Stability of Networked Control Systems Based on Free-Weighting Matrices Method
自动化专业英语王宏文全文翻译
自动化专业英语PART 1Electrical and Electronic Engineering BasicsUNIT 1A Electrical Networks ————————————3B Three-phase CircuitsUNIT 2A The Operational Amplifier ———————————5B TransistorsUNIT 3A Logical Variables and Flip-flop ——————————8B Binary Number SystemUNIT 4A Power Semiconductor Devices ——————————11B Power Electronic ConvertersUNIT 5A Types of DC Motors —————————————15B Closed-loop Control of DC DriversUNIT 6A AC Machines ———————————————19B Induction Motor DriveUNIT 7A Electric Power System ————————————22B Power System AutomationPART 2Control TheoryUNIT 1A The World of Control ————————————27B The Transfer Function and the Laplace Transformation —————29 UNIT 2A Stability and the Time Response —————————30B Steady State—————————————————31 UNIT 3A The Root Locus —————————————32B The Frequency Response Methods: Nyquist Diagrams —————33 UNIT 4A The Frequency Response Methods: Bode Piots —————34B Nonlinear Control System 37UNIT 5 A Introduction to Modern Control Theory 38B State Equations 40UNIT 6 A Controllability, Observability, and StabilityB Optimum Control SystemsUNIT 7 A Conventional and Intelligent ControlB Artificial Neural NetworkPART 3 Computer Control TechnologyUNIT 1 A Computer Structure and Function 42B Fundamentals of Computer and Networks 43UNIT 2 A Interfaces to External Signals and Devices 44B The Applications of Computers 46UNIT 3 A PLC OverviewB PACs for Industrial Control, the Future of ControlUNIT 4 A Fundamentals of Single-chip Microcomputer 49B Understanding DSP and Its UsesUNIT 5 A A First Look at Embedded SystemsB Embedded Systems DesignPART 4 Process ControlUNIT 1 A A Process Control System 50B Fundamentals of Process Control 52UNIT 2 A Sensors and Transmitters 53B Final Control Elements and ControllersUNIT 3 A P Controllers and PI ControllersB PID Controllers and Other ControllersUNIT 4 A Indicating InstrumentsB Control PanelsPART 5 Control Based on Network and InformationUNIT 1 A Automation Networking Application AreasB Evolution of Control System ArchitectureUNIT 2 A Fundamental Issues in Networked Control SystemsB Stability of NCSs with Network-induced DelayUNIT 3 A Fundamentals of the Database SystemB Virtual Manufacturing—A Growing Trend in AutomationUNIT 4 A Concepts of Computer Integrated ManufacturingB Enterprise Resources Planning and BeyondPART 6 Synthetic Applications of Automatic TechnologyUNIT 1 A Recent Advances and Future Trends in Electrical Machine DriversB System Evolution in Intelligent BuildingsUNIT 2 A Industrial RobotB A General Introduction to Pattern RecognitionUNIT 3 A Renewable EnergyB Electric VehiclesUNIT 1A 电路电路或电网络由以某种方式连接的电阻器、电感器和电容器等元件组成。
网络化控制``
Shanbin Li, PhD student NCS Research Group National Laboratory of Industrial Control Technology Zhejiang University
2013-9-10
ZJU-NLICT-NCS Research Group By Shanbin Li
11
Methodology One: Scheduling Part
Revise network MAC protocol
Ye, Walsh, Bushnell 2000 introduce a scheduling protocol PCSMA/CA (Prioritized-CSMACA) based on IEEE 802.11 wireless standard. For the first time, they proposed and validated several new algorithms, a constant penalty, an estimated error order and a lag first order for dynamically scheduling the traffic of wireless networked control systems.
Methodology Two: Control Part
Augmented deterministic discrete-time method
Halevi and Ray，1988 proposed a method to control a linear plant over periodic delay network Liou and Ray，1990 introduced a method to analyze the systems with multi-rate sampling sensor and control
网络控制系统的输出反馈保性能控制
网络控制系统的输出反馈保性能控制谢成祥;陈建平;胡维礼【摘要】研究了一类具有不确定时延的网络控制系统输出反馈保性能控制问题.将时延的不确定性建模为系统状态方程系数矩阵的不确定性,在输出反馈条件下,用状态观测器重构系统状态,将保性能控制问题转化为不确定离散系统的输出反馈鲁棒保性能控制问题.利用Lyapunov理论和矩阵不等式方法,得出了输出反馈保性能控制律的设计方法.仿真算例说明了设计方法的有效性.【期刊名称】《江苏科技大学学报（自然科学版）》【年(卷),期】2008(022)005【总页数】6页(P53-58)【关键词】网络控制系统;保性能控制;矩阵不等式;状态观测器;不确定时延【作者】谢成祥;陈建平;胡维礼【作者单位】南京理工大学,自动化学院,江苏,南京,210094;江苏科技大学,电子信息学院,江苏,镇江,212003;华中光电技术研究所,湖北,武汉,430074;南京理工大学,自动化学院,江苏,南京,210094【正文语种】中文【中图分类】TP2730 引言网络控制系统NCSs(Networked Control Systems)是涉及通信技术、计算机技术和控制技术的复杂系统,其分析和实现受到网络环境等因素的制约.网络诱导时延是NCS区别于传统控制系统的主要特征之一,不同的控制网络有不同的时延特性,可以是定常的、随机的、不确定的.时延的存在会使系统的性能下降,甚至使系统不稳定,因此是NCS研究中首先要考虑的核心问题.针对NCS中普遍存在的网络诱导时延,文献[1]采用互联网络拓朴结构的描述语言TOD(Topology Description),给出了保持系统稳定的时延范围,但保守性较大;文献[2]采用随机控制方法来设计控制器;考虑扰动的影响,文献[3]基于离散切换系统方法,对系统进行了稳定性分析和扰动衰减分析,这些方法都要求能够检测系统的全部状态.有些情况下,往往只能检测到被控对象的部分信息,难以实现全状态反馈,此时,可采用动态输出反馈控制器或状态观测器来解决这一问题.如文献[4]采用随机控制理论设计了NCS的随机输出反馈控制器,但要求事先知道网络诱导时延的分布规律. 在进行控制系统设计时,不仅要保证系统稳定,而且希望系统能够满足一定的性能指标要求,因此系统的保性能控制研究具有重要的意义.网络控制系统的保性能控制近年来已有研究.文献[5]在文献[6]数学模型的基础上,基于线性矩阵不等式LMIs(Linear Matrix Inequalities)的可行解给出了状态反馈网络控制系统的保性能控制律的设计方法;文献[7]基于动态输出反馈控制,设计了不确定时延网络控制系统的输出反馈保性能控制器.但文献[6]中给出的不确定时延网络控制系统的数学模型,其标称形式是不可控的,因而文献[5,7]中围绕该标称模型所得出的LMI将没有可行解,给控制器的设计带来了困难;文献[8]针对文献[6]中数学模型的不足提出了改进方案,并给出了状态反馈情况下的保性能控制存在的条件和保性能控制律的设计方法,但没有研究输出反馈情况下的保性能控制问题.本文针对一类不确定时延网络控制系统,通过选择合适的标称模型,将时延的不确定建模为范数有界的系统矩阵的不确定性,然后,在不能检测系统全部状态的情况下,构造状态观测器重构系统状态,基于Lyapunov理论和矩阵不等式方法研究了输出反馈网络控制系统的保性能控制问题,给出了保性能控制律存在的条件和保性能控制律的设计方法.1 网络控制系统的数学模型图1 网络控制系统结构Fig.1 Structure of networked control system图2 网络控制系统的信号时序Fig.2 Signal timing of NCS典型的网络控制系统结构如图1所示.图中分别表示被控对象的状态及其在控制器接收端的镜像；分别表示控制量及其在执行器接收端的镜像.由于网络的引入,信号的传输存在时延,用分别表示传感器到控制器和控制器到执行器的网络诱导时延. 对于图1 所示的网络控制系统作如下假设:1) 传感器节点由时间驱动,以固定的周期T(T> 0) 对被控对象采样, 并将数据(被控对象的状态量) 存放在单个数据包中发送到网络;2) 控制器节点为事件驱动, 采样数据到达时刻,计算控制量并输出;3) 执行器节点也为事件驱动,控制量到达时刻,执行相应的动作;4) 网络传输存在不确定时延,不考虑数据包丢失,控制回路总的时延且0≤τk≤T.大多数专门为控制设计的网络如CAN,满足以上假设.在上述假设下,控制系统中各信号的时序如图2所示,图中tk表示第k个采样时刻.考虑线性定常被控对象由以下状态方程描述y(t)=Cx(t)( 1 )其中x(t)∈Rn为对象状态,(t)∈Rm为对象输入,y(t)∈Rp为对象输出,A,B,C为适维矩阵.考虑网络诱导时延的影响,对应于图2中的信号时序,有所以包含网络的广义对象的离散数学模型可表示为xk+1=Gxk+Γ0(k)u(k)+Γ1(k)u(k-1), yk=Cxk( 2 )式中G=eAT,Γ0(k)=eAtBdt,Γ1(k)=eAtBdt.显然,Γ0(k)、Γ1(k)是时变的并且有Γ0(k)+Γ1(k)=H=eAtBdt因此式(2)可化为xk+1=Gxk+(H-Γ1(k))uk+Γ1(k)uk-1,yk=Cxk( 3 )不失一般性,假设矩阵A有一个为0的特征值，一个r重特征值,其余为互异特征值,即A=Λdiag(0,J1,J2)Λ-1式中J1是由非0互异特征值λ2,…,λn-r组成的对角块,J2是由r重特征值λ*对应的约当块,Λ为矩阵A的特征向量组成的矩阵. 这样,Γ1(k)可以表示为令D=Λdiag(α1,α2,…,αn),其中αn-r+1=…=αn=α*E=Λ-1B式中α1,…,αn-r,α*的选择使得FT(τk)F(τk)≤I成立.于是Γ1(k)=DF(τk)E( 4 )式中D,E均为定常矩阵.为方便起见,以下将F(τk)简记为F.NCS的对象离散模型可以转化为具有时滞的不确定性线性离散对象模型(式(3)).从式(3)可以发现,标称模型即为网络诱导时延为0时的离散模型,因此,只要[A,B]可控且采样周期合适,就能保证系统的可控性.针对NCS线性对象离散对象模型,设计输出反馈控制器,使网络闭环系统对于一定范围内的不确定传输时延鲁棒稳定,并使所选取的性能函数均小于某一上界,即把NCS的保性能控制问题转化为研究时滞的不确定离散系统的鲁棒保性能控制问题.2 NCS的输出反馈保性能控制若系统(3)的状态不能全部检测,可取系统基于状态观测器的输出进行反馈,此时,可以构造状态观测器来重构系统的状态,并利用观测器的状态来构成状态反馈.此时,控制器方程为( 5 )式中L∈Rn为待定的输出反馈增益向量.将式(3)和式(5)联立,可得输出反馈网络控制系统的闭环增广状态方程为zk+1=Mzk( 6 )式中定义1 取性能指标( 7 )对于所有满足式(4)的不确定性,如果输出反馈网络控制系统(式(6))渐近稳定,且系统的性能指标值不超过某个确定常数J*,则称相应的输出反馈控制器(式(5))是该系统的输出反馈保性能控制律,且J*为性能指标的上界.引理1 (Schur补)给定常数矩阵A,P=PT>0和Q=QT,则ATPA+Q<0成立,当且仅当或引理2[9] 设M,N,F为具有适当维数的实矩阵,其中F满足FTF≤I,那么存在常数ε>0,使得NTFTM+MFN≤ε-1NTN+εMMT定理1 对于系统(式(3))和性能指标(式(7)),若存在对称正定矩阵X∈Rn×n,矩阵K∈Rm×n,以及正常数ε,使得对于所有非零的xk和所有允许的不确定性(式(4)),矩阵不等式(式(8))成立,则控制律式(5)是系统(式(3))的一个保性能控制律,且性能指标的上界为( 8 )式中证明:取Lyapunov函数为其中P为对称正定矩阵.令V0为性能指标的上界,为保证上界存在,必须满足( 9 )将不等式(9)从k=0到k=∞叠加,考虑到系统渐近稳定时,V∞=0,可得(10)将式 (6)代入到式(9),可得(11)式中式(11)等效于MTPM-P+Π<0(12)由引理1,式(12)等效于(13)矩阵M中含有不确定项,考虑到式(4),有(14)于是,式(13)可以写成(15)由引理2,要使式(15)成立,只要存在正数ε,使得不等式(16)成立.再由引理1,式(16)等效于(17)又因为Π=ΘTΘ,所以,再次使用引理1,式(17)等效于(18)式(18)左乘、右乘diag(I,P-1,I,I)，并令X=P-1,即可得到式(8).证毕.式(8)是关于矩阵X,K和标量ε的双线性矩阵不等式,L可以根据极点配置方法事先确定.不等式(8)可在MATLAB环境下借助于PENBMI软件包求解[10].这种方法不仅给出了一个保性能控制律, 而且给出了保性能控制律的参数化设计方法.如果矩阵P选择为块对角矩阵的形式,仿照文献[7]中的证明方法,可以得到一个线性矩阵不等式,但该线性矩阵不等式只在被控对象开环稳定(即G的全部特征值都在单位圆内)时才可能有可行解,因此不具有一般性.3 仿真算例考虑如下的不稳定被控对象对象和观测器的初始状态均为取采样周期为Ts=0.5 s,网络诱导时延τk≤0.5 s且是随机不确定的,因A的特征值是0.1,-0.95,为保证取α1=0.512 7,α2=0.398 0,则可以计算出使用定理1,取L=[0.173 2 0.340 2],使得G-LC的极点为0.7,0.8.利用PENBMI求解矩阵不等式(7),得到K=[-1.903 9 -2.200 7],ε=0.006 5,图3 系统状态响应曲线Fig.3 System state response curve系统状态响应曲线如图3所示.4 结论本文首先改进了网络控制系统建模时网络诱导时延的处理方法,在网络诱导时延小于采样周期的条件下得到了网络控制系统的具有不确定性的离散化模型,其标称模型是可控的.在输出反馈的条件下,通过观测器重构系统状态,根据Lyapunov定理,使用矩阵不等式方法设计控制器,得到了网络控制系统的输出保性能控制律的设计方法.仿真算例表明了本文方法的有效性.参考文献(References)[1] Walsh G C,Ye H.Stability of networked control systems[J].IEEE Control Systems Magazine,2001,21(1):57-65.[2] Nilsson J Bernhardsson,Wittenmark B.Stochastic analysis and control of real-time systems with random time delays[J].Automatica,1998,34(1):57-64.[3] Lin H,Zhai G,Antsaklis P.Robust stability and disturbance attenuation analysis of a class of networked control systems[C]∥The 42nd IEEEConference on Decision and Control,2003:1182-1187.[4] 朱其新, 胡寿松. 网络控制系统的随机输出反馈控制[J]. 应用科学学报, 2004, 22(1): 71-75.Zhu Qixin, Hu Shousong. Stochastic output feedback control for networked control systems[J].Journal of Applied Science,2004, 22(1):71-75.(in Chinese)[5] 崔桂梅,穆志纯, 李晓理, 等. 网络控制系统保性能控制[J]. 北京科技大学学报,2006,28(6):595-599.Cui Guimei, Mu Zhichun, Li Xiaoli, et al. Guaranteed cost control for networked control systems[J].Journal of Beijing University of Science and Technology,2006, 28(6):595-599. (in Chinese)[6] 樊卫华, 蔡骅, 陈庆伟, 等. 时延网络控制系统的稳定性[J]. 控制理论与应用,2004,21(6):880-884.Fan Weihua, Cai Hua, Chen Qingwei, et al. Stability of networked control systems with time-delay[J].Control Theory and Application,2004, 21(6):880-884.(in Chinese)[7] 邱占芝,张庆灵,刘明. 不确定时延输出反馈网络化控制保性能控制[J]. 控制理论与应用,2007,24(2):274-278.Qiu Zhanzhi, Zhang Qingling, Liu Ming. Guaranteed performance control for output feedback networked control systems with uncertain time-delay[J].Control Theory and Application,2007, 24(2):274-278.(in Chinese) [8] 王罗莎,谢成祥. 时延网络控制系统的保行能控制研究[J].江苏科技大学学报,2007,21(6A):58-62.Wang Luosha, Xie Chengxiang. Research on guaranteed cost control oftime-delay networked control systems[J].Journal of Jiangsu University of Science and Technology,2007, 21(6A):58-62.(in Chinese)[9] 俞立. 鲁棒控制-线性矩阵不等式处理方法[M]. 北京:清华大学出版社,2002.[10] Henrion D, Lofberg J,Kocvara M, et al. Solving polynomial static output feedback problems with PENBMI[C]∥Proceedings of 44th IEEE Conference on CDC-ECC′05, 2005:7581-7586.。
基于马尔科夫链的网络控制系统调度
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用单速率调度ＲＭ方法，在单速率调度的有效性和网络控制 系统的稳定性约束下提出采样周期的优化调度方法，但这种 方法只能应用于有优先权分配功能的网络，如ＣＡＮ， ＤｅｖｉｃｅＮｅｔ等，且要求每个子控制系统的传输时延和最大允许 时延已知。文献【２Ｊ针对单ＣＰＵ多节点的连接方式，提出用
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欺骗攻击下网络化系统事件触发安全控制
Computer Engineering and Applications 计算机工程与应用2021，57（5）网络控制系统是将通信网络引入控制闭环的复杂分布式控制系统，其空间分布的传感器、控制器和执行器等系统组件经由共享通信网络传递信息。
该系统高度融合物理系统与信息系统，具有柔性高、成本低、安装和维护方便等优点[1]，广泛应用于智能电网、智慧交通、精准农业等领域。
在共享通信网络为网络控制系统带来诸多便利的同时，来自网络空间的恶意攻击使其面临重大的安全挑战。
针对网络控制系统的攻击大概分为拒绝服务攻击和欺骗攻击，拒绝服务攻击通过阻塞通信网络使有用数据包不能按时送达，欺骗攻击通过篡改数据包内容而破坏数据包的真实性和完整性[2]，虚假数据注入（False Data Injection ，FDI ）攻击为欺骗攻击的典型代表。
近年来，考虑网络攻击影响的安全控制系统研究引起了广泛关注，其中多数成果研究了拒绝服务攻击对网络控制系统的影响，仅有少量研究对FDI 攻击下的安全控制策略进行了探索。
例如：考虑传感器至控制器通道中具有伯努利分布的FDI 攻击影响，文献[3]设计了分布式脉冲控制器，保证了多智能体系统的均方有界一致性。
考虑文献[3]中随机FDI 攻击影响，文献[4]研究了基于采样数据的非线性多智能体系统均方一致性。
考虑传感器至控制器以及控制器至执行器通道中均存在欺骗攻击下网络化系统事件触发安全控制李富强1，2，郜丽赛2，郑宝周1，谷小青11.河南农业大学理学院，郑州4500022.上海大学机电工程与自动化学院，上海市电站自动化技术重点实验室，上海200444摘要：针对网络控制系统受欺骗攻击问题，提出了事件触发器与安全控制器协同设计策略。
提出了离散事件触发机制，克服了连续事件触发机制需要新增专用硬件且需要复杂计算避免芝诺现象局限。
建立了有机融合欺骗攻击、事件触发机制、网络诱导延时及外部扰动多约束参数的闭环系统时滞模型，推导出了欺骗攻击下事件触发控制系统渐近稳定的充分条件，得到了事件触发器与安全控制器的协同设计条件。
网络控制系统稳定性分析
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图 1.1
分布式控制系统结构示意图
最早出现的 DCS 系统是 Honeywell 公司 1975 年开发的 TDC-2000 系统。在 DCS 系统中，现地工作站、操作员站以及一些其它辅助工作站等功能不同的计算机通过通 信网络实现互连，共同分担系统的工作。其中现地工作站主要用于过程控制，操作员 站用于操作员监视过程控制的情况，其它辅助工作站主要用于数据记录、系统优化等 功能。然而，在 DCS 系统中各个控制单元之间只是一种“松散”的连接关系，这是因 为绝大部分基本的控制任务(传感器采集、控制律计算、控制输出执行等)均在各个现 地工作站上独立完成，用于控制的实时信号并未经过网络进行传输。在 DCS 系统中， 仅仅只有一些简单的开关信号、报警信息、监视信息等通过网络共享。 上世纪 90 年代以来，微芯片的发展日新月异，其价格也逐日降低，同时网络技 术在人们的生产、生活中广为使用，这一切都为发展网络化的控制系统提供了良好的 契机。时至今天，基于现场总线等控制网络的网络控制系统(Networked Control Systems 简称 NCSs)已在各行各业得到了广泛的应用。网络控制系统的典型结构与信息流一般 如下图所示(其中虚线表示实时控制信息的传输)：
Keywords:
NCSs, Switched system, Robust stability, LMI, Networked induced time-delay, Packets dropping.
II
独创性声明
本人声明所呈交的学位论文是我个人在导师指导下进行的研究工作及取得的研 究成果。尽我所知，除文中已经标明引用的内容外，本论文不包含任何其他个人或 集体已经发表或撰写过的研究成果。对本文的研究做出贡献的个人和集体，均已在 文中以明确方式标明。本人完全意识到，本声明的法律结果由本人承担。
网络控制系统的现状及发展前景
网络控制系统的现状及发展前景摘要网络控制系统领域的主要研究成果和将来可能遇到的挑战进行阐述。
网路控制系统主要分为网络控制、基于网络的控制和多智能体系统3个研究领域。
本文主要针对基于网络的控制系统中的时延、丢包、多包传输问题,介绍近年来网络控制系统分析与综合方面的研究成果和最新进展。
最后，论述了网络控制系统研究中尚待解决的问题，展望了网络控制系统未来的发展前景。
关键词：网络控制系统；网络时延；网络丢包；单包传输；多包传输Networked control systems: the current situation andthe prospect in futureZhao Yingwei Zhang Donglai(Harbin Institute of Technology Shenzhen Graduate School Shenzhen 518055) Abstract The paper reviews the major contributions and the possible future challenges in the emerging area of Networked Control Systems (NCSs). Activities in this field can be categorized as the control of networks, the control over networks, and the multi-agent system. Focusing on network delays, network packet dropout, and multiple-packet transmission in the area of control over networks, this paper reviews the research achievements and advances of analysis and synthesis in NCS. The pro b lems in NC S and the future d irectio ns in NCS are po inted o ut. Keywords networked control systems; network delay; network packet dropout; single-packet transmission; multiple-packet transmission1 引言随着计算机技术、网络通信技术和控制科学的日益发展与交叉渗透，控制系统的结构变得越来越复杂，空间分布越来越广，对系统控制性能的要求也越来越高。
