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系统的采样数据H_∞控制问题。特别地，假设输出算子是(-L)γ有界的，其中0&lt；γ&lt；1/2。例如，具有边界控制的扩散系统可以表示为具有这类输出算子的抛物型系统。对于具有理想采样器和零阶保持的抛物系统，目标是构造一个有限维离散时间镇定控制器，使反馈采样数据系统的L^2诱导范数小于给定的正数。为此，利用提升技术和变量变换，将无限维连续时间系统表示为无限维离散时间系统。基于具有有限维状态空间的无限维离散时间系统的降维模型，设计了包含剩余模滤波器的有限维控制器以提供期望的性能，并且将轴向色散系数足够小且可以忽略的系统作为控制对象。考虑轴向扩散时，具有边界输入的平行流换热器可用两个抛物线方程描述。另一方面，在轴向色散可以忽略的情况下，抛物型方程变成了双曲型方程。针对由双曲系统和输出反馈律组成的闭环系统，进行了稳定性分析。此外，还对双曲系统进行了能观测性、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.