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Nonlinear \(H_\infty\) control via measurement feedback using neural network

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Abstract

In this paper, we solve the nonlinear \(H_\infty\) optimal control with output feedback via the neural network (NN)–least squares method for the affine nonlinear system. The approach is based on successive approximate solution of two Hamilton–Jacobi–Isaacs (HJI) equations, which appear in the \(H_\infty\) optimal output feedback control. Successive approximation (SA) approach combined with neural network (NN) for updating control and disturbance inputs in the case of state-feedback control is first proposed to solve an HJI equation with two players. The obtained solution is then used to solve, with the SA-NN approach for updating disturbance input, an HJI equation with one player in the output feedback control problem. Simulations on the Translational oscillator with rotational actuator mechanical system are presented to illustrate the effectiveness of the proposed method.

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References

  1. Abu-Khalaf M, Lewis FL (2004) Nearly optimal state feedback contol of constrained nonlinear systems using a neural network HJB approch. Ann Rev Control 28:239–251

    Article  Google Scholar 

  2. Abu-Khalaf M, Lewis FL (2005) Nearly optimal contol law for nonlinear systems with sturating actuators using a neural network HJB approch. Automatica 41:779–791

    Article  MATH  Google Scholar 

  3. Abu-Khalaf M, Lewis FL, Huang J (2006) Policy iterations on the Hamilton JacobiIsaacs equation for \(H_\infty\) state feedback control with input saturation. IEEE Trans Autom Control 51(12):1989–1995

    Article  Google Scholar 

  4. Beard RW, Saridis GN, Wen JT (1997) Galerkin approximation of the generalized Hamiton–Jacobi–Bellman equation. Automatica 33(12):2159–2177

    Article  MathSciNet  MATH  Google Scholar 

  5. Beard RW, McLain TW (1998) Successive Galerkin approximation algorithms for nonlinear optimal and robust control. Int J Control 71(5):717–743

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen Z, Jagannathan S (2008) Generalized Hamilton–Jacobi–Bellman formulation-based neural network control of affine nonlinear discrete-time systems. IEEE Trans Neural Netw 19(1):90–106

    Article  Google Scholar 

  7. Cheng T, Lewis FL, Abu-Khalaf M (2007) A neural network solution for fixed-final time optimal control of nonlinear systems. Automatica 43:482–490

    Article  MathSciNet  MATH  Google Scholar 

  8. Ferreira HC, Sales RM, Da Rocha PH (2010) Galerkin method and weighting functions applied to nonlinear \(H_\infty\) control with output feedback. J Vib Control 16(12):1817–1843

    Article  MathSciNet  MATH  Google Scholar 

  9. Finlayson BA, Scriven LE (1966) The method of weighted residuals—a review. Appl Mech Rev 19(9):735–748

    Google Scholar 

  10. Isidori A, Astolfi A (1992) Disturbance attenuation and \(H_\infty\)-control via measurement feedback in nonlinear systems. IEEE Trans Autom Control 37(9):466–472

    Article  MathSciNet  MATH  Google Scholar 

  11. Isidori A, Kang W (1995) \(H_\infty\) control via measurement feedback for general nonlinear systems. IEEE Trans Autom Control 40(1):466–472

    Article  MathSciNet  MATH  Google Scholar 

  12. Mehraeen S, Dierks T, Jnagannathan S (2012) Zero-sum two player game theoretic formulation of affine nonlinear discrete-time systems using neural network. IEEE Trans Cybern

  13. Ortega MG, Vargas M, Vivas C, Rubio FR (2005) Robustness improvement of a nonlinear \(H_\infty\) controller for robot manipulators via saturation functions. J Robot Syst 22(8):421–437

    Article  MATH  Google Scholar 

  14. Park J, Chung W, Youngil Y (1998a) Analytic nonlinear \(H_\infty\) optimal control for robotic manipulators. Proc IEEE Int Conf Robot Autom, pp 2709–2715

  15. Park J, Chung W, Youngil Y (1998b) Analytic nonlinear \(H_\infty\) inverse-optimal control for Euler–Lagrange system. IEEE Trans Robot Autom 16(6):847–854

    Article  Google Scholar 

  16. Saridis GN, Lee CSG (1979) An pproximation theory of optimal control for trainable manipulators. IEEE Trans Syst Man Cybern 9(3):152–159

  17. Van der Schaft AJ (1992) \(L_2\) gain analysis of nonlinear systems and nonliear state feedback \(H_\infty\) control. IEEE Trans Autom Control 37(6):770–780

    Article  MATH  Google Scholar 

  18. Voulgaris P, Chen X (2001) A Taylor series approch for nonlinear \(H_\infty\) control applications. J Vib Control 7:51–72

    Article  MathSciNet  MATH  Google Scholar 

  19. Vrabie D, Pastravanu O, Abu-Khalaf M, Lewis FL (2009) Adaptive optimal control for continuous-time linear systems based on policy iteration. Automatica 45:477–484

    Article  MathSciNet  MATH  Google Scholar 

  20. Urs C, Cirillo R (1997) Nonlinear \(H_\infty\) control, derivation and implementation. IMRT, Report 31

  21. Wu H-N, Luo B (2012) Neural network based online simultaneous policy update algorithm for solving the HJI equation in nonlinear \(H_\infty\) control. IEEE Trans Neural Netw Learn Syst 23(12):1884–1895

    Article  Google Scholar 

  22. Lin W, Byrnes CI (1996) \(H_\infty\) control od discret-time nonlinear systems. IEEE Trans Autom Control 41(4):494–510

  23. Liu D, Li H, Wang D (2013) Neural-network-based zero-sum game for discrete-time nonlinear systems via iterative adaptive dynmic programming algorithm. Neurocomputing 110:92–100

    Article  Google Scholar 

  24. Huang Y, Liu D, Wei Q (2012) Generalized Hamilton–Jacobi–Isaacs formulation-based neural network \(H_\infty\) control for constrained input nonlinear systems. 19th international conference, ICONIP 2012, Doha, Qatar, November 12-15, 2012, proceedings, Part I. Springer, Berlin, Heidelberg, pp 218–225

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Acknowledgments

The authors would like to thank the referees and the editor for the detailed comments that have helped significantly improve the quality of presentation.

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Authors and Affiliations

  1. Computer Science Department, University Mustapha Stambouli of Mascara, Mascara, Algeria

    Rochdi Bachir Bouiadjra & Mohammed Salem

  2. Laboratory of Research in Industrial Computing and Networks, University Ahmed Benbella Oran 1, Oran, Algeria

    Rochdi Bachir Bouiadjra & Mohamed Fayçal Khelfi

  3. Laboratoire des télécommunications, Université 8 Mai 1945 Guelma, Guelma, Algeria

    Moussa Sedraoui

Authors
  1. Rochdi Bachir Bouiadjra
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  2. Mohamed Fayçal Khelfi
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  3. Mohammed Salem
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  4. Moussa Sedraoui
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Correspondence to Rochdi Bachir Bouiadjra.

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Technical Editor: Marcelo A. Savi.

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Bachir Bouiadjra, R., Khelfi, M.F., Salem, M. et al. Nonlinear \(H_\infty\) control via measurement feedback using neural network. J Braz. Soc. Mech. Sci. Eng. 39, 1109–1118 (2017). https://doi.org/10.1007/s40430-016-0597-4

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  • DOI: https://doi.org/10.1007/s40430-016-0597-4

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