网络环境下线性切换系统的事件触发及自触发控制
摘要摘要近年来，切换系统广泛应用于电力系统、机器人系统、交通系统等实际控制系统的建模与研究。
作为一类不可忽视的混杂系统，切换系统包括一系列连续或离散的子系统以及协调这些子系统如何切换的切换策略。
另一方面，随着网络控制系统的发展，不可避免地出现网络拥堵、带宽资源受限等问题，在这种背景下，事件触发控制应运而生。
系统根据触发条件来决定是否进行状态或输出的采样以及控制量的更新，这就在很大程度上减少了网络带宽的占用，从而节约了网络资源。
本文主要研究了几类切换系统的事件触发及自触发控制的相关问题，包含以下内容：首先，针对传感器和控制器分离的网络化切换系统，对传感器—控制器信道和控制器—执行器信道提出了双信道的异步动态事件触发机制。
与此同时，为系统设计了动态输出反馈控制器，使得闭环系统可以达到渐近稳定。
紧接着，给出了相应的自触发条件，并以F-18飞行器为例进行了仿真验证。
其次，针对同时包含连续和离散子系统的切换系统，设计了周期事件触发条件，结合基于模态的平均停留时间技术给出了保证闭环切换系统指数稳定的相应判据。
周期事件触发结合了周期触发和事件触发的优点，自然地避免了Zeno现象的发生。
最后，针对包含部分稳定子系统和部分不稳定子系统的切换系统，依据基本时间单元思想设计了一种事件触发机制，保证下一次的触发时刻在上一次触发的一个基本时间单元之后。
采用基于时间调度的Lyapunov函数方法和基于模态的平均停留时间方法给出了能够使系统指数稳定的充分条件。
关键词：切换系统事件触发控制自触发控制平均停留时间渐近稳定ABSTRACTIn recent years,switched system has been widely applied in modeling and research of power system,robot system,traffic system and other practical control systems.As a class of important hybrid systems,switched systems include a series of continuous-time or discrete-time subsystems and a rule which orchestrates the switches among them.On the other hand,with the development of networked control system,network congestion and bandwidth limitation occur inevitably.In such a case,event-triggered control emerges and it can largely reduce the occupation of bandwidth and save net-work resources.The related problems of event-triggered and self-triggered control for switched systems are investigated in this thesis.The main contents are listed below: First,for switched systems where the sensor and controller are not co-located, a dual channel asynchronous dynamic event-triggered mechanism is designed for the sensor-controller channel and the controller-actuator channel.At the same time,the dynamic output feedback controller is designed so that the closed-loop system can be asymptotically stable.Then,the corresponding self-triggered conditions are put for-ward.The simulation verification is carried out with the example of F-18aircraft.Second,periodic event-triggered condition is designed for switched systems which are composed of both continuous-time and discrete-time subsystems.Based on the mode-dependent average dwell time,a criterion is presented to ensure the exponential stability of switched systems.Periodic event-triggered control,which combines the advantages of periodic control and event-triggered control,can avoid Zeno behavior.Finally,for switched systems with partial unstable subsystem,we design an event-triggered mechanism involving elementary time unit.This ensures the next triggering instant is followed by a unit of latest triggering instant.The sufficient conditions for the exponential stability of the system are given according to the time-scheduled Lyapunov function and the mode-dependent average dwell time approach.KEY WORDS：Switched Systems，Event-Triggered Control，Self-Triggered Control，Average Dwell Time，Asymptotic Stability目录摘要 (I)ABSTRACT (III)目录 (V)符号表 (VII)第1章绪论 (1)1.1研究背景与意义 (1)1.2基础知识简介 (3)1.2.1切换系统 (3)1.2.2事件触发及自触发控制 (7)1.2.3Zeno现象 (8)1.2.4相关引理 (9)1.3文章的主要工作和结构安排 (9)第2章切换系统的双信道事件触发及自触发控制 (11)2.1系统模型与问题描述 (12)2.2主要结果及证明 (15)2.3数值仿真及分析 (20)2.4本章小结 (25)第3章包含连续/离散子系统的切换系统事件触发及自触发控制 (27)3.1系统模型与问题描述 (27)3.2主要结果及证明 (30)3.3数值仿真及分析 (35)3.4本章小结 (36)第4章包含部分不稳定子系统的切换系统事件触发及自触发控制 (39)4.1系统模型与问题描述 (40)4.2主要结果及证明 (42)V天津大学硕士学位论文4.3数值仿真及分析 (48)4.4本章小结 (51)第5章总结与展望 (53)5.1总结 (53)5.2展望 (54)参考文献 (55)发表论文和参加科研情况说明 (59)致谢 (61)VI符号表符号表N：正整数集R+：正实数集R n：n维实数空间R m×n：m×n维实矩阵空间I/0：适当维数的单位矩阵/零矩阵A T：矩阵A的转置A−1：矩阵A的逆λmax(A)：矩阵A的最大特征值λmin(A)：矩阵A的最小特征值A>0：矩阵A是正定的A≥0：矩阵A是半正定的∗：矩阵中的对称块∥·∥：欧几里德范数K={1,...,N}：1至N的正整数集合He[A] A+A Tx(t+k )=lim t→t+kx(t)x(t−k )=lim t→t−kx(t)第1章绪论第1章绪论人类正处在一个飞速发展的时代，社会进步、科技发展日新月异，社会生产力的提高也对控制理论的发展提出了新的要求与挑战。
网络控制系统PPT课件
第十五章 网络控制系统
• 15.1 网络控制系统概述 • 21世纪是一个网络化的时代，网络的普遍性决定了其在生活中的广泛应用。对网络系统的研究最早始于20
在网络控制系统中，模拟信号在通过网络信道传输之前必须要量化后才能传输，因此信号量化在网络控制 系统中有着广泛的应用。通过量化器的作用，使得系统实时输出信号被转化为有限长度的数字信号，因而 可能会影响系统的控制精度和稳定性。量化器按量化方式分主要有两大类；均匀量化器和非均匀量化器。 • 15.5.3基于状态观测的量化反馈稳定性分析 • 考虑到网络控制系统中状态不一定能测量到，因此分析带有状态观测器的量化反馈控制稳定性具有现实 意义。假设带有状态观测器的量化反馈网络控制系统如图所示，其中被控对象的状态空间模型为
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第十五章 网络控制系统
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• 量化反馈控制系统结构框图
• 15.6 网络控制系统控制器设计
• 15.6.1 控制器设计方法
• 针对网络诱导的时延，有不少补偿控制设计方法，其中延迟整形技术是其中一种。 所谓延迟整形是指将网络上时变的时延通过“整形器”转化为固定的时延。系统 时延经过整形后，控制律的设计问题就转化为一般的采样数据控制问题。延迟整 形方法是研究可变时延条件下NCSs稳定性问题的一种简便方法。
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自动化专业英语教程(王宏文)第二版全文翻译
《自动化专业英语教程》-王宏文-全文翻译PART 1Electrical and Electronic Engineering BasicsUNIT 1A Electrical Networks ————————————3B Three-phase CircuitsUNIT 2A The Operational Amplifier ———————————5B TransistorsUNIT 3A Logical Variables and Flip-flop ——————————8B Binary Number SystemUNIT 4A Power Semiconductor Devices ——————————11B Power Electronic ConvertersUNIT 5A Types of DC Motors —————————————15B Closed-loop Control of DC DriversUNIT 6A AC Machines ———————————————19B Induction Motor DriveUNIT 7A Electric Power System ————————————22B Power System AutomationPART 2Control TheoryUNIT 1A The World of Control ————————————27B The Transfer Function and the Laplace Transformation —————29 UNIT 2A Stability and the Time Response —————————30B Steady State—————————————————31 UNIT 3A The Root Locus —————————————32B The Frequency Response Methods: Nyquist Diagrams —————33 UNIT 4A The Frequency Response Methods: Bode Piots —————34B Nonlinear Control System 37UNIT 5 A Introduction to Modern Control Theory 38B State Equations 40UNIT 6 A Controllability, Observability, and StabilityB Optimum Control SystemsUNIT 7 A Conventional and Intelligent ControlB Artificial Neural NetworkPART 3 Computer Control TechnologyUNIT 1 A Computer Structure and Function 42B Fundamentals of Computer and Networks 43UNIT 2 A Interfaces to External Signals and Devices 44B The Applications of Computers 46UNIT 3 A PLC OverviewB PACs for Industrial Control, the Future of ControlUNIT 4 A Fundamentals of Single-chip Microcomputer 49B Understanding DSP and Its UsesUNIT 5 A A First Look at Embedded SystemsB Embedded Systems DesignPART 4 Process ControlUNIT 1 A A Process Control System 50B Fundamentals of Process Control 52UNIT 2 A Sensors and Transmitters 53B Final Control Elements and ControllersUNIT 3 A P Controllers and PI ControllersB PID Controllers and Other ControllersUNIT 4 A Indicating InstrumentsB Control PanelsPART 5 Control Based on Network and InformationUNIT 1 A Automation Networking Application AreasB Evolution of Control System ArchitectureUNIT 2 A Fundamental Issues in Networked Control SystemsB Stability of NCSs with Network-induced DelayUNIT 3 A Fundamentals of the Database SystemB Virtual Manufacturing—A Growing Trend in AutomationUNIT 4 A Concepts of Computer Integrated ManufacturingB Enterprise Resources Planning and BeyondPART 6 Synthetic Applications of Automatic TechnologyUNIT 1 A Recent Advances and Future Trends in Electrical Machine DriversB System Evolution in Intelligent BuildingsUNIT 2 A Industrial RobotB A General Introduction to Pattern RecognitionUNIT 3 A Renewable EnergyB Electric VehiclesUNIT 1A 电路电路或电网络由以某种方式连接的电阻器、电感器和电容器等元件组成。
时延离散网络系统的均方指数稳定控制
时延离散网络系统的均方指数稳定控制姚合军【摘要】针对带有随机时延和数据包丢失的不确定离散网络系统,得到了系统的均方指数稳定控制器设计策略.通过把网络诱导时延和数据包丢失看作满足Bernoulli 分布的等价时延,并结合等价时延在不同区间上的概率取值,建立了更加切合实际的网络系统数学模型,并在此基础上,结合Lyapunov稳定性理论,对网络系统的均方指数稳定性进行分析,同时给出了输出反馈鲁棒控制器设计方法.仿真算例说明了该方法的有效性.【期刊名称】《科学技术与工程》【年(卷),期】2019(019)011【总页数】5页(P178-182)【关键词】网络控制系统;时延;指数均方稳定;随机;数据丢失【作者】姚合军【作者单位】安阳师范学院数学与统计学院,安阳455000【正文语种】中文【中图分类】TP273网络系统是指由通信网络连接传感器，执行器和控制器的控制系统[1]。
与传统的点对点控制系统相比较,网络控制系统具有连线少,高性能,低成本,易于维护等优点。
因此,近二十年来，对网络控制系统的研究成果不断出现，网络控制系统已经成为控制界研究的重要分支之一[2—6]。
由于通信信号是通过网络进行传输，因此不可避免地会在系统中存在时延，数据包丢失，网络拥塞等现象，从而使得对网络系统的研究难度不断加大[7—12]。
信息在传感器、执行器和控制器之间传输过程中,以及控制器中的计算过程中不可避免地会出现网络诱导时延。
众所周知时延的存在常常会使系统性能变差,甚至不稳定[13—16]。
另一个区别于传统点对点系统的是由于不确定性和外部干扰的影响,网络系统中常常会存在数据丢失现象。
为分析网络诱导时延和丢包对系统性能影响,高会军给出了一个网络控制系统镇定的新方法,并给出了使闭环系统稳定的控制器设计方法[17]。
Huang通过分析随机时延与数据包丢失对系统的影响，建立了随机网络控制系统的数学模型，并通过设计系统的状态观测器，给出了系统的动态输出反馈控制器设计方法[18]。
孙富春简介pdf
孙富春简介pdf孙富春简历孙富春，清华大学计算机科学与技术系教授，博士生导师，国家863计划专家组成员，国家自然基金委重大研究计划“视听觉信息的认知计算”指导专家组成员，计算机科学与技术系学术委员会副主任, 智能技术与系统国家重点实验室常务副主任; 兼任国际刊物《IEEE Trans. on Fuzzy Systems》，《Mechatronics》和《International Journal of Control, Automation, and Systems》副主编或大区主编，《International Journal of Computational Intelligence Systems》和《Robotics and Autonumous Systems》编委；兼任国内刊物《中国科学F：信息科学》和《自动化学报》编委；兼任中国人工智能学会认知系统与信息处理专业委员会主任，IEEE CSS智能控制技术委员会委员。
98年3月在清华大学计算机应用专业获博士学位。
98年1月至2000年1月在清华大学自动化系从事博士后研究，2000年至今在计算机科学与技术系工作。
工作期间获得的主要奖励有：2000年全国优秀博士论文奖，2001年国家863计划十五年先进个人，2002年清华大学“学术新人奖”，2003年韩国第十八届Choon-Gang 国际学术奖一等奖第一名，2004年教育部新世纪人才奖，2005年清华大学校先进个人，2006年国家杰出青年基金。
获奖成果5项，两项分别获2010年教育部自然科学奖二等奖（排名第一）和2004年度北京市科学技术奖（理论类）二等奖（排名第一）、一项获2002年度教育部提名国家科技进步二等奖（排名第二）、三项获省部级科技进步三等奖。
译书一部，专著两部，在国内外重要刊物发表或录用论文150余篇，其中在IEE、IEEE汇刊、Automatica等国际重要刊物发表论文90余篇，80余篇论文收入SCI，SCI期刊他人引用700余次，200多篇论文收入EI，有两篇论文曾被评为国内二级学会的最佳优秀论文奖。
Stability robustness of networked control systems with respect to packet loss
Automatica43(2007)1243–1248/locate/automaticaBrief paperStability robustness of networked control systemswith respect to packet lossଁShawn Hu,Wei-Yong Yan∗Department of Electrical and Computer Engineering,Curtin University of Technology,GPO Box U1987,Perth6845,AustraliaReceived16March2005;received in revised form14August2006;accepted18December2006AbstractThis paper is concerned with stability analysis of discrete-time networked control systems over a communication channel subject to packet loss whose behavior is modeled by an i.i.d Bernoulli process with a packet dropping probability bounded by a constant.A necessary and sufficient condition for stability is obtained.A packet dropping margin is introduced as a measure of stability robustness of a system against packet dropping,and a formula for it is derived.A design method is proposed for achieving a large margin subject to a constraint that the system has a set of prescribed nominal closed-loop poles.᭧2007Elsevier Ltd.All rights reserved.Keywords:Networked control systems;Packets;Stability robustness;Probability;Pole placement1.IntroductionNetworked control systems(NCSs)are systems with a feedback loop closed via a real-time shared media network. The study of such systems has received increasing attention in recent years because of ubiquity of the Internet(Chow& Tipsuwan,2001).The advantages of NCSs over conventional or hardwired control include reduction of system wiring and ease of installation.On the other hand,such factors as bandwidth constraints,packet delays,and packet dropping often affect the performance of an NCS or even cause instability.There has been some work on the effects of these factors.For example, in Tatikonda and Mitter(2004),Tatikonda,Sahai,and Mitter (2004),Liu,Elia,and Tatikonda(2004),control problems were formulated and studied for an NCS subject to bandwidth con-straints.A stability analysis of NCSs with time delay was pre-sented in Zhang,Branicky,and Phillips(2001)and a method to obtain a maximum allowable delay bound was provided in ଁThis paper was not presented at any IFAC meeting.This paper was recommended for publication in revised form by Associate Editor Ioannis Paschalidis under the direction of Editor Ian Petersen.∗Corresponding author.Tel.:+61892667931;fax:+61892662584.E-mail addresses:xiaolin.hu@.au(S.Hu),w.yan@.au(W.-Y.Yan).0005-1098/$-see front matter᭧2007Elsevier Ltd.All rights reserved. doi:10.1016/j.automatica.2006.12.020Kim,Lee,Kwon,and Park(2003).LQG controller design prob-lems of NCSs with delay were studied in Nilsson,Bernhardsson, and Wittenmark(1998),and Hu and Zhu(2003).Stability of NCSs with a communication network subject to packet dropping has received much attention as well.In the continuous-time case,an NCS was modeled as an asynchronous dynamical system with rate constraints on events(Zhang et al., 2001),where the network was treated as a switch that closes at a certain rate corresponding to a packet dropping probability (PDP).It was shown there that a sufficiently fast sampling rate can guarantee stability of the system.In Montestruque and Antsaklis(2003,2004),the authors considered stability of a model-based NCS i.e.an NCS with an additional model used for estimating the plant state between transmission times and generating a control signal.A stability condition was obtained for the Try-Once-Discard network protocol in Walsh,Ye,and Bushnell(1999).In the discrete-time case,the packet dropping process of a communication network is usually modeled as either an i.i.d Bernoulli process or a Markov chain,which is more general. With such a characterization of the packet dropping process, an NCS can be viewed as a special jump system.The mean-square stability of jump systems has been well studied,see e.g.Ji,Chizeck,Feng,and Loparo(1991),Feng,Loparo,Ji,1244S.Hu,W.