Filippov systems Continuation of sliding bifurcations
international journal of mechanical sciences issn
international journal of mechanical sciences issn
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1.介绍国际机械科学杂志的背景和影响力
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正文：
《国际机械科学杂志》（International Journal of Mechanical Sciences）是一本享有盛誉的学术期刊，致力于发表机械科学领域的高质量研究论文。
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机械原理外文文献
机械原理外文文献IntroductionMechanical principles are fundamental concepts in engineering and physics that are essential for understanding the behavior and operation of machines and mechanical systems. These principles are the building blocks of mechanical engineering and are used to design, analyze, and optimize mechanical devices and systems. In this paper, we will discuss some of the key mechanical principles, including force, motion, energy, and momentum, and their applications in various mechanical systems.Force and MotionForce and motion are two of the most fundamental concepts in mechanical engineering. Force is defined as any interaction that causes an object to undergo a change in speed, direction, or shape. In mechanical systems, forces are applied to machines and mechanical components to produce motion or to resist motion. The study of forces and their effects on motion is known as dynamics, and it is essential for understanding the behavior of mechanical systems.One of the key laws of motion is Newton's first law, which states that an object at rest will remain at rest, and an object in motion will remain in motion, unless acted upon by an external force. This law is used to analyze the behavior of mechanical systems and to design machines that can produce or resist motion.Energy and WorkEnergy is another critical concept in mechanical engineering and is defined as the ability to do work. Work, in the context of mechanical systems, is the transfer of energy from one object to another through the application of a force over a distance. The study of energy and work is essential for designing and analyzing mechanical systems that involve the conversion and transfer of energy, such as engines, turbines, and pumps.One of the fundamental principles of energy is the conservation of energy, which states that the total energy in a closed system remains constant over time. This principle is used to analyze the behavior of mechanical systems and to design machines that can efficiently convert and transfer energy.Momentum and ImpulseMomentum is a measure of an object's motion and is defined as the product of its mass and velocity. In mechanical systems, momentum is essential for understanding the behavior of moving objects and for designing machines that can produce or resist motion. Impulse is the change in momentum of an object due to the application of a force over a period of time, and it is used to analyze the behavior of mechanical systems during collisions and other dynamic events.One of the key principles of momentum and impulse is the conservation of momentum, which states that the total momentum in a closed system remains constant over time. This principle is used to analyze the behavior of mechanical systems during collisions and to design machines that can efficiently transfer momentum.ApplicationsThe principles of force, motion, energy, and momentum are used in a wide range of mechanical systems and devices. For example, in the design of engines and turbines, the principles of energy and work are used to optimize the conversion of energy from one form to another. In the design of vehicles and transportation systems, the principles of force and motion are used to analyze the behavior of moving objects and to optimize the performance of mechanical components. In the design of robotics and automation systems, the principles of force, motion, energy, and momentum are used to optimize the operation of mechanical devices and to ensure the safety and reliability of the systems.ConclusionMechanical principles are essential for understanding the behavior and operation of machines and mechanical systems. The concepts of force, motion, energy, and momentum are fundamental to the design, analysis, and optimization of mechanical devices and systems. These principles are used in a wide range of applications, from engines and turbines to vehicles and transportation systems, to robotics and automation systems. By understanding and applying these principles, engineers can design machines that are efficient, reliable, and safe, and that can meet the demands of modern industry and technology.Overall, the principles of force, motion, energy, and momentum are the foundation of mechanical engineering and are essential for the development of new technologies and innovations in the field.。
