Robust stability criteria for systems with interval time-varying delay and
基于事件触发脉冲控制的变时滞系统稳定性研究
考虑以下非线性变时滞微分方程：
DOI: 10.12677/dsc.2021.101002
14
动力系统与控制
郑焕南，孙文
x (t) = −Cx (t) + Af (x (t)) + Bg (x (t −τ (t))) +U (t)
(1)
= 式中 x (t ) ( x1 (t ),, xn (t ))T ∈ Rn 是方程的状态向量，对角矩阵 C = diag (c1,, cn ) 为正定矩阵，A，B 为
假设 1 [11]对 ∀z1, z2 ∈ Rn 非线性函数 f (⋅), g (⋅) 满足 Lipschitz 条件： f ( z1 ) − f ( z2 ) ≤ u1 z1 − z2 , g ( z1 ) − g ( z2 ) ≤ u2 z1 − z2 ,
u1,u2 > 0 为 Lipschitz 常数。
Dynamical Systems and Control 动力系统与控制, 2021, 10(1), 13-23 Published Online January 2021 in Hans. /journal/dsc https:///10.12677/dsc.2021.101002
基于事件触发脉冲控制的变时滞系统 稳定性研究
郑焕南，孙 文
长江大学信息与数学学院，湖北 荆州
收稿日期：2020年11月30日；录用日期：2020年12月23日；发布日期：2020年12月31日
摘要
本文研究了事件触发脉冲控制下变时滞系统的稳定性问题。首先我们设计了一种由系统状态决定脉冲作 用时刻的事件触发脉冲控制，证明了系统不存在Zeno现象。其次我们得到了变时滞系统在提出的事件触 发脉冲控制下指数稳定的充分条件。最后仿真实验结果验证了理论方案的有效性。
一种改进的最优PID参数自整定控制方法
一种改进的最优PID参数自整定控制方法作者：于洪国,王平来源：《现代电子技术》2010年第19期摘要:为了解决大滞后系统控制难度大的问题,针对非线性、大时变、大延迟的控制对象,设计一种带嵌入式函数的最优PID参数自整定控制方法,通过仿真实验与ISTTE最优准则下最优PID参数自整定控制方法进行了比较,给出仿真波形及其分析,通过仿真实验结果表明该控制方法不仅具有PID控制器高精度、稳定性、鲁棒性高的优点,另外可以缩短系统的调节时间,有效地抑制系统的超调,从而可以有效改善大滞后系统的控制效果。
关键词:大滞后系统; 嵌入式函数; 最优PID参数; 自整定中图分类号:TN919-34文献标识码:A文章编号:1004-373X(2010)19-0162-03Improved Method of Optimal PID Parameter Self-adjusting ControlYU Hong-guo1, WANG Ping2(1. Research Center for Automatic Control, Binzhou University, Binzhou 256600, China;2. College of Information and Control Engineering, China University of Petroleum, Dongying 257061, China)Abstract: In order to solve the difficulty control problem of big lagging systems, an optimal PID parameter self-adjusting control method with embedded function is designed. The control method aims at the systems with nonlinear, large time-method is compared with the optimal PID parameter self-adjusting control methods which conforms to ISTTE optimal standard by a simulation experiment. The simulation wave patterns and analysis are given. The experiment results indicate that the improved control method possess a lot of advantages such as the properties of high-accuracy, high-stability and high-robust. The control method can shorten the sdjusting time of the system and restrain overshoot effectively, and improve the control effect of the big lagging system.Keywords: big lagging system; embedded function; optimal PID parameter; self-adjusting0 引言在一些复杂工业过程中,许多对象具有时滞特性。
Stability and stabilization of nonlinear system-Chapter 8非线性系统稳定性和稳定化
Chapter8Open ProblemsIn this chapter,we would like to give a list of open and unanswered problems in Mathematical Control Theory.The solutions of these open problems will be very important for the development of modern nonlinear control theory.Expectedly novel mathematical analysis and synthesis tools need to be developed to address these challenging problems.The interested reader should also consult the book[3]for other significant and important open problems in Mathematical Control Theory.Open Problem#1Under what conditions WIOS implies IOS?A qualitative characterization of the IOS property for abstract control systems as discussed in this book has not been available yet.For systems described by ODEs, many qualitative characterizations of the ISS and IOS properties are provided in [21–23].Moreover,Theorem4.1in Chap.4gives a complete qualitative character-ization of the WIOS property:“0-GAOS”+“RFC”+“the continuity with respect to initial conditions and external inputs”implies WIOSA similar qualitative characterization for the IOS property in a general context of abstract dynamical systems as discussed in this book will be very important for control designs and applications.Open Problem#2Development of small-gain techniques for dynamical systems described by Partial Differential Equations(PDEs).Small-gain results have been well studied forfinite-dimensional nonlinear sys-tems described by ordinary differential,or difference,equations(see,e.g.,[8–10] and references therein).However,as of today,there is little research devoted to the development of small-gain techniques for nonlinear systems described by Partial Differential Equations(PDEs).We believe that the small-gain results provided in the present book(Theorems5.1and5.2in Chap.5)will pave the road for the appli-cation of small-gain results to systems described by PDEs.I.Karafyllis,Z.-P.Jiang,Stability and Stabilization of Nonlinear Systems,381 Communications and Control Engineering,DOI10.1007/978-0-85729-513-2_8,©Springer-Verlag London Limited2011Open Problem#3Formulas for the Coron–Rosier methodology.Theorem6.1in Chap.6is an existence-type result.Although its proof is con-structive,it cannot be easily applied for feedback design purposes.The creation of formulas for the Coron–Rosier approach will be very significant for control pur-poses,since the Coron–Rosier approach can allow nonconvex control sets and does not require additional properties for the Control Lyapunov Function.The signifi-cance of the solution of this open problem is also noted in[5].Open Problem#4When is a nonlinear,time-varying,time-delay system stabiliz-able?We have recently provided a positive answer to the above question when the sys-tem only involves state-delay[13].A complete answer to the question