Delay-dependent robust H∞ control for uncertain systems with time-varying state-delay

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.。

第37卷　第5期2009年9月河南师范大学学报(自然科学版)J ournal of Henan N ormal Universit y(N atural Science)　V ol.37　N o.5　Se pt.2009 文章编号:1000-2367(2009)05-0011-03不确定线性时滞系统新的鲁棒稳定性分析陈永刚,白春阳(河南科技学院数学系,河南新乡453003)摘　要:考虑了一类含有时变时滞的不确定线性系统的鲁棒稳定性问题.基于新构造的L yapunov函数和自由权矩阵方法,得到了较小保守的时滞相关的稳定性准则.数值例子说明了所得结果的有效性和较小保守性.关键词:线性系统;稳定性;时变时滞;线性矩阵不等式中图分类号:TP273文献标识码:A在许多实际系统中,时滞现象是经常存在的.时滞的存在往往会导致系统的不稳定性和系统性能变差,因此在过去20年内,时滞系统的稳定性分析和控制综合问题受到很多学者关注[1-5].通过利用新的L ya2 p unov函数,本文进一步考虑了含有不确定参数的线性时滞系统的鲁棒稳定性问题.所构造的L yap unov函数将充分利用系统状态x(t),x(t-δ1(t)),x(t-h(t)),x(t-δ2(t))和x(t-h)的信息.而且为了减小可能的保守性,引入了自由权矩阵来表示x(t),x(t-δ1(t),x(t-h(t)),x(t-δ2(t)和x(t-)之间的关系.所得结果用线性矩阵不等式表示,该结果能很好的用Matlab中的L M I工具箱求解.最后,两个数值例子说明了所得的稳定性准则改进了文献[4-5]中的结果.1　问题描述考虑如下一类含有时变时滞的不确定线性系统:x(t)=(A+ΔA)x(t)+(B+ΔB)x(t-h(t)),t>0;x(s)=φ(s),Πs∈[-h,0],(1)其中x(t)∈R n是系统状态向量,A,B是具有相应维数的已知实常矩阵,φ(t)是初始状态函数,h(t)是时变时滞且满足0Φh(t)Φh和h(t)Φμ.ΔA和ΔB表示不确定参数且满足:[ΔAΔB]=D F(t)[E a E b],其中D,E a,E b是具有相应维数的已知常矩阵,F(t)是未知矩阵且满足F T(t)F(t)ΦI.2　主要结果首先考虑系统(1)不含参数不确定性的情形,即下面标称系统的稳定性:x(t)=A x(t)+B x(t-h(t)),t>0;x(s)=φ(s),Πs∈[-h,0].(2)定理1　对于给定的常数h和μ,系统(3)是渐近稳定的,如果存在对称正定矩阵P>0,Q i>0,i=1, 2,3,4,R>0和矩阵M j,N j,U j,V j,j=1,2,3,4,5,使得下面的线性矩阵不等式成立:Ξ1=(Ωij)5×5hΨT R h2Mh2N3-hR0033-h2R0333-h2R<0,Ξ2=(Ωij)5×5hΨT Rh2Uh2V3-hR0033-h2R0333-h2R<0,(3)收稿日期:2009-02-22基金项目:河南科技学院自然科学基础研究计划项目(6053)作者简介:陈永刚(1981-),男,河南西平人,河南科技学院讲师,研究方向:时滞系统,神经网络.其中3表示矩阵中对称位置中元素的转置,Ψ=[A　0　B　0　0],且Ω11=PA+A T P+Q1+Q2+Q3+Q4+M1+MT1,Ω12=M T2-M1+N1,Ω13=P B+M T3-N1+U1,Ω14=M T4-U1+V1,Ω15=M T5-V1,Ω22=-(1-μ2)Q1-M2-M T2+N2+N T2,Ω23=-M T3+N T3-N2+U2,Ω24=N T4-M T4-U2+V2,Ω25=N T5-M T5-V2,Ω33=-(1-μ)Q2-N3-N T3+U3+U T3,Ω34=-N T4+U T4-U3+V3,Ω35=-N T5+U T5-V3,Ω44=-(1-μ2)Q3-U4-U T4+V4+V T4,Ω45=-U T5+V T5-V4,Ω55=-Q4-V5-V T5,M=[M T1M T2M T3M T4M T5], N=[N T1　N T2　N T3　N T4　N T5],U=[U T1　U T2　U T3　U T4　U T5],V=[V T1　V T2　V T3　V T4　V T5].证明　选取标称系统(2)的L yap unov函数为:V(t)=x T(t)Px(t)+∫t t-δ1(t)x T(s)Q1x(s)d s+∫t t-h(t)x T(s)Q2x(s)d s+∫t t-δ2(t)x T(s)Q3x(s)d s+∫t t-h x T(s)Q4x(s)d s+∫0-h∫t t+θ x T(s)R x(s)d s dθ,δ1(t)=h(t)2,δ2(t)=h(t)+h2.