论文标题
热方程的有限维控制：Dirichlet致动和点测量
Finite-dimensional control of the heat equation: Dirichlet actuation and point measurement
论文作者
论文摘要
最近，为1D热方程式引入了有限维观察者的控制器，其中至少一个观测或对照算子是有界的。在本文中，我们首次使用这样的控制器来管理1D热方程式，两个操作员都不受限。我们考虑Dirichlet致动和点测量，并通过动态扩展使用模态分解方法。我们建议对全阶闭环系统进行直接的Lyapunov方法，其中有限维状态与状态傅立叶膨胀的无限维尾相结合，并提供LMI，以找到控制器尺寸和产生的指数衰减率。我们进一步研究了在采样数据测量下对控制器的采样数据实现。我们在时间上使用基于Wirtinger的，不连续的Lyapunov功能，以补偿有限维状态的采样。为了补偿无限维尾巴中的采样，我们使用了一种新型的Halanay的不平等，这适用于跳跃不连续性的Lyapunov功能，这些功能不会在跳跃中生长。数值示例证明了该方法的效率。
Recently finite-dimensional observer-based controllers were introduced for the 1D heat equation, where at least one of the observation or control operators is bounded. In this paper, for the first time, we manage with such controllers for the 1D heat equation with both operators being unbounded. We consider Dirichlet actuation and point measurement and use a modal decomposition approach via dynamic extension. We suggest a direct Lyapunov approach to the full-order closed-loop system, where the finite-dimensional state is coupled with the infinite-dimensional tail of the state Fourier expansion, and provide LMIs for finding the controller dimension and the resulting exponential decay rate. We further study sampled-data implementation of the controller under sampled-data measurement. We use Wirtinger-based, discontinuous in time, Lyapunov functionals which compensate sampling in the finite-dimensional state. To compensate sampling in the infinite-dimensional tail, we use a novel form of Halanay's inequality, which is appropriate for Lyapunov functions with jump discontinuities that do not grow in the jumps. Numerical examples demonstrate the efficiency of the method.
