A composite pseudospectral method for solving multi-delay optimal control problems involving piecewise constant delay functions

Abstract An adaptive numerical method for solving multi-delay optimal control problems with piecewise constant delay functions is introduced. The proposed method is based on composite pseudospectral method using the well-known Legendre–Gauss–Lobatto points. In this approach, the main problem converts to a mathematical optimization problem whose solution is much more easier than the original one. The necessary conditions of optimality associated to nonlinear piecewise constant delay systems are derived. The method is easy to implement and provides very accurate results.

[1]  Yadollah Ordokhani,et al.  Optimal Control of Delay Systems by Using a Hybrid Functions Approximation , 2012, J. Optim. Theory Appl..

[2]  H. Maurer,et al.  Optimal control problems with delays in state and control variables subject to mixed control–state constraints , 2009 .

[3]  Huaguang Zhang,et al.  Novel Weighting-Delay-Based Stability Criteria for Recurrent Neural Networks With Time-Varying Delay , 2010, IEEE Transactions on Neural Networks.

[4]  Hamid Reza Marzban,et al.  Optimal control of linear multi-delay systems with piecewise constant delays , 2016, IMA J. Math. Control. Inf..

[5]  H. Marzban,et al.  Solution of piecewise constant delay systems using hybrid of block‐pulse and Chebyshev polynomials , 2011 .

[6]  Gamal N. Elnagar,et al.  Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems , 1998, Comput. Optim. Appl..

[7]  H. R. Marzban,et al.  An efficient discretization scheme for solving nonlinear optimal control problems with multiple time delays , 2016 .

[8]  D. V. Senthilkumar,et al.  Dynamics of Nonlinear Time-Delay Systems , 2011 .

[9]  H. R. Marzban,et al.  SOLUTION OF NONLINEAR DELAY OPTIMAL CONTROL PROBLEMS USING A COMPOSITE PSEUDOSPECTRAL COLLOCATION METHOD , 2010 .

[10]  M. Newman,et al.  Interpolation and approximation , 1965 .

[11]  M. Namik Oguztöreli,et al.  Time-lag control systems , 1966 .

[12]  HuaGuang Zhang,et al.  New delay-dependent criterion for the stability of recurrent neural networks with time-varying delay , 2009, Science in China Series F: Information Sciences.

[13]  H. Banks,et al.  Hereditary Control Problems: Numerical Methods Based on Averaging Approximations , 1978 .

[14]  B. M. Mohan,et al.  Orthogonal Functions in Systems and Control , 1995 .

[15]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[16]  W. Ray,et al.  The optimal control of processes containing pure time delays—I necessary conditions for an optimum , 1970 .

[17]  Llan Y. Lee.  Numerical solution of time-delayed optimal control problems with terminal inequality constraints , 1993 .

[18]  Hamid Reza Marzban,et al.  A composite pseudospectral method for optimal control problems with piecewise smooth solutions , 2017, J. Frankl. Inst..

[19]  Hal L. Smith,et al.  An introduction to delay differential equations with applications to the life sciences / Hal Smith , 2010 .

[20]  Hamid Reza Marzban,et al.  Numerical treatment of non-linear optimal control problems involving piecewise constant delay , 2016, IMA J. Math. Control. Inf..

[21]  Vladimir Borisovich Kolmanovskiĭ,et al.  Control of Systems with Aftereffect , 1996 .

[22]  Mohsen Razzaghi,et al.  Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials , 2004, J. Frankl. Inst..

[23]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[24]  Gamal N. Elnagar,et al.  Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems , 1998 .

[25]  I. Michael Ross,et al.  Costate Estimation by a Legendre Pseudospectral Method , 1998 .

[26]  Gamal N. Elnagar,et al.  Numerical solution of time-delayed functional differential equation control systems , 2001 .

[27]  Huaguang Zhang,et al.  Global Asymptotic Stability of Recurrent Neural Networks With Multiple Time-Varying Delays , 2008, IEEE Transactions on Neural Networks.

[28]  Wen-Liang Chen,et al.  A general procedure of solving the linear delay system via block pulse functions , 1982 .

[29]  Helmut Maurer,et al.  THEORY AND APPLICATIONS OF OPTIMAL CONTROL PROBLEMS WITH MULTIPLE TIME-DELAYS , 2013 .

[30]  A. Halanay.  Optimal Controls for Systems with Time Lag , 1968 .

[31]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[32]  W. H. Ray,et al.  On the optimal control of systems having pure time delays and singular arcsI, Some necessary conditions for optimality† , 1972 .

[33]  I. M. Ross,et al.  Convergence of pseudospectral discretizations of optimal control problems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[34]  T. Guinn.  Reduction of delayed optimal control problems to nondelayed problems , 1976 .

[35]  W. Hager,et al.  An hp‐adaptive pseudospectral method for solving optimal control problems , 2011 .

[36]  H. R. Marzban,et al.  Optimal control of linear multi-delay systems based on a multi-interval decomposition scheme , 2016 .

[37]  Hamid Reza Marzban,et al.  Costate Computation by an Adaptive Pseudospectral Method for Solving Optimal Control Problems with Piecewise Constant Time Lag , 2016, J. Optim. Theory Appl..

[38]  H. R. Marzban,et al.  Solution of linear optimal control problems with time delay using a composite Chebyshev finite difference method , 2013 .

[39]  Harvey Thomas Banks.  Approximation of nonlinear functional differential equation control systems , 1979 .

[40]  Alireza Nazemi,et al.  Solving optimal control problems of the time-delayed systems by Haar wavelet , 2016 .

[41]  Gamal N. Elnagar,et al.  The pseudospectral Legendre method for discretizing optimal control problems , 1995, IEEE Trans. Autom. Control..

[42]  Wen-Liang Chen,et al.  Analysis of piecewise constant delay systems via block-pulse functions , 1981 .