of when the nonlinear time-varying system with both state and input delays is stabilizable re-mains open and requires deeper investigation.Nonetheless,it should be mentioned that sufficient,but not necessary,conditions for the solution of the stabilization prob-lem with input delays are proposed in the recent work of Krsti´c[14–16](also see [11]).To our knowledge,a necessary and sufficient condition for stabilizability is missing even for linear time-varying systems with input delays.Open Problem#5Application of small-gain results for distributed feedback design of large-scale nonlinear systems.Large-scale systems are abundant in variousfields of science and engineering and have gained increasing attention due to emerging engineering and biomedical applications.Examples of these applications are from smart grids with green and re-newable energy sources,modern transportation networks,and biological networks. There has been some success with the use of decentralized control strategy for both linear and nonlinear large-scale systems;see[7,19]and many references therein. Clearly more remains to be accomplished in this excitingfield.We feel that small-gain is a very appropriate tool for addressing some of these modern-day challenges. The small-gain results of the present book(Theorems5.1and5.2in Chap.5)make a preliminary step forward toward studying some complex large-scale systems be-yond the past literature of decentralized systems and control.Open Problem#6Extension of the discretization approach for autonomous sys-tems.The discretization approach for Lyapunov functionals was described in Chap.2 (Propositions2.4and2.5).However,as remarked in Chap.2,the discretization ap-proach requires good knowledge of some approximation of the solution map,and its use has been restricted to time-varying systems with special structure(see[1,17, 18]).An extension of the discretization approach for autonomous systems wouldbe an important contribution in stability theory because such a result would al-low the use of positive definite functions with non sign-definite derivative.The re-quired extension of the discretization approach must utilize appropriate differential inequalities in the same spirit as the classical Lyapunov’s approach(without requir-ing knowledge of the solution map or a system with special structure).The recent work in[12]is an attempt in this research direction(see also references therein). However,the problem is still completely“untouched.”Open Problem#7Application of feedback design methodologies to other mathe-matical problems.In this book,we have seen the applications of certain tools of modern nonlinear control theory to problems arising from mathematics and economics.Particularly, we have seen•applications of small-gain results to game theory(see Sect.5.5in Chap.5),•applications to numerical analysis(see Sect.7.3).We believe that feedback design methodologies can be applied with success to other areas of mathematical sciences.Fixed Point Theory(see[6])and Optimization Theory can be benefited by the application of certain tools of modern nonlinear con-trol theory.Corollary5.4in Chap.5already shows that small-gain results can have serious consequences in Fixed Point Theory.Further connections between Fixed Point Theory and Stability Theory are provided by the work of Burton(see[4]and references therein)but are in the opposite direction from what we propose,that is, the work of Burton applies results from Fixed Point Theory to Stability Theory.The efforts for the solution of problems in Game Theory,Numerical Analysis, Fixed Point Theory,and Optimization Theory will necessarily demand the creation of novel results in stability theory and feedback stabilization theory.Therefore,the application of modern nonlinear control theory to other areas of applied mathe-matics will result to a“knowledge feedback mechanism”between Mathematical Control Theory and other areas in mathematics!Open Problem#8Integral input-to-state stability(for short,iISS)in complex dy-namical systems.The external stability results of this book are exclusively targeted at extensions of Sontag’s ISS property and its variants to a very general context of complex dynamic systems.That is,we want to address a wide class of dynamical systems which may not satisfy the semigroup property,motivated by important examples of hybrid sys-tems,switched systems,and time-delay systems.It remains an open and important, but interesting,question to know how much we could do with the iISS property introduced in[2,20].References1.Aeyels,D.,Peuteman,J.:A new asymptotic stability criterion for nonlinear time-variant dif-ferential equations.IEEE Transactions on Automatic Control43(7),968–971(1998)2.Angeli,D.,Sontag,E.D.,Wang,Y.:A characterization of integral input-to-state stability.IEEETransactions on Automatic Control45(6),1082–1097(2000)3.Blondel,V.D.,Megretski,A.(eds.):Unsolved Problems in Mathematical Systems and ControlTheory.Princeton University Press,Princeton(2004)4.Burton,T.A.:Stability by Fixed Point Theory for Functional Differential Equations.Dover,Mineola(2006)5.Coron,J.