### 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 L_{2} 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 language | English (US) |
---|---|

Pages (from-to) | 123-136 |

Number of pages | 14 |

Journal | Optimal Control Applications and Methods |

Volume | 27 |

Issue number | 3 |

DOIs | |

State | Published - May 2006 |

Externally published | Yes |

### Fingerprint

### 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 Applications and Methods*,

*27*(3), 123-136. https://doi.org/10.1002/oca.769

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

Research output: Contribution to journal › Article

*Optimal Control Applications and Methods*, vol. 27, no. 3, pp. 123-136. https://doi.org/10.1002/oca.769

}

TY - JOUR

T1 - Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials

AU - Deshmukh, Venkatesh

AU - Ma, Haitao

AU - Butcher, Eric

PY - 2006/5

Y1 - 2006/5

N2 - 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.

AB - 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.

KW - Chebyshev polynomials

KW - Convergence conditions

KW - Periodic differential delay equation

KW - Quadratic cost function

UR - http://www.scopus.com/inward/record.url?scp=33745797065&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745797065&partnerID=8YFLogxK

U2 - 10.1002/oca.769

DO - 10.1002/oca.769

M3 - Article

AN - SCOPUS:33745797065

VL - 27

SP - 123

EP - 136

JO - Optimal Control Applications and Methods

JF - Optimal Control Applications and Methods

SN - 0143-2087

IS - 3

ER -