In this article, we address the design and analysis of multivariable Newton-based extremum seeking (ES) for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations (PDEs). Multi-input systems with distinct diffusion coefficients in each individual input channel are dealt with. The phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation-based (averaging-based) estimate of the Hessian's inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point. The stability analysis is carried out using backstepping transformation and averaging in infinite dimensions, capturing the infinite-dimensional state due the diffusion PDEs. In addition, the generalization of the results for different classes of parabolic (reaction-advection-diffusion) PDEs, wave equations and/or first-order hyperbolic (transport-dominated) PDEs is also discussed. As for ordinary differential equations case, the proposed Newton approach removes the dependence of the algorithm's convergence rate on the unknown Hessian of the nonlinear map to be optimized, being user-assignable unlike the gradient algorithm. A numerical example illustrates the performance of our multivariable ES for compensating PDE-based systems.

中文翻译：

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偏微分方程动态系统的多变量极值搜索


在本文中，我们针对受扩散偏微分方程 (PDE) 控制的驱动动力学的静态局部二次映射进行基于多变量牛顿的极值搜索 (ES) 的设计和分析。处理每个单独输入通道中具有不同扩散系数的多输入系统。抖动信号的相位补偿被处理为轨迹生成问题，并且包含具有 Hessian 逆矩阵的基于扰动（基于平均）估计的多变量扩散反馈控制器，允许获得局部指数收敛结果到一个小邻域最佳点。使用反步变换和无限维平均进行稳定性分析，捕获由于扩散偏微分方程而产生的无限维状态。此外，还讨论了不同类别的抛物线（反应-平流-扩散）偏微分方程、波动方程和/或一阶双曲（输运主导）偏微分方程的结果的推广。对于常微分方程情况，所提出的牛顿方法消除了算法收敛速度对待优化非线性映射的未知 Hessian 矩阵的依赖，与梯度算法不同，用户可以指定。数值示例说明了我们的多变量 ES 用于补偿基于 PDE 的系统的性能。