Robust Feedback Control of Nonlinear PDEs by Numerical Approximation of High-Dimensional Hamilton-Jacobi-Isaacs Equations
Robust Feedback Control of Nonlinear PDEs by Numerical Approximation of High-Dimensional Hamilton-Jacobi-Isaacs Equations
复制标题
通过高维 Hamilton-Jacobi-Isaacs 方程的数值逼近实现非线性偏微分方程的鲁棒反馈控制
DOI：
10.1137/19m1262139
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
K. Kunisch
中科院分区：
文献类型：
--
作者：
D. Kalise;Sudeep Kundu;K. Kunisch
提出了一种非线性偏微分方程鲁棒最优反馈控制器的综合方法。我们的方法认为，通过伪谱配置方法的无穷维控制系统的近似，导致高维非线性动力学。对于降阶模型，我们构造了一个鲁棒反馈控制的基础上的$\cH_{\infty}$控制方法，这需要解决相关的高维Hamilton-Jacobi-Isaacs非线性偏微分方程。Isaacs PDE的维数是通过控制系统的可分离表示和相应的值函数的多项式逼近来解决的。我们的方法被证明是有效的非线性动力学的鲁棒稳定的尺寸$d\approximat12 $。我们评估了一类非线性抛物型偏微分方程，包括非线性对流和反应项的鲁棒性和最优性特征的设计。所提出的设计产生了一个反馈控制器，可实现最佳稳定和干扰抑制特性，并沿着提供了参数不确定性下偏微分方程鲁棒控制的建模框架。
We propose an approach for the synthesis of robust and optimal feedback controllers for nonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systems by a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For the reduced-order model, we construct a robust feedback control based on the $\cH_{\infty}$ control method, which requires the solution of an associated high-dimensional Hamilton-Jacobi-Isaacs nonlinear PDE. The dimensionality of the Isaacs PDE is tackled by means of a separable representation of the control system, and a polynomial approximation ansatz for the corresponding value function. Our method proves to be effective for the robust stabilization of nonlinear dynamics up to dimension $d\approx 12$. We assess the robustness and optimality features of our design over a class of nonlinear parabolic PDEs, including nonlinear advection and reaction terms. The proposed design yields a feedback controller achieving optimal stabilization and disturbance rejection properties, along with providing a modelling framework for the robust control of PDEs under parametric uncertainties.
克服状态相关的 Hamilton-Jacobi 方程维数灾难的算法
影响因子：
4.1
作者：
Chow, Yat Tin;Darbon, Jérôme;Osher, Stanley;Yin, Wotao
通讯作者：
Yin, Wotao