Cell Mapping Techniques for Tuning Dynamical Systems

In this chapter, to be dedicated in the 90th birthday of Professor C. S. Hsu, several computational schemes are presented for the optimal tuning of the global behavior of nonlinear dynamical systems. Specifically, the maximization of the size of domains of attraction associated with invariants in parametrized dynamical systems is addressed. Cell Mapping (CM) techniques are used to estimate the size of the domains for different parameter values, and such size function is then maximized via several optimization methods. First, a genetic algorithm is tested whose performance shows to be good for determining global maxima at the expense of high computational cost. Secondly, an iterative scheme based on a Stochastic Approximation procedure (the Kiefer–Wolfowitz algorithm) is evaluated showing acceptable performance at low cost. Finally, several schemes combining neural network based estimations and model-based optimization procedures are addressed with promising results. The performance of the methods is illustrated with some applications including the well-known van der Pol equation with standard parametrization, and the tuning of a controller for saturated systems.

[1]  Sujeet Shenoi,et al.  Incorporating cell map information in fuzzy controller design , 1994, Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference.

[2]  C. S. G. Lee,et al.  Robotics: Control, Sensing, Vision, and Intelligence , 1987 .

[3]  Sujeet Shenoi,et al.  Cell mapping for controller design and evaluation , 1997 .

[4]  Tingshu Hu,et al.  An analysis and design method for linear systems subject to actuator saturation and disturbance , 2002, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[5]  Michael A. Cohen,et al.  The construction of arbitrary stable dynamics in nonlinear neural networks , 1992, Neural Networks.

[6]  Jian-Xue Xu,et al.  Domains of attraction for multiple limit cycles of coupled van der pol equations by simple cell mapping , 1985 .

[7]  M.H. Hassoun,et al.  Fundamentals of Artificial Neural Networks , 1996, Proceedings of the IEEE.

[8]  H. Flashner,et al.  A computational approach for studying domains of attraction for non-linear systems , 1988 .

[9]  Roger W. Brockett,et al.  A light weight rotary double pendulum: maximizing the domain of attraction , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[10]  Pedro J. Zufiria,et al.  Computational Schemes for Optimizing Domains of Attraction in Dynamical Systems , 2011 .

[11]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[12]  P. J. Zufiria,et al.  The adjoining cell mapping and its recursive unraveling, part II: Application to selected problems , 1993 .

[13]  Ronald C. Arkin,et al.  An Behavior-based Robotics , 1998 .

[14]  Henryk Flashner,et al.  A numerical method for computing domains of attraction for dynamical systems , 1988 .

[15]  Harold J. Kushner,et al.  Stochastic Approximation Algorithms and Applications , 1997, Applications of Mathematics.

[16]  C. S. Hsu,et al.  Cell-to-Cell Mapping , 1987 .

[17]  R. Seydel.  From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis , 1988 .

[18]  C. Hsu.  A discrete method of optimal control based upon the cell state space concept , 1985 .

[19]  Frank L. Lewis,et al.  Optimal Control , 1986 .

[20]  Tingshu Hu,et al.  Conjugate Lyapunov functions for saturated linear systems , 2005, Autom..

[21]  Pedro J. Zufiria,et al.  Adaptive cellular integration of linearly implicit differential equations , 1999 .

[22]  Tomás Martínez-Marín,et al.  Optimal control of non‐linear systems through hybrid cell‐mapping/artificial‐neural‐networks techniques , 1999 .

[23]  Kevin L. Moore,et al.  Maximal Domains of Attraction in a Hopfield Neural Network with Learning , 1993, 1993 American Control Conference.

[24]  P. J. Zufiria,et al.  The adjoining cell mapping and its recursive unraveling, part I: Description of adaptive and recursive algorithms , 1993, Nonlinear Dynamics.

[25]  P. J. Zufiria,et al.  Improved Optimal Control Methods Based Upon the Adjoining Cell Mapping Technique , 2003 .

[26]  Lawrence. Davis,et al.  Handbook Of Genetic Algorithms , 1990 .