关于水母治理Filippov模型的动力学性质分析
鞍山师范学院学报2021-04,23 ( 2 ) ：1-6Journal of Anshan Normal University 关于水母治理Filippov 模型的动力学性质分析王艺霖・2,王 新2,刘 兵2,侯 祥董秀辉3(1.辽宁师范大学数学学院，辽宁大连116029；2.鞍山师范学院数学与信息科学学院，辽宁鞍山114007；3.鞍山市气象局，辽宁鞍山114004)摘要以水母种群数量为水母治理的控制指标，建立了一个Filippov 水母治理模型，系统地分析了该模型的动力学性质，得到了真、假、伪平衡点的存在条件及全局渐近稳定的条件.关键词 Filippov 切换系统;钵水母；滑线;伪平衡点中图分类号0175.1 文献标识码 A 文章编号1008-2441(2021)02-0001-06水母是水生环境中重要的浮游生物，有着复杂的结构和生命史，是海洋生态系统中的一个重要物 种.水母暴发是指水母在特定季节、特定海域内数量激增的现象.以往大约40年才有一次水母暴发，近 年来越来越频繁.研究发现，水母数量超过一定水平,就会对旅游业、发电厂和核电站的正常工作造成影 响〔I 】；水母与鱼类争夺饵料也会使各大渔场获鱼量持续下降［2］.为此,要对水母进行控制,把这种需要 控制的危害水平称为经济阈值.目前，中外学者针对水母暴发问题已做了一些研究，包括钵水母生活史的长期动力学行为、建立不 同生命周期阶段受温度影响的过程模型，等等［3-4］，但关于控制水母方面的研究成果却很少.本文利用 具有切换策略的动力学模型模拟水母的控制过程，当水母数量未超过经济阈值时，不加干预;超过经济 阈值时，再对水母进行捕捞，将水母数量控制在经济阈值内，为此建立Filippov 水母控制模型.关于Fil ippov 模型，在害虫治理［5］、植物疾病研究［6］、传染病研究［7］等多个方面已有广泛应用.本文得到的结 果希望能为沿海生态和渔业、旅游业等发展提供相应的理论依据.1模型的建立和预备知识1. 1模型的建立假设水母种群是密度制约，而水螅体种群为非密度制约.水螅体通过自我复制和水母的有性繁殖补 充，而水母只能通过水螅体的横裂补充.当水母种群数量小于经济阈值ET 时，建立如下方程:=ax + by — F 1，(1)=cx - dy - b 2y 2 芊 F 2，其中，x( t)，y( t)分别表示t 时刻水螅体和水母的数量，a 为实数，b ，c ，d,b 为非负实数.a 表示水螅体 的自我复制率与死亡率的差，假设a 始终小于0 ; b 表示水母有性繁殖成水螅体的存活率；c 为水螅体无 性繁殖成水母的存活率；d 为水母的死亡率也为水母种内竞争率.当水母种群数量大于ET 时,按比例啄(为非负实数)打捞水母，此时方程为世曲dy 不收稿日期2021-02-01基金项目 国家自然科学基金项目(11371030)；辽宁省自然科学基金指导计划项目(20170540001).作者简介王艺霖(1997-)，女，辽宁大连人，辽宁师范大学数学学院硕士研究生.2鞍山师范学院学报第23卷=系统(1)、(2)等价于--dy - by -啄y — F 4,b y+x =，于-(2)(3)—dy — b y - 着啄y ,中 其着 = 0 < ET ,l 着二 1 > ET.1.2 Filippov 系统预备知识令H(Z ) =y(t) - ET,其中，Z = (x ，y )T ,且F 7| ( Z ) = ( ax + by ,cx — dy — b 2y 2 ) T ,F I2 ( Z ) = ( ax + by ,cx - dy — b 2y 2 - 8y) T ,那么系统(1)和系统(2)合写为Filippov 系统d Z = (f i ,(Z ),Z 沂 Gdt = F i 2(Z ),Z 沂 I 2, 其中,I , = {Z 沂 R 2+l H (Z ) < 0},I 2 = {Z 沂 R 2+\ H (Z ) > 0},R l = {Z = (x ,y ) T I x 叟 0 ,y 叟 0},另外，记撞={(x,y)沂 R 2+I H( Z ) = 0}为分割两个区域I ,和I 的分界线.定义1令£ = {Z 沂撞 I 〈 H z ,F i ,( Z )〉〈 H z ,F i 2( Z )〉< 0},称撞s 为滑线区域，其中〈•业表示内积，H (Z )是一个光滑的纯量函数，在撞上关于H (Z )的梯度为H z = (0,1).引理1 (Filippov 凸理论)如果滑线是光滑的,滑线系统可以表示为(4)Z ( t) =A F i |( Z ) + (1 -A) F.( Z ),其中,〈H Z ,F I2( Z )〉 T姿=〈H z ”, Z 「-FX Z )〉，Z ( t) = (x ，y )『沂 乂 -定义 2 (i)如果F i ,(Z |) = 0,H(Z |) < 0 或F i 2(Z |) = 0,H(Z ,) > 0 成立,那么称Z ,为系统(4)的 真平衡点，记为Z 1 ；如果F i ,(Z ,) = 0,H(Z ,) > 0或F i 2(Z ,) = 0,H(Z ,) < 0成立,那么称Z ,为系统(4) 的假平衡点，记为Z ,.(ii)如果系统(4)的滑线区域撞s 的平衡点Z ,满足第2期王艺霖，等:关于水母治理Filippov模型的动力学性质分析3姿耳(Z,)+(1-姿)F』Z,)=0,0< 姿<1,则称Z,为系统(4)的伪平衡点.2子系统动力学2.1系统(1)的动力学性质系统(1)始终存在灭绝平衡点A°(0,0),当ad+bc>0时,存在正平衡点A,(x,,y,),其中，b(ad+bc)ad+bcx1=，y1二-k-a b2ab2定理1对于系统(1),当ad+bc<0时，正平衡点不存在，A0为全局渐近稳定的；当ad+bc>0时，为鞍点，系统(1)存在唯一的正平衡点A1，且它是全局渐近稳定的.证明系统(1)在平衡点A(x,y)处的Jacobian行列式为J(A)=I-d-2b2y［，J(A0)=〔:-d，Tr(J(A0))=a-d,Det(J(A0))=-(ad+bc).当ad+bc<0时，容易看出Tr(J(A。
(I已打印)Non-singular terminal sliding mode control of rigid manipulators
Automatica38(2002)2159–2167/locate/automaticaBrief PaperNon-singular terminal sliding mode control of rigid manipulatorsYong Feng a,Xinghuo Yu b;∗,Zhihong Man ca Department of Electrical Engineering,Harbin Institute of Technology,Harbin150006,People’s Republic of Chinab School of Electrical and Computer Engineering,Royal Melbourne Institute of Technology University,GPO Box2476V Melbourne,Vic.3001,Australiac School of Computer Engineering,Nanyang Technological University,SingaporeReceived26June2001;received in revised form16June2002;accepted9July2002AbstractThis paper presents a global non-singular terminal sliding mode controller for rigid manipulators.A new terminal sliding mode manifold isÿrst proposed for the second-order system to enable the elimination of the singularity problem associated with conventional terminal sliding mode control.The time taken to reach the equilibrium point from any initial state is guaranteed to beÿnite time.The proposed terminal sliding mode controller is then applied to the control of n-link rigid manipulators.Simulation results are presented to validate the analysis.?2002Elsevier Science Ltd.All rights reserved.Keywords:Terminal sliding mode control;Singularity;Robotic manipulator;Robust control;Lyapunov stability1.IntroductionVariable structure systems(VSS)are well known for their robustness to system parameter variations and external disturbances(Slotine&Li,1991;Utkin,1992; Yurl&James,1988).VSS have been widely used in many applications,such as