Theoretical and Numerical Study on Sampled-Data Control of Parabolic Distributed Parameter Systems

研究了具有无界输出算子的抛物型系统的采样H_∞控制问题。特别地，假设输出算子是（-L）^γ-有界的，其中0<γ<1/2。例如，具有边界控制的扩散系统可以被公式化为具有这种类型的输出算子的抛物系统。针对具有理想采样器和零阶保持器的抛物系统，设计了一个有限维离散时间镇定控制器，使得反馈采样系统的L^2诱导范数小于给定的正数.为此，通过使用提升技术和变量变换，将无限维连续时间系统配制为无限维离散时间系统。基于无限维离散时间系统的有限维状态空间降阶模型，设计了一个包含剩余模式滤波器的有限维控制器，并将轴向色散系数足够小且可以忽略的系统作为控制对象，以获得理想的控制性能.考虑轴向扩散时，有边界输入的平行流换热器可由两个抛物型方程描述。另一方面，在轴向色散可以忽略的情况下，抛物型方程变成双曲型方程。本研究针对由双曲型系统与输出回授律所组成的闭回路系统，进行稳定性分析。此外，还对双曲系统进行了能观性和能达性等动力学分析。

This research is concerned with sampled-data H_∞ control of parabolic systems with unbounded output operators. Especially, the output operator is assumed to be (-L)^γ-bounded, where 0<γ<1/2. For example, diffusion systems with boundary control can be formulated as parabolic systems with output operators of such a type. For the parabolic system with an ideal sampler and a zero-order hold, the aim is to construct a finite-dimensional discrete-time stabilizing controller that makes the L^2 -induced norm of the feedback sampled-data system less than a given positive number. For that purpose, the infinite-dimensional continuous-time system is formulated as an infinite-dimensional discrete-time system by using a lifting technique and a variable transformation. Based on a reduced-order model with a finite-dimensional state space for the infinite-dimensional discrete-time system, a finite-dimensional controller containing a residual mode filter is designed to provide the desirable performance.Moreover, systems whose axial dispersion coefficients are sufficiently small and can be neglected are treated as control objects. A parallel-flow heat exchanger with boundary inputs is described by two parabolic equations when the axial dispersion is taken into consideration. On the other hand, the parabolic equations become hyperbolic equations in the case where the axial dispersion can be neglected. In this research, the stability analysis is carried out, for the closed-loop system which consists of the hyperbolic system and an output feedback law. In addition, the dynamical analysis such as observability and reachability is performed for the hyperbolic system.