Journal of Dynamical and Control Systems

, Volume 24, Issue 4, pp 563–576 | Cite as

Backstepping State Feedback Regulator Design for an Unstable Reaction-Diffusion PDE with Long Time Delay

  • Jian-Jun GuEmail author
  • Jun-Min Wang


We consider the output regulation of an unstable reaction-diffusion PDE in the presence of regulator delay and unmatched disturbances, which are generated by an exosystem. The systematic design procedure of a backstepping state feedback regulator is first presented by mapping the reaction-diffusion PDE cascaded with a transport equation into an error system, which is shown to be exponentially stable with a prescribed rate in a suitable Hilbert space. The regulator design relies on solving regulator equations, and the solvability condition of the regulator equations is then characterized by a transfer function and eigenvalues of the exosystem. Finally, the numerical simulations are provided to illustrate the effect of the regulator.


Reaction-diffusion PDE Time delay Output regulation Backstepping Stability 

Mathematics Subject Classification (2010)

93A20 93C20 93D15 



This work was supported by the National Natural Science Foundation of China (Grant No. 61673061).


  1. 1.
    Adams RA, Fournier JJF. Sobolev spaces. Amsterdam: Elsevier/Academic Press; 2003.zbMATHGoogle Scholar
  2. 2.
    Ahmed-Ali T, Giri F, Krstic M, Lamnabhi-Lagarrigue F. Adaptive observer for a class of output- delayed systems with parameter uncertainty - a PDE based approach. In IFAC-PapersOnLine 2016;49(13):158–163.CrossRefGoogle Scholar
  3. 3.
    Ailon A, Gil MI. Stability analysis of a rigid robot with output-based controller and time delay. Syst Control Lett 2000;40(1):31–5.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anderson RJ, Spong MW. Bilateral control of teleoperators with time delay. IEEE Trans Automat Control. 1989;34(5):494–501.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aulisa E, Gilliam D. A practical guide to geometric regulation for distributed parameter systems. Monographs and research notes in mathematics. Boca Raton: CRC Press; 2016.zbMATHGoogle Scholar
  6. 6.
    Biberovic E, Iftar A, Ozbay H. A solution to the robust flow control problem for networks with multiple bottlenecks. In: Proceedings of the 40th IEEE conference on decision and control. Orlando; 2001. p. 2303–2308.Google Scholar
  7. 7.
    Chentouf B, Wang JM. Stabilization of a one-dimensional dam-river system: nondissipative and noncollocated case. J Optim Theory Appl. 2007;134(2):223–39.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chentouf B, Wang JM. A Riesz basis methodology for proportional and integral output regulation of a one-dimensional diffusive-wave equation. SIAM J Control Optim. 2008;47(5):2275–302.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Deutscher J. A backstepping approach to the output regulation of boundary controlled parabolic PDEs. Automatica. 2015;57:56–64.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Deutscher J, Kerschbaum S. Backstepping design of robust state feedback regulators for second order hyperbolic PIDEs. In IFAC- PapersOnLine 2016;49(8):80–85.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Han X, Fridman E, Spurgeon SK. Sliding-mode control of uncertain systems in the presence of unmatched disturbances with applications. Internat J Control. 2010; 83(12):2413–26.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hou M, Zítek P, Patton RJ. An observer design for linear time-delay systems. IEEE Trans Automat Control. 2002;47(1):121–5.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Krstic M. Control of an unstable reaction-diffusion PDE with long input delay. Syst Control Lett. 2009;58:773–82.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Krstic M, Smyshlyaev A. Boundary control of PDEs: a course on backstepping designs. Philadelphia: SIAM; 2008.CrossRefzbMATHGoogle Scholar
  15. 15.
    Lasiecka I, Triggiani R. 2000. Control theory for partial differential equations: continuous and approximation theories II: Abstract hyperbolic-like systems over a finite time horizon, vol. 75. Cambridge: Cambridge University Press;Google Scholar
  16. 16.
    Liu XF, Xu GQ. Output-based stabilization of Timoshenko beam with the boundary control and input distributed delay. J Dyn Control Syst. 2016;22(2):347–67.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Smyshlyaev A, Krstic M. Closed-form boundary state feedbacks for a class of 1-d partial integro-differential equations. IEEE Trans Automat Control. 2004;49(12): 2185–202.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tsubakino D, Hara S. Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages. Automatica. 2015;53:179–87.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang JM, Guo BZ, Krstic M. Wave equation stabilization by delays equal to even multiples of the wave propagation time. SIAM J Control Optim. 2011;49(2):517–54.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhao DX, Wang JM. Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks. J Dyn Control Syst. 2012;18(2): 269–95.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsChangshu Institute of TechnologyJiangsuPeople’s Republic of China

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