Optimal control of time-varying linear delay systems based on the Bezier curves

In this paper, time-delay control systems with quadratic performance are solved by applying the least square method on the Bezier control points. The approximation process is done in two steps. First, the time interval is divided into $$2k$$2k subintervals, then in each subinterval the trajectory and control functions are approximated by the Bezier curves. We have chosen the Bezier curves as piecewise polynomials of degree $$n$$n and determined the Bezier curves on any subinterval by $$n+1$$n+1 control points. By considering a least square optimization problem, the control points can be found, then the Bezier curves that approximate the action of control and trajectory can be computed as well. Some numerical examples are given to verify the efficiency of the proposed method.

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

[2]  J. Perez,et al.  An Optimal Regulator for Linear Systems with Multiple State and Input Delays , 2007, 2007 American Control Conference.

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

[4]  L. Wrobel,et al.  Cubic Bezier splines for BEM heat transfer analysis of the 2-D continuous casting problems , 2002 .

[5]  Yun Q. Shi,et al.  Image and Video Compression for Multimedia Engineering , 1999 .

[6]  S. Lefschetz.  Contributions to the theory of nonlinear oscillations , 1950 .

[7]  Ganti Prasada Rao,et al.  Optimal control of linear systems with delays in state and control via Walsh functions , 1983 .

[8]  O uztöreli.  A Time Optimal Control Problem for Systems Described by Differential Difference Equations , 1963 .

[9]  N. N. Krasovskii,et al.  On the analytic construction of an optimal control in a system with time lags , 1962 .

[10]  Hussein Jaddu,et al.  Spectral method for constrained linear-quadratic optimal control , 2002, Math. Comput. Simul..

[11]  Thomas W. Sederberg,et al.  Least squares methods for solving differential equations using Bézier control points , 2004 .

[12]  W. Rudin.  Principles of mathematical analysis , 1964 .

[13]  E. Nakamae,et al.  Application of the Bézier curve to data interpolation , 1982 .

[14]  A. W. Knapp.  Basic Real Analysis , 2005 .

[15]  S. Somali,et al.  Least squares methods for solving singularly perturbed two-point boundary value problems using Bézier control points , 2008, Appl. Math. Lett..

[16]  Xing Tao Wang.  Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions , 2007, J. Frankl. Inst..

[17]  M. H. Farahi,et al.  The Bezier Control Points Method for Solving Delay Differential Equation , 2012 .

[18]  Frank Zeilfelder,et al.  Developments in bivariate spline interpolation , 2000 .

[19]  Mortaza Gachpazan,et al.  Solving of time varying quadratic optimal control problems by using Bézier control points , 2011 .

[20]  N. N. Krasovskii,et al.  Optimal processes in systems with time lag , 1963 .

[21]  Kok Lay Teo,et al.  An Optimization Approach to State-Delay Identification $ $ , 2010, IEEE Transactions on Automatic Control.

[22]  Charlie C. L. Wang,et al.  Computer aided geometric design of strip using developable Bézier patches , 2008, Comput. Ind..

[23]  J. V. Beltran,et al.  Bézier Solutions of the Wave Equation , 2004, ICCSA.

[24]  Michiel van de Panne,et al.  A numerically efficient and stable algorithm for animating water waves , 2002, The Visual Computer.

[25]  Kok Lay Teo,et al.  Optimal control of switched systems with time delay , 2006, Appl. Math. Lett..

[26]  Gerald Farin,et al.  Curves and surfaces for computer aided geometric design , 1990 .

[27]  H. Banks,et al.  Optimal control of linear time-delay systems , 1969 .

[28]  Matthias Heinkenschloss,et al.  A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems , 2005 .

[29]  V. G. Boltyanskiy.  The Maximum Principle in the Theory of Optimal Processes. , 1961 .

[30]  D. Ross.  Optimal control of systems described by differential-difference equations , 1969 .

[31]  Rudolf Winkel,et al.  Generalized Bernstein Polynomials and Bézier Curves: An Application of Umbral Calculus to Computer Aided Geometric Design , 2001, Adv. Appl. Math..

[32]  Michael V. Basin,et al.  Optimal control for linear systems with multiple time delays in control input , 2006, IEEE Transactions on Automatic Control.