-Y.Yan/Automatica43(2007)1243–1248and Chizeck(1992),Costa and Fragoso(1993),Ghaouli and Aitrami(1996),and Fang and Loparo(2002).As a conse-quence,stability results in Ji et al.(1991)and Costa and Fragoso (1993)naturally lead to a necessary and sufficient condition for stability of an NCS.Expressed in terms of a set of linear matrix inequalities,this condition has been used to check the stability of an NCS when the controller is constructed using the LQG method(Azimi-Sadjadi,2003).It has also been used for designing an H∞controller in Seiler and Sengupta(2005).An Markov chain for describing the packet dropping process has been considered for other purposes in e.g.Ling and Lemmon (2003b),Zhang,Ding,Frank,and Sader(2004),and Gupta, Spanos,Hassibi,and Murray(2005).More specifically,a for-mula for the power spectral density was presented in Ling and Lemmon(2003b),the problem of designing an observer-based residual generator was discussed in Zhang et al.(2004),and the problem of designing a decoder and an encoder for optimal LQG control was considered in Gupta et al.(2005).When the packet dropping process is modeled by an i.i.d process with a given PDP,a necessary and sufficient condition for mean-square stability of NCSs can be found in Ling and Lemmon (2003a).In the one-dimensional case,a stabilization scheme was provided in Hadjicostis and Touri(2002).In this paper,by modeling the packet dropping of a com-munication network as an i.i.d Bernoulli process,we carry on stability analysis of a discrete-time NCS with static state feed-back where the PDP of the communication network is bounded by a known upper bound.It can easily be argued that it makes more sense to assume the availability of an upper bound of the PDP than the exact value of the PDP,which is assumed to be known in the literature such as Azimi-Sadjadi(2003)and Ling and Lemmon(2003a).Another aim of the paper is to address stability robustness of the system with respect to packet drop-ping by introducing a quantity termed a packet dropping mar-gin(PDM),which is defined to be the maximum PDP that an NCS can tolerate before becoming unstable.A formula for the margin is derived in the paper as well.Finally,we discuss how a controller can be designed to achieve a large margin subject to the constraint that the resulting NCS has certain guaranteed transient performance.A design method based on a robust pole placement technique is proposed.The paper is organized as follows.In Section2,an NCS is described.Section3presents a stability analysis of the NCS. Section4deals with the problem of designing a state feedback law to achieve a suboptimal PDM subject to performance spec-ifications on nominal closed-loop poles.Conclusions are given in Section5.2.System descriptionA typical NCS as depicted in Fig.1consists of three com-ponents:a nominal plant to be controlled,a network such as the Internet,and a controller.In this paper,it is assumed that the nominal plant is described byx k+1=Ax k+B1u k+B2w k,(2.1)Fig.1.A networked control system.where x k∈R n is the state,u k∈R m is the control input,w k∈R r is an external input,and A,B1,B2are constant matrices with appropriate dimensions.We assume that the full state of the plant is transmitted to the controller in the form of packets through the network.The network is assumed to be modeled byˆx k= k x k,(2.2) where k is an i.i.d Bernoulli random process.At any time in-stant k, k has two possible values0and1.The value0indicates that the state vector x k is completely lost during transmission while the value1indicates that x k is successfully transmitted. The probability that k=0,commonly termed a PDP,is invari-ant with time k and measures how reliable a network is.Though it is a standard assumption in the literature that the PDP of an NCS is known,this assumption may be too strong due to the following observations:1.The complexity of the network environment often makes itdifficult to determine the exact PDP associated with an NCS.2.For an NCS with n 2,even if it is stable for a given PDP,the system may become unstable when the PDP reduces due to the network improvement,which will be shown by an example in Section3.As such,in this paper we assume the availability of an upper bound on the PDP instead of the exact PDP.Moreover,it is clear that any packet dropping occurring at the plant input can be treated as the packet dropping occurring at the plant output as far as the closed-loop system is canceled. Therefore,it suffices to consider the case where packet dropping happens only at the plant output.Unlike a conventional controller,a controller in an NCS re-ceives plant information as input from a network which may be shared by any other type of devices.In this paper,the controller in Fig.1is assumed to be of the state feedback formu k=Kˆx k,(2.3) whereˆx k is the received state vector at time instant k and K∈R m×n is a feedback gain.Because of the assumption on the network,it is seen that the received state vectorˆx k is either equal to the transmitted state vector x k or zero.It is convenient to refer to the combination of the nominal plant and the network in Fig.1as a networked plant,and the eigenvalues of A+B1K in(2.1)–(2.3)as the nominal closed-loop poles of the NCS.In the literature,the stability of an NCS is commonly defined as follows,see e.g.Feng et al.(1992),S.Hu,W.-Y.Yan/Automatica43(2007)1243–12481245Ghaouli and Aitrami(1996),Ling and Lemmon(2003a),and Montestruque and Antsaklis(2003).Definition1.The networked plant(2.1)–(2.2)is said to be ms-stabilized(stabilized in the mean-square sense)by the controller (2.3)if there holds lim k→∞E{ x k 2}=0for any initial state x0∈R n and w k=0.The NCS(2.1)–(2.3)is said to be nominallystable if it is ms-stable for the PDP equal to0.3.Stability analysisThe main purpose of this section is to derive a necessary and sufficient condition for an NCS to be stable in the case where only an upper bound of the PDP is known.The condition will lead to the introduction of a new concept of PDM,which can be used to quantitatively measure the degree of stability robustness of the system with respect to packet dropping.First we need the following lemma which gives two equiva-lent necessary and sufficient conditions for stability of an NCS in the case of a known PDP.Thefirst condition was given with-out proof in Ling and Lemmon(2003a)and the second condi-tion has not been obtained before.Lemma2.The networked plant(2.1)–(2.2)with the PDP equal to a constant is ms-stabilized by(2.3)if and only if either of the following two conditions holds:(i)[ A⊗A+(1− )A K⊗A K]<1,(3.1) where⊗denotes a Kronecker product and (·)is the spec-tral radius of a matrix,andA K=A+B1K.(3.2) (ii)For any given symmetric positive definite matrix Q,there isa unique symmetric positive definite solution to the equationP= A T P A+(1− )A T K P A K+Q.(3.3)Proof.Due to the paper length limitation,we only prove the condition in(i)as the other condition in(ii)can be derived fromfirst principles.First note that when w k=0,the state covariance k+1can be expressed ask+1=E[x k+1x T k+1]=A E[x k x T k]A T+A E[x k x T k]E[ k]K T B T1+B1K E[ k]E[x k x T k]A T+B1K E[ 2k]E[x k x T k]K T B T1(3.4)i.e.k+1= A k A T+(1− )(A+BK1) k(A+BK1)T(3.5)due toE[ k]=E[ 2k]=1− .With the identitycs(MY N)=(N T⊗M)cs(Y)one obtainscs( k)=H k cs( 0),(3.6) whereH= A⊗A+(1− )A K⊗A Kand cs(·)is the column vector formed by the columns of a matrix.Thus,it follows from(3.1)that cs( k)converges to zero as k tends to infinity.By definition,the system(2.1)–(2.3) is stable.To prove the necessity of(3.1),let us assume that the system (2.1)–(2.3)is stable.Then it is easy to show that the state covariance matrix will converge to zero for any complex initial state,which implieslimk→∞H k cs( 0)=0(3.7)for any semi-positive Hermitian matrix 0.Since a Hermitian matrix can be expressed as the difference between two semi-positive Hermitian matrices,Eq.(3.7)holds for any Hermitian matrix 0as well.Furthermore,the equation remains true even for any complex matrix 0due to the decomposition0= 1−j 2,where both 1and 2are Hermitian.Therefore,it follows that H k converges to zero,which is equivalent to (H)<1.Remark3.Based on the above lemma,the following obser-vations can be made:•The NCS is nominally stable if and only if the nominal closed-loop poles of the NCS are strictly inside the unit circle.•B2has no effect on the stability of the NCS(2.1)–(2.3).•In the one-dimensional case,the stability condition(3.1)can be simplified as| (A2−A2K)+A2K|<1.(3.8)It should be pointed out that the stability of the system for a given PDP does not necessarily imply the stability for any smaller PDP,which may result when the network becomes more reliable.This can be seen by considering the NCS withA=0.5−10−1,B1=1002,K=−0.21−10.9.This system is ms-stable both for the PDP equal to0and for the PDP equal to0.8,but it is unstable when the PDP equals0.4. As such,it is important to address the following two questions. Under what condition is the system ms-stable for any PDP less than a given bound?What is the largest PDP bound?To answer these questions,let us introduce what will be called the PDM, which can serve as a measure of stability robustness of an NCS with respect to packet dropping.1246S.Hu,W.-Y.Yan /Automatica 43(2007)1243–1248Definition 4.The PDM of the NCS (2.1)–(2.3)is the largest positive bound such that the system is ms-stable for any PDP less than .Obviously,the NCS has to be nominally stable for the PDM to exist.In the one-dimensional case,the following formula for the PDM can be easily derived from (3.8):PDM =⎧⎨⎩1−A 2K A 2−A 2K when |A |>|A K |,∞when |A | |A K |.(3.9)To obtain a formula for the PDM in the general case,the fol-lowing lemma is needed.Lemma 5.Given two constant matrices M 1,M 2∈R n ×n with(M 1)<1,there holdssup { >0; (M 1+ M 2)<1,∀ ∈(0, )}=1 (U),(3.10)where (·)is the largest positive eigenvalue of a matrix andU =(M 1⊗M 2+M 2⊗M 1)(I −M 1⊗M 1)−1M 2⊗M 2(I −M 1⊗M 1)−10.(3.11)Proof.See the Appendix.Theorem 6.If the NCS (2.1)–(2.3)is nominally stable ,its PDMis given by PDM =1,(3.12)whereV =(S ⊗ˆS +ˆS ⊗S)(I −S ⊗S)−1ˆS ⊗ˆS (I −S ⊗S)−10,S =A K ⊗A K ,ˆS=A ⊗A −A K ⊗A K .(3.13)Proof.Notice that A ⊗A +(1− )A K ⊗A K =S + ˆS.Moreover, (S)<1if and only if (A K )<1.Thus,the theorem follows from Lemma 5.Remark 7.It is clear that the NCS (2.1)–(2.3)is ms-stable for any PDP less than if and only if PDM.4.Constrained optimization of PDMA distinct feature of an NCS is that the state information is transmitted across a network.When there is no packet loss during the transmission,the NCS is simply a conventional con-trol system whose behavior is largely determined by the nom-inal closed-loop poles.Therefore,it makes sense to specify or prescribe the performance of an NCS in terms of its nominal closed-loop poles.On the other hand,it is worth noting that the state feedback law which achieves desired closed-loop poles is not unique.The PDM introduced in Section 3measures the maximum amount of packet loss that an NCS can tolerate before becoming unstable.The larger the margin,the less sensitive the NCS to packet loss.While the PDM is readily computable via the formula (3.12)for a given controller,its dependence on the controller is so complicated that it is hard to directly maximize the PDM through the design of a controller by using (3.12).In this section,we seek a state feedback law which not only places the nominal closed-loop poles at given locations but also optimizes the PDM in a certain sense.Specifically,we consider the following problem:Problem 1.For the NCS (2.1)–(2.3),find a state feedback gain K ∈R m ×n which maximizes the PDM and places the nominal closed-loop poles at given locations.To solve this problem,we need the following lemma which reveals the relationship between the PDM and a condition num-ber associated with the feedback gain.Lemma 8.If the NCS (2.1)–(2.3)is nominally stable ,there holdsPDM 1− 2(A K )22(A K ) A 22− 2(A K ),(4.1)where A K =A +B 1K , · 2and 2(·)denote the spectral normand the spectral condition number of a matrix ,respectively .Proof.The lemma obviously holds if 22(A K ) A 22−2(A K )<0.It suffices to prove the lemma in the case where22(A K ) A 22− 2(A K )>0.To this end,we show that for anygiven PDP ,the networked plant (2.1)–(2.2)is ms-stabilized by (2.3)if there holds<1− 2(A K )22(A K ) A 22− 2(A K ).