-M.:Control and Nonlinearity.Mathematical Surveys and Monographs,vol.136.AMS,Providence(2007)6.Granas,A.,Dugundji,J.:Fixed Point Theory.Springer Monographs in Mathematics.Springer,New York(2003)7.Jiang,Z.P.:Decentralized control for large-scale nonlinear systems:A review of recent results.Dynamics of Continuous,Discrete and Impulsive Systems11,537–552(2004).Special Issue in honor of Prof.Siljak’s70th birthday8.Jiang,Z.P.:Control of interconnected nonlinear systems:a small-gain viewpoint.In:deQueiroz,M.,Malisoff,M.,Wolenski,P.(eds.)Optimal Control,Stabilization,and Nonsmooth Analysis.Lecture Notes in Control and Information Sciences,vol.301,pp.183–195.Springer, Heidelberg(2004)9.Jiang,Z.P.,Mareels,I.M.Y.:A small-gain control method for nonlinear cascaded systems withdynamic uncertainties.IEEE Transactions on Automatic Control42,292–308(1997)10.Jiang,Z.P.,Teel,A.,Praly,L.:Small-gain theorems for ISS systems and applications.Mathe-matics of Control,Signals,and Systems7,95–120(1994)11.Karafyllis,I.:Stabilization by means of approximate predictors for systems with delayed in-put.To appear in SIAM Journal on Control and Optimization12.Karafyllis,I.:Can we prove stability by using a positive definite function with non sign-definite derivative?Submitted to Nonlinear Analysis Theory,Methods and Applications 13.Karafyllis,I.,Jiang,Z.P.:Necessary and sufficient Lyapunov-like conditions for robustnonlinear stabilization.ESAIM:Control,Optimization and Calculus of Variations(2009).doi:10.1051/cocv/2009029,pp.1–42,August200914.Krsti´c,M.:Delay Compensation for Nonlinear,Adaptive,and PDE Systems.Systems&Con-trol:Foundations&Applications.Birkhäuser,Boston(2009)15.Krsti´c,M.:Input delay compensation for forward complete and feedforward nonlinear sys-tems.IEEE Transactions on Automatic Control55,287–303(2010)16.Krsti´c,M.:Lyapunov stability of linear predictor feedback for time-varying input delay.IEEETransactions on Automatic Control55,554–559(2010)17.Peuteman,J.,Aeyels,D.:Exponential stability of slowly time-varying nonlinear systems.Mathematics of Control,Signals and Systems15,42–70(2002)18.Peuteman,J.,Aeyels,D.:Exponential stability of nonlinear time-varying differential equationsand partial averaging.Mathematics of Control,Signals and Systems15,202–228(2002)19.Siljak,D.:Decentralized Control of Complex Systems.Academic Press,New York(1991)20.Sontag,E.D.:Comments on integral variants of ISS.Systems Control Letters3(1–2),93–100(1998)21.Sontag,E.D.,Wang,Y.:On characterizations of the input-to-state stability property.Systemsand Control Letters24,351–359(1995)22.Sontag,E.D.,Wang,Y.:New characterizations of the input-to-state stability.IEEE Transac-tions on Automatic Control41,1283–1294(1996)23.Sontag,E.D.,Wang,Y.:Lyapunov characterizations of input to output stability.SIAM Journalon Control and Optimization39,226–249(2001)。
Robust Control
Robust ControlRobust control is a critical aspect of engineering and technology,particularly in the field of control systems. It refers to the ability of a system to maintain stability and performance in the presence of uncertainties and variations. This is crucial in ensuring the reliability and safety of various engineering systems, including aircraft, automobiles, industrial processes, and robotics. In this response, we will explore the importance of robust control from various perspectives, including its significance in different industries, the challenges and limitations associated with it, and the advancements and future prospects in this field. From an engineering perspective, robust control plays a vital role in ensuring the stability and performance of complex systems. In industries such as aerospace and automotive, where safety is of utmost importance, robust control techniques are essential for dealing with uncertainties and disturbances that can affect the behavior of the system. For example, in aircraft control systems, robust control is used to ensure that the aircraft remains stable and responsive to pilot commands, even in the presence of external disturbances such as turbulence or engine failures. Similarly, in automotive systems, robust control is crucial for maintaining the stability and handling of vehicles, especially in challenging road conditions or during emergency maneuvers. Moreover, robust control is also essential in industrial processes, where the stability and performance of control systems can have a significant impact on the efficiency and safety of the entire operation. For instance, in chemical plants and manufacturing facilities, robust control techniques are used to ensure the stability of various processes, such as temperature control, pressure regulation, and flow management. By incorporating robust control strategies, engineers can mitigate the effects of uncertainties and disturbances, thereby improving the overall reliability and safety of the industrial processes. Despite its significance, robust control also poses several challenges and limitations. One of the primary challenges is the complexity of modeling uncertainties and variations in real-world systems. Engineering systems are often subject to various sources of uncertainties,including parameter variations, external disturbances, and modeling