A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 9 Issue 6
Jun.  2022

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 7.847, Top 10% (SCI Q1)
    CiteScore: 13.0, Top 5% (Q1)
    Google Scholar h5-index: 64, TOP 7
Turn off MathJax
Article Contents
M. Ballesteros, R. Q. Fuentes-Aguilar, and I. Chairez, “Exponential continuous non-parametric neural identifier with predefined convergence velocity,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 6, pp. 1049–1060, Jun. 2022. doi: 10.1109/JAS.2022.105650
Citation: M. Ballesteros, R. Q. Fuentes-Aguilar, and I. Chairez, “Exponential continuous non-parametric neural identifier with predefined convergence velocity,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 6, pp. 1049–1060, Jun. 2022. doi: 10.1109/JAS.2022.105650

Exponential Continuous Non-Parametric Neural Identifier With Predefined Convergence Velocity

doi: 10.1109/JAS.2022.105650
Funds:  The work was supported by the National Polytechnic Institute (SIP-20221151, SIP-20220916)
More Information
  • This paper addresses the design of an exponential function-based learning law for artificial neural networks (ANNs) with continuous dynamics. The ANN structure is used to obtain a non-parametric model of systems with uncertainties, which are described by a set of nonlinear ordinary differential equations. Two novel adaptive algorithms with predefined exponential convergence rate adjust the weights of the ANN. The first algorithm includes an adaptive gain depending on the identification error which accelerated the convergence of the weights and promotes a faster convergence between the states of the uncertain system and the trajectories of the neural identifier. The second approach uses a time-dependent sigmoidal gain that forces the convergence of the identification error to an invariant set characterized by an ellipsoid. The generalized volume of this ellipsoid depends on the upper bounds of uncertainties, perturbations and modeling errors. The application of the invariant ellipsoid method yields to obtain an algorithm to reduce the volume of the convergence region for the identification error. Both adaptive algorithms are derived from the application of a non-standard exponential dependent function and an associated controlled Lyapunov function. Numerical examples demonstrate the improvements enforced by the algorithms introduced in this study by comparing the convergence settings concerning classical schemes with non-exponential continuous learning methods. The proposed identifiers overcome the results of the classical identifier achieving a faster convergence to an invariant set of smaller dimensions.

     

