We focus on the Lyapunov-based output feedback control problem for a class of distributed parameter systems with spatiotemporal dynamics described by input-affine linear and semilinear dissipative partial differential equations (DPDEs). The control problem is addressed via model order reduction. Galerkin projection is applied to discretize the DPDE and derive low-dimensional reduced order models (ROMs). The empirical basis functions needed for this discretization are recursively computed using adaptive proper orthogonal decomposition (APOD). To update the basis functions during process operation, APOD needs measurements of the system state’s complete profile (called snapshots) at revision times. This paper’s main objective is to minimize the demand for snapshots from the spatially distributed sensors by the control structure while maintaining closed-loop stability and performance. A control Lyapunov function is defined, and its value is monitored as the system evolves. Only when the value violates a closed-loop stability threshold, snapshots are requested for a brief period by APOD after which the ROM is updated, and the controller is reconfigured. The proposed approach is applied to stabilize the Kuramoto–Sivashinsky equation.

我们专注于一类具有输入仿射线性和半线性耗散偏微分方程 (DPDE) 描述的时空动态的分布式参数系统的基于 Lyapunov 的输出反馈控制问题。控制问题通过模型降阶来解决。Galerkin 投影用于离散化 DPDE 并导出低维降阶模型 (ROM)。这种离散化所需的经验基函数是使用自适应适当正交分解 (APOD) 递归计算的。为了在过程操作期间更新基函数，APOD 需要在修订时测量系统状态的完整配置文件（称为快照）。本文的主要目标是最大限度地减少控制结构对空间分布传感器快照的需求，同时保持闭环稳定性和性能。定义了一个控制李雅普诺夫函数，并随着系统的发展监测其值。仅当该值违反闭环稳定性阈值时，APOD 才会在短时间内请求快照，然后更新 ROM，并重新配置控制器。所提出的方法用于稳定 Kuramoto-Sivashinsky 方程。