(4.2)Let A ⊗A be a perturbation onto (1− )A K ⊗A K .ByBauer–Fike theorem,for any given ˆ∈spec [(1− )A K ⊗A K + A ⊗A ],there exists a ∈spec (A K ⊗A K )such that |(1− ) −ˆ| 2(A K ⊗A K ) A ⊗A 2,(4.3)where spec (·)is the spectrum of a matrix.It is easy to showthat 2(A K ⊗A K )= 22(A K )and A ⊗A 2= A 22.Then,weobtain|ˆ |−(1− )| | 22(A K ) A 22(4.4)implying that|ˆ | [ 22(A K ) A 22−| |]+| |.(4.5)From (4.2)and 2(A K ) | |,we have[ 22(A K ) A 22−| |]+| |<1.(4.6)With (4.5),we have |ˆ|<1.As ˆ is arbitrary,the networked plant (2.1)–(2.2)is ms-stabilized by (2.3).S.Hu,W.-Y.Yan /Automatica 43(2007)1243–12481247Table 1The packet dropping margin and p p 12345PDM0.100.2740.3370.3350.335It can be seen that the lower bound of the PDM in (4.1)is positive when the nominal plant (2.1)is unstable.In addition,this bound only depends on the condition number of A K if the nominal closed-loop poles,i.e.the eigenvalues of A K ,are prescribed in advance.More precisely,the bound is inversely proportional to the condition number 2(A K ).This latter ob-servation suggests that a state feedback law which achieves a large PDM while placing the nominal closed-loop poles at de-sired locations can be found by solving the problem of mini-mizing 2(A K )subject to the constraint spec (A K )= ,where is a given set of self-conjugate complex numbers of cardi-nality n .Such a problem can be solved by using an ODE-based algorithm in Lam and Yan (1997)as it is essentially a robust pole placement problem,which has been well discussed in the literature.The following example shows that the proposed algorithm is effective.Example 4.1.Consider the networked plant (2.1)–(2.2)withA =⎡⎢⎣1.70.41.8−2−0.8−3.1−3.2−1.5−1.2⎤⎥⎦,B 1=⎡⎢⎣0.5000.510.5⎤⎥⎦and the desired nominal closed-loop poles −0.2±0.2j and −0.6.We invoke the algorithm in Lam and Yan (1997)with G 0=[111111]as the starting point,where the corresponding K 0yields a PDM of 0.108.The algorithm results in a final feedbackgain K ∗=[−2.1475.071−0.5762.221−3.0816.287]which yields a PDM of 0.335.Table 1shows the variation of the PDM associated with G p for p =1,2,3,4,5.5.ConclusionsIn this paper,we discussed the stability of an NCS subjectto packet loss during the transmission of plant information to the controller over a network.A necessary and sufficient condi-tion for stability was derived in the case where an upper bound of the PDP is given.In addition,the PDM was introduced to measure the degree of stability robustness of an NCS with re-spect to packet dropping.A formula for computing the PDM was obtained.An optimization method was proposed for de-signing a state feedback law to achieve a large PDM subject to the constraint that the NCS has prescribed nominal closed-loop poles.It goes without saying that all the obtained results can easily be extended to the case where the transmission of a control signal is also subject to packet dropping.AcknowledgmentsThis work has been partially supported by an ARC research grant.The constructive comments and suggestions from the anonymous reviewers and the Associate Editor are very much appreciated.Appendix A.Proof of Lemma 5First,we define∗ sup { >0; (M 1+ M 2)<1,∀ ∈(0, )}, ∗ min { >0; (M 1+ M 2)=1}.(A.1)Due to continuity and (M 1+ ∗M 2)=1,we have ∗ ∗.From the definition of ∗,it is clear that ∗ ∗.Thus,we obtain ∗= ∗.Since M 1+ M 2is a real matrix and any eigenvalue of (M 1+ M 2)⊗(M 1+ M 2)is the product of two eigenvalues of M 1+ M 2,it can be seen that∗=min { >0;det [I −(M 1+ M 2)⊗(M 1+ M 2)]=0},(A.2)where det (·)is the determinant of a matrix.Notice that I −(M 1+ M 2)⊗(M 1+ M 2)= 2( 2 1+ 2+ 3),(A.3)where =1/ ,1=I −M 1⊗M 1,2=−M 2⊗M 1−M 1⊗M 2,3=−M 2⊗M 2.Let = 2 1+ 2+ 3.Since 2=0,we obtain ∗=1max { >0;∃w =0s .t . w =0}.(A.4)Let be a fixed positive number for which there exists w =0such that w =0.Then,we have [− −1100][0w ]=0.With theidentity − −1100 = I 0 1I0I I 2− −1100− 3 × I 10I = I 0 1I ( U 1−U 2)× I 10I ,(A.5)where U 1=[0I I 2]and U 2=[ −1100− 3],it is easy to see thatthe equation w =0is equivalent to ( U 1−U 2)v =0i.e.U −11U 2v = v where v =[ 1w w ].Therefore, satisfies w =0for some w =0if and only if is an eigenvalue of the matrix U −11U 2.From (A.4),it follows that ∗=1/ (U −11U 2)=1/ (U).ReferencesAzimi-Sadjadi, B.(2003).Stability of networked control systems in the presence of packet losses.Proceedings of the 42nd IEEE conference on decision and control (pp.676–681).1248S.Hu,W.-Y.Yan/Automatica43(2007)1243–1248Chow,M.Y.,&Tipsuwan,Y.,(2001).Network-based control systems:A tutorial.In Proceedings of IECON’01,the27th annual conference of the IEEE Industrial Electronics Society:(V ol.3,pp.1593–1602).Costa,O.L.V.,&Fragoso,M.D.(1993).Stability results for discrete-time linear systems with Markovian jumping parameters.Journal of Mathematical Analysis and Applications,179,154–178.Fang,Y.,&Loparo,K.A.(2002).Stabilization of continuous-time jump linear systems.IEEE Transactions on Automatic Control,47,1590–1603. Feng,X.,Loparo,K. A.,Ji,Y.,&Chizeck,H.J.(1992).Stochastic stability properties of jump linear systems.IEEE Transactions on Automatic Control,37,38–52.Ghaouli,L.E.,&Aitrami,M.(1996).Robust state-feedback stabilization of jump linear systems via LMIs.International Journal of Robust and Nonlinear Control,6,1015–1022.Gupta,V.,Spanos,D.,Hassibi,B.,&Murray,R.M.(2005).On LQG control across a stochastic packet-dropping link.Proceedings of the American Control Conference’01(pp.360–365).Hadjicostis,C.N.,&Touri,R.,(2002).Feedback control utilizing packet dropping network links.In Proceedings of the CDC2002,the41st IEEE Conference on Decision and Control.(V ol.2,pp.1205–1210).Hu,S.,&Zhu,Q.X.(2003).Stochastic optimal control and analysis of stability of networked control systems with long delay.Automatica,39, 1877–1884.Ji,Y.,Chizeck,H.J.,Feng,X.,&Loparo,K. A.(1991).Stability and control of discrete-time jump linear systems.Control Theory and Advanced Technology,2,247–270.Kim,D.-S.,Lee,Y.S.,Kwon,W.H.,&Park,H.S.(2003).Maximum allowable delay bounds of networked control systems.Control Engineering Practice,11,1301–1313.Lam,J.,&Yan,W.-Y.(1997).Pole assignment with optimal spectral conditioning.System and Control Letters,29,241–253.Ling,Q.,&Lemmon,M.D.(2003a).Optimal dropout compensation in networked control systems.Proceedings of the42nd IEEE conference on decision and control(pp.670–675).Hawaii:Maui.Ling,Q.,Lemmon,M.D.,(2003b).Soft real-time scheduling of networked control systems with dropouts governed by a Markov chain.In Proceedings of the American Control Conference(V ol.6,pp.4845–4850).Liu,J.,Elia,N.,&Tatikonda,S.(2004).Capacity-achieving feedback scheme forflat fading channels with channel state information.Proceedings of the American the control conference’04(pp.3593–3598).MA:Boston. Montestruque,L.A.,&Antsaklis,P.,(2003).Stochastic stability for model-based networked control systems.In Proceedings of the American Control Conference’03.(V ol.5,pp.4119–4124).Montestruque,L. A.,&Antsaklis,P.(2004).Stability of model-based networked control systems with time-varying transmission times.IEEE Transactions on Automatic Control,49,1562–1572.Nilsson,J.,Bernhardsson,B.,&Wittenmark,B.(1998).Stochastic analysis and control of real-time systems with random time delays.Automatica, 34,57–64.Seiler,P.,&Sengupta,R.(2005).An H∞approach to networked control.IEEE Transactions on Automatic Control,50,356–364.Tatikonda,S.,&Mitter,S.(2004).Control under communication constraints.IEEE Transactions on Automatic Control,49,1056–1068. Tatikonda,S.,Sahai,A.,&Mitter,S.(2004).Stochastic linear control overa communication channel.IEEE Transactions on Automatic Control,49,1549–1560.Walsh,G.C.,Ye,H.,&Bushnell,L.,(1999).Stability analysis of networked control systems.In Proceedings of the American Control Conference’99.(V ol.4,pp.2876–2880).Zhang,P.,Ding,S.X.,Frank,P.M.,&Sader,M.,(2004).Fault detection of networked control systems with missing measurements.In Proceedings of thefifth Asian Control Conference.(V ol.2,pp.1258–1263).Zhang,W.,Branicky,M.S.,&Phillips,S.M.(2001).Stability of networked control systems.IEEE Control Systems Magazine,21,84–89.Shawn Hu received his B.Eng.degree in Ma-terial Science from Beijing Polytechnic Univer-sity,M.EngSc.in Telecommunications and Net-working from Curtin University of Technology.Now he is a Ph.D.student in the Departmentof Electrical and Computer Engineering,CurtinUniversity of Technology.His research interestsinclude stability theory,stability robustness andoptimization in networked controlsystems.Wei-Yong Yan received the B.S.degree inMathematics from Nankai University,Tianjin,in1983,the M.S.degree in Systems Sciencefrom Academia Sinica.Beijing.in1986,andthe Ph.D.degree in Systems Engineering fromthe Australian National University,Canberra,in1990.From1990to1992,he worked as a ResearchFellow in the Department of Systems Engi-neering,the Australian National University.Hewas a Lecturer in Applied Mathematics at theUniversity of Western Australia from1993to 1994prior to joining the Nanyang Technological University in Singapore first as a Lee Kuan Yew Fellow and later as a Senior Lecturer in the School of Electrical and Electronic Engineering.Since1998,he has been a Senior Lecturer in the Department of Electrical and Computer Engineering at Curtin University of Technology,Perth,Australia.His current research interests are in the areas of control and signal processing.。
具有量化和时延的不确定系统的鲁棒预测控制
（School of Automation，Qingdao University，Qingdao 266071，China）
Abstract: In order to solve the problem of time delay and quantization in the process of data transmission
method is verified by numerical simulation.
Keywords: delay；quantization；uncertain systems；robust predictive control；sector⁃bounded
网 络 控 制 系 统（Networked Control Systems,NCS）
u 0 > 0。p 为量化密度，p 比较小时，量化级数也小，量
化器较为粗糙。根据量化级数的定义，对数量化器
可以表示为：
ìu i , 1 u i < v < 1 u i ,v > 0
ïï 1 + δ
1-δ
f (v) = í0,v = 0
ïï
î-f (-u),v < 0
其中，δ =
1-p
。量化器的量化误差可以定义为
化问题可以描述为：
min
max
J(k) = min
max
(J 0 (k) + J1(k))
U(k)
U(k)
Δ(k)
Δ(k)
（7）
其 中 ，J 0 (k) = xT (k| k)Qx(k| k) +[(I + Δ(k))× u(k| k)]T
Stability Analysis of Networked Control Systems
438IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, MAY 2002Stability Analysis of Networked Control SystemsGregory C. Walsh, Hong Ye, and Linda G. BushnellAbstract—We introduce a novel control network protocol, tryonce-discard (TOD), for multiple-input–multiple-output (MIMO) networked control systems (NCSs), and provide, for the first time, an analytic proof of global exponential stability for both the new protocol and the more commonly used (statically scheduled) access methods. Our approach is to first design the controller using established techniques considering the network transparent, and then to analyze the effect of the network on closed-loop system performance. When implemented, an NCS will consist of multiple independent sensors and actuators competing for access to the network, with no universal clock available to synchronize their actions. Because the nodes act asynchronously, we allow access to the network at anytime but we assume each access occurs before a prescribed deadline, known as the maximum allowable transfer interval. Only one node may access the network at a time. This communication constraint imposed by the network is the main focus of the paper. The performance of the new, TOD protocol and the statically scheduled protocols are examined in simulations of an automotive gas turbine and an unstable batch reactor. Index Terms—Limited communications, networked control systems (NCSs), stability.Fig. 1. Schematic diagram of a complicated control system. In this diagram, the network is found between sensors and the controller.I. INTRODUCTION N MANY complicated control systems, such as manufacturing plants, vehicles, aircraft, and spacecraft, serial communication networks are employed to exchange information and control signals between spatially distributed system components, like supervisory computers, controllers, and intelligent input–output (I/O) devices (e.g., smart sensors and actuators). Each of the system components connected directly to the network is denoted as a node. When a control loop is closed via the serial communication channel, we label it a networked control system (NCS). The serial communication channel, which multiplexes signals from the sensors to the controller and/or from the controller to the actuators, serves many other uses besides control (see Fig. 1). In contrast to widely used computer networks, an NCS is concerned primarily with the quality of real-time reliable service. NCSs are being adopted in many application areas for a number of reasons [16] including their low cost, reduced weight, and power requirements, simple installation and maintenance, and higher reliability. However, using a networkManuscript received January 7, 2000; revised September 11, 2001. Manuscript received in final form December 4, 2001. Recommended by Associate Editor R. Middleton. This work was supported in part by the NSF under Grant ECS-97-02717 and the ARO under Grant DAAG55-98-D-0002. G. C. Walsh is on leave from the University of Maryland and is with Cyra Technologies, Oakland, CA 94621 USA (e-mail: greg.walsh@). H. Ye is with Delphi Communication Systems, Inc., Maynard, MA 01754 USA (e-mail: hye@). L. G. Bushnell is with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195-2500 USA (e-mail: bushnell@ ). Publisher Item Identifier S 1063-6536(02)03421-8.Ipresents some new analytical challenges because the network imposes a communication constraint: only one sensor can report its measurements at a time. Furthermore, the lack of a universal clock and the presence of noncontrol related traffic makes assumptions about constant sampling intervals unrealistic in many applications. In this paper, an access deadline, or maximum allowable transfer interval, , is used in its place to ensure absolute stability of an NCS. Ample research papers in analyzing and scheduling the real-time network traffic have been published [2], [6], [21], [24], [25]. The significance of combining communication constraints and control specifications has not apparently been addressed in these papers. We propose and analyze a new scheduling algorithm to determine the transmission order of multiple sensor nodes in an NCS based on need. The new scheduling algorithm efficiently allocates network resources to multiple smart sensors and maintains good closed-loop control system performance. Some researchers noticed the detrimental effects of network-induced randomly time-varying delay on the stability of feedback control systems [17], [18]. However, all previous research is confined to the one packet transmission problem, i.e., all system outputs are lumped and sent out in one packet, and as a consequence, there is no competition between smart sensors of an NCS [1], [4], [9], [11], [22], [23]. No general explicit stability condition has been obtained in the literature even for one packet transmission case. This paper presents, for the first time, an analytic proof of global stability for an NCS with general multiple-packet transmission in addition to providing a global stability condition for the special one packet transmission problem. The augmented state space method and jump linear control system method are two significant methods proposed in the literature for analyzing and designing an NCS. The former one reduced the problem to a finite dimensional discrete-time control by augmenting the system model to include past values of plant1063-6536/02$17.00 © 2002 IEEEIEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, MAY 2002439input and output (i.e., delayed variables) as additional states [4], [5]. A necessary and sufficient condition for system stability was established only for the special case of periodic delays. This technique is very useful for developing control laws to improve the performance of an NCS [12], [13], [19], [20] except that it fails to give a general stability condition for random delay. In [11] and [14], distributed linear feedback control systems with random communication delays were modeled as a jump linear control systems, in which random variation of system delays corresponds to randomly varying structure of the state-space representation. Necessary and sufficient conditions were found for zero-state mean-square exponential stability of the considered class of systems. This method requires that the transition probability matrix is known a priori. Furthermore, both methods were limited to the one packet transmission problem. The occurrence of transmission events on the network is timevarying and often modeled as a random process, e.g., Poisson process, and the resulting times that pass between each access to the network are independent and have an exponential distribution. The stochastic Lyapunov function method [8], [9] holds much promise for determining almost-sure stability and control system performance. This paper’s approach, however, is to provide guarantees by employing transmission deadlines. The results presented here are absolute instead of almost sure. This paper is organized as follows. The dynamic model for the NCS and our new TOD protocol are described in Section II. In particular, we model the effect of different scheduling technique as a finite error bound imposed on the system. In Section III, absolute stability conditions are derived for both a multiple-packet transmission system and a single packet transmission system. The results of numerical simulations are presented in Section IV. Conclusions are stated in Section V. II. MODELING OF A NCS The NCS model is shown in Fig. 2. It consists of three main with state and parts: the plant ; the controller with state output and output , and the network, with state , consisting of the most recently reported and , Without loss of generality we have asversions of . Outputs measured locally at an actuator can be sumed incorporated directly into the controller and do not require treatment in our model. If such outputs are needed elsewhere, the actuator node can also be considered a smart sensor. Because of is available to the conthe network, only the reported output is availtroller and its prediction processes, similarly, only able to the actuators on the plant. Commonly used local area is globally known and networks support broadcast, hence in such a case the controller itself may be physically distributed. To focus on the effect of network competition on the stability of an NCS, we make the following assumptions. The control law is designed in advance without considering the presence of the network. The controller dynamics are considered continuous and sampling delay is ignored, because the access interval of the NCS to the network is much larger than the processing period of the controller and smart sensors. Once access to a particular sensor node is granted, data is assumed to be transmittedFig. 2. Configuration of a networked control system.instantly, since most of the NCS is connected by a local area network with very high data rate and a physical range less than 100 m. The communication medium is error-free based on the lower error rate of modern high-speed communication systems and the higher reliability offered by many error detection and correction technologies. No observation noise exists. All matrices in the paper have compatible dimensions and the standard Euclidean norm will be used unless noted otherwise. We label the network-induced error and the combined state of the controller and plant . The state of the entire NCS is given by and between transmission instances the dynamics of the NCS can be summarized as (1) whereDefine the matrix such that . Any prediction or . filtering process can be used to improve the estimate of Such predicting and filtering will add extra states and dynamics and . which we incorporate in matrices , and hence the dynamics reduce Without a network, . It is assumed that the controller has been to is Hurwitz. Consedesigned ignoring the network, hence quently there exists a unique symmetric positive definite matrix such that (2) and ,( Define the constants eigenvalue). Since we are modeling the network as a perturbation on the system, choosing the right-hand side of (2) equal to is desirable for maximizing the tolerable perturbation bound [10, p. 206]. is mainly deThe behavior of the network-induced error termined by the architecture of the NCS and the scheduling strategy. In the special case of one-packet transmission, there is only one node transmitting control data on the network, thereis set to zero at each transmission time. fore the entire vector and/or For multiple nodes, transmitting measured outputs440IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, MAY 2002computed inputs , the transmission order of the nodes depends on the scheduling strategy chosen for the NCS. In other words, the scheduling strategy decides which components of are set to zero at the transmission times. Dynamic and static scheduler are two major scheduling strategies. Both of them will be analyzed for NCS implementation. A dynamic scheduler determines the network schedule while the system runs. Unlike dynamic scheduling processor time in real-time control, however, the information needed to decide which node should be granted access to the network is not centrally located. Based on the characteristics of the real-time NCS, we propose a novel protocol, try-once-discard (TOD) protocol, which employs dynamic scheduling, allocating network resource based on the need. In TOD, the node with the greatest weighted error from the last reported value will win the competition for the network resource. We call the scheduling technique maximum-error-first (MEF) and the protocol TOD because if a data packet fails to win the competition for network access, it is discarded and new data is used next time. Such a method is vulnerable to noise. In practice, the sensor nodes must employ some sort of filtering to prevent a channel with a large noise signal from dominating the network. This protocol can be realized by using the flexible low-level software implementation and the mature hardware technology of controller area network (CAN), which is specifically designed for bitwise arbitration. How does the TOD protocol work? Without loss of generality, assume there are nodes competing, each one may be associated with one or multiple plant inputs and outputs. In the TOD protocol, the priority level of each node’s message is propor, which is a -dimensional subvector tional to the norm of with representing the number of plant or of controller outputs transmitted by node . The weights assigned to error signals are assumed already built into the output matrix. At every transmission time, the node with the highest priority (or greatest weighted error) gets transmitted. If two or more messages have equal priority, a prespecified ordering of the nodes will be imposed to resolve the collision. Today, static scheduling is the most common methodology, in which the order (or pattern) of transmission is decided in advance and fixed during system operation. We label such a scheme static scheduler, which is typically implemented by polling or by token ring. Though the schedule is fixed, some nodes may be granted access multiple times before others get any access. If a transmission pattern is of length , every consecutive visits form a repeated cycle. In one cycle, all nodes are visited at least once. The pattern length is called the periodicity of the static scheduler. In order to characterize the behavior of the scheduling algo, we introduce a constant . rithms and their relation with The existence of will be proved in later section by the Lipschitz condition of the differential equations (1), i.e., over a short will be bounded by a conperiod of time the growth in error stant , which is dependent on the system characteristics and exists initial conditions. Assume the transmission deadline . The bound is defined such that as the maximum allowable transfer interval used to guaranteethe absolute stability of NCS. The size of does not affect the bound of . The following two lemmas characterize the scheduling algorithms. Lemma 1 (Dynamic