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

机译：通过Chebyshev多项式最优控制参激线性时滞微分系统。

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摘要

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{sub}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 nonlinear optimization programme. The third approach computes the Chebyshev coefficients of the control vector by maximizing a linear decay rate of the L{sub}∞ 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.
机译：Chebyshev多项式在解决与具有恒定延迟的一般线性时变系统相关的有限水平最优控制问题中的使用在文献中是众所周知的。本文对具有时间周期系数和恒定延迟的动力系统的有限水平控制进行了改进。控制的运动微分方程根据系统矩阵的Chebyshev系数，延迟和当前状态向量以及输入向量转换为代数递归关系。考虑了三种不同的方法。第一种方法是通过最小化有限水平或有限时间间隔序列上的二次成本函数来计算控制矢量的切比雪夫系数。然后提出了两个收敛条件，以根据间隔内受控状态的振荡来改善优化轨迹的性能。第二种方法是通过最大化处于线性匹配和二次收敛条件下的状态的Chebyshev系数的L {sub} 2范数的二次衰减率，来计算控制矢量的Chebyshev系数。每个间隔中的控制矢量是通过制定非线性优化程序来计算的。第三种方法是通过最大化处于线性匹配和线性收敛条件下的状态的Chebyshev系数的L {sub}∞范数的线性衰减率，来计算控制矢量的Chebyshev系数。通过为有限控制范围内的延迟Mathieu方程设计调节控制器来说明所提出的技术。

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