Predictor Feedback: Time-Varying, Adaptive, and Nonlinear

  • Miroslav Krstic
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)


We present a tutorial introduction to methods for stabilization of systems with long input delays—the so-called “predictor feedback” techniques. The methods are based on techniques originally developed for boundary control of partial differential equations using the “backstepping” approach. We start with a consideration of linear systems, first with a known delay and then subject to a small uncertainty in the delay. Then we study linear systems with constant delays that are completely unknown, which requires an adaptive control approach. For linear systems, we also present a method for compensating arbitrarily large but known time-varying delays. Next, we consider nonlinear control problems in the presence of arbitrarily long input delays. Finally, we close with a design for general nonlinear systems with delays that have a general dependency on the system state.


Automatic Control Control Letter Input Delay Smith Predictor Adaptive Control Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Sipahi, R., Niculescu, S.-I., Abdallah, C.T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay. Control Systems Magazine 31, 38–65 (2011)MathSciNetGoogle Scholar
  2. 2.
    Smith, O.J.M.: A controller to overcome dead time. ISA 6, 28–33 (1959)Google Scholar
  3. 3.
    Manitius, A.Z., Olbrot, A.W.: Finite spectrum assignment for systems with delays. IEEE Trans. on Automatic Control 24, 541–553 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Kwon, W.H., Pearson, A.E.: Feedback stabilization of linear systems with delayed control. IEEE Trans. on Automatic Control 25, 266–269 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Artstein, Z.: Linear systems with delayed controls: a reduction. IEEE Trans. on Automatic Control 27, 869–879 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Zhong, Q.-C.: Robust Control of Time-delay Systems. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  7. 7.
    Michiels, W., Niculescu, S.-I.: Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach. SIAM (2007)Google Scholar
  8. 8.
    Mondie, S., Michiels, W.: Finite spectrum assignment of unstable time-delay systems with a safe implementation. IEEE Trans. on Automatic Control 48, 2207–2212 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhong, Q.-C.: On distributed delay in linear control laws—Part I: Discrete-delay implementation. IEEE Transactions on Automatic Control 49, 2074–2080 (2006)CrossRefGoogle Scholar
  10. 10.
    Zhong, Q.-C., Mirkin, L.: Control of integral processes with dead time—Part 2: Quantitative analysis. IEE Proc. Control Theory & Appl. 149, 291–296 (2002)CrossRefGoogle Scholar
  11. 11.
    Krstic, M., Smyshlyaev, A.: Boundary Control of PDEs: A Course on Backstepping Designs. SIAM (2008)Google Scholar
  12. 12.
    Vazquez, R., Krstic, M.: Control of Turbulent and Magnetohydrodynamic Channel Flows. Birkhäuser (2007)Google Scholar
  13. 13.
    Krstic, M.: Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Birkhäuser (2009)Google Scholar
  14. 14.
    Krstic, M.: On compensating long actuator delays in nonlinear control. IEEE Transactions on Automatic Control 53, 1684–1688 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Krstic, M.: Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch. Automatica 44, 2930–2935 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Krstic, M.: Compensating actuator and sensor dynamics governed by diffusion PDEs. Systems and Control Letters 58, 372–377 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Krstic, M.: Compensating a string PDE in the actuation or sensing path of an unstable ODE. IEEE Transactions on Automatic Control 54, 1362–1368 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Krstic, M.: Input delay compensation for forward complete and feedforward nonlinear systems. IEEE Transactions on Automatic Control 55, 287–303 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Krstic, M.: Lyapunov stability of linear predictor feedback for time-varying input delay. IEEE Transactions on Automatic Control 55, 554–559 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Krstic, M.: Control of an unstable reaction-diffusion PDE with long input delay. Systems and Control Letters 58, 773–782 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Krstic, M.: Compensation of infinite-dimensional actuator and sensor dynamics: Nonlinear and delay-adaptive systems. IEEE Control Systems Magazine 30, 22–41 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Bresch-Pietri, D., Krstic, M.: Adaptive trajectory tracking despite unknown input delay and plant parameters. Automatica 45, 2075–2081 (2009)CrossRefGoogle Scholar
  23. 23.
    Bresch-Pietri, D., Krstic, M.: Delay-adaptive predictor feedback for systems with unknown long actuator delay. IEEE Transactions on Automatic Control 55, 2106–2112 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Bekiaris-Liberis, N., Krstic, M.: Delay-adaptive feedback for linear feedforward systems. Systems and Control Letters 59, 277–283 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Bekiaris-Liberis, N., Krstic, M.: Compensating the distributed effect of a wave PDE in the actuation or sensing path of MIMO LTI Systems. Systems & Control Letters 59, 713–719 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Bekiaris-Liberis, N., Krstic, M.: Stabilization of linear strict-feedback systems with delayed integrators. Automatica 46, 1902–1910 (2010)zbMATHCrossRefGoogle Scholar
  27. 27.
    Bekiaris-Liberis, N., Krstic, M.: Lyapunov stability of linear predictor feedback for distributed input delay. IEEE Transactions on Automatic Control 56, 655–660 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Bekiaris-Liberis, N., Krstic, M.: Compensating distributed effect of diffusion and counter-convection in multi-input and multi-output LTI systems. IEEE Transactions on Automatic Control 56, 637–642 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Teel, A.R.: Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Transactions on Automatic Control 43, 960–964 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Datko, R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimization 26, 697–713 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Ortega, R., Lozano, R.: Globally stable adaptive controller for systems with delay. Internat. J. Control 47, 17–23 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Niculescu, S.-I., Annaswamy, A.M.: An Adaptive Smith-Controller for Time-delay Systems with Relative Degree n* ≥ 2. Systems and Control Letters 49, 347–358 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Jankovic, M.: Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems. IEEE Trans. on Automatic Control 46, 1048–1060 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Germani, A., Manes, C., Pepe, P.: Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability. International Journal of Robust and Nonlinear Control 13, 909–937 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Karafyllis, I.: Finite-time global stabilization by means of time-varying distributed delay feedback. SIAM Journal of Control and Optimization 45, 320–342 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Mazenc, F., Bliman, P.-A.: Backstepping design for time-delay nonlinear systems. IEEE Transactions on Automatic Control 51, 149–154 (2006)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Mazenc, F., Mondie, S., Francisco, R.: Global asymptotic stabilization of feedforward systems with delay at the input. IEEE Trans. Automatic Control 49, 844–850 (2004)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Smyshlyaev, A., Krstic, M.: Closed form boundary state feedbacks for a class of 1D partial integro-differential equations. IEEE Trans. on Automatic Control 49(12), 2185–2202 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan Diego, La JollaUSA

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