Scheduler Error Bound): Given a dynamic (TOD) network scheduler starting at time with nodes competing, maximum allowable transfer interval , maximum seconds strictly bounded by . growth in error in , . Then, for any time Proof: There are at least transmissions in the interval . Let be the last transmission times with , , and be the nodes that got transmitted at those times, respectively. nodes being transmitted are disSuppose the first th transmitted node was also transmitted tinct, and the . Then , for before, say at time . Since node was transmitted both at and , we have with denote the instant right before transmission. By the construction of the dynamic scheduler (TOD), at transmission time, node has the greatest error. As a consequence, and , for all , . . So Since , when or , we have the worst case error bound . for the dynamic scheduler Lemma 2 (Static Scheduler Error Bound): Given a static network scheduler starting at time , with integer periodicity , , maximum growth in maximum allowable transfer interval seconds strictly bounded by . Then, for error in , . any time Proof: The integer periodicity allows at most nodes . The schedcompeting. Assume there are nodes, uler design also implies that each node is visited at least once during every consecutive transmissions. At least once cycle (or transmissions) is completed during . Let be the last transmission the interval , , and times with be the nodes that were transmitted at those times, , for . Since respectively. Then , we have the set . So for any time , . The worst case error bound of the dynamic scheduler is the same as that of a special case of the static scheduler, i.e., all nodes are visited equally. The bound is conservative for both represents a deadline. But scheduling algorithms, because for the same transmission times distribution, the error bound for dynamic scheduler will be better than that of the static scheduler, because it grants access to the node with the greatest error.III. STABILITY OF NETWORKED CONTROL SYSTEMS Two stability theorems for general multiple-packet and one-packet transmissions are derived in this section. Both theorems are derived based on Lyapunov’s second method and treat the network induced-error term as a vanishing perturbation [10, p. 204].IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, MAY 2002441We first consider the stability of a general multiple-packet transmission NCS with either a static or dynamic (TOD) scheduling algorithm. Theorem 1 (Stability of a General NCS): Given an NCS whose continuous dynamics are described by (1) with nodes operating under TOD, or with integer periodicity under static , scheduling, and a maximum allowable transfer interval, which is less than the minimum ofare met. The state vector changes discretely, but always decreases in magnitude when jumping, so at a transition time the bound cannot be jumped over. At time , either or or both. Suppose we have and , then the case , and hence we have(3) and becauseThen, the NCS is globally exponentially stable. Proof: Consider any initial time and the associated ini. If , then for all tial state we have and , even if there are no transmissions, since we are at an equilibrium point of the system. For . this reason, without loss of generality, we assume At time , nothing is known about the magnitude of the error . , at least transmissions of the netHowever, at time , we have from work have occurred. For any in Bellman–Gronwall where is the induced norm of . Since we have chosen , we have . This bound is conservative as it holds even in the case where there are no trans, missions of data. At any transmission time hence . we have that and refers to the limits from the left and The notation right, respectively. Because of this bound, we have that forSince at time , , is strictly negative. Even can only reduce instantaneously if is a transition time, in size hence will remain strictly negative. and thereCertainly then at time , for all including time . We fore now consider the remaining possibility that at time , we have . We can conclude not a transition timeNote thatHence the maximum possible growth between transmission in this time interval is strictly bounded by as , where is the condition number of . Depending on the type of protocol utilized, either Lemma 1 or , the error Lemma 2 may be applied to verify that at time where the constant is bounded by is . We have selected so that is smaller than both and . , Furthermore, we have that at time . Consequently, using for we have that with . Certainly , . If at any time we , then with the have equal to . constant we have both At time and . We now prove by contradiction that , both conthese two conditions imply that for all any of these two conditions hold. If at any time which is ditions fail, then there exists a time the first time either one or both conditions have failed. In in, of minimum length seconds, both conditions tervalbecause has been chosen so that . We now have the conditions of Lemma 1 or Lemma 2 applying to the interval , since the maximum growth of in seconds in the is limited by . The lemmas interval , manifesting our contradicindicate that , . tion. We conclude then View the control system as perturbed by the bounded error . If we write , signal evolves according to the then the state starting at time variation of parameters formula:The zero state term is the solution of the differential equation with zero initial conditions, that . At time we have is, . We know that for all time, , , since if we consequently, and by (3), had equality, then . As is bounded, thenBy choice of , we have therefore the zero state term is strictly less thanand .442IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, MAY 2002GivenHurwitz, there exists a time . Finally, andsuch thatb)where . Then by induction . The closed-loop NCS is then exponentially stable. One-packet transmission is a special case of multiple-packet transmission. In the following, a general sufficient stability condition of an NCS with one-packet transmission is derived in a way different from the above proof. This derivation gives a less conservative bound on . , In the one-packet transmission case, for , , . The system equation can be written where as . Under normal operation, as , will go to zero because will track closely. Two lemmas are introduced to prove the theorem. It should be noted that Lemma 3 is a variation of the commonly used Bellman–Gronwall Lemma. The proof of Lemma 4 follows that of Lemma 3. and Lemma 3 (Bellman–Gronwall Lemma [7]): Given nonnegative, piecewise continuous and differentiable funcsatisfies tions of time . If the function , then . nonnegative piecewise condifferentiable. If the func, , , , then is Because sume (4) is a constant vector in , as, then . In order to derive the relation between and , we fix the final time and let the initial time be . So changeable, i.e., withthen the origin is a globally exponentially stable equilibrium point of the NCS. , define , Proof: When . The system equation can be written asthenandwhere stands for vector norm or induced matrix norm. Using Bellman–Gronwall Lemma 3, we getand Lemma 4: Given tinuous functions of time with satisfies tion then . Proof: Let differentiable andSinceLet, then , whose state transition matrix is . ThereforeSince . Substitute , then . Thus,, then Using Lemma 4, we get, . Theorem 2 (Single-Packet Transmission Stability): Let be a globally exponentially stable equilibrium point of the nonnetworked system. If the maximum transmission interval satisfies: a) Let , . Since , thenIEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, MAY 2002443Let. Sincethenbetween dynamic and static scheduling are explored in a simulation of an automotive gas turbine, because the analysis also does not differentiate between the approaches. A token ring scheduler alternating access between the two nodes and the dynamic TOD scheduling algorithm are compared, using a Poisson packet arrival model. A. Unstable Batch ReactorUsing this and inequalities (4), we deriveThe first example, an unstable batch reactor [3, p. 62], is a coupled two-input–two-output NCS. Based on the linearized process modelSince is a globally exponentially stable equilibrium , there exists point of the nonnetworked system a unique symmetric positive definite matrix , satisfying the . Lyapunov equation: be a Lyapunov function of the nonLet networked system satisfying the following inequalities:a proportional-plus-integral controller The derivative of along the trajectories of the perturbed . system satisfies Since the equation shown at the bottom of the page is true, for . So, then satisfies the inequalities a) and b), in any when only when . transmission interval, The origin is a globally exponentially stable equilibrium point of the NCS. IV. SIMULATION RESULTS AND DISCUSSION We explore the application of networking technology to two example systems, an unstable batch reactor and an automotive gas turbine. The network is considered transparent for the purpose of controller design. The two models are taken from the literature and the reported controller is used without modification. The bounds on the maximum allowable transfer interval derived in the theorems are very conservative, as both examples demonstrate. The simulations explore not only better estimates for the bound but also the impact of different packet arrival models on the system performance. Our experience suggests that the commonly used Poisson packet arrival model is unlikely to accurately reflect the traffic on a control network, as most packets are relatively short and frequent, and because TOD control traffic does not use queues. Alternate packet arrival models are compared in the unstable batch reactor simulation. The differencesis designed in advance to stabilize the feedback system and achieve good performance. need to be transmitted to Only the system outputs and the controller via the network, each with its associated node. In the simulations, the network model is placed between the output of the plant and the input of the controller. A Poisson process is used to model the packet arrival events. In with mean unit time, the probability of transmission events occurring , where stands for the expectation of is , number of events occurring in unit time. Its inverse, is the average transfer interval length. The system remains stable for around 0.06 s. The theoreton the linearized system from Theorem 2 is ical bound of around 10 s. This discrepancy is due to the conservative nature of the Bellman–Gronwall Lemma. In Fig. 3, we show both s, and an unstable a stable trajectory of the system, for s. trajectory for The transmission intervals in an NCS were modeled as random variables because of the effect of bursty traffic on the network. In our former simulation, the access to the network was modeled as the Poisson process. Two other models are proposed. The first, which we refer to as the spiked Poisson,444IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, MAY 2002Fig. 3. Resulting trajectories of the batch reactor system with one-packet transmission. Note the different magnitude scales.disallows any packets arrivals before , and places half of the transmission interval times at exactly , the others, spread like a Poisson process. This emulates the controllers receiving access half of the time after a precise time interval, and on occasion being interrupted by other traffic on the network. Arbitrarily long delays are possible. The second model, which we refer to as the MATI (maximum allowable transfer interval) model, disallows arbitrarily long delays. Some mechanism in is always met. the network ensures that a deadline of Such a model is more consistent with the theorems. Fig. 4 compares the three different packet arrival models. Monte Carlo simulation is used because of the random nature of the network traffic. For each , a number of simulations were run and for each we checked if the control specifications (such as overshoot, rise time and settling time) were met. A constant transfer interval simulation was also run as a point of comparison. Notice that using the popular constant transfer interval (constant delay) network model would delude the control system designer into believing he or she need less network bandwidth than is actually required to meet control specifications. Of the three probabilistic models chosen, the plain Poisson arrival model shows the worst behavior, while the spiked Poisson and the MATI model are of comparable performance, close to but more conservative than the constant delay model. The simulation results reveal that the theoretical bound, on the order of tens of nanoseconds, is conservative since with average transfer inms, all simulation results pass the test. terval B. Automotive Gas Turbine The two-shaft automotive gas turbine is basically a coupled two-input two-output system [15, p. 249]. The two system outputs to be controlled are gas generator speed and inlet-turbine temperature. The two input variables are fuel pump excitationFig. 4. Comparison of differing packet arrival models. Plant: Batch reactor, protocol: TOD.and nozzle actuator excitation. The linearized plant model is shown in the equation at the bottom of the next page. The controller is designed in advance to reduce cross-channel interaction (or to realize “diagonal dominant”), to remove steady-state error and increase the system response speed without considering the network effectsThe controller for a two-shaft gas turbine has many functions, for example, engine health monitoring (EHM) is currently of great interest. In the simulation, we show that the performance。
