Extensive theoretical/numerical comparative studies on <i>H</i> <sub>2</sub> and generalised <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 generalised plants and LTI controllers) and studies their <i>H</i> <sub>2</sub> norms from the viewpoint of impulse responses and generalised <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 generalised <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 generalised <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 generalised <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 generalised <i>H</i> <sub>2</sub> norms. Through theoretical and numerical studies, it is shown that the two generalised <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 summarise, this paper clarifies that the five control performance measures are mutually related with each other but they are also intrinsically different from each other.
本文针对线性时不变（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）展开研究，从脉冲响应视角分析其H₂范数，并从L₂到L_∞的诱导范数视角分析其广义H₂范数。本文首先基于现有两类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，本文通过定义在采样区间[0, h)上的矩阵函数G(τ)，实现了三类H₂范数定义的统一处理。接下来，本文研究广义H₂范数：其中输出的L_∞范数分别被定义为向量值函数在空间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在[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范数的数值计算与对比，本文还讨论了将G(τ)与F(θ)的处理拓展至闭区间[0, h]的合理方案。通过理论与数值分析，本文证明：尽管当将单输出连续时间LTI系统作为LTI采样数据系统的特例时，上述五类定义完全等价，但两类广义H₂范数均与LTI采样数据系统的三类H₂范数均不相等。综上，本文明确了五项控制性能指标之间存在相互关联，但本质上又彼此迥异。
