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Control of Uncertain Systems

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Springer Handbook of Automation

Part of the book series: Springer Handbooks ((SHB))

Abstract

Novel direct adaptive robust state and output feedback controllers are presented for the output tracking control of a class of nonlinear systems with unknown system dynamics and disturbances. Both controllers employ a variable-structure radial basis function (RBF) network that can determine its structure dynamically to approximate unknown system dynamics. Radial basis functions are added or removed online in order to achieve the desired tracking accuracy and prevent to network redundancy. The raised-cosine RBF is employed to enable fast and efficient training and output evaluation of the RBF network. The direct adaptive robust output feedback controller is constructed by utilizing a high-gain observer to estimate the tracking error for the controller implementation. The closed-loop systems driven by the variable neural direct adaptive robust controllers are actually switched systems.

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Abbreviations

2-D:

two-dimensional

DAROFC:

direct adaptive robust output feedback controller

DARSFC:

direct adaptive robust state feedback controller

GRBF:

Gaussian RBF

MIMO:

multi-input multi-output

RBF:

radial basis function

RCRBF:

raised-cosine RBF

SISO:

single-input single-output

References

  1. S.H. Żak: Systems and Control (Oxford Univ. Press, New York 2003)

    Google Scholar 

  2. D.W. Clarke: Self-tuning control. In: The Control Handbook, ed. by W.S. Levine (CRC, Boca Raton 1996)

    Google Scholar 

  3. A.S.I. Zinober: Deterministic Control of Uncertain Systems (Peregrinus, London 1990)

    MATH  Google Scholar 

  4. R.A. DeCarlo, S.H. Żak, G.P. Matthews: Variabble structure control of nonlinear multivariable systems: A tutorial, Proc. IEEE 76(3), 212–232 (1988)

    Article  Google Scholar 

  5. V.I. Utkin: Sliding Modes in Control and Optimization (Springer, Berlin 1992)

    MATH  Google Scholar 

  6. C. Edwards, S.K. Spurgeon: Sliding Mode Control: Theory and Applications (Taylor Francis, London 1998)

    Google Scholar 

  7. S. Gutman: Uncertain dynamical systems – a Lyapunov min-max approach, IEEE Trans. Autom. Control 24(3), 437–443 (1979)

    Article  MATH  Google Scholar 

  8. M.J. Corless, G. Leitmann: Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Autom. Control 26(5), 1139–1144 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. J.B. Pomet, L. Praly: Adaptive nonlinear regulation: Estimation from the Lyapunov equation, IEEE Trans. Autom. Control 37(6), 729–740 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. F.-C. Chen, C.-C. Liu: Adaptively controlling nonlinear continuous-time systems using multilayer neural networks, IEEE Trans. Autom. Control 39(6), 1306–1310 (1994)

    Article  MATH  Google Scholar 

  11. M. Krstic, I. Kanellakopoulos, P.V. Kokotovic: Nonlinear and Adaptive Control Design (Wiley, New York 1995)

    Google Scholar 

  12. K.S. Narendra, A.M. Annaswamy: Stable Adaptive Systems (Prentice Hall, Englewood Cliffs 1989)

    Google Scholar 

  13. K.J. Åström, B. Wittenmark: Adaptive Control (Addison-Wesley, Reading 1989)

    MATH  Google Scholar 

  14. S.S.  Sastry, A. Isidori: Adaptive control of linearizable systems, IEEE Trans. Autom. Control 34(11), 1123–1131 (1989)

    Google Scholar 

  15. I. Kanellakopoulos, P.V. Kokotovic, R. Marino: An extended direct scheme for robust adaptive nonlinear control, Automatica 27(2), 247–255 (1991)

    Google Scholar 

  16. I. Kanellakopoulos, P.V. Kokotovic, A.S. Morse: Systematic design of adaptive controllers for feedback linearizable systems, IEEE Trans. Autom. Control 36(11), 1241–1253 (1991)