errors. Addressing these uncertainties and designing robust controllers that caneffectively handle them requires a deep understanding of the system dynamics and robust control theory. Furthermore, the implementation of robust controlstrategies can be challenging, especially in systems with nonlinear dynamics or time-varying parameters. Designing robust controllers that can guarantee stability and performance under such conditions requires advanced mathematical tools and computational techniques. Another limitation of robust control is the trade-off between performance and robustness. In many cases, achieving robustness may come at the expense of performance, and vice versa. Designing a controller that can effectively balance these trade-offs is a non-trivial task, as it requires careful consideration of the system's specifications and operational requirements. Moreover, the robustness of a control system is often evaluated under specific assumptions and operating conditions, which may not always reflect the real-world scenarios. As a result, there is a need for robust control techniques that can adapt to varying operating conditions and uncertainties, while still maintaining the desired level of performance. Despite these challenges and limitations, significant advancements have been made in the field of robust control, paving the way for new opportunities and future prospects. One of the key advancements is the development of advanced control techniques, such as H-infinity control, mu-synthesis, and robust model predictive control. These techniques offer more robustness and performance guarantees compared to traditional control methods, making them well-suited for complex and uncertain systems. Furthermore, the integration of advanced computational tools, such as optimization algorithms and machine learning, has enabled engineers to design robust controllers that can adapt to changing operating conditions and uncertainties. Moreover, the emergence of new technologies, such as cyber-physical systems and Internet of Things (IoT), has opened up new avenues for applying robust control in various domains. These technologies enable the integration of sensors, actuators, and control systemsinto interconnected networks, allowing for real-time monitoring and control of complex systems. By leveraging the capabilities of these technologies, engineers can develop robust control solutions that are more adaptive and responsive to uncertainties, thereby enhancing the reliability and safety of critical systems. In conclusion, robust control is a fundamental aspect of engineering andtechnology, with far-reaching implications for various industries and applications. From aerospace and automotive systems to industrial processes and beyond, theability to maintain stability and performance in the presence of uncertainties is crucial for ensuring the reliability and safety of complex systems. While robust control poses several challenges and limitations, significant advancements and future prospects offer new opportunities for addressing these challenges andfurther advancing the field. As engineers continue to push the boundaries ofrobust control, the potential for enhancing the performance and reliability of critical systems remains promising.。
改进Lyapunov泛函下变时滞系统稳定性分析
改进Lyapunov泛函下变时滞系统稳定性分析王新伟;张颖;戎玉密【期刊名称】《深圳大学学报（理工版）》【年(卷),期】2017(034)002【摘要】We provide a new delay-range-dependent criterion for interval time-varying delay linear systems.By using Jensen inequality method and reciprocally convex combination technique,the upper bound of the derivative of our improved Lyapunov function can be estimated more tightly.And then newless conservative stability criteria arederived.Numerical examples are given to illustrate the effectiveness and the improvementof the proposed criterion.%针对具有区间变时滞的线性系统,提出一种新的稳定性判据.基于改进Lyapunov泛函,采用Jensen积分不等式和倒数凸组合技术,对Lyapunov泛函导数中的积分项进行界定,获得更紧的时滞上界,从而得到保守性更低的稳定性判据.通过数值实例验证所提出的稳定性判据有效.【总页数】7页(P181-187)【作者】王新伟;张颖;戎玉密【作者单位】深圳信息职业技术学院机电工程学院,广东深圳518172;哈尔滨工业大学深圳研究生院,广东深圳518055;哈尔滨工业大学深圳研究生院,广东深圳518055【正文语种】中文【中图分类】TP13【相关文献】1.变时滞不确定Lurie系统的时滞分布依赖鲁棒稳定性分析 [J], MUKHIJA Pankaj;KAR Indra Narayan;BHATT Rajeudra Kumar Purushottam2.基于增广Lyapunov泛函的不确定时滞系统的时滞相关鲁棒H∞控制 [J], 刘小梅;王天成;李圣涛3.时滞系统稳定性分析——齐次多项式Lyapunov泛函方法 [J], 刘兴文4.改进的多时滞Lurie控制系统时滞相关绝对稳定性分析 [J], 白宇;徐兆棣;邓超;刘畅5.基于改进的Lyapunov泛函的时变时滞系统稳定判据 [J], 李佳; 陈刚; 肖会芹因版权原因，仅展示原文概要，查看原文内容请购买。
非参数不确定多智能体系统一致性误差跟踪学习控制
非参数不确定多智能体系统一致性误差跟踪学习控制严求真;孙明轩;李鹤【摘要】This paper presents a consensus-error-tracking iterative learning control method to tackle the consensus problem for a class of leader-following non-parametric uncertain multi-agent systems, which perform a given repetitive task over a finite interval with arbitrary initial error. The iterative learning controllers are designed by applying Lyapunov synthesis. As the iteration increases, each following multi-agent’s consensus-error can track its desired consensus-error trajectory, and the all following multi-agents’ states perfectly track the leader’s state on the specified interval. The robust learning technique is applied to deal with the nonparametric uncertainties, and the hyperbolic tangent function is used to design feedback terms, in order to compensate the cycle-varying but bounded uncertainty. Numerical results demonstrate the effectiveness of the learning control scheme.%针对一类在有限时间区间上执行重复任务的主−从型非参数不确定多智能体系统，提出一致性误差跟踪学习控制方法，用于解决在任意初始误差情形下的一致性问题。
