Advertisement

Nonlinear Dynamics

, Volume 76, Issue 4, pp 2041–2058 | Cite as

Distributed proportional plus second-order spatial derivative control for distributed parameter systems subject to spatiotemporal uncertainties

  • Jun-Wei Wang
  • Han-Xiong Li
  • Huai-Ning Wu
Original Paper

Abstract

In this paper, a robust distributed control design based on proportional plus second-order spatial derivative (P-sD\(^2\)) is proposed for exponential stabilization and minimization of spatial variation of a class of distributed parameter systems (DPSs) with spatiotemporal uncertainties, whose model is represented by parabolic partial differential equations with spatially varying coefficients. Based on the Lyapunov’s direct method, a robust distributed P-sD\(^2\) controller is developed to not only exponentially stabilize the DPS for all admissible spatiotemporal uncertainties but also minimize the spatial variation of the process. The outcome of the robust distributed P-sD\(^2\) control problem is formulated as a spatial differential bilinear matrix inequality (SDBMI) problem. A local optimization algorithm that the SDBMI is treated as a double spatial differential linear matrix inequality (SDLMI) is presented to solve this SDBMI problem. Furthermore, the SDLMI optimization problem can be approximately solved via the finite difference method and the existing convex optimization techniques. Finally, the proposed design method is successfully applied to feedback control problem of the FitzHugh–Nagumo equation.

Keywords

Distributed parameter systems Robust control Exponential stability Spatiotemporal uncertainty Linear matrix inequalities (LMIs) 

Notes

Acknowledgments

This work was supported in part by the National Basic Research Program of China (973 Program) (2012CB720003), in part by the National Natural Science Foundation of China under Grants 61121003, 51175519, and 91016004, and in part by the Beijing Youth Fellowship Program (YETP0378). The authors gratefully acknowledge the helpful comments and suggestions of the anonymous reviewers, which have improved the presentation.

