Google Scholar
  53. 338.
    Sussmann, H.J., Sontag, E.D., Yang, Y.: A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Autom. Control 39(12), 2411–2425 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  54. 348.
    Tarbouriech, S., Gomes da Silva Jr., J.M.: Admissible polyhedra for discrete-time linear systems with saturating controls. In: American Control Conference, Albuquerque, USA, vol. 6, pp. 3915–3919 (1997) Google Scholar
  55. 349.
    Tarbouriech, S., Gomes da Silva Jr., J.M.: Synthesis of controllers for continuous-time delay systems with saturating controls via LMIs. IEEE Trans. Autom. Control 45(1), 105–111 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  56. 353.
    Tarbouriech, S., Gomes da Silva Jr., J.M., Garcia, G.: Stability and disturbance tolerance for linear systems with bounded controls. In: European Control Conference, Porto, Portugal, pp. 3219–3224 (2001) Google Scholar
  57. 354.
    Tarbouriech, S., Garcia, G., Gomes da Silva Jr., J.M.: Robust stability of uncertain polytopic linear time-delay systems with saturating inputs: an LMI approach. Comput. Electr. Eng. 28, 157–169 (2002) zbMATHCrossRefGoogle Scholar
  58. 395.
    Wredenhagen, G.F., Bélanger, P.R.: Piecewise-linear LQ control for systems with input constraints. Automatica 30(3), 403–416 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  59. 399.
    Wu, F., Grigoriadis, K.M., Packard, A.: Anti-windup controller design using linear parameter-varying control methods. Int. J. Control 73(12), 1104–1114 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 400.
    Wu, F., Lin, Z., Zheng, Q.: Output feedback stabilization of linear systems with actuator saturation. IEEE Trans. Autom. Control 52(1), 123–128 (2007) MathSciNetCrossRefGoogle Scholar
  61. 410.
    Zheng, Q., Wu, F.: Output feedback control of saturated discrete-time linear systems using parameter-dependent Lyapunov functions. Syst. Control Lett. 57, 896–903 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  62. 417.
    Zhou, B., Zheng, W.Y., Duan, G.-R.: An improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation. Automatica 47, 306–315 (2011) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Sophie Tarbouriech
    • 1
    Email author
  • Germain Garcia
    • 2
  • João Manoel Gomes da SilvaJr.
    • 3
  • Isabelle Queinnec
    • 1
  1. 1.Laboratoire Analyse et Architecture des Systèmes (LAAS)CNRSToulouse CX 4France
  2. 2.Laboratoire Analyse et Architecture des Systèmes (LAAS)CNRSToulouse CX 4France
  3. 3.Departamento de Engenharia ElétricaUniversidade Federal do Rio Grande do SulPorto AlegreBrazil

Personalised recommendations