Assessment of He's Homotopy Perturbation Method for Optimal Control of Linear Time-Delay Systems

This paper renders an optimal regulator for a system with state time-delay and quadratic performance index. First, by means of Maximum Principle and necessary optimal conditions, a continues-time two-point boundary-value problem (TPBVP) including both time-delay and time-advance is derived. Then, using Homotopy Perturbation Method (HPM), the obtained TPBVP is transformed into a sequence of linear time-invariant TPBVP without any time-delay or timeadvance. Solving the resulted linear TPBVP sequence in a recursive manner, concludes that the optimal control law in the form of rapid convergent series. At the end, an iterative algorithm with low complexity in computation and fast convergence rate is proposed to achieve an accurate enough suboptimal control law. It's worth noting that simulation based on application of a harmonic oscillator is conducted to demonstrate efficiency of this method.

[1]  Mehdi Dehghan,et al.  Solution of delay differential equations via a homotopy perturbation method , 2008, Math. Comput. Model..

[2]  Saeid Abbasbandy,et al.  Application of He’s homotopy perturbation method to functional integral equations , 2007 .

[3]  Gong-You Tang,et al.  OPTIMAL TRACKING CONTROL FOR DISCRETE TIME‐DELAY SYSTEMS WITH PERSISTENT DISTURBANCES , 2006 .

[4]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[5]  Arman Sargolzaei,et al.  Optimal Control of Nonlinear Systems Using the Homotopy Perturbation Method: Infinite Horizon Case , 2010, J. Digit. Content Technol. its Appl..

[7]  Randal W. Beard,et al.  Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation , 1997, Autom..

[8]  Rein Luus,et al.  Use of Piecewise Linear Continuous Optimal Control for Time-Delay Systems , 1995 .

[9]  F. Khellat,et al.  Optimal Control of Linear Time-Delayed Systems by Linear Legendre Multiwavelets , 2009 .

[10]  Oskar von Stryk,et al.  Direct and indirect methods for trajectory optimization , 1992, Ann. Oper. Res..

[11]  R Bellman,et al.  On the Theory of Dynamic Programming. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

[12]  G. Y. Tang,et al.  Optimal Control of Nonlinear Time-Delay Systems with Persistent Disturbances , 2007 .

[13]  Ji-Huan He.  A coupling method of a homotopy technique and a perturbation technique for non-linear problems , 2000 .

[14]  Jann N. Yang,et al.  Application of Optimal Control Theory to Civil Engineering Structures , 1975 .

[15]  Audrey Hermant,et al.  Optimal control of the atmospheric reentry of a space shuttle by an homotopy method , 2011 .

[16]  Mo M. Jamshidi,et al.  A computational algorithm for large-scale nonlinear time-delay systems , 1984, IEEE Transactions on Systems, Man, and Cybernetics.