(4) V(t)沿着系统(2)对t求导可得:V(t)Φ2x T(t)P x(t)+x T(t)(Q1+Q2+Q3+Q4)x(t)-(1-μ2)x T(t-δ1(t))Q1x(t-δ1(t))-(1-μ)x T(t-h(t))Q2x(t-h(t))-(1-μ2)x T(t-δ2(t))Q3x(t-δ2(t))-x T(t-h)Q4x(t-h)+h x T(t)R x(t)-∫t t-h x T(s)R x(s)d s.(5)根据Leibniz2Newton公式,对于具有相应维数的矩阵M,N,U和V,可得下面方程成立:2ξT(t)M T[x(t)-x(t-δ1(t))-∫t t-δ1(t) x(s)d s]=0,(6)2ξT(t)N T[x(t-δ1(t))-x(t-h(t))-∫t-δ1(t)t-h(t) x(s)d s]=0,(7)2ξT(t)U T[x(t-h(t))-x(t-δ2(t))-∫t-h(t)t-δ2(t) x(s)d s]=0,(8)2ξT(t)V T[x(t-δ2(t))-x(t-h)-∫t-δ2(t)t-h x(s)d s]=0,(9)其中M,N,U,和V和定理1中定义相同,且ξT(t)=[x T(t)　x T(t-δ1(t)　x T(t-h(t))　x T(t-δ2(t)　x T(t-h)].利用不等式±2a T bΦa T Qa+b T Q-1b(a,b为实向量,Q为正定矩阵),可得:-2ξT(t)M T∫t t-δ1(t) x(s)d sΦh(t)2ξT(t)M T R-1Mξ(t)+∫t t-δ1(t) x T(s)R x(s)d s,(10)-2ξT(t)N T∫t-δ1(t)t-h(t) x(s)d sΦh(t)2ξT(t)N T R-1Nξ(t)+∫t-δ1(t)t-h(t) x T(s)R x(s)d s,(11)-2ξT(t)U T∫t-h(t)t-δ2(t) x(s)d sΦh-h(t)2ξT(t)U T R-1Uξ(t)+∫t-h(t)t-δ2(t) x T(s)R x(s)d s,(12)-2ξT(t)V T∫t-δ2(t)t-h x(s)d s<h-h(t)2ξT(t)V T R-1Vξ(t)+∫t-δ2(t)t-h x T(s)R x(s)d s.(13)把方程(6)-(9)左端加到V(t),并利用(10)-(13)式可得:V(t)ΦξT(t)[(Ωij)5×5+hΨT RΨ+h(t)2M T R-1M+h(t)2N T R-1N+h-h(t)2U T R-1U+h-h(t)2V T R-1]ξ(t)=h(t)hξT(t)[(Ωij)5×5+hΨT RΨ+h2M T R-1M+h2N T R-1]ξ(t)+h-h(t)hξT(t)・[(Ωij)5×5+hΨT RΨ+h2U T R-1U+h2V T R-1]ξ(t),(14)21河南师范大学学报(自然科学版) 2009年其中(Ωij )5×5,Ψ与定理1中的定义相同.由上式可知,如果矩阵不等式(Ωij )5×5+hΨT R Ψ+h 2M T R -1M +h2N T R-1N <0和(Ωij )5×5+hΨT R Ψ+h 2U T R -1U +h 2V T R -1V 成立,可得 V (t )<0,从而能够保证标称系统(3)渐近稳定.由舒尔补性质可知,线性矩阵不等式(4)和(5)分别等价于(Ωij )5×5+h ΨT R Ψ+h2M T R-1M +h2N T R-1N <0和(Ωij )5×5+hΨTR Ψ+h2U T R -1U +h2V T R -1V.证毕.利用一般的处理不确定参数的方法[2-4],很容易得到系统(1)的鲁棒稳定性准则.定理2　对于给定的常数h 和μ,系统(1)是鲁棒渐近稳定的,如果存在正定矩阵P >0,Q i >0,i =1,2,3,4,R >0,矩阵M j ,N j ,U j ,V j ,j =1,2,3,4,5,以及常数ε>0使得下面的线性矩阵不等式成立:Ξ1 H ε1E T3-ε1I33-ε1I<0,Ξ2H ε2E T3-ε2I33-ε2I<0,(15)其中Ξ1,Ξ2和定理1中相同,且 H =[H TP 0　0　0　0　h H TR 0　0]T, E=[E a 0　E b 0　0　0　0　0].3　数值例子例1　考虑标称系统(2),其中:A=-2　0　0-0.9,B=-1　0-1-1,μ=1.文献[4]和[5]给出的允许时滞界分别为1.345和1.868.利用本文定理1,可得更大的时滞界1.960.例2　考虑不确定时滞系统(1),其中相关参数定义如下:A =-0.5-2　1-1,B =-0.5-1　0　0.6,H =1001,E a =0.2000.2,E b =0.2000.6.当μ=0.5和μ=0.9时,文献[4]中给出的允许时滞界为0.3420和0.3378.利用本文定理2,容易得到时滞界为0.4006和0.3998.显然,对于这个例子,本文结果较文献[4]降低了保守性.参　考　文　献[1]　Gu K ,Kharitonov V L ,Chen J.Stability of time 2delay systems [M ].Boston :Birkh user ,2003.