ABSTRACT This paper addresses the L1 adaptive control problem for general Partial Differential Equation (PDE) systems. Since direct computation and analysis on PDE systems are difficult and time-consuming, it is preferred to transform the PDE systems into Ordinary Differential Equation (ODE) systems. In this paper, a polynomial interpolation approximation method is utilized to formulate the infinite dimensional PDE as a high-order ODE first. To further reduce its dimension, an eigenvalue-based technique is employed to derive a system of low-order ODEs, which is incorporated with unmodeled dynamics described as bounded-input, bounded-output (BIBO) stable. To establish the equivalence with original PDE, the reduced-order ODE system is augmented with nonlinear time-varying uncertainties. On the basis of the reduced-order ODE system, a dynamic state predictor consisting of a linear system plus adaptive estimated parameters is developed. An adaptive law will update uncertainty estimates such that the estimation error between predicted state and real state is driven to zero at each time-step. And a control law is designed for uncertainty handling and good tracking delivery. Simulation results demonstrate the effectiveness of the proposed modeling and control framework.

中文翻译：

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一般偏微分方程 (PDE) 系统的 L1 自适应控制

摘要 本文解决了一般偏微分方程 (PDE) 系统的 L1 自适应控制问题。由于对 PDE 系统的直接计算和分析既困难又耗时，因此最好将 PDE 系统转换为常微分方程 (ODE) 系统。在本文中，首先利用多项式插值近似方法将无限维偏微分方程表示为高阶常微分方程。为了进一步降低其维度，采用基于特征值的技术来推导低阶 ODE 系统，该系统与描述为有界输入、有界输出 (BIBO) 稳定的未建模动力学相结合。为了建立与原始 PDE 的等价性，降阶 ODE 系统增加了非线性时变不确定性。在降阶 ODE 系统的基础上，开发了一个由线性系统和自适应估计参数组成的动态状态预测器。自适应律将更新不确定性估计，以便预测状态和真实状态之间的估计误差在每个时间步长为零。控制律是为不确定性处理和良好的跟踪交付而设计的。仿真结果证明了所提出的建模和控制框架的有效性。