Scheduling of networked control systems
February 2001
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In summary, designers choosing to CAN devices range from stand-alone use a networked control system archiCAN interface chips such as Intel’s 82527 tecture are motivated not by perfor(see Fig. 1 for the authors’ application), mance, but by cost, maintenance, and Microchip’s MCP2510, and Philips’ reliability gains. The use of serial comSJA1000 to microcontrollers withorks in a control sysgrated on-chip CAN interface, such as tem is such a clear win that it is now Motorola’s 68HC05, Siemens’ (or widely supported from the device to Infineon’s) SAB-C167CR, and Texas Inthe system level. The next section outstruments’ TMS320C24x series DSP. lines some application issues with The serial communication channel, NCSs and narrows the discussion to which multiplexes signals from the senscheduling. Scheduling control netsors to the controller and/or from the controller to the actuators, serves Figure 1. A networked control system work traffic is the focus of the following many other uses besides control (see smart node designed by the authors circa section. The last section reports simuFig. 2). Each of the system components 1996. The card is composed of a Motorola lation and physical experiments. connected directly to the network is 68HC11 with an Intel 82527 CAN interface denoted a physical node. Logical subdi- chip. Application Issues vision is also common, so, for example, Control networks differ from data netalthough the high speed CAN network in an automobile might works in important ways, such as having short, frequent packhave 50 logical nodes, physically only a half a dozen devices ets with real-time requirements. Consider the generic NCS may be connected. Control networks are typically local area problem in Fig. 2, where several sampled continuous- time networks, and while hierarchical collections of networks are (vector-valued) outputs, data records, are sent through a sinfound, control loops are closed locally. Important exceptions gle serial communications channel and reconstructed by a to this rule exist. For example, in teleoperation systems, su- smart actuator, which then uses the resulting data image to pervisory control of discrete-event systems, and sensor net- compute the control action. The computation may be carried works control loops are often closed over wide area out remotely and the commands also sent over the network, networks. A control network should have at least three although this is not shown in the figure. A control network imnodes; with two nodes there is no reason to use a network. pacts the closed-loop performance by creating differences bePlant outputs from different nodes are often coupled, and tween the data records and their associated remote images, outputs on different time scales are also found. Congestion is and the performance of a well-designed NCS degrades gracea common issue. fully in the presence of congestion on the network. In a net-
stability of networked control systems
lay) that occurs while exchanging data among devices connected to the shared medium. This delay, either constant (up to jitter) or time varying, can degrade the performance of control systems designed without considering the delay and can even destabilize the system. Next, the network can be viewed as a web of unreliable transmission paths. Some packets not only suffer transmission delay but, even worse, can be lost during transmission. Thus, how such packet
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February 2001
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dropouts affect the performance of an NCS is an issue that must be considered. Another issue is that plant Physical Plant outputs may be transmitted using multiple network packets (so-called multiple-packet transmissio
状态时滞的网络控制系统的保成本控制器设计
Vol.33,No.2ACTA AUTOMATICA SINICA February,2007 Guaranteed Cost Controller Design of NetworkedControl Systems with State DelayXIE Jin-Song1F AN Bing-Quan1Young Sam Lee2YANG Jin3Abstract This paper is concerned with the state-feedback guaranteed cost controller design for a class of networked control systems (NCSs)with state-delay.A new model of the NCSs is provided under consideration of the network-induced delay.A sufficient condition for the existence of a guaranteed cost controller for NCSs is presented by a set of linear matrix inequalities(LMIs).A method,which can transform non-convex to the convex,is applied.Accordingly,a numerical algorithm is proposed to obtain the lower bound.Theoretical analysis through an example shows the effectiveness of the method.Key words Networked control systems,guaranteed cost control,linear matrix inequalities1IntroductionThe networked control systems(NCSs)have recentlybeen studied by more and more researchers since their lowcost,reduced weight and power requirements,simple in-stallation and maintenance and high reliability[1,2].In anNCS,one of the important issues to treat is the effect ofthe network-induced delay on the system performance.Forthe NCSs with different scheduling protocols,the network-induced delay may be constant,time-varying,or even ran-dom variable[3].The guaranteed cost control of uncertain systems wasfirst put forward by[4]and studied by a lot of researchers,which is to design a controller to robustly stabilize the un-certain system and guarantee an adequate level of perfor-mance.The guaranteed cost control approach has recentlybeen extended to the uncertain time-delay systems,for thestate feedback cases,see[5∼7];for the output feedbackcase,see[8].In this paper,the author s intention is to design a guar-anteed cost controller based on the network delay for aclass of uncertain time-delay.Sufficient condition for theexistence of a guaranteed cost state-feedback controller isestablished in terms of matrix inequalities.At the sametime,the maximum allowable valueτmax of the network-induced and guaranteed cost bound are obtained and theguaranteed cost control strategy is proven by a numericalexample.The remainder of this paper is organized as follows:Sec-tion2describes the problem formulation.In Section3,details of the modeling of networked control systems arediscussed.Section4obtains the guaranteed cost controllerdesign and illuminates the method which switches a non-convex problem to a convex.Section5introduces a numer-ical example.Section6presents conclusions.2Problem formulationConsider a class of linear uncertain system with time-delay in the state described by the following equations8 < :x (t)=[A+∆A]x(t)+[A d+∆A d]x(t−d)+[B+∆B]u(t)x(t)=ϕ(t),t∈[−d,0](1)Received April12,2005;in revised form February27,2006 Supported by National Natural Science Foundation of P.R.China (70371011),Shanghai Leading Academic Discipline Project(T0502) 1.Center of Research on Transports System,University of Shang-hai for Science&Technology,Shanghai200093,P.R.China 2. School of Electrical Engineering,Inha University,Inchon,Korea 3. College of Science,University of Shanghai for Science&Technology, Shanghai200093,P.R.ChinaDOI:10.1360/aas-007-0170where x(t)∈R n is the state vector,u(t)∈R m is the control input vector,A,A d∈R n×n and B∈R n×m are known con-stant real matrices,∆A,∆A d and∆B are matrix-valued functions of appropriate dimension parameter uncertain-ties in the system model.ϕ(t)is a given continuous vector-valued initial function and subsection differential on[−d,0]. The parameter uncertainties considered are assumed to be norm bounded and satisfy[∆A∆A d∆B]=DF(t)[E E d E b](2) where D,E,E d,E b are known constant real matrices of ap-propriate dimensions that represent the structure of uncer-tainties,and F(t)∈R i×j is an uncertain matrix function with Lebesgue measurable elements and satisfiesF T(t)F(t)≤I(3) in which I denotes the identity matrix of appropriate di-mension.Associated with system(1)is the cost functionJ=Z∞[x T(t)Qx(t)+u T(t)Ru(t)]d t(4)where Q and R are given positive-definite symmetric ma-trices.Definition[9].For the uncertain system(1),if there exist a control law u∗(t)and a positive scalar J∗such that for all admissible uncertainties the closed-loop system is stable and the closed-loop value of the cost function(4) satisfies J≤J∗,then J∗is said to be a guaranteed cost and u∗(t)is said to be a guaranteed cost control law for the uncertain system(1).The objective of this paper is to develop a procedure for designing a memoryless state feedback guaranteed cost control lawu(t)=Kx(t)(5) for the linear uncertain time-delay system(1).3Modeling of networked con-trol systemsIn an NCS,suppose that the sensor is clock-driven,the controller and actuator are event-driven and the data is transmitted with a single-packet.Then the real input u(t) realized through zero-order hold in(1)is a piecewise con-stant function.Furthermore,if we consider the effect of the network-induced delay and network packet dropout on the NCSs,then the real control system(1)with(5)can be rewritten as8<:x (t)=[A+∆A]x(t)+[A d+∆A d]x(t−d)+[B+∆B]u(t),t∈[i k h+τk,i k+1h+τk+1) u(t+)=Kx(t−τk),t∈{i k h+τk,···}(6)No.2Xie Jin-Song et al.:Guaranteed Cost Controller Design of Networked Control Systems with State Delay171 where h is the sampling period,i k(k=1,2,···)are someintegers and{i1,i2,···}⊆{1,2,···},τk is the time de-lay,which denotes the time from the instant i k h when thesensor nodes sample sensor data from a plant to the in-stant when actuators transfer data to the plant.Obviously,S∞k=1[i k h+τk,i k+1h+τk+1)=[0,∞).According to(2),system(6)can be rewritten as the following equivalent form x (t)=[A+DF(t)E]x(t)+[A d+DF(t)E d]x(t−d)+[B+DF(t)E b]Kx(i k h),t∈[i k h+τk,i k+1h+τk+1)(7) and(4)can be rewritten asJ=Z∞x T(t)Qx(t)d t+∞X k=1Z ik+1h+τk+1i k h+τkx T(i k h)K T RKx(i k h)d t(8)4Main resultsThe following Lemmas will be used.Lemma1.For any vectors a,b and positive-definite matrix X,there exists±2a T b≤a T X−1a+b T XbLemma2.Given appropriate dimension matrices D,E and symmetric Y,the matrix inequalityY+DF E+E T F T D T<0holds for all F satisfying F T F≤I if and only if there exists a constantε>0such thatY+εDD T+ε−1E T E<0First we present a sufficient condition for the existence of memoryless state feedback guaranteed cost control laws for an uncertain time-delay system.Theorem1.u(t)=Kx(t),(K=Y X−T)is a guar-anteed cost controller if there exist symmetric positive-definite matrices P>0, S>0, T>0,and appropriate dimension matrices X,Y, N i(i=1,2,3,4),and a scalar ε>0for any given scalarsτ,ρi(i=2,3,4),and matri-ces Q>0,R>0such that(9)holds.At the same time the guaranteed cost J∗satisfies(11).