一种含扰动项的非线性系统执行器故障估计方法
收稿日期：2020-01-11修回日期：2020-03-11基金项目：2019年辽宁省教育厅科学研究经费资助项目（L201906）作者简介：孙延修（1981-），男，河北邯郸人，硕士，副教授。
研究方向：非线性系统观测器。
*摘要：现代控制系统的安全性与可靠性是各领域研究的热点之一，系统故障诊断与估计的方法越来越引起人们的重视。
针对一类含有外部扰动的非线性系统，研究了系统执行器故障估计问题。
通过设计一种增广系统观测器对原系统中的执行器故障进行估计；考虑到系统中的非线性扰动项，利用线性矩阵不等式（LMI ）方法给出了观测器存在的充分条件并保证误差系统渐近稳定。
同时，通过设定性能指标减少了外部扰动对执行器故障估计的影响；通过数值算例验证了执行器故障估计方法的有效性，表明该估计方法能够较好的对系统中的执行器故障进行鲁棒估计。
关键词：外部扰动，非线性系统，执行器故障，线性矩阵不等式，状态观测器中图分类号：TP391.9文献标识码：ADOI ：10.3969/j.issn.1002-0640.2021.03.007引用格式：孙延修.一种含扰动项的非线性系统执行器故障估计方法［J ］.火力与指挥控制，2021，46（3）：38-42.一种含扰动项的非线性系统执行器故障估计方法*孙延修（沈阳工学院基础课部，辽宁抚顺113122）A Method of Actuator Fault Estimation forNonlinear System With Disturbance TermSUN Yan-xiu（Basic Course Department ，Shenyang Institute of Technology ，Fushun 113122，China ）Abstract ：The safety and reliability of modern control system is one of the hotspots in variousfields ，more and more attention has been paid to the fault diagnosis and estimation of the system.For a class of nonlinear systems with external disturbances ，the problem of actuator fault estimation is studied.Firstly ，an augmented system observer is designed to estimate actuator faults in the original system ；Secondly ，considering the nonlinear disturbance term in the system ，the sufficient conditions for the existence of the observer are given by using the LMI method to ensure the asymptotic stability of the error system ，at the same time ，the influence of external disturbance on actuator fault estimation is reduced by setting performance index.Finally ，the effectiveness of the actuator fault estimation method is verified by a numerical example ，which shows that the method can estimate the actuator fault in thesystem robustly.Key words ：external disturbance ，nonlinear system ，actuator failure ，linear matrix inequality ；state observerCitation format ：SUN Y X.A method of actuator fault estimation for nonlinear system with disturbance term ［J ］.Fire Control &Command Control ，2021，46（3）：38-42.0引言目前，控制系统对稳定性及可靠性的要求越来越高。
Robust Control
Robust ControlRobust control is a critical concept in the field of engineering and technology, particularly in the realm of control systems. It refers to the ability of a control system to maintain stable and satisfactory performance in the face of various uncertainties and disturbances. This is a highly important aspect ofcontrol system design, as real-world systems are often subject to a wide range of uncertainties, such as changes in operating conditions, variations in parameters, and external disturbances. In this response, we will explore the significance of robust control from various perspectives, including its importance in different engineering applications, its challenges and limitations, and the strategies and techniques used to achieve robust control. From an engineering perspective,robust control plays a crucial role in ensuring the stability and performance of control systems in a variety of applications. For instance, in the field of aerospace engineering, where systems are often subject to unpredictable environmental conditions and disturbances, robust control is essential forensuring the safety and reliability of aircraft and spacecraft. Similarly, in the realm of automotive engineering, robust control is vital for the stability and performance of vehicle control systems, particularly in the presence of uncertainties such as varying road conditions and external disturbances. In industrial automation and manufacturing, robust control is also critical for maintaining the stability and efficiency of production processes, where uncertainties in operating conditions and disturbances can have significantimpacts on product quality and process reliability. One of the key challenges in achieving robust control is the inherent uncertainties and variations present in real-world systems. These uncertainties can arise from various sources, including variations in system parameters, modeling errors, external disturbances, and environmental changes. Traditional control design techniques, which rely onprecise mathematical models and assumptions of known system dynamics, may not be sufficient to address these uncertainties. As a result, robust control requiresthe development of innovative strategies and techniques that can effectivelyhandle and mitigate the effects of uncertainties on system performance. This often involves the use of advanced control design methods, such as robust control theory,adaptive control, and model predictive control, which are specifically tailored to address uncertainties and variations in system dynamics. In addition to uncertainties, another significant challenge in achieving robust control is the trade-off between performance and robustness. In many cases, improving the robustness of a control system may come at the expense of its performance, andvice versa. For example, increasing the robustness of a control system to handle uncertainties and disturbances may