Advertisement

The Bernstein Polynomials Based Globally Optimal Nonlinear Model Predictive Control

  • Bhagyesh V. Patil
  • Ashok KrishnanEmail author
  • Foo Y. S. Eddy
  • Ahmed Zidna
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

Nonlinear model predictive control (NMPC) has shown considerable success in the control of a nonlinear systems due to its ability to deal directly with nonlinear models. However, the inclusion of a nonlinear model in the NMPC framework potentially results in a highly nonlinear (usually ‘nonconvex’) optimization problem. This paper proposes a solution technique for such optimization problems. Specifically, this paper proposes an improved Bernstein global optimization algorithm. The proposed algorithm contains a Newton-based box trim operator which extends the classical Newton method using the geometrical properties associated with the Bernstein polynomial. This operator accelerates the convergence of the Bernstein global optimization algorithm by discarding those regions of the solution search space which do not contain any solution. The utility of this improved Bernstein algorithm is demonstrated by simulating an NMPC problem for tracking multiple setpoint changes in the reactor temperature of a continuous stirred-tank reactor (CSTR) system. Furthermore, the performance of the proposed algorithm is compared with those of the previously reported Bernstein global optimization algorithm and a conventional sequential-quadratic programming based sub-optimal NMPC scheme implemented in MATLAB.

References

  1. 1.
    Patil, B.V., Bhartiya, S., Nataraj, P.S.V., Nandola, N.N.: Multiple-model based predictive control of nonlinear hybrid systems based on global optimization using the Bernstein polynomial approach. J. Process Control 22(2), 423–435 (2012)CrossRefGoogle Scholar
  2. 2.
    Cizniar, M., Fikar, M., Latifi, M.A.: Design of constrained nonlinear model predictive control based on global optimisation. In: 18th European Symposium on Computer Aided Process Engineering-ESCAPE 18, pp. 1–6 (2008)Google Scholar
  3. 3.
    Doyle, J.C., Francis, B.A., Tannenbaum, A.R.: Feedback Control Theory. Dover Publications, USA (2009)Google Scholar
  4. 4.
    Germin Nisha, M., Pillai, G.N.: Nonlinear model predictive control with relevance vector regression and particle swarm optimization. J. Control. Theory Appl. 11(4), 563–569 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Grüne, L., Pannek, J.: Nonlinear Model Predictive Control, pp. 43–66. Springer, London (2011)CrossRefGoogle Scholar
  6. 6.
    Hansen, E.R., Walster, G.W.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New York (2005)Google Scholar
  7. 7.
    Inga J. Wolf, Marquardt, W.: Fact NMPC schemes for regulatory and economic NMPC\(-\) A review. J. Process Control 44, 162–183 (2016)Google Scholar
  8. 8.
    Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models. CRC Press, USA (2007)zbMATHGoogle Scholar
  9. 9.
    Long, C., Polisetty, P., Gatzke, E.: Nonlinear model predictive control using deterministic global optimization. J. Process Control 16(6), 635–643 (2006)CrossRefGoogle Scholar
  10. 10.
    \(\mathring{\rm A\rm str\ddot{\rm o}}\)m, K.J., Wittenmark, B.: Computer-Controlled Systems: Theory and Design, 3rd edn. Dover Publications, USA (2011)Google Scholar
  11. 11.
    Patil, B.V., Maciejowski, J., Ling, K.V.: Nonlinear model predictive control based on Bernstein global optimization with application to a nonlinear CSTR. In: IEEE Proceedings of 15th Annual European Control Conference, pp. 471–476. Aalborg, Denmark (2016)Google Scholar
  12. 12.
    Ratschek, H., Rokne, J.: New Computer Methods for Global Optimization. Ellis Horwood Publishers, Chichester, England (1988)zbMATHGoogle Scholar
  13. 13.
    Rawlings, J.B., Mayne, D.Q., Diehl, M.M.: Model Predictive Control: Theory, Computation, and Design, 2nd edn. Nob Hill Publishing, USA (2017)Google Scholar
  14. 14.
    Stahl, V.: Interval methods for bounding the range of polynomials and solving systems of nonlinear equations. Ph.D. thesis, Johannes Kepler University, Linz (1995)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Bhagyesh V. Patil
    • 1
  • Ashok Krishnan
    • 1
    • 2
    Email author
  • Foo Y. S. Eddy
    • 2
  • Ahmed Zidna
    • 3
  1. 1.Cambridge Centre for Advanced Research and Education in SingaporeSingaporeSingapore
  2. 2.School of Electrical and Electronic EngineeringNanyang Technological UniversityNanyangSingapore
  3. 3.LGIPMUniversité de LorrainLorrainFrance

Personalised recommendations