Further results on the L1 analysis of sampled-data systems via kernel approximation approach
Further results on the L1 analysis of sampled-data systems via kernel approximation approach
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DOI：
10.1080/00207179.2016.1144239
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发表时间：
2016-02
影响因子：
2.1
通讯作者：
Jung Hoon Kim;T. Hagiwara
中科院分区：
文献类型：
--
作者：
Jung Hoon Kim;T. Hagiwara
本文给出了两种分析采样数据系统L1的方法，即计算采样数据系统的L∞诱导范数。这是通过在采样数据系统的设置中开发我们所称的核近似方法来实现的。我们首先考虑了采样数据系统的提升处理，并给出了其输入输出关系的算子理论表示。进一步应用快速提升技术，将抽样区间[0，h]分成等宽的M个子区间，给出了L∞诱导范数的计算方法。与以前发展的一种类似的方法称为输入近似方法不同，我们使用了核逼近的思想，其中输入算子的核函数和输出算子的保持函数由分段常量或分段线性函数来逼近。进一步证明了分段常数逼近和分段线性逼近的逼近误差分别以1/M和1/M2的速度收敛到0。与现有的用分段常数或分段线性函数逼近输入算子的输入函数(而不是核函数)的方法相比，核逼近方法具有更好的计算结果。更准确地说，尽管核逼近方法的收敛速度与输入逼近方法的收敛速度在定性上相同，但新发展的核逼近方法比输入逼近方法的逼近误差在数量上有所改善，特别是在采用分段线性逼近方案的情况下。最后给出了一个数值算例，验证了该方法的核逼近方法的有效性。
ABSTRACT This paper gives two methods for the L1 analysis of sampled-data systems, by which we mean computing the L∞-induced norm of sampled-data systems. This is achieved by developing what we call the kernel approximation approach in the setting of sampled-data systems. We first consider the lifting treatment of sampled-data systems and give an operator theoretic representation of their input/output relation. We further apply the fast-lifting technique by which the sampling interval [0, h) is divided into M subintervals with an equal width, and provide methods for computing the L∞-induced norm. In contrast to a similar approach developed earlier called the input approximation approach, we use an idea of kernel approximation, in which the kernel function of an input operator and the hold function of an output operator are approximated by piecewise constant or piecewise linear functions. Furthermore, it is shown that the approximation errors in the piecewise constant approximation or piecewise linear approximation scheme converge to 0 at the rate of 1/M or 1/M2, respectively. In comparison with the existing input approximation approach, in which the input function (rather than the kernel function) of the input operator is approximated by piecewise constant or piecewise linear functions, we show that the kernel approximation approach gives improved computation results. More precisely, even though the convergence rates in the kernel approximation approach remain qualitatively the same as those in the input approximation approach, the newly developed former approach could lead to quantitatively improved approximation errors than the latter approach particularly when the piecewise linear approximation scheme is taken. Finally, a numerical example is given to demonstrate the effectiveness of the kernel approximation approach with this scheme.