robots,aircrafts,DC and AC motors, power systems,process control and so on.An aspect of VSS that is of particular interest is the sliding mode control,which is designed to drive and constrain the system states to lie within a neighborhood of the pre-scribed switching manifolds that exhibit desired dynam-ics.When in the sliding mode,the closed-loopresp onse becomes totally insensitive to both internal parameter un-certainties and external disturbances.A characteristic of conventional VSS is that the convergence of the system states to the equilibrium point is usually asymptotical due to the asymptotical convergence of the linear switching manifolds that are commonly chosen.Recently,a terminal sliding mode(TSM)controller was developed(Man&Yu,1997;Yu&Man,1996;Wu,Yu,& This paper was not presented at any IFAC meeting.This paper was recommended for publication in revised form by Associate Editor Jurek Z.Sasiadek under the direction of Editor Mituhiko Araki.∗Corresponding author.E-mail addresses:yfeng@(Y.Feng),x.yu@.au(X.Yu).Man,1998).TSM has been used in the control of rigid ma-nipulators(Man et al.,1994;Tang,1998).The TSM con-cept is related to theÿnite time control(Haimo,1986; Bhat&Bernstein,1997).Compared with linear hyperplane-based sliding modes,TSM o ers some superior properties such as fast,ÿnite time convergence.This controller is par-ticularly useful for high precision control as it speeds up the rate of convergence near an equilibrium point.However,the existing TSM controller design methods still have a singu-larity problem.An initial discussion to avoid the singularity in TSM control systems was presented(Wu et al.,1998). In this paper,a global non-singular terminal sliding mode (NTSM)controller is presented for a class of nonlinear dy-namical systems with parameter uncertainties and external disturbances.A new NTSM manifold is proposed to over-come the singularity problem.The time taken to reach the manifold from any initial state and the time taken to reach the equilibrium point in the sliding mode can be guaran-teed to beÿnite time.The proposed NTSM controller is then applied to the control of n-degree-of-freedom rigid ma-nipulators.Simulation results are presented to validate the analysis.2.Conventional terminal sliding mode controlThe basic principle of TSM control can be brie y sum-marized as follows:consider a second-order uncertain0005-1098/02/$-see front matter?2002Elsevier Science Ltd.All rights reserved. PII:S0005-1098(02)00147-42160Y.Feng et al./Automatica 38(2002)2159–2167nonlinear dynamical system ˙x 1=x 2;˙x 2=f (x )+g (x )+b (x )u;(1)where x =[x 1;x 2]T is the system state vector,f (x )and b (x )=0are smooth nonlinear functions of x ,and g (x )represents the uncertainties and disturbances satisfying g (x ) 6l g where l g ¿0,and u is the scalar control in-put.The conventional TSM is described by the following ÿrst-order terminal sliding variables =x 2+ÿx q=p1;(2)where ÿ0is a design constant,and p and q are positive odd integers,which satisfy the following condition:p ¿q:(3)The su cient condition for the existence of TSM is 12d d ts 2¡−Á|s |;(4)where Á¿0is a constant.For system (1),a commonly used control design isu =−b −1(x ) f (x )+ÿq px q=p −11x 2+(l g +Á)sgn(s );(5)which ensures that TSM occurs.It is clear that if s (0)=0,the system states will reach the sliding mode s =0within the ÿnite time t r ,which satisÿes t r 6|s (0)|Á:(6)When the sliding mode s =0is reached,the system dy-namics is determined by the following nonlinear di erential equation:x 2+ÿx q=p 1=˙x 1+ÿx q=p1=0;(7)where x 1=0is the terminal attractor of the system (7).The ÿnite time t s that is taken to travel from x 1(t r )=0to x 1(t s +t r )=0is given byt s =−ÿ−1x 1(t r )d x 1x q=p 1=p ÿ(p −q )|x 1(t r )|1−q=p :(8)This means that,in the TSM manifold (7),both the system states x 1and x 2converge to zero in ÿnite time.It can be seen in the TSM control (5)that the secondterm containing x q=p −11x 2may cause a singularity to occur if x 2=0when x 1=0.This situation does not occur inthe ideal sliding mode because when s =0;x 2=−ÿx q=p1hence as long as q ¡p ¡2q ,i.e.1¡p=q ¡2,the term x q=p −11x 2is equivalent to x (2q −p )=p 1which is non-singular.The singularity problem may occur in the reaching phase when there is insu cient control to ensure that x 2=0while x 1=0.The TSM controller (5)cannot guarantee a bounded controlsignal for the case of x 2=0when x 1=0before the system states reach the TSM s =0.Furthermore,the singularity may