result in slower response times or reduced control accuracy. Conversely, optimizing a control system for high performance may make it more susceptible to uncertainties and disturbances. Balancing these trade-offs is a complex and delicate task that requires careful consideration of the specific requirements and constraints of the application at hand. Engineers and control system designers must carefully weigh the trade-offs between performance and robustness to achieve an optimal balance that meets the desired performanceand stability criteria. To address these challenges, engineers and researchers have developed a wide range of techniques and strategies for achieving robust control in various engineering applications. These include robust control design methods, such as H-infinity control, mu-synthesis, and loop shaping, which are specifically tailored to handle uncertainties and variations in system dynamics. These methods leverage advanced mathematical tools and optimization techniques to design control systems that are inherently robust to uncertainties and disturbances. Additionally, techniques such as adaptive control and modelpredictive control enable control systems to adapt and adjust their behavior inreal time to accommodate changes in system dynamics and operating conditions, further enhancing their robustness and performance. Furthermore, the emergence of advanced technologies, such as machine learning and artificial intelligence, has opened up new opportunities for achieving robust control in complex and uncertain environments. These technologies enable control systems to learn from data and experience, and to adapt their behavior based on real-time observations and feedback. By leveraging machine learning and AI techniques, control systems can enhance their robustness and adaptability to uncertainties and disturbances, ultimately improving their performance and reliability in real-world applications. In conclusion, robust control is a critical concept in the field of engineeringand technology, with far-reaching implications for the stability and performance of control systems in various applications. Achieving robust control poses significant challenges, including handling uncertainties, balancing trade-offs between performance and robustness, and developing effective strategies and techniques for robust control design. However, with the advancement of control theory, advanced mathematical tools, and emerging technologies, engineers and researchers are well-equipped to address these challenges and to develop innovative solutions for achieving robust control in complex and uncertain environments. As the demand for high-performance and reliable control systems continues to grow in various engineering applications, the pursuit of robust control will remain a central focus of research and development in the field of control systems engineering.。
Robust Control
Robust ControlRobust control is a critical concept in engineering and technology,particularly in the field of control systems. It pertains to the ability of a system to maintain stable and satisfactory performance despite uncertainties and variations in its parameters and environment. This is a significant concern in various engineering applications, such as aerospace, automotive, robotics, and industrial automation, where the performance and safety of the system must be guaranteed under diverse operating conditions. One of the key perspectives in understanding robust control is its relevance in ensuring the stability and reliability of control systems. In many real-world scenarios, control systems are subjected to uncertainties and disturbances, which can significantly impact their performance. Robust control techniques, such as H-infinity control and mu-synthesis, provide methodologies to design controllers that can effectively handle these uncertainties, thereby ensuring stability and performance robustness. This is particularly crucial in safety-critical applications, where the consequences of system failure can be catastrophic. Moreover, robust control plays a pivotal role in addressing the challenges posed by nonlinearity and time-varying dynamics in control systems. Traditional control design methods based on linear models may not suffice to capture the complex behavior of many real-world systems. Robust control techniques offer a systematic framework to account for these nonlinearities and variations, enabling the development of controllers that exhibit resilience to such dynamics. By doing so, robust control facilitates the deployment of control systems in diverse applications with varying operating conditions, without necessitating extensive redesign efforts. Another significant perspective to consider is the impact of robust control on the advancement of emerging technologies. With the rapid evolution of autonomous systems, smart manufacturing, and unmanned vehicles, there is a growing demand for control solutions that can adapt to dynamic and uncertain environments. Robust control methodologies provide a pathway to address these challenges, offering a level of assurance in the performance and safety of these advanced