Weighting Function Estimation in Distributed-Parameter Systems

  • Henry E. Lee
  • D. W. C. Shen


An accelerated stochastic approximation algorithm is developed for the identification of weighting function associated with boundary control in one-dimensional linear distributed-parameter systems. The weighting function is assumed to be variable separable, and each variable is approximated by a finite number of orthonormal polynomials. In the absence of noise, this algorithm will converge in a finite number of steps. For adaptive control, on-line weighting function estimators are developed which use the optimal control function as input. These estimators are functional gradient algorithms based on least square approach. They can be used for estimating weighting function associated with either boundary or distributed control.


Weighting Function Orthonormal Polynomial Stochastic Approximation Algorithm Partial Differential Equation Model Optimal Control Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. J. Perdreauville and R. E. Goodson, “Identification of Systems Described by Partial Differential Equations,” Trans. ASME, J. of Basic Engr., Ser. D, Vol. 88, No. 2, pp. 463–468, 1966.CrossRefGoogle Scholar
  2. 2.
    P. L. Collins and H. C. Khatri, “Identification of Distributed Parameter Systems Using Finite Differences,” Trans. ASME, J. of Basic Engr., Ser. D, Vol. 91, No. 2, pp. 239–245, 1969.CrossRefGoogle Scholar
  3. 3.
    G. N. Saridis and P. C. Badavas, “Identifying Solutions of Distributed Parameter Systems by Stochastic Approximation,” Proc. of 7th IEEE Symp. on Adaptive Processes, December 1968.Google Scholar
  4. 4.
    G. S. Levy, “Accelerated Gradient Pattern Recognition: An Application to System Identification,” M.S. Thesis, Department of Electrical Engineering, McGill University, 1968.Google Scholar
  5. 5.
    H. S. Wilf and A. Ralston, Mathematical Methods for Digital Computers, Wiley and Sons, 1960.Google Scholar
  6. 6.
    A. Dvoretzky, “On Stochastic Approximation,” Proc. Third Berkeley Symp. on Math. Stat. and Prob., Vol. 1, 1956.Google Scholar
  7. 7.
    E. I. Axelband, “The Structure of the Optimal Tracking Problem for Distributed-Parameter Systems,” IEEE Trans. on Automatic Control, Vol. AC-13, pp. 50–56, 1968.CrossRefGoogle Scholar
  8. 8.
    L. V. Kantarovich, “Functional Analysis and Applied Mathematics,” Uspekhi Mat. Nauk, Vol. 3, p. 89, 1948.Google Scholar
  9. 9.
    H. E. Lee, “Identification and Optimal Control of Stochastic Distributed-Parameter Systems,” Ph.D. Dissertation, Univ. of Pennsylvania, 1970.Google Scholar

Copyright information

© Plenum Press, New York 1971

Authors and Affiliations

  • Henry E. Lee
    • 1
  • D. W. C. Shen
    • 2
  1. 1.Westinghouse Electric Corp.AnnapolisUSA
  2. 2.Univ. of PennsylvaniaPhiladelphiaUSA

Personalised recommendations