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Data-driven stabilization of unknown nonlinear dynamical systems using a cognition-based framework

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Abstract

In this paper, a cognitive stabilizer concept is introduced. The framework acts as an adaptive discrete control approach. The aim of the cognitive stabilizer is to stabilize a specific class of unknown nonlinear MIMO systems. The cognitive stabilizer is able to gain useful local knowledge of the system assumed as unknown. The approach is able to define autonomously suitable control inputs to stabilize the system. The system class to be considered is described by the following assumptions: unknown input/output behavior, fully controllable, stable zero dynamics, and measured state vector. The cognitive stabilizer is realized by its four main modules: (1) “perception and interpretation” using system identifier for the system local dynamic online identification and multi-step-ahead prediction; (2) “expert knowledge” relating to the quadratic stability criterion to guarantee the stability of the considered motion of the controlled system; (3) “planning” to generate a suitable control input sequence according to a certain cost function; (4) “execution” to generate the optimal control input in a corresponding feedback form. Each module can be realized using different methods. Two realizations will be stated in this paper. Using the cognitive stabilizer, the control goal can be achieved efficiently without an individual control design process for different kinds of unknown systems. Numerical examples (e.g., a chaotic nonlinear MIMO system–Lorenz system) demonstrate the successful application of the proposed methods.

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References

  1. Ahle, E., Söffker, D.: Interaction of intelligent and autonomous systems part-ii realization of cognitive technical systems. Math. Comput. Model. Dyn. Syst. 14(1), 319–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barmish, B.: Necessary and sufficient conditions for quadratic stability of an uncertain system. J. Optim. Theory Appl. 46(4), 399–408 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bukkems, B., Kostic, D., de Jager, B., Steinbuch, M.: Learning-based identification and iterative learning control of direct-drive robots. IEEE Trans. Control Syst. Technol. 13, 537–549 (2005)

    Article  Google Scholar 

  4. Cacciabue, P.: Modelling and Simulation of Human Behaviour in System Control. Springer, London (1998)

    Book  Google Scholar 

  5. Campi, M., Savaresi, S.: Direct nonlinear control design: the virtual reference feedback tuning approach. IEEE Trans. Autom. Control 51, 14–27 (2006)

    Article  MathSciNet  Google Scholar 

  6. Gill, P., Murray, W., Wright, M.: Practical Optimization. Academic press, London (1981)

    MATH  Google Scholar 

  7. Girard, A., Rasmussenand, C., Murray-Smith, R.: Multiple-step ahead prediction for nonlinear dynamic systems—a gaussian process treatment with propagation of the uncertainty. Adv. Neural Inf. Process. Syst. 15, 529–536 (2003)

    Google Scholar 

  8. Haykin, S.: Neural Networks. Prentice Hall international, New Jersey (1999)

    MATH  Google Scholar 

  9. Hjalmarsson, H., Gevers, M., Gunnarsson, S., Lequin, O.: Iterative feedback tuning: theory and applications. IEEE Control Syst. 18(4), 26–41 (1998)

    Article  Google Scholar 

  10. Hou, Z., Jin, S.: Data-driven model-free adaptive control for a class of mimo nonlinear discrete-time systems. IEEE Trans. Neural Netw. 22, 2173–2188 (2011)

    Article  Google Scholar 

  11. Jiang, Y., Jiang, Z.: Robust adaptive dynamic programming and feedback stabilization of nonlinear systems. IEEE Trans. Neural Netw. Learn. Syst. 25, 882–893 (2014)

    Article  Google Scholar 

  12. Kocijan, J., Girard, A., Banko, B., Murray-Smith, R.: Dynamic systems identification with gaussian processes. Math. Comput. Model. Dyn. Syst. 11(4), 411–424 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kwok, C., Fox, D., Meila, M.: Real-time particle filters. Proc. IEEE 92, 469–484 (2004)

  14. Liu, D., Wang, D., Zhao, D.: Adaptive dynamic programming for optimal control of unknown nonlinear discrete-time systems. In: 2011 IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning, pp. 242–249 (2011)

  15. Meiser, S.: Points location in arrangements of hyperplanes. Inf. Comput. 106, 286–303 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Murray, J., Cox, C., Lendaris, G., Saeks, R.: Adaptive dynamic programming. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 32, 140–153 (2002)

    Article  Google Scholar 

  17. Niu, B., Zhao, J.: Barrier Lyapunov functions for the output tracking control of constrained nonlinear switched systems. Syst. Control Lett. 62(10), 963–971 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nowak, X., Söffker, D.: A new model-free stability-based cognitive control method. In: Proceedings of the ASME 2014 Dynamic Systems and Control (DSC) conference, vol. 3, p. V003T47A002 (2014)

  19. Precup, R., Rădac, M., Tomescu, M., Petriu, E., Preitl, S.: Stable and convergent iterative feedback tuning of fuzzy controllers for discrete-time siso systems. Expert Syst. Appl. 40(1), 188–199 (2013)

    Article  Google Scholar 

  20. Rasmussen, C., Williams, C.: Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2006)

    MATH  Google Scholar 

  21. Rojas, J., Flores-Alsina, X., Jeppsson, U., Vilanova, R.: Application of multivariate virtual reference feedback tuning for wastewater treatment plant control. Control Eng. Pract. 20(5), 499–510 (2012)

    Article  Google Scholar 

  22. Shen, X., Söffker, D.: A model-free stability-based adaptive control method for unknown nonlinear systems. In: Proceedings of the ASME 2012 Dynamic Systems and Control (DSC) Conference, vol. 1, pp. 65–73 (2012)

  23. Siegelmann, H., Horne, B., Giles, C.: Computational capabilities of recurrent narx neural networks. IEEE Trans. Syst. Man Cybern. Part B Cybern. 27(2), 208–215 (1997)

    Article  Google Scholar 

  24. Strube, G., Habel, C., Hemforth, B., Konieczny, L., Becker, B.: Kognition, in Einführung in die künstliche Intelligenz, 2nd edn. Addison-Wesley, Bonn, Germany (1995)

    Google Scholar 

  25. Vernon, D., Metta, G., Sandini, G.: A survey of artificial cognitive systems: implications for the autonomous development of mental capabilities in computational agents. IEEE Trans. Evolut. Comput. 11(2), 151–180 (2007)

    Article  Google Scholar 

  26. Xu, J., Qu, Z.: Robust iterative learning control for a class of nonlinear systems. Automatica 34(8), 983–988 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xu, J., Tan, Y.: Linear and Nonlinear Iterative Learning Control. Springer, London (2003)

    MATH  Google Scholar 

  28. Zhang, C., Li, J.: Adaptive iterative learning control of non-uniform trajectory tracking for strict feedback nonlinear time-varying systems with unknown control direction. Appl. Math. Model. 39(10–11), 2942–2950 (2015)

    Article  MathSciNet  Google Scholar 

  29. Zhang, F., Söffker, D.: A data-driven quadratic stability condition and its application for stabilizing unknown nonlinear systems. Nonlinear Dyn. 77(3), 877–889 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Chair of Dynamics and Control, University of Duisburg-Essen, Lotharstrasse 1-21, 47048, Duisburg, Germany

    Xi Nowak & Dirk Söffker

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  1. Xi Nowak
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  2. Dirk Söffker
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Correspondence to Xi Nowak.

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Nowak, X., Söffker, D. Data-driven stabilization of unknown nonlinear dynamical systems using a cognition-based framework. Nonlinear Dyn 86, 1–15 (2016). https://doi.org/10.1007/s11071-016-2868-0

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  • DOI: https://doi.org/10.1007/s11071-016-2868-0

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