Extensive Theoretical/Numerical Comparative Studies on <i>H</i><sub>2</sub> and Generalized <i>H</i><sub>2</sub> Norms in Sampled-Data Systems
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This paper is concerned with linear time-invariant (LTI) sampled-data systems (by which we mean sampled-data systems with LTI generalized plants and LTI controllers) and studies their <i>H</i><sub>2</sub> norms from the viewpoint of impulse responses and generalized <i>H</i><sub>2</sub> norms from the viewpoint of the induced norms from <i>L</i><sub>2</sub> to <i>L</i><sub>∞</sub>. A new definition of the <i>H</i><sub>2</sub> norm of LTI sampled-data systems is first introduced through a sort of intermediate standpoint of those for the existing two definitions. We then establish unified treatment of the three definitions of the <i>H</i><sub>2</sub> norm through a matrix function <i>G</i>(τ) defined on the sampling interval [0, <i>h</i>). This paper next considers the generalized <i>H</i><sub>2</sub> norms, in which two types of the <i>L</i><sub>∞</sub> norm of the output are considered as the temporal supremum magnitude under the spatial 2-norm and ∞-norm of a vector-valued function. We further give unified treatment of the generalized <i>H</i><sub>2</sub> norms through another matrix function <i>F</i>(θ) which is also defined on [0, <i>h</i>). Through a close connection between <i>G</i>(τ) and <i>F</i>(θ), some theoretical relationships between the <i>H</i><sub>2</sub> and generalized <i>H</i><sub>2</sub> norms are provided. Furthermore, appropriate extensions associated with the treatment of <i>G</i>(τ) and <i>F</i>(θ) to the closed interval [0, <i>h</i>] are discussed to facilitate numerical computations and comparisons of the <i>H</i><sub>2</sub> and generalized <i>H</i><sub>2</sub> norms. Through theoretical and numerical studies, it is shown that the two generalized <i>H</i><sub>2</sub> norms coincide with neither of the three <i>H</i><sub>2</sub> norms of LTI sampled-data systems even though all the five definitions coincide with each other when single-output continuous-time LTI systems are considered as a special class of LTI sampled-data systems. To summarize, this paper clarifies that the five control performance measures are mutually related with each other but they are also intrinsically different from each other.
本文聚焦线性时不变（linear time-invariant, LTI）采样数据系统（即广义被控对象与控制器均为LTI的采样数据系统），从脉冲响应（impulse response）视角分析其H₂范数，并从L₂到L_∞的诱导范数（induced norm）视角研究其广义H₂范数。首先，本文依托现有两类H₂范数定义的中间视角，提出了LTI采样数据系统H₂范数的全新定义。随后，通过定义于采样区间[0, h)上的矩阵函数（matrix function）G(τ)，本文实现了三类H₂范数定义的统一处理。本文继而针对广义H₂范数展开研究，其中将输出的L_∞范数分别定义为：矢量值函数在空间2范数与∞范数下的时域峰值幅值。本文进一步通过另一个同样定义于[0, h)上的矩阵函数F(θ)，完成了广义H₂范数的统一处理。借助G(τ)与F(θ)之间的紧密联系，本文推导得到了H₂范数与广义H₂范数之间的若干理论关系。此外，为便于H₂范数与广义H₂范数的数值计算（numerical computation）与对比分析，本文讨论了将G(τ)与F(θ)的处理拓展至闭区间（closed interval）[0, h]的适配方法。通过理论与数值研究，本文表明：即便将单输出连续时间LTI系统（continuous-time LTI systems）作为LTI采样数据系统的特殊子类时，五类定义彼此等价，但两类广义H₂范数与LTI采样数据系统的三类H₂范数均不重合。综上，本文阐明了五类控制性能指标（control performance measures）既相互关联，又存在本质区别。
