{"title":"分数阶离散线性系统的最大输出集","authors":"A. El Bhih, Y. Benfatah, A. Ghazaoui, M. Rachik","doi":"10.23939/mmc2022.02.262","DOIUrl":null,"url":null,"abstract":"In this paper, we consider a linear discrete-time fractional-order system defined by \\[\\Delta ^{\\alpha }x_ {k+1}=Ax_k+B u_k, \\quad k \\geq 0, \\quad x_{0} \\in \\mathbb{R}^{n};\\] \\[y_{k}=Cx_k, \\quad k \\geq 0,\\] where $A$, $B$ and $C$ are appropriate matrices, $x_{0}$ is the initial state, $\\alpha$ is the order of the derivative, $y_k$ is the signal output and $u_k=K x_k$ is feedback control. By defining the fractional derivative in the Grunwald–Letnikov sense, we investigate the characterization of the maximal output set, $\\Gamma(\\Omega)=\\lbrace x_{0} \\in \\mathbb{R}^{n}/y_{i} \\in \\Omega,\\forall i \\geq 0 \\rbrace$, where $\\Omega\\subset\\mathbb{R}^{p}$ is a constraint set; and, by using some hypotheses of stability and observability, we prove that $\\Gamma(\\Omega)$ can be derived from a finite number of inequations. A powerful algorithm approach is included to identify the maximal output set; also, some appropriate algorithms and numerical simulations are given to illustrate the theoretical results.","PeriodicalId":37156,"journal":{"name":"Mathematical Modeling and Computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On the maximal output set of fractional-order discrete-time linear systems\",\"authors\":\"A. El Bhih, Y. Benfatah, A. Ghazaoui, M. Rachik\",\"doi\":\"10.23939/mmc2022.02.262\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we consider a linear discrete-time fractional-order system defined by \\\\[\\\\Delta ^{\\\\alpha }x_ {k+1}=Ax_k+B u_k, \\\\quad k \\\\geq 0, \\\\quad x_{0} \\\\in \\\\mathbb{R}^{n};\\\\] \\\\[y_{k}=Cx_k, \\\\quad k \\\\geq 0,\\\\] where $A$, $B$ and $C$ are appropriate matrices, $x_{0}$ is the initial state, $\\\\alpha$ is the order of the derivative, $y_k$ is the signal output and $u_k=K x_k$ is feedback control. By defining the fractional derivative in the Grunwald–Letnikov sense, we investigate the characterization of the maximal output set, $\\\\Gamma(\\\\Omega)=\\\\lbrace x_{0} \\\\in \\\\mathbb{R}^{n}/y_{i} \\\\in \\\\Omega,\\\\forall i \\\\geq 0 \\\\rbrace$, where $\\\\Omega\\\\subset\\\\mathbb{R}^{p}$ is a constraint set; and, by using some hypotheses of stability and observability, we prove that $\\\\Gamma(\\\\Omega)$ can be derived from a finite number of inequations. A powerful algorithm approach is included to identify the maximal output set; also, some appropriate algorithms and numerical simulations are given to illustrate the theoretical results.\",\"PeriodicalId\":37156,\"journal\":{\"name\":\"Mathematical Modeling and Computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Modeling and Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.23939/mmc2022.02.262\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Modeling and Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.23939/mmc2022.02.262","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 2
摘要
本文考虑一个由\[\Delta ^{\alpha }x_ {k+1}=Ax_k+B u_k, \quad k \geq 0, \quad x_{0} \in \mathbb{R}^{n};\]、\[y_{k}=Cx_k, \quad k \geq 0,\]定义的线性离散分数阶系统,其中$A$、$B$和$C$为适当矩阵,$x_{0}$为初始状态,$\alpha$为导数阶数,$y_k$为信号输出,$u_k=K x_k$为反馈控制。通过定义Grunwald-Letnikov意义上的分数阶导数,我们研究了最大输出集$\Gamma(\Omega)=\lbrace x_{0} \in \mathbb{R}^{n}/y_{i} \in \Omega,\forall i \geq 0 \rbrace$的特征,其中$\Omega\subset\mathbb{R}^{p}$是约束集;并利用稳定性和可观测性的一些假设,证明了$\Gamma(\Omega)$可以由有限个不等式导出。提出了一种识别最大输出集的有效算法;并给出了相应的算法和数值模拟来验证理论结果。
On the maximal output set of fractional-order discrete-time linear systems
In this paper, we consider a linear discrete-time fractional-order system defined by \[\Delta ^{\alpha }x_ {k+1}=Ax_k+B u_k, \quad k \geq 0, \quad x_{0} \in \mathbb{R}^{n};\] \[y_{k}=Cx_k, \quad k \geq 0,\] where $A$, $B$ and $C$ are appropriate matrices, $x_{0}$ is the initial state, $\alpha$ is the order of the derivative, $y_k$ is the signal output and $u_k=K x_k$ is feedback control. By defining the fractional derivative in the Grunwald–Letnikov sense, we investigate the characterization of the maximal output set, $\Gamma(\Omega)=\lbrace x_{0} \in \mathbb{R}^{n}/y_{i} \in \Omega,\forall i \geq 0 \rbrace$, where $\Omega\subset\mathbb{R}^{p}$ is a constraint set; and, by using some hypotheses of stability and observability, we prove that $\Gamma(\Omega)$ can be derived from a finite number of inequations. A powerful algorithm approach is included to identify the maximal output set; also, some appropriate algorithms and numerical simulations are given to illustrate the theoretical results.