  • loading
  • [1]
    L. Ljung, System Identification: Theory for the User. 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999.
    [2]
    R. Haber and L. Keviczky, Nonlinear System Identification: Input-Output Modeling Approach. Netherlands: Springer, 1999.
    [3]
    L. Ljung, “Some aspects on nonlinear system identification,” IFAC Proc. Volumes, vol. 39, no. 1, pp. 553–564, Jan. 2006. doi: 10.3182/20060329-3-AU-2901.00085
    [4]
    S. A. Billings, “Identification of nonlinear systems: A survey,” IEEE Proc. D-Control Theory Appl., vol. 127, no. 6, pp. 272–285, Nov. 1980. doi: 10.1049/ip-d.1980.0047
    [5]
    J. Roll, A. Nazin, and L. Ljung, “Nonlinear system identification via direct weight optimization,” Automatica, vol. 41, no. 3, pp. 475–490, Mar. 2005. doi: 10.1016/j.automatica.2004.11.010
    [6]
    S. Jagannathan and F. L. Lewis, “Identification of nonlinear dynamical systems using multilayered neural networks,” Automatica, vol. 32, no. 12, pp. 1707–1712, Dec. 1996. doi: 10.1016/S0005-1098(96)80007-0
    [7]
    O. Nelles, Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Berlin Heidelberg: Springer-Verlag, 2001.
    [8]
    K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw., vol. 2, no. 5, pp. 359–366, Jul. 1989. doi: 10.1016/0893-6080(89)90020-8
    [9]
    N. E. Cotter, “The Stone-Weierstrass theorem and its application to neural networks,” IEEE Trans. Neural Netw., vol. 1, no. 4, pp. 290–295, Dec. 1990. doi: 10.1109/72.80265
    [10]
    K. Hornik, “Approximation capabilities of multilayer feedforward networks,” Neural Netw., vol. 4, no. 2, pp. 251–257, Mar. 1991. doi: 10.1016/0893-6080(91)90009-T
    [11]
    J. Park and I. W. Sandberg, “Universal approximation using radial-basis-function networks,” Neural Comput., vol. 3, no. 2, pp. 246–257, Jun. 1991. doi: 10.1162/neco.1991.3.2.246
    [12]
    G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signal. Syst., vol. 2, no. 4, pp. 303–314, Dec. 1989. doi: 10.1007/BF02551274
    [13]
    A. S. Poznyak, E. N. Sanchez, and W. Yu, Differential Neural Networks for Robust Nonlinear Control (Identification, State Estimation and Trajectory Tracking). New Jersey: World Scientific, 2001.
    [14]
    C. P. Bechlioulis and G. A. Rovithakis, “Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems,” Automatica, vol. 45, no. 2, pp. 532–538, Feb. 2009. doi: 10.1016/j.automatica.2008.08.012
    [15]
    C. P. Bechlioulis and G. A. Rovithakis, “Prescribed performance adaptive control for multi-input multi-output affine in the control nonlinear systems,” IEEE Trans. Automat. Control, vol. 55, no. 5, pp. 1220–1226, May 2010. doi: 10.1109/TAC.2010.2042508
    [16]
    Y. B. Huang, J. Na, X. Wu, X. Q. Liu, and Y. Guo, “Adaptive control of nonlinear uncertain active suspension systems with prescribed performance,” ISA Trans., vol. 54, pp. 145–155, Jan. 2015. doi: 10.1016/j.isatra.2014.05.025
    [17]
    G. Chowdhary, T. Yucelen, M. Mühlegg, and E. N. Johnson, “Concurrent learning adaptive control of linear systems with exponentially convergent bounds,” Int. J. Adapt. Control Signal Process., vol. 27, no. 4, pp. 280–301, Apr. 2013. doi: 10.1002/acs.2297
    [18]
    R. Kamalapurkar, J. R. Klotz, and W. E. Dixon, “Concurrent learning-based approximate feedback-Nash equilibrium solution of N-player nonzero-sum differential games,” IEEE/CAA J. Autom. Sinica, vol. 1, no. 3, pp. 239–247, Jul. 2014. doi: 10.1109/JAS.2014.7004681
    [19]
    E. B. Kosmatopoulos, M. A. Christodoulou, and P. A. Ioannou, “Dynamical neural networks that ensure exponential identification error convergence,” Neural Netw., vol. 10, no. 2, pp. 299–314, Mar. 1997. doi: 10.1016/S0893-6080(96)00060-3
    [20]
    R. Q. Fuentes-Aguilar and I. Chairez, “Adaptive tracking control of state constraint systems based on differential neural networks: A barrier lyapunov function approach,” IEEE Trans. Neural Netw. Learn. Syst., vol. 31, no. 12, pp. 5390–5401, Dec. 2020. doi: 10.1109/TNNLS.2020.2966914
    [21]
    K. Q. Gu, J. Chen, and V. L. Kharitonov, Stability of Time-Delay Systems. Boston, MA: Birkhäuser, 2003.
    [22]
    K. Q. Gu and S. I. Niculescu, “Survey on recent results in the stability and control of time-delay systems,” J. Dyn. Syst. Meas. Control, vol. 125, no. 2, pp. 158–165, Jun. 2003. doi: 10.1115/1.1569950
    [23]
    D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Optimal Control: Deterministic and Stochastic Systems. Berlin, Heidelberg: Springer, 2012.
    [24]
    V. Gaitsgory, L. Grüne, and N. Thatcher, “Stabilization with discounted optimal control,” Syst. Control Lett., vol. 82, pp. 91–98, Aug. 2015. doi: 10.1016/j.sysconle.2015.05.010
    [25]
    H. K. Khalil, Nonlinear Systems. 3rd ed. Upper Saddle River: Prentice Hall, 2002.
    [26]
    K. I. Funahashi, “On the approximate realization of continuous mappings by neural networks,” Neural Netw., vol. 2, no. 3, pp. 183–192, Dec. 1989. doi: 10.1016/0893-6080(89)90003-8
    [27]
    A. R. Barron, “Approximation and estimation bounds for artificial neural networks,” Mach. Learn., vol. 14, no. 1, pp. 115–133, Jan. 1994.
    [28]
    S. Haykin, Neural Networks: A Comprehensive Foundation. New York: IEEE, 1994.
    [29]
    S. Elfwing, E. Uchibe, and K. Doya, “Sigmoid-weighted linear units for neural network function approximation in reinforcement learning,” Neural Netw., vol. 107, pp. 3–11, Nov. 2018. doi: 10.1016/j.neunet.2017.12.012
    [30]
    M. B. Stinchcombe, “Neural network approximation of continuous functionals and continuous functions on compactifications,” Neural Netw., vol. 12, no. 3, pp. 467–477, Apr. 1999. doi: 10.1016/S0893-6080(98)00108-7
    [31]
    B. Igelnik and Y. H. Pao, “Stochastic choice of basis functions in adaptive function approximation and the functional-link net,” IEEE Trans. Neural Netw., vol. 6, no. 6, pp. 1320–1329, Nov. 1995. doi: 10.1109/72.471375
    [32]
    D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas. 2nd ed. Princeton, NJ: Princeton University Press, 2009.
    [33]
    A. S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers. vol. 1. Deterministic Techniques. Amsterdam: Elsevier, 2008.
    [34]
    A. Poznyak, A. Polyakov, and V. Azhmyakov, Attractive Ellipsoids in Robust Control. Berlin: Springer, 2014.
    [35]
    M. W. Spong, J. De Schutter, H. Bruyninckx, and J. T. Y. Wen, Control of Robots and Manipulators. CRC Press, 2018.

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(9)

    Article Metrics

    Article views (403) PDF downloads(47) Cited by()

    Highlights

    • This paper addresses the design of an exponential function-based learning law for differential neural networks
    • Two novel adaptive algorithms with predefined exponential convergence rate adjust the weights of the neural network
    • The application of the invariant ellipsoid method yields to obtain an algorithm to reduce the volume of the convergence region for the identification error
    • The proposed identifiers overcome the results of the non-exponential identifier achieving a faster convergence to an invariant set of smaller dimensions

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return