论文标题
双表示和$ h _ {\ infty} $ - 部分微分方程的最佳控制
Dual Representations and $H_{\infty}$-Optimal Control of Partial Differential Equations
论文作者
论文摘要
我们考虑$ h _ {\ infty} $ - 线性偏微分方程（PDES）类的最佳状态反馈控制，该类允许部分积分方程（PIE）表示。虽然线性矩阵不等式通常用于最佳控制普通微分方程（ODE），但缺乏通用状态空间表示和合适的双重形式可防止将这种方法应用于PDE的最佳控制。具体而言，对于ODE，控制器的合成问题是在状态空间中定义的，并且二元性用于解决该合成问题的双线性。最近，提出了派代表为线性PDE系统的通用状态空间表示。在本文中，我们表明，任何由PIE代表的PDE系统都允许具有相同稳定性和I/O属性的双派。这一结果使我们能够将稳定和最佳的状态反馈控制问题重新制定为对积极部分积分（PI）操作员锥的凸优化。然后，操作员倒置公式允许我们为原始PDE系统构建反馈收益。通过应用于PDE的最佳控制中的几个规范问题来验证结果。
We consider $H_{\infty}$-optimal state-feedback control of the class of linear Partial Differential Equations (PDEs) class, which admit a Partial Integral Equation (PIE) representation. While linear matrix inequalities are commonly used for optimal control of Ordinary Differential Equations (ODEs), the absence of a universal state-space representation and suitable dual form prevents such methods from being applied to optimal control of PDEs. Specifically, for ODEs, the controller synthesis problem is defined in state-space, and duality is used to resolve the bilinearity of that synthesis problem. Recently, the PIE representation was proposed as a universal state-space representation for linear PDE systems. In this paper, we show that any PDE system represented by a PIE admits a dual PIE with identical stability and I/O properties. This result allows us to reformulate the stabilizing and optimal state-feedback control problems as convex optimization over the cone of positive Partial Integral (PI) operators. Operator inversion formulae then allow us to construct feedback gains for the original PDE system. The results are verified through application to several canonical problems in optimal control of PDEs.