    Google Scholar 

  17. D. Seto, A.M. Annaswamy, J. Baillieul: Adaptive control of nonlinear systems with a triangular structure, IEEE Trans. Autom. Control 39(7), 1411–1428 (1994)

    Google Scholar 

  18. A. Kojic, A.M. Annaswamy: Adaptive control of nonlinearly parameterized systems with a triangular structure, Automatica 38(1), 115–123 (2002)

    Google Scholar 

  19. M. Krstic, P.V. Kokotovic: Adaptive nonlineaer design with controller-identifier separation and swapping, IEEE Trans. Autom. Control 40(3), 426–460 (1995)

    Google Scholar 

  20. A.S. Morse, D.Q. Mayne, G.C. Goodwin: Applications of hysteresis switching in parameter adaptive control, IEEE Trans. Autom. Control 34(9), 1343–1354 (1992)

    Google Scholar 

  21. E.B. Kosmatopoulos, P.A. Ioannou: A switching adaptive controller for feedback linearizable systems, IEEE Trans. Autom. Control 44(4), 742–750 (1999)

    Google Scholar 

  22. E.B. Kosmatopoulos, P.A. Ioannou: Robust switching adaptive control of multi-input nonlinear systems, IEEE Trans. Autom. Control 47(4), 610–624 (2002)

    Google Scholar 

  23. J.S. Reed, P.A. Ioannou: Instability analysis and robust adaptive control of robotic manipulators, IEEE Trans. Robot. Autom. 5(3), 381–386 (1989)

    Article  Google Scholar 

  24. M.M. Polycarpou, P.A. Ioannou: A robust adaptive nonlinear control design, Automatica 32(3), 423–427 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. R.A. Freeman, M. Krstic, P.V. Kokotovic: Robustness of adaptive nonlinear control to bounded uncertainties, Automatica 34(10), 1227–1230 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Xu, P.A. Ioannou: Robust adaptive control for a class of MIMO nonlinear systems with guaranteed error bounds, IEEE Trans. Autom. Control 48(5), 728–742 (2003)

    Article  MathSciNet  Google Scholar 

  27. R.M. Sanner, J.J.E. Slotine: Gaussian networks for direct adaptive control, IEEE Trans. Neural Netw. 3(6), 837–863 (1992)

    Article  Google Scholar 

  28. L.-X. Wang: Stable adaptive fuzzy control of nonlinear systems, IEEE Trans. Fuzzy Syst. 1(2), 146–155 (1993)

    Article  Google Scholar 

  29. C.-Y. Su, Y. Stepanenko: Adaptive control of a class of nonlinear systems with fuzzy logic, IEEE Trans. Fuzzy Syst. 2(4), 285–294 (1994)

    Article  Google Scholar 

  30. M.M. Polycarpou: Stable adaptive neutral control scheme for nonlinear systems, IEEE Trans. Autom. Control 41(3), 447–451 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. L.-X. Wang: Stable adaptive fuzzy controllers with application to inverted pendulum tracking, IEEE Trans. Syst. Man Cybern. B 26(5), 677–691 (1996)

    Article  Google Scholar 

  32. J.T. Spooner, K.M. Passino: Stable adaptive control using fuzzy systems and neural networks, IEEE Trans. Fuzzy Syst. 4(3), 339–359 (1996)

    Article  Google Scholar 

  33. C.-H. Wang, H.-L. Liu, T.-C. Lin: Direct adaptive fuzzy-neutral control with state observer and supervisory controller for unknown nonlinear dynamical systems, IEEE Trans. Fuzzy Syst. 10(1), 39–49 (2002)

    Article  Google Scholar 

  34. E. Tzirkel–Hancock, F. Fallside: Stable control of nonlinear systems using neural networks, Int. J. Robust Nonlin. Control 2(1), 63–86 (1992)

    Article  MATH  Google Scholar 

  35. A. Yesildirek, F.L. Lewis: Feedback linearization using neural networks, Automatica 31(11), 1659–1664 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. B.S. Chen, C.H. Lee, Y.C. Chang: H∞ tracking design of uncertain nonlinear SISO systems: Adaptive fuzzy approach, IEEE Trans. Fuzzy Syst. 4(1), 32–43 (1996)