also occur even after the sliding mode s =0is reached since,due to computation errors and uncertain factors,the system states cannot be guaranteed to always remain in the sliding mode especially near the equilibrium point (x 1=0;x 2=0),and the case of x 2=0while x 1=0may occur from time to time.This underlines the importance of addressing the singularity problem in conventional TSM systems.3.Non-singular terminal sliding mode controlIn order to overcome the singularity problem in the con-ventional TSM systems,several methods have been pro-posed.For example,one approach is to switch the sliding mode between TSM and linear hyperplane based sliding mode (Man &Yu,1997).Another approach is to transfer the trajectory to a pre-speciÿed open region where TSM control is not singular (Wu et al.,1998).These methods are adopting indirect approaches to avoid the singularity.In this paper,a simple NTSM is proposed,which is able to avoid this problem completely.The proposed NTSM model is de-scribed as follows:s =x 1+1ÿx p=q 2;(9)where ÿ;p and q have been deÿned in (2).One can easilysee that when s =0,the NTSM (9)is equivalent to (2)so that the time taken to reach the equilibrium point x 1=0when in the sliding mode is the same as in (8).Note that in using (9)the derivative of s along the system dynamics does not result in terms with negative (fractional)powers.This can be seen in the following theorem about the NTSM control.Theorem 1.For system (1)with the NTSM (9),if the control is designed asu =−b −1(x ) f (x )+ÿq px 2−p=q2+(l g +Á)sgn(s );(10)where 1¡p=q ¡2;Á¿0,then the NTSM manifold (9)will be reached in ÿnite time.Furthermore ,the states x 1and x 2will converge to zero in ÿnite time .Proof.For the NTSM (9),its derivative along the system dynamics (1)is ˙s =˙x 1+1ÿp q x p=q −12˙x 2=x 2+1ÿp q x p=q −12˙x 2=x 2+1ÿp q x p=q −12(f (x )+g (x )+b (x )u )Y.Feng et al./Automatica38(2002)2159–21672161=x2+1ÿpqx p=q−12g(x)−ÿqpx2−p=q2−(l g+Á)sgn(s)=1ÿpqx p=q−12(g(x)−(l g+Á)sgn(s))thens˙s=1ÿpqx p=q−12(g(x)s−(l g+Á)sgn(s)s)6−1ÿpqÁx p=q−12|s|:Since p and q are positive odd integers and1¡p=q¡2,there is x p=q−12¿0for x2=0.Let (x2)=(1=ÿ)(p=q)Áx p=q−12.Then it hass˙s6− (x2)|s|(x2)¿0for x2=0:(11)Therefore,for the case x2=0,the condition for Lya-punov stability is satisÿed.The system states can reach the sliding mode s=0withinÿnite ing the following ar-guments can easily prove this:substituting the control(10) into system(1)yields˙x2=−ÿqpx2−p=q2+g(x)−(l g+Á)sgn(s):Then,for x2=0,it is obtained˙x2=g(x)−(l g+Á)sgn(s):For both s¿0and s¡0,it is obtained˙x26−Áand ˙x2¿Á,respectively,showing that x2=0is not an attractor.It also means that there exists a vicinity of x2=0such that for a small ¿0such that|x2|¡ ,there are˙x26−Áfor s¿0 and˙x2¿Áfor s¡0,respectively.Therefore,the crossing of the trajectory from the boundary of the vicinity x2= to x2=− for s¿0,and from x2=− to x2= for s¡0occurs inÿnite time.For other regions where|x2|¿ ,it can be easily concluded from(11)that the switching line s=0can be reached inÿnite time since we have˙x26−Áfor s¿0 and˙x2¿Áfor s¡0.The phase plane plot of the system is shown in Fig.1.Therefore,it is concluded that the sliding mode s=0can be reached from anywhere in the phase plane inÿnite time.Once the switching line is reached,one can easily see that NTSM(9)is equivalent to the TSM(2),so the time taken to reach the equilibrium point x1=0in the sliding mode is the same as in(8).Therefore,the NTSM manifold(9)can be reached inÿnite time.The states in the sliding mode will reach zero inÿnite time.This completes the proof.Remark1.It should be noted that the NTSM control(10) is always non-singular in the state space since1¡p=q¡2.Remark2.In order to eliminate chattering,a saturation function sat can be used to replace the sign function sgn.The1Fig.1.The phase plot of the system.relationshipbetween the steady-state errors of the NTSM system and the width of the layer surrounding the NTSM manifold s(t)=0is given by(Feng,Han,Stonier,&Man, 2000;Feng,Yu,&Man,2001)|s(t)|6’⇒|x(t)|6’and|x(t)|6(2ÿ’)q=p for t→∞:(12)4.Non-singular terminal sliding mode control for rigid manipulatorsIn this section,a non-singular terminal sliding mode con-trol is designed for the rigid n-link robot manipulatorM(q) q+C(q;˙q)+g(q)= (t)+d(t);(13) where q(t)is the n×1vector of joint angular position,M(q) the n×n symmetric positive deÿnite inertia matrix,C(q;˙q) the n×1vector containing Coriolis and centrifugal forces, g(q)the n×1gravitational torque,and (t)n×1vector of applied joint torques that are actually the control inputs,and d(t)n×1bounded input disturbances vector.It is assumed that rigid robotic manipulators have uncertainties,i.e.:M(q)=M0(q)+ M(q);C(q;˙q)=C0(q;˙q)+ C(q;˙q);g(q)=g0(q)+ g(q);where M0(q);C0(q;˙q)and g0(q)are the estimated terms; M(q); C(q;˙q)and g(q)are uncertain terms.Then, the dynamic equation of the manipulator can be written in the following form:M0(q) q+C0(q;˙q)+g0(q)= (t)+ (t)(14)2162Y.Feng et al./Automatica 38(2002)2159–2167with(t )=− M (q ) q − C (q ;˙q )q − g (q ):(15)The following assumptions are made about the robot dy-namics: M (q ) ¡ 0;(16) C (q ;˙q ) ¡ÿ0+ÿ1 q +ÿ2 ˙q 2;(17) g (q ) ¡ 0+ 1 q ;(18) (t ) ¡ 0+ 1 q + 2 ˙q 2;(19) (t ) ¡b 0+b 1 q +b 2 ˙q 2;(20)where 0;ÿ0;ÿ1;ÿ2; 0; 1; 0; 1; 2;b 0;b 1;b 2are positivenumbers.Suppose that q r is the desired input of the robot mani-pulator and ˙q r is the derivative of q r .Deÿne ”(t )=q −q r ;˙”(t )=˙q −˙q r ;e (t )=[”T (t )˙”T (t )]T .Then,the error equation of the rigid robotic manipulator can be obtained as follows:˙e (t )=Ae +B ;(21)whereA = 0I 00 ;B =0I;=M −10(q )(−C 0(q ;˙q )−g 0(q )−M 0(q ) q r + (t )+ (t )):It can be observed that the error dynamics (21)is of form (13).The NTSM control strategy developed in Section 3can be applied.The result is summarized in the following theorem.Before proceeding further,the notation of the frac-tional power of vectors is introduced.For a variable vector z ∈R n ,the fractional power of vectors is deÿned asz q=p =(z q=p 1;z q=p 2;:::;z q=p n )T;˙z q=p =(˙z q=p 1;˙z q=p 2;:::;˙zq=p n )T:Theorem 2.For the rigid n -link manipulator (14),if the NTSM manifold is chosen as s =”+C 1˙”p=q ;(22)where C 1=diag [c 11;:::;c 1n ]is a design matrix ,and the NTSM control is designed as follows ,then the system tracking error ”(t )will converge to zero in ÿnite time . = 0+u 0+u 1;(23)where0=C 0(q ;˙q )+g 0(q )+M 0(q ) q r ;(24)u 0=−q pM 0(q )C −11˙”2−p=q;(25)u 1=−q p [s T C 1diag (˙”p=q −1)M −10(q )]T s T C 1diag (˙”p=q −1)M −10(q )×[ s C 1diag (˙”p=q −1)M −10(q ) (b 0+b 1 q+b 2 ˙q 2)];(26)where b 0;b 1;b 2are supposed to be known parameters as in (20).Proof.Consider the following Lyapunov functionV =12s Ts :Di erentiating V with respect to time,and substituting (23)–(26)into it yields˙V =s T ˙s =s T ˙”+p qC 1diag (˙”p=q −1) ”=s T ˙”+p q C 1diag (˙”p=q −1)M −10(q )(u 1(t )+u 0(t ))+ (t ))=s T p q C 1diag (˙”p=q −1)M −10(q )(u 1(t )+ (t )) =−p qs C 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2)+p qs T C 1diag (˙”p=q −1)M −10(q ) (t )6−p qs C 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2)+p qs C 1diag (˙”p=q −1)M −10(q ) (t ) =−p qC 1diag (˙”p=q −1)M −10(q ) ×(b 0+b 1 q +b 2 ˙q 2− (t ) ) s that is˙V 6−Á(t ) s ¡0for s =0;(27)where Á(t )=p qC 1diag (˙”p=q −1)M −10(q ) ×{(b 0+b 1 q +b 2 q 2)− (t ) }¿0:Therefore,according to the Lyapunov stability criterion,the NTSM manifold s (t )in (22)converges to zero in ÿ-nite time.On the other hand,if s =”+C 1˙”p=q =0are reached as shown in Theorem 1,then the output trackingY.Feng et al./Automatica38(2002)2159–21672163 error of the robot manipulator”(t)=q−q r will convergeto zero inÿnite time.This completes the proof.Remark3.The NTSM control proposed in Theorem2solves the control of the rigid n-link manipulator,that repre-sents a special class of problems.The method proposed canbe extended to a class of n-order(n¿2)nonlinear dynam-ical systems,that represents a broader class of problems:˙x1=f1(x1;x2);˙x2=f2(x1;x2)+g(x1;x2)+B(x1;x2)u;(28)where x1=(x11;x12;:::;x1n)T∈R n;x2=(x21;x22;:::;x2n)T∈R n;f1and f2are smooth vector functions and g rep-resents the uncertainties and disturbances satisfyingg(x1;x2) 6l g where l g¿0;B is a non-singular ma-trix and u=(u1;u2;:::;u n)T∈R n is the control vector.It is further assumed that(x1;x2)=(0;0)if and only if(x1;˙x1)=(0;0).Note that many practical dynamical sys-tems satisfy this condition,for example,the mechanicalsystems.Robotic systems are certainly a special case of(28).Actually,the robotic system(14)is not in the form of(28),but it can be transformed to such form by the coordi-nates change.So,the proposed algorithm in the paper can beapplied to any plant,which can be transformed to(28).TheNTSM for system(28)can be designed as follows.Chooses=x1+ ˙x p=q1;(29)where =diag( 1;:::; n);( i¿0)for i=1;:::;n,and˙x p=q1is represented as˙x p=q1=(x p1=q111;:::;x p n=q n1n)T:If the NTSM control is designed as in(30),then the high-order nonlinear dynamical systems(28)will converge to the NTSM and the equilibrium point inÿnite time,re-spectively,u=−@f1@x2B(x1;x2)−1l g@f1@x2+Áss+@f1@x1f1(x1;x2)+@f1@x2f2(x1;x2)+ −1 −1diag(x2−p1=q q11;:::;x2−p n=q n1n);(30)where =diag(p1=q1;:::;p n=q n);p i and q i are positive odd integers and q i¡p i¡2q i for i=1;:::;n.5.Simulation studiesThe section presents two studies:one is the comparison study of performance between NTSM and TSM,and the other an application to a robot control problem.