(i k+1−i k)h+τk+1≤τ,k=1,2, (10)J≤ϕT(0)X−1 P X−Tϕ(0)+Z0−d ϕT(α)X−1 SX−Tϕ(α)dα+Zτ0Z0−βϕ T(α)X−1 T X−Tϕ (α)dαdβ=J∗(11)where8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:Ω11= S+ N1+ N1T−AX T−XA T+εDD TΩ21= N2−XA T d−ρ2AX T+ερ2DD TΩ22=− S−ρ2XA T d−ρ2A d X T+ερ22DD TΩ31=− N1T+ N3−Y T B T−ρ3AX T+ερ3DD TΩ32=− N2T−ρ2Y T B T−ρ3A d X T+ερ2ρ3DD TΩ33=− N3− N3T−ρ3BY−ρ3Y T B T+ερ23DD TΩ41= N4+X−ρ4AX T+ P+ερ4DD TΩ42=ρ2X−ρ4A d X T+ερ2ρ4DD TΩ43=− N4+ρ3X−ρ4BY+ερ3ρ4DD TΩ44=τ T+ρ4X+ρ4X T+ερ24DD TProof.Construct a Lyapunov functional asV(t)=x T(t)P x(t)+Zt−dx T(α)Sx(α)dα+ZτZ t−βx T(α)T x (α)dαdβ(12)where P>0,S>0,T>0.Since x(t)−x(i k h)−R ti k hx (α)dα=0and by(7),onecan see that any arbitrary matrices N i,M i(i=1,2,3,4)ofappropriate dimensions satisfy2[x T(t)N1+x T(t−d)N2+x T(i k h)N3+x T(t)N4]×[x(t)−x(i k h)−Zti k hx (α)dα]=0(13)2[x T(t)M1+x T(t−d)M2+x T(i k h)M3+x T(t)M4]×[−¯Ax(t)−¯A d x(i k h)−¯BKx(i k h)+x (t)]=0(14)where¯A=A+DF(t)E,¯A d=A d+DF(t)E d,¯B=B+DF(t)E b.Taking the time derivative of V(t)for t∈[i k h+τk,i k+1h+τk+1),and using(13)and(14)yieldsV (t)=2x T(t)P x (t)+x T(t)Sx(t)+τx T(t)T x (t)−x T(t−d)Sx(t−d)−Zt−τx T(α)T x (α)dα+2[x T(t)N1+x T(t−d)N2+x T(i k h)N3+x T(t)N4]×[x(t)−x(i k h)−Z ti k hx (α)dα]+2[x T(t)M1+x T(t−d)M2+x T(i k h)M3+x T(t)M4]×[−¯Ax(t)−¯A d x(i k h)−¯BKx(i k h)+x (t)](15)Ω=266666666664Ω11∗∗∗∗∗∗∗Ω21Ω22∗∗∗∗∗∗Ω31Ω32Ω33∗∗∗∗∗Ω41Ω42Ω43Ω44∗∗∗∗τ N1Tτ N2Tτ N3Tτ N4T−τ T∗∗∗00Y00−R−1∗∗X T00000−Q−1∗EX T E d X T E b X T0000−εI377777777775<0(9)172ACTA AUTOMATICA SINICA Vol.33 From(10)it can be seen that whent∈[i k h+τk,i k+1h+τk+1)−Ztt−τx (α)T x (α)dα≤−Zti k hx (α)T x (α)dα(16)Using Lemma1,we can show that−2[x T(t)N1+x T(t−d)N2+x T(i k h)N3+x T(t)N4]×Zt i k h x (α)dα≤τe T(t)NT−1N T e(t)+Zti k hx T(α)T x (α)dα(17)wheree T(t)=[x T(t)x T(t−d)x T(i k h)x T(t)]N T=[N T1N T2N T3N T4]Combining(15),(16)and(17),we obtainV (t)≤e T(t)¯Ωe(t)−x T(t)Qx(t)−x T(i k h)K T RKx(i k h)t∈[i k h+τk,i k+1h+τk+1)(18) where¯Ω=26664¯Ω11∗∗∗∗¯Ω21¯Ω22∗∗∗¯Ω31¯Ω32¯Ω33∗∗¯Ω41¯Ω42¯Ω43¯Ω44∗τN T1τN T2τN T3τN T4−τT377758 >>> >>> >>> >>> >>> < >>> >>> >>> >>> >>> :¯Ω11=S+N1+N T1−M1¯A−¯A T M T1+Q¯Ω21=N2−M2¯A−¯AdT M T1¯Ω22=−S−M2¯Ad−¯A d T M T2¯Ω31=−N T1+N3−K T¯B T M T1−M3¯A¯Ω32=−N T2−K T¯B T M T2−M3¯A d¯Ω33=−N3−N T3−M3¯BK−K T¯B T M T3+K T RK ¯Ω41=N4+M T1−M4¯A+P¯Ω42=M T2−M4¯A d¯Ω43=−N4+M T3−M4¯BK¯Ω44=τT+M4+M T4So if¯Ω<0,then(18)implies V (t)<0.Using Lemma2,we obtain¯Ω=¯Y+M TdF M e+M T e F M d<0⇔¯Y+εM T d M d+ε−1M T e M e<0(19) where¯Y=26664¯Y11∗∗∗∗¯Y21¯Y22∗∗∗¯Y31¯Y32¯Y33∗∗¯Y41¯Y42¯Y43¯Y44∗τN T1τN T2τN T3τN T4−τT377758>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:¯Y11=S+N1+N T1−M1A−A T M T1+Q¯Y21=N2−M2A−A TdM T1¯Y22=−S−M2Ad−A T d M T2¯Y31=−N T1+N3−K T B T M T1−M3A¯Y32=−N T2−K T B T M T2−M3A d¯Y33=−N3−N T3−M3BK−K T B T M T3+K T RK¯Y41=N4+M T1−M4A+P¯Y42=M T2−M4A d¯Y43=−N4+M T3−M4BK¯Y44=τT+M4+M T4M d=−[(M1D)T(M2D)T(M3D)T(M4D)T0]M e=[E E d E b K00]In the sequel,by using Schur complement and defining:M=M1,M2=ρ2M1,M3=ρ3M1,M4=ρ4M1,X=M−1,Y=KX T, P=XP X T, S=XSX T, T=XT X T, N i=XN i X T(i=1,2,3,4),and pre-,post-multiplying both sides with diag(X X X X X I I I)and its transpose,we have(19)⇔Ω<0,thus proof ofV (t)<0.From(18)we can seex T(t)Qx(t)+x T(i k h)K T RKx(i k h)<−V (t)(20)integrating(20)from i k h+τk to i k+1h+τk+1,and usingS∞k=1[i k h+τk,i k+1h+τk+1)=[0,∞),V(∞)=0then weget the(11).Thus we complete the proof of Theorem1.Given d,in order to obtain a controller u(t)=Y X−T x(t),which achieves the least guaranteed cost valueJ∗,we have to solve the following minimization problemMinimize J1+J2+J3subject to(9)(21)whereJ1=ϕT(0)X−1 P X−Tϕ(0)J2=Z0−dϕT(α)X−1 SX−Tϕ(α)dαJ3=ZτZ0−βϕ T(α)X−1 T X−Tϕ (α)dαdβ(22)However,it is noted that the terms J1,J2,J3are notconvex functions of X and P, S, T.As a result,unfortu-nately,we can notfind in general the global minimum ofthe above minimization problem using a convex optimiza-tion algorithm[10].However if we can afford more compu-tational efforts,we can obtain a guaranteed cost controllerachieving a suboptimal guaranteed cost,say J∗so,using aniterative algorithm presented in[11].In the sequel,let us derive the upper bounds on the costfunctions J1,J2,J3.Before starting the problem,let us de-fine matrix valuesΠ1,Π2,Π3,such thatΠ1=ϕ(0)ϕT(0)Π2=Z0−dϕ(α)ϕT(α)dαΠ3=ZτZ0−βϕ (α)ϕ T(α)dαdβ(23)To derive the upper bound on J1,let us introduce a newvariableΛ=ΛT such thatX−1 P X−T<Λ(24)No.2Xie Jin-Song et al.:Guaranteed Cost Controller Design of Networked Control Systems with State Delay173 By Schur complement,(24)is equivalent to»−ΛX−1 X−T− P−1–<0(25)By introducing new values M,¯P,the condition(25)can be replaced by»−ΛM M T−¯P –<0,M=X−1,¯P= P−1(26)Assuming(26),we can conclude that the following in-equality holdsϕT(0)X−1 P X−Tϕ(0)<tr(Π1Λ)(27) To derive the upper bound on J2,let us introduce a new variableΠ=ΠT such thatX−1 SX−T<Π(28) By Schur complement,(28)is equivalent to»−ΠX−1 X−T− S−1–<0(29)By introducing a new value¯S,the condition(29)can bereplaced by»−ΠMM T−¯S –<0,¯S= S−1(30)Assuming(30),we can conclude that the following in-equality holdsZ0−dϕT(α)X−1 SX−Tϕ(α)dα<tr(Π2Π)(31)To derive the upper bound on J3,let us introduce a new variableΞ=ΞT such thatX−1 T X−T<Ξ(32)By Schur complement,(32)is equivalent to»−ΞX−1 X−T− T−1–<0(33)By introducing a new value¯T,the condition(33)can bereplaced by»−ΞMM T−¯T –<0,¯T= T−1(34)Assuming(34),we can conclude that the following in-equality holdsZτ0Z0−βϕ T(α)X−1 T X−Tϕ (α)dαdβ<tr(Π3Ξ)(35)For some constant J,assumetr(Π1Λ)+tr(Π2Π)+tr(Π3Ξ)<J(36) Combining these facts,we can construct a feasibility problem as followsFind P, S, T,X,¯P,¯S,¯T,M,Λ,Π,Ξ,Y, N i(i=1,2,3,4),εSubject to P>0, S>0, T>0,(9)(26)(30)(34)(36)(37)Given d and J,if the above problem has a solution,we can say that there exists a controller u(t)=Y X−T x(t) which guarantees the cost function(4)is less than J∗. Note that the conditions(26)(30)(34)still include non-linear condition,eg.¯P= P−1.However,using the idea in a cone complementary linearization algorithm[11],the above feasibility problem can be solved iteratively.Now, we suggest the following nonlinear minimization problem involving LMI conditions instead of the original non-convex minimization problem in(21)Minimize tr( P¯P+ S¯S+ T¯T+XM)Subject to P>0, S>0, T>0,(9)(36)»−ΛMM T−¯P–<0,»−ΠMM T−¯S–<0»−ΞMM T−¯T–<0,»P II¯P–≥0»S II¯S–≥0,»T II¯T–≥0»X II M–≥0(38)If the solution of the above minimization problem is4n, that is,tr( P¯P+ S¯S+ T¯T+XM)=4n[11],we can say from Theorem1that the system(6)with the controller u(t)=Y X−T x(t)is uniformly asymptotically stable with the guaranteed cost J∗.Although it is still impossible to alwaysfind the globally optimal solution,the proposed nonlinear optimization problem is easier to solve than the original non-convex minimization problem in(21).Actu-ally,utilizing the linearization method in[11],we can easily find a suboptimal minimum of the guaranteed cost using an iterative algorithm presented in the following. Algorithm.Step1.Choose a sufficiently large initial J such that there exists a feasible solution in(38).Set J so=J.Step2.Find a feasible set( P0,¯P0, S0,¯S0, T0,¯T0,X0, M0)satisfying LMIs in(38).Set k=1.Step3.Solve the following LMI problem for the vari-ables( P,¯P, S,¯S, T,¯T,X,M):Minimize tr( P k¯P+ S k¯S+ T k¯T+X k M+ P¯P k+ S¯S k+ T¯Tk+XM k)subject to LMIs in(38).Set P k+1= P, S k+1= S, T k+1= T,X k+1=X,¯P k+1=¯P,¯Sk+1=¯S,¯T k+1=¯T,M k+1=M.Step4.If the condition(24),(28)and(32)are all satis-fied,then set J so=J and return to Step2after decreasing J to some extent.If the condition(24),(28)and(32)are not satisfied within a specified number of iterations,say k max,then exit.Otherwise set k=k+1and go to Step3. 5Numerical exampleExample.Consider the following uncertain time-delay system:x (t)=»1+0.33F(t)0.42F(t)0.53F(t)−2+0.67F(t)–x(t)+»0.50.320−0.5–x(t−d)+»0.470.75–u(t)where d=1,x1(t)=3e t+1−1,x2(t)=0,for t∈[−1,0]174ACTA AUTOMATICA SINICA Vol.33and F2(t)≤1.We consider the cost function(11),set Q=0.5I,R=0.1,ρ2=2,ρ3=3,ρ4=4and use the algorithm in the above,then obtain Table1with d given.From Table1:when d declines a little,the maximum al-lowable valueτmax of the network-induced increases a lot, even so,the system will be asymptotically stable as before. Accordingly,the value J so drops a lot.Of course,we can give Q,R,ρ2,ρ3,ρ4other values,relativeτmax,J so,K will be obtained respectively.Table1When d given then obtainingτmax,J so,Kdτmax J so K10.15469.98[-2.690-1.096]0.50.45126.63[-1.2950.120]6ConclusionIn this paper,wefirstly model NCSs with network delay for a class of uncertain time-delay systems.Based on the model,a guaranteed cost controller design is presented and a sufficient condition for the existence of a guaranteed cost state-feedback for a class of uncertain time-delay systems is given and an algorithm involving a convex optimization is applied to construct a controller with a suboptimal guar-anteed cost such that the system can be stabilized for all admissible uncertainties.At the same time a simulated ex-ample is given to show the effectiveness of the algorithm.References1Walsh G C,Ye H.Scheduling of networked control systems.IEEE Control Systems Magazine,2001,21(1):57∼652Walsh G C,Ye H,Bushnell L G.Stability analysis of net-worked control systems.IEEE Transactions on Control Sys-tems Technology,2002,10(3):438∼4463Zhang W,Branicky M S,Phillips S M.Stability of net-worked control systems.IEEE Control Systems Magazine, 2001,21(1):84∼994Chang S S L,Peng T K C.Adaptive guaranteed cost control of systems with uncertain parameters.IEEE Transactions on Automatica Control,1972,17(4):474∼4835Petersen I R,McFarlane D C.Optimal guaranteed cost con-trol andfiltering for uncertain linear system.IEEE Transac-tions on Automatica Control,1994,39(9):1971∼19776Yu L,Chu J.An LMI approach to guaranteed cost control of linear uncertain time-delay systems.Automatica,1999, 35(6):1155∼11597Mahmoud M S.Control of uncertain state-delay systems: guaranteed cost approach.IMA Journal of Mathematical Control and Information,2001,18(1):109∼1288Esfahani S H,Petersen I R.An LMI approach to the output-feedback guaranteed cost control for uncertain time-delay systems.In:Proceedingss of the37th IEEE Conference on Decision and Control,1998,2:1358∼13639Nian X,Feng J.Guaranteed cost control of a linear uncertain system with multiple time-varying delay:an LMI approach.IEE Proceedings:Control Theory and Applications,2003, 150:17∼2110Ghaoui L E,Oustry F,AitRami M.A cone complementar-ity linearization algorithm for static output-feedback and related problems.In:Proceeding of the American Control Conference Arlington,1997.42(8):1171∼117611Young Sam Lee,Young Soo Moon,Wook Hyun Kwon.Delay-dependent guaranteed cost control for uncertain state-delayed systems.In:Proceedings of American Control Con-ference,2001,5:3376∼3381XIE Jin-Song Received his B.S.andM.S.degrees from Nanjing Normal Univer-sity in1996and2005,respectively.Nowhe is a Ph.D.candidate of the College ofManagement,University of Shanghai forScience&Technology.His research in-terests include transportation system engi-neering,transportation planning,and in-telligent transportation systems.Corre-sponding author of this paper.E-mail:jin-songxie@F AN Bing-Quan Professor and the di-rector of Center of Research on TransportsSystem,University of Shanghai for Science&Technology.His research interests in-clude system engineering,complex trans-portation system analysis,network opti-mization,network trafficflow,transporta-tion planning,and transportation impactanalysis.E-mail:crts-usst@Young Sam Lee Received his Ph.D.de-gree in School of Electrical Engineeringand Computer Science from Seoul NationalUniversity,Seoul,Korea in2003.His re-search interests include time delay systems,receding horizon control,robotics,and em-bedded systems.He is currently with theSchool of Electrical Engineering at InhaUniversity,Inchon,Korea,since September2004.E-mail:lys@inha.ac.krYANG Jin Received her B.S.and M.S.degrees from Nan-jing Normal University in2000and2003,respectively.Now she is a Ph.D candidate of the College of Science,University of Shanghai for Science&Technology.Her research interests include graph theory and combinational optimization.E-mail: yangjin78@。
非线性系统的网络化控制
非线性系统的网络化控制非线性系统的网络化控制摘要随着控制技术与网络信息技术的不断融合，网络化控制系统已经应用于智能交通控制、楼宇自动化和航天器等许多领域。
它的优点有可以资源共享，便于安装、扩展与维护，较高的可靠性。
但也使得数据传输出现了随机时变延时，数据丢失和数据时序颠倒等问题，对系统性能和稳定性造成了不利影响。
本文针对非线性系统的网络化控制进行了关于稳定性、预测控制的分析。
关键词非线性网络化控制；延时；丢包0引言从20世纪90年代NCS出现以来，在国民经济和国防建设等领域迅速得到了应用，随着计算机网络技术的发展，网络和控制结合的技术也日趋成熟，并在实际的工业控制，机器人控制，远程控制等方面得到了广泛应用。
目前研究非线性系统已经有了不少成果，但因为非线性系统本身有着非常复杂的特性，已提出的方法仍有着其局限性，对于一切非线性系统都适用的方法还没有找到。
经典的研究方法主要包括相平面法、描述函数法和李亚普诺夫方法等，后来又出现了微分几何、微分代数和预测方法等理论[13]-。
对于非线性网络化系统的控制，主要有两种方法对非线性系统进行精确线性化，利用线性网络化控制方法对原始的非线性网络化系统进行研究。
另外，还有一种思路就是直接对非线性系统的网络化控制问题进行非线性化方面的研究，此部分的相关文献还比较少，应用的效果也比较局限[46]-。
本文旨在针对非线性网络化控制系统做出分析。
1网络化控制系统的稳定性分析1.1网络化控制系统中的基本问题对稳定性的影响通信协议、驱动方式、单包传输和多包传输，以及网络调度等问题属于网络化控制系统的基本概念范畴，可以引起网络诱导延时、数据包丢失、时序错乱和网络拥塞等出现不同情况，而网络诱导延时、数据包丢失、时序错乱和网络拥塞等问题则又会导致网络化控制系统性能的降低，甚至引起失稳[]7。
网络诱导延时会降低系统的性能，会减小网络化系统的稳定范围，进而影响系统稳定性。
在保证稳定性的前提下，可以有一个比较小的丢包数范围，但如果超过就会导致系统出现失稳现象。