    Article  Google Scholar 

  37. S.S. Ge, C.C. Hang, T. Zhang: A direct method for robust adaptive nonlinear control with guaranteed transient performance, Syst. Control Lett. 37(5), 275–284 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Seshagiri, H.K. Khalil: Output feedback control of nonlinear systems using RBF neural networks, IEEE Trans. Neural Netw. 11(1), 69–79 (2000)

    Article  Google Scholar 

  39. M.I. EI–Hawwary, A.L. Elshafei, H.M. Emara, H.A. Abdel Fattah: Output feedback control of a class of nonlinear systems using direct adaptive fuzzy controller, IEE Proc. Control Theory Appl. 151(5), 615–625 (2004)

    Article  Google Scholar 

  40. Y. Lee, S.H. Żak: Uniformly ultimately bounded fuzzy adaptive tracking controllers for uncertain systems, IEEE Trans. Fuzzy Syst. 12(6), 797–811 (2004)

    Article  Google Scholar 

  41. C.-C. Liu, F.-C. Chen: Adaptive control of non-linear continuous-time systems using neural networks – general relative degree and MIMO cases, Int. J. Control 58(2), 317–335 (1993)

    Article  MATH  Google Scholar 

  42. R. Ordóñez, K.M. Passino: Stable multi-input multi-output adaptive fuzzy/neural control, IEEE Trans. Fuzzy Syst. 7(3), 345–353 (2002)

    Article  Google Scholar 

  43. S. Tong, J.T. Tang, T. Wang: Fuzzy adaptive control of multivariable nonlinear systems, Fuzzy Sets Syst. 111(2), 153–167 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  44. Y.-C. Chang: Robust tracking control of nonlinear MIMO systems via fuzzy approaches, Automatica 36, 1535–1545 (2000)

    Article  MATH  Google Scholar 

  45. Y.-C. Chang: An adaptive H∞ tracking control for a class of nonlinear multiple-input-multiple-output (MIMO) systems, IEEE Trans. Autom. Control 46(9), 1432–1437 (2001)

    Article  MATH  Google Scholar 

  46. S. Tong, H.-X. Li: Fuzzy adaptive sliding-mode control for MIMO nonlinear systems, IEEE Trans. Fuzzy Syst. 11(3), 354–360 (2003)

    Article  Google Scholar 

  47. H.K. Khalil, F. Esfandiari: Semiglobal stabilization of a class of nonlinear systems using output feedback, IEEE Trans. Autom. Control 38(9), 1412–1115 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  48. H.K. Khalil: Robust servomechanism output feedback controllers for a class of feedback linearizable systems, Automatica 30(10), 1587–1599 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  49. H.K. Khalil: Adaptive output feedback control of nonlinear systems represented by input–output models, IEEE Trans. Autom. Control 41(2), 177–188 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  50. B. Aloliwi, H.K. Khalil: Robust adaptive output feedback control of nonlinear systems without persistence of excitation, Automatica 33(11), 2025–2032 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  51. S. Tong, T. Wang, J.T. Tang: Fuzzy adaptive output tracking control of nonlinear systems, Fuzzy Sets Syst. 111(2), 169–182 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  52. J.-X. Xu, Y. Tan: Nonlinear adaptive wavelet control using constructive wavelet networks, IEEE Trans. Neural Netw. 18(1), 115–127 (2007)

    Article  MathSciNet  Google Scholar 

  53. S. Fabri, V. Kadirkamanathan: Dynamic structure neural networks for stable adaptive control of nonlinear systems, IEEE Trans. Neural Netw. 7(5), 1151–1166 (1996)

    Article  Google Scholar 

  54. G.P. Liu, V. Kadirkamanathan, S.A. Billings: Variable neural networks for adaptive control of nonlinear systems, IEEE Trans. Syst. Man Cybern. B 39(1), 34–43 (1999)