technologies. By incorporating robust control principles into the design and development of such systems, engineers can enhance their resilience to unforeseen conditions, thereby accelerating thedeployment and adoption of these innovative technologies. Furthermore, the integration of robust control techniques with modern control paradigms, such as adaptive and learning-based control, presents an intriguing avenue for research and development. By combining the strengths of robust control in handling uncertainties with the capabilities of adaptive and learning-based approaches to adjust to changing environments, it is possible to create control systems that exhibit a high degree of flexibility and robustness. This convergence of methodologies holds promise in addressing complex control challenges in dynamic and uncertain environments, opening new possibilities for applications in fields such as healthcare, energy, and environmental monitoring. In conclusion, robust control stands as a cornerstone in the realm of control systems, offering indispensable tools and methodologies to address the challenges posed by uncertainties, variations, and nonlinearities. Its significance spans across diverse domains, from ensuring the stability and reliability of critical engineering systems to enabling the advancement of emerging technologies. As the landscape of engineering and technology continues to evolve, the role of robust control is poised to expand, driving innovation and resilience in the design and deployment of next-generation control systems.。
一类新的Takagi-Sugeno模糊时滞系统的稳定性准则
一类新的Takagi-Sugeno模糊时滞系统的稳定性准则贾茹;汪刚;宋华东【摘要】本文提出一种新的时滞划分方法—变时滞划分法,以解决连续延时Takagi-Sugeno模糊系统的稳定性和镇定性问题.不同于已有的文献,用可变参数将时变时滞区间[0,d(t)]划分为若干个可变子区间,并得出模糊时滞系统的新的时滞相关稳定性准则.本文提出的新方法能充分利用时滞子区间的内部信息,因此新的时滞相关稳定性准则比以往结果具有更小的保守性.基于Lyapunov稳定性理论,以线性矩阵不等式形式给出T--S模糊系统的新的时滞相关稳定性准则,并将稳定性和镇定性研究结果扩展到具有不确定参数的T--S模糊系统.仿真实例证明了本文方法降低保守性的有效性.%A new method,namely,the variable delay partitioning method,is firstly developed to solve the problems of stability analysis and stabilization for continuous time-delay Takagi-Sugeno fuzzysystems.Different from previous results,the delay interval[0,d(t)]is partitioned into some variable subintervals by employing variable delay partitioning method.Thus,new delay-dependent stability criteria for fuzzy time-delay systems are derived by applying this variable delay partitioning method.The proposed method can make full use of the variable subintervals information,so the new delay-dependent stability criteria are less conservative than previous results.Based on the Lyapunov stability theory,a new delay-dependent stability criterion for T--S fuzzy systems is given in the form of linear matrix inequalities(LMIs).Both the stability and stabilization results are further extended to fuzzy time-delay systems withtime-varying parameter uncertainties.Illustrative examples are provided to demonstrate the effectiveness for conservatism reduction.【期刊名称】《控制理论与应用》【年(卷),期】2018(035)003【总页数】7页(P317-323)【关键词】Takagi-Sugeno(T--S)模糊系统;变时滞划分;时滞相关稳定性;稳定性准则;不确定性【作者】贾茹;汪刚;宋华东【作者单位】东北大学信息科学与工程学院,辽宁沈阳110003;东北大学信息科学与工程学院,辽宁沈阳110003;中国机械(装备)集团公司沈阳仪表科学研究院,辽宁沈阳110043【正文语种】中文【中图分类】TP2731 引言(Introduction)大多数物理系统和工业过程都可以用T--S(Takagi-Sugeno)模型近似描述,以T--S 模型为基础的控制已经成为解决复杂非线性系统分析与综合问题的有效方法.T--S 模糊模型将全局非线性系统通过模糊划分建立多个简单的线性关系,通过局部线性输入输出关系的IF-THEN规则,实现在任意精度上逼近任何光滑的非线性系统.这种结合使得T--S模糊系统和线性系统相似,因此可以利用线性系统的丰富成果得到它的稳定性分析和综合.在过去的十几年里,T--S模糊系统稳定性的研究受到了学者们的广泛关注,并获得了大量成果[1-2].例如稳定性分析[3]、观测器设计[4]、H∞控制器设计[5-6]等.由于时滞现象往往存在于大多数的动态系统当中,如化学工艺、冶金工艺、生物系统等,时滞现象的存在是造成系统性能变差甚至于不稳定的重要因素之一,所以,对时滞系统的稳定性分析不仅具有重要的理论意义,而且还具有很强的实用价值.目前,有两种有效的方法处理时滞问题：自由加权矩阵法[7]和增广Lyapunov函数法[8-9].但是,在自由加权矩阵法中,由于Lyapunov函数中-·x(s)dx项仅考虑了时滞函数d(t)的上限和下限而略显保守,式中d(t)∈[0,τ].在增广Lyapunov函数法中,文献[9]使用了一个新的增广Lyapunov函数,其中包含了一个更广义的含有状态向量的积分项,从而降低了系统的保守性.近来,一种对时滞进行等分的方法被提出来[10],通过利用时滞内部的信息来减小系统的保守性.文献[11]借助于输入输出法研究了T--S模糊时滞系统的稳定性并进行了控制器设计.在具有时变时滞d(t)的T--S模糊系统中,通常将时滞区间[0,d(t)]看成是一个单一的区间.本文研究的目的是如何利用[0,d(t)]里的信息来进一步降低系统的保守性.通过将区间[0,d(t)]划分为两个子区间[0,md(t)]和[md(t),d(t)],或K+1个子区间[0,m1d(t)],[m1d(t),m2d(t)],···,[mKd(t),d(t)],式中：m∈(0,1),m1＜m2＜···＜mK,即变时滞划分法,研究T--S模糊时滞系统的稳定性.该方法具有以下特点：1)将时滞函数d(t)划分为若干个子区间,而不是作为整体来处理,因此,可以充分利用[0,d(t)]区间的更多内部信息;2)[0,d(t)]的区间划分可变,而不是等区间划分,因此具有更多灵活性,更适合处理时变时滞函数d(t);3)稳定性结果与时滞划分参数有关,当时滞划分参数改变时,相应的稳定性准则也会不同,通过优化时滞划分参数,可以大大降低系统的保守性.本文基于变时滞划分方法研究了T--S模糊时滞系统的稳定性问题,基于Lyapunov 稳定性理论,以线性矩阵不等式形式给出了T--S模糊系统的新的时滞相关稳定性准则,并进一步研究了带有不确定性参数的T--S模糊系统的稳定性.仿真结果证明了本文方法的有效性.2 系统描述(System description)考虑一类具有不确定参数的非线性时滞系统,其T--S模型为Rulei：IFz1(t)isMi1andz2(t)isMi2and ···andzp(t)isMip,THEN式中：x(t)∈Rn是状态向量,u(t)∈Rp是控制输入变量,Mij(i=1,2,···,r,j=1,2,···,p)是模糊集,r是IF-THEN规则数,z(t)=[z1(t)z2(t) ···zp(t)]是已知的前件变量,φ(t)是[-τ,0]上的初始函数,时变时滞d(t)满足式中τ＞0和µ是两个标量.假设前件变量z(t)与控制输入变量u(t)无关,Ai,Adi,Bi是具有适当维数的常数矩阵,ΔAi(t),ΔAdi(t),ΔBi(t)是范数有界的不确定性且满足式中：Dai,Ddi,Dbi,Nai,Ndi,Nbi是具有适当维数的常数矩阵,F(t)是具有Lebesgue可测元的未知函数矩阵且满足通过中心平均解模糊化,乘积推理机和单值模糊器,式(1)可以描述为式中：Mij(zj(t))是前件变量zj(t)隶属于模糊集Mij的隶属度函数,对任意t有式(5)对应的标称系统可表示为利用并行分布补偿(parallel distributed compensation,PDC)原理设计模糊控制器是一种简便有效的方法.对于式(1)所示的系统,其相应的模糊状态反馈控制规则为Rulei：IFz1(t)isMi1andz2(t)isMi2and ···andzp(t)isMip,THEN式中Gi是模糊状态反馈控制器的增益矩阵.模糊状态反馈控制器的模型为将式(9)代入式(5)和式(7),分别得到带有不确定参数的闭环系统和标称闭环系统.带有不确定参数的闭环系统为标称闭环系统为下面,首先给出推导需要的定理1和引理1.