[2]　Wu M ,He Y ,She J H ,et al.Delay 2dependent criteria for robust stability of time 2varying delay systems indent [J ].Automatica ,2004,40(8):1435-1439.[3]　陈永刚,陈科委,毕卫萍.一类中立型时滞系统的时滞依赖保性能控制[J ].河南师范大学学报(自然科学版),2006,34(4):28-31.[4]　He Y ,Wang Q G ,Xie L.Furt her improvement of free 2weighting matrices technique for systems wit h time 2varying delay [J ].IEEETrans Autom Control ,2007,52(2):293-299.[5]　Park P ,K o J W.Stability and robust stability for systems wit h a time 2varying delay [J ].Automatica ,2007,43(10):1855-1858.N e w Robust Stability Analysis for U ncertain Linear Time 2Delay SystemsC H EN Y ong 2gang ,BA I Chun 2yang(Depart ment of Mat hematics ,Henan Institute of Science and Technology ,Xinxiang 453003,China )Abstract :This paper considers the robust stability problem for uncertain linear systems with time 2varying delay.Basedon the new constructed L yapunov f unctional and f ree weight matrix method ,the less conservative delay 2dependent stability cri 2teria are obtained.Numerical examples are given to show the effectiveness and less conservativeness of the obtained results.K ey w ords :linear systems ;stability ;time 2varying delay ;linear matrix inequality (L MI )31第5期 陈永刚等:不确定线性时滞系统新的鲁棒稳定性分析。

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。

参数不确定史密斯预估器的自适应控制丁晓迪;崔宝同【摘要】The adaptive control for Smith predictor with uncertain parameters was studied.Since model mismatch is caused by uncertain parameters,the plant can no longer track the original referencemodel.Traditional Smith predictor combined with the algorithm of model reference adaptive control (MRAC)was adopted to regulate the plant.The uncertainty of gain and time con-stant were adjusted using the control algorithm,the uncertain structural perturbations were regulated through choosing better adaptive law.The adaptive law was obtained using Lyapunov function.The effectiveness of the proposed method was verified by simulation using MATLAB.%研究参数不确定史密斯预估器的自适应控制问题。

不确定性参数引起系统的模型失配，控制对象无法追踪原始的参考模型，采用模型参考自适应控制算法对控制对象进行调节；增益和时间常数的不确定性可通过模型失配时的控制算法进行补偿，结构扰动的不确定性采用合适的自适应率进行自调节；选取李雅普诺夫函数，求出自适应律。