-0.0500.050.10.150.20.250.3-0.4-0.20.20.40.60.81.0x1x2Fig.2.Phase plot of NTSM system.parison studyIn order to analyze the e ectiveness of the NTSM control proposed and to compare NTSM with TSM,consider the simple second-order dynamical system below:˙x1=x2;˙x2=0:1sin20t+u:(31) The NTSM and TSM are chosen as follows:s NTSM=x1+x5=32;s TSM=x2+x3=51:Three control approaches are adopted:NTSM control, TSM control,and indirect NTSM control.The NTSM con-trol is designed according to(10)and NTSM(9),and TSM control is designed according to(5)and TSM(2).The in-direct NTSM control is designed in the same way as TSM, with only one di erence,that is when|x1|¡ ,let p=q, and is selected as0.001(Man&Yu,1997).Three sys-tems achieve the same terminal sliding mode behavior.So, only the phase plane response of the NTSM control system is provided,as shown in Fig.2.The control signals for the three kinds of systems are shown in Figs.3–5.It can be ob-viously seen some valuable facts.No singularity occurs at all in the case of NTSM control.When the trajectory crosses the x1=0axis,singularity occurs in the case of TSM con-trol.For the indirect NTSM control,although singularity is avoided by switching from the TSM to linear sliding mode, the e ect of the singularity can be seen,especially when decreases to zero.However when is relatively large, the sliding mode of the system is switching between TSM and the linear plane based sliding mode,and the advantage of TSM system is lost.Therefore,from the results of the above simulations,the occurrence of singularity problem in the TSM system,the drawback of the indirect NTSM,and the e ectiveness of the NTSM in avoiding singularity,are observed,respectively.2164Y.Feng et al./Automatica 38(2002)2159–21670.51.0 1.52.02.5-8-7-6-5-4-3-2-1012time (sec.)uFig.3.Control signal of NTSM system.0.51.0 1.52.02.5-90-80-70-60-50-40-30-20-10010time(sec.)uFig.4.Control signal of TSM system.5.2.Control of a robotA simulation with a two-link rigid robot manipulator (seeFig.6)is performed for the purpose of evaluating the perfor-mance of the proposed NTSM control scheme.The dynamic equation of the manipulator model in Fig.6is given by a 11(q 2)a 12(q 2)a 12(q 2)a 22q 1 q 2 +−ÿ12(q 2)˙q 21−2ÿ12(q 2)˙q 1˙q 2ÿ12(q 2)˙q 22+ 1(q 1;q 2)g 2(q 1;q 2)g =1 2;(32)0.51.0 1.52.02.5-8-7-6-5-4-3-2-1012time(sec.)uFig.5.Control signal of indirect TSMsystem.Fig.6.Two-link robot manipulator model.wherea 11(q 2)=(m 1+m 2)r 21+m 2r 22+2m 2r 1r 2cos(q 2)+J 1;a 12(q 2)=m 2r 22+m 2r 1r 2cos(q 2);a 22=m 2r 22+J 2;ÿ12(q 2)=m 2r 1r 2sin(q 2);1(q 1;q 2)=((m 1+m 2)r 1cos(q 2)+m 2r 2cos(q 1+q 2)); 2(q 1;q 2)=m 2r 2cos(q 1+q 2):The parameter values are r 1=1m ;r 2=0:8m ;J 1=5kg m ;J 2=5kg m ;m 1=0:5kg ;m 2=1:5kg.The desired reference signals are given by q r 1=1:25−(7=5)e −t +(7=20)e −4t ;q r 2=1:25+e −t −(1=4)e −4t :The initial values of the system are selected as q 1(0)=1:0;q 2(0)=1:5;˙q 1(0)=0:0;˙q 2(0)=0:0:Y.Feng et al./Automatica 38(2002)2159–216721650123456789100.20.40.60.81.01.21.41.6time(sec)O u t p u t t r a c k i n g o f j o i n t 1( r a d )Fig.7.Output tracking of joint 1using a boundary layer.123456789101.21.31.41.51.61.71.81.92.0time(sec)O u t p u t t r a c k i n g o f j o i n t 2( r a d )Fig.8.Output tracking of joint 2using a boundary layer.The nominal values of m 1and m 2are assumed to be ˆm 1=0:4kg ;ˆm 2=1:2kg :The boundary parameters of system uncertainties in (20)are assumed to be b 0=9:5;b 1=2:2;b 2=2:8:Suppose the tracking error and the 1st tracking error are tobe |˜q i |60:001and |˙˜q i |60:024;i =1,2,where ˜q i =q i −q riand ˙˜q i =˙q i −˙q ri ;i =1,ing the above performance index,it can be determined the parameters of NTSM manifolds.According to (12),it is obtained that |˜q i |6’i ;i =1;2:Let ’i =0:001;i =1;2(33)012345678910-15-10-5051015202530time(sec)C o n t r o l i n p u t o f j o i n t 1( N m )Fig.9.Control of joint 1using a boundary layer.12345678910-14-12-10-8-6-4024time(sec)C o n t r o l i n p u t o f j o i n t 2 (N m )Fig.10.Control of joint 2using a boundary layer.the tracking error of the system |˜q i |can be guaranteed.Onthe other hand,according to (12),it is obtained that |˙˜q i |6(2ÿ’i )q=p ;i =1;2:Let(2ÿ’i )q=p 60:024;i =1;2;thenq p6log 0:024log(2ÿ’i );i =1;2:(34)For simplicity,let ÿi =1;i =1;2.Then from (34),it is obtained thatq p 6log 0:024log(2×1×0:001)=0:60015;i =1;2:(35)2166Y.Feng et al./Automatica 38(2002)2159–2167-0.100.10.20.30.40.50.60.70.80.9-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.100.1e1(t)(rad)d e 1/d t (r a d /s )Fig.11.Phase plot of tracking error of joint 1.