    Google Scholar 

  55. Y. Lee, S. Hui, E. Zivi, S.H. Żak: Variable neural adaptive robust controllers for uncertain systems, Int. J. Adapt. Control Signal Process. 22(8), 721–738 (2008)

    Article  MATH  Google Scholar 

  56. J. Lian, Y. Lee, S.H. Żak: Variable neural adaptive robust control of uncertain systems, IEEE Trans. Autom. Control 53(11), 2658–2664 (2009)

    Article  Google Scholar 

  57. J. Lian, Y. Lee, S.D. Sudhoff, S.H. Żak: Variable structure neural network based direct adaptive robust control of uncertain systems, Proc. Am. Control Conf. (Seattle 2008) pp. 3402–3407

    Google Scholar 

  58. J. Lian, J. Hu, S.H. Żak: Adaptive robust control: A switched system approach, IEEE Trans. Autom. Control, to appear (2010)

    Google Scholar 

  59. J.P. Hespanha, A.S. Morse: Stability of switched systems with average dwell-time, Proc. 38th Conf. Decis. Control (Phoenix 1999) pp. 2655–2660

    Google Scholar 

  60. R.J. Schilling, J.J. Carroll, A.F. Al–Ajlouni: Approximation of nonlinear systems with radial basis function neural network, IEEE Trans. Neural Netw. 12(1), 1–15 (2001)

    Article  Google Scholar 

  61. J. Lian, Y. Lee, S.D. Sudhoff, S.H. Żak: Self-organizing radial basis function network for real-time approximation of continuous-time dynamical systems, IEEE Trans. Neural Netw. 19(3), 460–474 (2008)

    Article  Google Scholar 

  62. M. Jankovic: Adaptive output feedback control of nonlinear feedback linearizable systems, Int. J. Adapt. Control Signal Process. 10, 1–18 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  63. T. Zhang, S.S. Ge, C.C. Hang: Stable adaptive control for a class of nonlinear systems using a modified Lyapunov function, IEEE Trans. Autom. Control 45(1), 129–132 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  64. T. Zhang, S.S. Ge, C.C. Hang: Design and performance analysis of a direct adaptive controller for nonlinear systems, Automatica 35(11), 1809–1817 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  65. B. Yao: Integrated direct/indirect adaptive robust control of SISO nonlinear systems in semi-strict feedback form, Proc. Am. Control Conf., Vol. 4 (Denver 2003) pp. 3020–3025

    Google Scholar 

  66. B. Yao: Indirect adaptive robust control of SISO nonlinear systems in semi-strict feedback forms, Proc. 15th IFAC World Congr. (Barcelona 2002) pp. 1–6

    Google Scholar 

  67. F. Esfandiari, H.K. Khalil: Output feedback stabilization of fully linerizable systems, Int. J. Control 56(5), 1007–1037 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  68. N.A. Mahmoud, H.K. Khalil: Asymptotic regulation of minimum phase nonlinear systems using output feedback, IEEE Trans. Autom. Control 41(10), 1402–1412 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  69. B. Yao: Lecture Notes from Course on Nonlinear Feedback Controller Design (School of Mechanical Engineering, Purdue University, West Lafayette 2007)

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Avenue, 47907-2035, West Lafayette, IN, USA

    Jianming Lian PhD

  2. School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Avenue, 47907-2035, West Lafayette, IN, USA

    Stanislaw H. Żak PhD

Authors
  1. Jianming Lian PhD
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  2. Stanislaw H. Żak PhD
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Correspondence to Jianming Lian PhD or Stanislaw H. Żak PhD .

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Editors and Affiliations

  1. PRISM Center, and School of Industrial Engineering, Purdue University, 315 N. Grant Street, 47907, West Lafayette, IN, USA

    Shimon Y. Nof

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Lian, J., Żak, S.H. (2009). Control of Uncertain Systems. In: Nof, S. (eds) Springer Handbook of Automation. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78831-7_11

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  • DOI: https://doi.org/10.1007/978-3-540-78831-7_11

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