基于梯度动态优化的边界控制抛物型偏微分方程输出镇定

Zhigang Ren, Chao Xu, Qun Lin, R. Loxton
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引用次数: 1

摘要

提出了一种新的边控抛物型偏微分方程稳定的控制综合方法。在该方法中,最优边界控制以二次核函数的积分状态反馈形式表示,其中二次系数为待优化的决策变量。我们引入了一个系统代价函数来惩罚状态和核大小,然后根据辅助“状态”PDE的解推导出代价函数的梯度。在此基础上,利用序列二次规划等基于梯度的优化技术解决输出镇定问题。所得到的最优边界控制保证了在温和条件下的闭环稳定性。新方法的主要优点是协态偏微分方程是标准形式的,可以很容易地用有限差分法求解。而传统的边界控制抛物型偏微分方程的控制综合方法(即LQ控制和反演方法)需要求解非标准riccati型和klein - gordon型偏微分方程。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Output stabilization of boundary-controlled parabolic PDEs via gradient-based dynamic optimization
This paper proposes a new control synthesis approach for the stabilization of boundary-controlled parabolic partial differential equations (PDEs). In the proposed approach, the optimal boundary control is expressed in integral state feedback form with quadratic kernel function, where the quadratic's coefficients are decision variables to be optimized. We introduce a system cost functional to penalize both state and kernel magnitude, and then derive the cost functional's gradient in terms of the solution of an auxiliary “costate” PDE. On this basis, the output stabilization problem can be solved using gradient-based optimization techniques such as sequential quadratic programming. The resulting optimal boundary control is guaranteed to yield closed-loop stability under mild conditions. The primary advantage of our new approach is that the costate PDE is in standard form and can be solved easily using the finite difference method. In contrast, the traditional control synthesis approaches for boundary-controlled parabolic PDEs (i.e., the LQ control and backstepping approaches) require solving non-standard Riccati-type and Klein-Gorden-type PDEs.
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