定理1 给定具有适当维数的常数矩阵χ1,χ2,α,则不等式成立的充要条件是式中：σ∈[0,β],β＞0.证 1)必要性证明：令矩阵ϖ1和ϖ2满足则有即由于β＞0,所以2)充分性证明：当σ∈[0,β],有分别令σ=0和σ=β,可得证毕.引理1[12] 假设M,E,F是具有适当维数的实矩阵,F满足FTF≤I,那么,对任意的实数ε＞0,有MFE+ETFTMT ≤ εMMT+ε-1ETE.3 主要结果及证明(Main results and proofs)本节将时变时滞区间[0,d(t)]划分为2个动态区间,即[0,md(t)],[md(t),d(t)](0＜m＜1),并构建适当的Lyapunov-Krasovskii泛函,得到了标称闭环系统和带有不确定参数的闭环系统的稳定性准则.定理2 对于给定的参数0＜m＜1,如果存在适当维数的矩阵式中：m为已知参数,P,Y,Q,R,Z,W等参数为未知参数,满足线性矩阵不等式式中：则系统(11)是渐近稳定的.式中控制器选择式(9).证构建Lyapunov-Krasovskii函数式中：其中：计算V(t)沿着系统(11)对t的导数,得所以式中：其中M=-(1-mµ)Q(t-md(t)).根据牛顿-莱布尼兹公式,对适当维数的矩阵M,N,S有所以,综合式(22)-(25),可以得到式中：由于Z＞0,所以式(26)的后3项小于0.如果那么.又因为式(26)可以写作式中根据定理1,令那么,当d(t)=τ和d(t)=0时,Lijlk＜0等价于应用Schur补引理[13-14],式(28)等价于式(17),式(29)等价于式(18).因此,如果式(17)和式(18)成立,那么˙V(t)＜0,即式(11)是渐近稳定的.证毕.定理3 对于给定的参数0＜m＜1,如果存在适当维数的矩阵标量,和,满足线性矩阵不等式式中：Υijlk和Fijlk同定理2中的定义,则系统(10)是渐近稳定的.式中控制器选择式(9). 证对比系统(10)和系统(11)发现,如果把系统(11)中的Ai,Bi,Adi换成Ai+ΔAi(t),Bi+ΔBi(t),Adi+ΔAdi(t),就可以得到系统(10).因此,可以通过定理2推导系统(10)的稳定性.将式(17)-(18)中的Ai,Bi,Adi替换成Ai+ΔAi(t),Bi+ΔBi(t),Adi+ΔAdi(t),根据式(3),可以得到和根据引理1,式(32)-(33)成立分别等价于和由Schur补引理[13-14],式(34)和式(30)等价,式(35)和式(31)等价. 证毕.实际上,划分变量m可以取多个值mi(i=1,2,···,K),将区间分为更多的子区间,进而获得更小的保守性.本文将在以后的文献中研究该问题.4 数值实例(Examples)利用MATLAB中的LMI工具箱,通过仿真实验可以找到最大允许时滞τ的上限.为了和现有文献比较,证明本文提出方法的有效性,本文给出以下实例.例1 考虑具有不确定性的T--S模糊系统式中：并且Na1=[0 0],Na2=[-0.05 0],Ndi=[0 0],Nbi=0.03,i=1,2.表1为在保证系统稳定的条件下,系统(36)允许的最大时滞τmax对比.表1 本文与部分参考文献结果比较Table 1 Comparison results with several references由表1可以明显看出,本文提出的方法具有更大的时滞上限,从而具有更小的保守性.仿真中,划分变量m的求解方法见参考文献[20].5 结论(Conclusions)本文应用变时滞划分法研究了T--S模糊时滞系统的稳定性和镇定性,并以LMI形式给出了稳定性准则.通过将时滞区间划分为不等分子区间,找到划分变量m的最优解,可以获得更大的时滞允许上限,从而具有更小的保守性.仿真结果表明,同已有文献对比,时滞最大值明显提高,证明了本文方法具有更好的结果.参考文献(References)：[1]DENG Z H,CHOI K S,CHUNG F L,et al.Scalable TSK fuzzy modeling for very large datasets using minimal-enclosing-ball approximation[J].IEEE Transactoins on Fuzzy Systems,2011,19(2)：210-226.[2]CHEN B S,WU C H.Robust optimal reference-tracking design method for stochastic synthetic biology systems：T--S fuzzy approach[J].IEEE Transactions on Fuzzy Systems,2010,18(6)：1144-1159.[3]CHEN Guoyang,LI Ning,LI Shaoyuan.Stability analysis and design for a class of Takagi-Sugeno fuzzy control systems[J].ControlTheory&Applications,2010,27(3)：310-316.(陈国洋,李柠,李少远.一类T--S模糊控制系统的稳定性分析及设计[J].控制理论与应用,2010,27(3)：310-316.) [4]ZHANG H G,YANG J,SU C Y.T--S fuzzy-model-based robustH∞design for networked control systems with uncertainties[J].IEEE Transactions on Industry Applications,2007,3(4)：289-301.[5]LIN C,WANG Q G,LEE T H.Improvement on observer-basedH∞control for T--S fuzzy systems[J].Automatica,2005,41(9)：1651-1656.[6]ZHANG H G,LUN S X,LIU D R.FuzzyH∞f i lter design for a class of nonlinear discrete-time systems with multiple time delays[J].IEEE Transactions on Fuzzy Systems,2007,15(3)：453-469.[7]SOUZA F O,MOZELLI L A,PALHARES R M.On stability and stabilization of T--S fuzzy time-delayed systems[J].IEEE Transactions on Fuzzy Systems,2009,17(6)：1450-1455.[8]ZHANG H G,GONG D W,CHEN B,et al.Synchronization for coupled neural networks with interval delay：a novel augmented Lyapunov-Krasovskii functional method[J].IEEE Transactions on Neural Networks and Learning Systems,2013,24(1)：58-70.[9]YIN Zuoyou,ZHANG Huaguang.Robust tolerant control for nonlinear systems with uncertainties and time delays based on fuzzymodels[J].Control Theory&Applications,2009,26(6)：683-686.(尹作友,张化光.基于模糊模型的非线性不确定时滞系统的H∞鲁棒容错控制[J].控制理论与应用,2009,26(6)：683-686.)[10]WANG G,LIU J H,LU S X.Stability analysis and stabilization for fuzzy hyperbolic time-delay system based on delay partitioningapproach[J].Neurocomputing,2016,214(4)：555-566.[11]ZHAO L,GAO H J,KARIMI H R.Robust stability and stabilization of uncertain T--S fuzzy systems with time-varying delay：an inputoutput approach[J].IEEE Transactions on Fuzzy Systems,2013,21(5)：883-897. [12]WANG Y,XIE L H,DE SOUZA C E.Robust control of a class of uncertain nonlinear systems[J].Systems&Control Letters,1992,19(2)：139-149. [13]BOYD S,EL GHAOUI L,FERON E.Linear Matrix Inequalities in System and Control Theory[M].Philadelphia,PA：SIAM,1994.[14]YU Li.Robust Control—Linear Matrix Inequality ProcessingMethod[M].Beijing：Tsinghua University Press,2002.(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.)[15]WU H N,LI H X.New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with timevaryingdelay[J].IEEE Transactions on Fuzzy Systems,2007,15(3)：482-493.[16]ZHAO Y,GAO H J,LAM J,et al.Stability and stabilization of delayed T--S fuzzy systems：a delay partitioning approach[J].IEEE Transactions on Fuzzy Systems,2009,17(4)：750-762.[17]ZHANG Z,LIN C,CHEN B.New stability and stabilization conditions for T--S fuzzy systems with time delay[J].Fuzzy Sets Systems,2015,263(1)：82-91.[18]ZHAO L,GAO H J,KARIMI H R.Robust stability and stabilization of uncertain T--S fuzzy systems with time-varying delay：an inputoutput approach[J].IEEE Transactions on Fuzzy Systems,2013,21(5)：883-897. [19]TSAI S H,CHEN Y A,LO J C.A novel stabilization condition for a class of T--S fuzzy time-delay systems[J].Neurocomputing,2016,175(1)：223-232.[20]ZHANG H G,LIU Z W,HUANG G B,et al.Novel weightingdelay-based stability criteria for recurrent neural networks with timevaryingdelay[J].IEEE Transactions on Neural Networks,2009,21(1)：91-106.。