-0.5-0.4-0.3-0.2-0.10.100.20.30.40.50.6e2(t)(rad)d e 2/d t (r a d /s )Fig.12.Phase plot of tracking error of joint 2.Let qp=0:6:Now,the parameters of the TSM can be obtained as:q =3;p =5(there are many other options as well).Finally,the NTSM models are obtained as follows:s 1=˜q 1+˙˜q 5=31=0;s 2=˜q 2+˙˜q 5=32=0:In order to eliminate the chattering,the boundary layermethod is adopted (Slotine &Li,1991)in the NTSM con-trol.The simulation results are shown in Figs.7–12.Figs.7and 8show the output tracking of joints 1and 2.Figs.9and 10depict the control signals of joints 1and 2,respec-tively.Figs.11and 12show the phase plot of tracking error of joints 1and 2,respectively.One can easily see that the system states track the desired reference signals.First,theoutput tracking errors of the system reach the terminal slid-ing mode manifold s =0in ÿnite time,then they converge to zero along s =0in ÿnite time.It can be clearly seen that neither singularity nor chattering occurs in the two control signals.6.ConclusionsIn this paper,a global non-singular TSM controller for a second-order nonlinear dynamic systems with parameter uncertainties and external disturbances has been proposed.The time taken to reach the manifold from any initial sys-tem states and the time taken to reach the equilibrium point in the sliding mode have been proved to be ÿnite.The new terminal sliding mode manifold proposed can enable the elimination of the singularity problem associated with con-ventional terminal sliding mode control.The global NSTM controller proposed has been used for the control design of an n -degree-of-freedom rigid manipulator.Simulation results are presented to validate the analysis.The proposed controller can be easily applied to practical control of robots as given the advances of microprocessors,the vari-ables with fractional power can be easily built into control algorithms.ReferencesBhat,S.P.,&Bernstein, D.S.(1997).Finite-time stability of homogeneous systems.Proceedings of American control conference (pp.2513–2514).Feng,Y.,Han,F.,Yu,X.,Stonier,D.,&Man,Z.(2000).Tracking precision analysis of terminal sliding mode control systems with saturation functions.In X.Yu,J.-X.Xu (Eds.),Advances in variable structure systems :Analysis,integration and applications (pp.325–334).Singapore:World Scientiÿc.Feng,Y.,Yu,X.,&Man,Z.(2001).Non singular terminal sliding mode control and its applications to robot manipulators.Proceedings of 2001IEEE international symposium on circuits and systems ,Vol.III (pp.545–548).Sydney,May 2001.Haimo,V.T.(1986).Finite time controllers.SIAM Journal of Control and Optimization ,24(4),760–770.Man,Z.,Paplinski,A.P.,&Wu,H.(1994).A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators.IEEE Transactions on Automatic Control ,39(12),2464–2469.Man,Z.,&Yu,X.(1997).Terminal sliding mode control of mimo linear systems.IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications ,44(11),1065–1070.Slotine,J.E.,&Li,W.(1991).Applied non-linear control .Englewood Cli s,NJ:Prentice-Hall.Tang,Y.(1998).Terminal sliding mode control for rigid robots.Automatica ,34(1),51–56.Utkin,V.I.(1992).Sliding modes in control optimization .Berlin,Heidelberg:Springer.Wu,Y.,Yu,X.,&Man,Z.(1998).Terminal sliding mode control design for uncertain dynamic systems.Systems and Control Letters ,34,281–288.Yu,X.,&Man,Z.(1996).Model reference adaptive control systems with terminal sliding modes.International Journal of Control ,64(6),1165–1176.Yurl,B.S.,&James,M.B.