Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials

Venkatesh Deshmukh, Haitao Ma, Eric Butcher

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

The use of Chebyshev polynomials in solving finite horizon optimal control problems associated with general linear time-varying systems with constant delay is well known in the literature. The technique is modified in the present paper for the finite horizon control of dynamical systems with time periodic coefficients and constant delay. The governing differential equations of motion are converted into an algebraic recursive relationship in terms of the Chebyshev coefficients of the system matrices, delayed and present state vectors, and the input vector. Three different approaches are considered. The first approach computes the Chebyshev coefficients of the control vector by minimizing a quadratic cost function over a finite horizon or a finite sequence of time intervals. Then two convergence conditions are presented to improve the performance of the optimized trajectories in terms of the oscillation of controlled states within intervals. The second approach computes the Chebyshev coefficients of the control vector by maximizing a quadratic decay rate of the L2 norm of Chebyshev coefficients of the state subject to linear matching and quadratic convergence conditions. The control vector in each interval is computed by formulating a non-linear optimization programme. The third approach computes the Chebyshev coefficients of the control vector by maximizing a linear decay rate of the L norm of Chebyshev coefficients of the state subject to linear matching and linear convergence conditions. The proposed techniques are illustrated by designing regulation controllers for a delayed Mathieu equation over a finite control horizon.

Original languageEnglish (US)
Pages (from-to)123-136
Number of pages14
JournalOptimal Control Applications and Methods
Volume27
Issue number3
DOIs
StatePublished - May 2006
Externally publishedYes

Fingerprint

Delay-differential Systems
Chebyshev Polynomials
Chebyshev
Optimal Control
Vector Control
Linear Systems
Polynomials
Convergence Condition
Finite Horizon
Coefficient
Time varying systems
Decay Rate
Cost functions
Equations of motion
Interval
Dynamical systems
Differential equations
Trajectories
Mathieu Equation
Norm

Keywords

  • Chebyshev polynomials
  • Convergence conditions
  • Periodic differential delay equation
  • Quadratic cost function

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Applied Mathematics
  • Control and Optimization
  • Management Science and Operations Research

Cite this

Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials. / Deshmukh, Venkatesh; Ma, Haitao; Butcher, Eric.

In: Optimal Control Applications and Methods, Vol. 27, No. 3, 05.2006, p. 123-136.

Research output: Contribution to journalArticle

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