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
收藏资源简介：
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.
本文聚焦于线性时不变（LTI）采样数据系统（即带有LTI广义被控对象与LTI控制器的采样数据系统），分别从冲激响应视角研究其H₂范数，以及从L₂到L_∞的诱导范数视角研究其广义H₂范数。本文首先借助介于现有两种H₂范数定义之间的中间视角，提出了LTI采样数据系统H₂范数的全新定义。随后，通过定义在采样区间[0, h)上的矩阵函数G(τ)，实现了三种H₂范数定义的统一处理。接下来，本文讨论广义H₂范数：其中将输出的L_∞范数分别定义为向量值函数在空间2-范数与∞-范数下的时域上确界幅值。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范数的数值计算与对比，本文还讨论了将G(τ)与F(θ)的处理扩展至闭区间[0, h]的相关可行方案。通过理论与数值研究表明：即便将单输出连续时间LTI系统视作LTI采样数据系统的特殊类别时，五种定义彼此等价，但LTI采样数据系统的两种广义H₂范数，均与三种H₂范数互不重合。综上，本文阐明了这五种控制性能指标既彼此关联，又存在本质区别。