References

  1. 1.
    Ray, W.H.: Advanced Process Control. McGraw-Hill, New York (1981)Google Scholar
  2. 2.
    Christofides, P.D.: Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. MA, Birkhäuser, Boston (2001)CrossRefGoogle Scholar
  3. 3.
    Li, H.-X., Qi, C.: Modeling of distributed parameter systems for applications-A synthesized review from time-space separation. J. Process Control 20, 891–901 (2010)CrossRefGoogle Scholar
  4. 4.
    Dorato, P.: A historical review of robust control. IEEE Control Syst. Mag. 7, 44–47 (1987)Google Scholar
  5. 5.
    Green, M., Limebeer, D.: Linear Robust Control. Prentice-Hall Inc, Upper Saddle River (1995)MATHGoogle Scholar
  6. 6.
    Freeman, R.A., Kokotovic, P.V.: Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Birkhauser, Boston (1996)CrossRefMATHGoogle Scholar
  7. 7.
    Krstic, M., Deng, H.: Stabilization of Nonlinear Uncertain Systems. Springer, London (1998)MATHGoogle Scholar
  8. 8.
    Curtain, R.F., Zwart, H.J.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York (1995)CrossRefMATHGoogle Scholar
  9. 9.
    Christofides, P.D.: Control of nonlinear distributed process systems: recent developments and challenge. AIChE J. 47, 514–518 (2001)CrossRefGoogle Scholar
  10. 10.
    Padhi, R., Ali, Sk F.: An account of chronological developments in control of distributed parameter systems. Annu. Rev. Control 33, 59–68 (2009)Google Scholar
  11. 11.
    Balas, M.J.: Feedback control of linear diffusion processes. Int. J. Control 29, 523–534 (1979)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Curtain, R.F.: Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input. IEEE Trans. Autom. Control 27, 98–104 (1982)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Christofides, P.D.: Robust control of parabolic PDE systems. Chem. Eng. Sci. 53, 2949–2965 (1998)CrossRefGoogle Scholar
  14. 14.
    Dubljevic, S., Christofides, P.D.: Predictive control of parabolic PDEs with boundary control actuation. Comput. Chem. Eng. 61, 6239–6248 (2006)Google Scholar
  15. 15.
    Wu, H.-N., Li, H.-X.: Adaptive neural control design for nonlinear distributed parameter systems with persistent bounded disturbances. IEEE Trans. Neural Netw. 20, 1630–1644 (2009)CrossRefGoogle Scholar
  16. 16.
    Hagen, G., Mezic, I.: Spillover stabilization in finite-dimensional control and observer design for dissipative evolution equations. SIAM J. Control Optim. 42, 746–768 (2003)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Chen, B.-S., Chang, Y.-T.: Fuzzy state-space modeling and robust observer-based control design for nonlinear partial differential systems. IEEE Trans. Fuzzy Syst. 17, 1025–1043 (2009)Google Scholar
  18. 18.
    Padhi, R., Balakrishnan, S.: Optimal dynamic inversion control design for a class of nonlinear distributed parameter systems with continuous and discrete actuators. IET Control Theory Appl. 1, 1662–1671 (2007)CrossRefGoogle Scholar
  19. 19.
    Bamieh, B., Paganini, F., Dahleh, M.: Distributed control of spatially invariant systems. IEEE Trans. Autom. Control 47, 1091–1107 (2002)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Demetriou, M.: Model reference adaptive control of slowly time-varying parabolic distributed parameter systems. In Proceedings of the 33rd Conference on Decision and Control, pp. 775–780 . Lake Buena Vista, 14–16 Dec (1994)Google Scholar
  21. 21.
    Orlov, Y., Dochain, D.: Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor. IEEE Trans. Autom. Control 47, 1293–1304 (2002)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Hagen, G., Mezic, I., Bamieh, B.: Distributed control design for parabolic evolution equations: application to compressor stall control. IEEE Trans. Autom. Control 49, 1247–1258 (2004)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Orlov, Y., Pisano, A., Usai, E.: Continuous state-feedback tracking of an uncertain heat diffusion process. Syst. Control Lett. 59, 754–759 (2010)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Orlov, Y., Pisano, A., Usai, E.: Exponential stabilization of the uncertain wave equation via distributed dynamic input extension. IEEE Trans. Autom. Control 56, 212–217 (2011)Google Scholar
  25. 25.
    Wang, J.-W., Wu, H.-N., Li, H.-X.: Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters. Automatica 48, 569–576 (2012)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Wang, J.-W., Wu, H.-N., Li, H.-X.: Distributed proportional-spatial derivative control of nonlinear parabolic systems via fuzzy PDE modeling approach. IEEE Trans. Syst. Man Cybern Part B 42, 927–938 (2012) Google Scholar
  27. 27.
    Boyd, S., Ghaoui, L., Feron, E., et al.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)CrossRefMATHGoogle Scholar
  28. 28.
    Gahinet, P., Nemirovskii, A., Laub, A.J., et al.: LMI Control Toolbox for Use with Matlab. The Math. Works Inc., Natic (1995)Google Scholar
  29. 29.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefMATHGoogle Scholar
  30. 30.
    Balas, M.J.: Finite-dimensional control of distributed parameter systems by Galerkin approximation of infinite dimensional controller. J. Math. Anal. Appl. 114, 17–36 (1986)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Toker, O., Ozbay, H.: On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In: Proceedings of American Control Conference, pp. 2525–2526. Seattle (1995)Google Scholar
  32. 32.
    Goh, K., Turan, L., Safonov, M., et al.: Biaffine matrix inequality properties and computational methods. In: Proceedings of American Control Conference, pp. 850–855. Baltimore (1994)Google Scholar
  33. 33.
    Cao, Y.-Y., Lam, J., Sun, Y.-X.: Static output feedback stabilization: an ILMI approach. Automatica 34, 1641–1645 (1998)CrossRefMATHGoogle Scholar
  34. 34.
    Burden, R., Faires, J.: Numerical Analysis, 7th edn. Brooks/Cole, New York (2000)Google Scholar
  35. 35.
    Shaked, U., Suplin, V.: A new bounded real lemma representation for the continuous-time case. IEEE Trans. Autom. Control 46, 1420–1426 (2001)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Wu, H.-N., Wang, J.-W., Li, H.-X.: Exponential stabilization for a class of nonlinear parabolic PDE systems via fuzzy control approach. IEEE Trans. Fuzzy Syst. 20, 318–329 (2012)CrossRefGoogle Scholar
  37. 37.
    Wu, H.-N., Wang, J.-W., Li, H.-X.: Fuzzy boundary control design for a class of nonlinear parabolic distributed parameter systems. IEEE Trans. Fuzzy Syst. Available online (2013). doi: 10.1109/TFUZZ.2013.2269698
  38. 38.
    Keener, J., Sneyd, J.: Mathematical Physiology. Springer, New York (1998)MATHGoogle Scholar
  39. 39.
    Theodoropoulos, K., Qian, Y.-H., Kevrekidis, I.: ”Coarse” stability and bifurcation analysis using timesteppers: a reaction diffusion example. Proc. Natl. Acad. Sci. 97, 9840–9843 (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.School of Automation and Electrical EngineeringUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  2. 2.Department of Systems Engineering and Engineering ManagementCity University of Hong KongHong Kong SARPeople’s Republic of China
  3. 3.State Key Laboratory of High Performance Complex ManufacturingCentral South UniversityChangshaChina
  4. 4.Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical EngineeringBeihang University (Beijing University of Aeronautics and Astronautics)BeijingPeople’s Republic of China

Personalised recommendations