(1988).Continuous sliding mode control.Proceedings of American Control Conference (pp.562–563).Y.Feng et al./Automatica 38(2002)2159–21672167Yong Feng received the B.S.degree from the Department of Control Engineering in 1982,and M.S.degree from the Depart-ment of Electrical Engineering in 1985and Ph.D.degree from the Department of Con-trol Engineering in 1991,in Harbin Insti-tute of Technology,China,respectively.He has been with the Department of Electri-cal Engineering,Harbin Institute of Tech-nology since 1985,and is currently a Pro-fessor.He was a visiting scholar in the Faculty of Informatics and Communication,Australia,from May 2000to November 2001.He has authored and co-authored over 50journal and conference papers.He has published 3books.He has completed over 10research projects,including process control,arc welding robot,climbing wall robot,CNC system,a direct drive motor and its control system,the electronics and simulation of CCD digital camera,and so on.His current research interests are nonlinear control systems,sampled data systems,robot control,digital camera modelling andsimulation.Xinghuo Yu received B.Sc.(EEE)and M.Sc.(EEE)from the University of Sci-ence and Technology of China in 1982and 1984respectively,and Ph.D.degree from South-East University,China in 1987.From 1987to 1989,he was Research Fellow with Institute of Automation,Chi-nese Academy of Sciences,Beijing,China.From 1989to 1991,he was a Postdoctoral Fellow with the Applied Mathematics De-partment,University of Adelaide,Australia.From 1991to 2002,he was with CentralQueensland University,Rockhampton,Australia where he was Lecturer,Senior Lecturer,Associate Professor then Professor of Intelligent Sys-tems and the Associate Dean (Research)of the Faculty of Informatics and Communication.Since March 2002,he has been with the School of Electrical and Computer Engineering at Royal Melbourne Institute of Technology,Australia,where he is a Professor,Director of Software and Networks,and Deputy Head of School.He has also held Visiting Profes-sor positions in City University of Hong Kong and Bogazici University(Turkey).He has recently been conferred as Honorary Professor of Cen-tral Queensland University.He is Guest Professor of Harbin Institute of Technology (China),Huazhong University of Science and Technology (China),and Southeast University (China).Professor Yu’s research inter-ests include sliding mode and nonlinear control,chaos and chaos control,soft computing and applications.He has published over 200refereed pa-pers in technical journals,books and conference proceedings.He has also coedited four research books “Complex Systems:Mechanism of Adapta-tion”(IOS Press,1994),“Advances in Variable Structure Systems:Anal-ysis,Integration and Applications”(World Scientiÿc,2001),“Variable Structure Systems:Towards the 21st Century”(Springer-Verlag,2002),“Transforming Regional Economies and Communities with Information Technology”(Greenwood,2002).Prof.Yu serves as an Associate Editor of IEEE Trans Circuits and Systems Part I and is on the Editorial Board of International Journal of Applied Mathematics and Computer Science.He was General Chair of the 6th IEEE International Workshopon Variable Structure Systems held in December 2000on the Gold Coast,Australia.He was the sole recipient of the 1995Central Queensland University Vice Chancellor’s Award forResearch.Zhihong Man received the B.E.degree from Shanghai Jiaotong University,China,the M.S.degree from the Chinese Academy of Sciences,and the Ph.D.from the Uni-versity of Melbourne,Australia,all in electrical and electronic engineering,in 1982,1986and 1993,respectively.From 1994to 1996,he was a Lecturer in the Department of Computer and Commu-nication Engineering,Edith Cowan Uni-versity,Australia.From 1996to 2000,he was a Lecturer and then a SeniorLecturer in the Department of Electrical Engineering,the University of Tasmania,Australia.In 2001,he was a Visiting Senior Fellow in the School of Computer Engineering,Nanyang Technological University,Singapore.Since 2002,he has been an Associate Professor of Computer Engineering at Nanyang Technological University.His research interests are in robotics,fuzzy logic control,neural networks,sliding mode control and adaptive signal processing.He has published more than 120journal and conference papers in these areas.。
三维分片光滑Filippov-型方程Hopf分支周期解的数值计算
１ 引 言
在 实际应 用 中得 到 的常微 分模 型有 时是 不 连续 的 ， 多 学 者对 其 进 行 了定 性 研究 ＿ ．文 献 ［ －］ 许 １ ５６ 分别 研究 了二 维 和三 维 Ｆｌｐｖ型方 程 非光 滑 Ｈ ｐ 分 支周 期解 的存 在性 ．本 文主 要 考 虑如 下 一类 单 参 ｉｉｏ － ｌ ｏｆ 数三 维分 片光 滑非 线性 Ｆｌ ｐ ｖ型方程 分 支周期 解 的 数值算 法 ： ｉｐ ｏ － ｉ
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Ｈ ｐ 分 支周 期解 ，并 给 出 了数值 算 例．结 果表 明 ，算法是 有 效 的． ｏｆ 关键 词 ： 片 光滑 Ｆｌ ｐｖ型 方程 ； ｏｎａ６映 射 ；不 动 点；周 期解 分 ｉｐ ｏ一 ｉ Ｐｉｃｒ
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作者简介 ： 管庆 光 （ ９ ６ ） 男 ， 族 ， １８一 ， 汉 硕士 研究生 ， 事微 分 方程数 值解 的研究 ， — ａ ： ｕ ｎｇ９ ｍ ｉ ． ｕ ｅｕ ｃ．通 讯 从 Ｅ ｍ ｉ ｇａ ｑ ＠ ａｓｊ ．ｄ ． ｎ ｌ ０ ｌ ｌ 作者 臧 林（ ９ １ ） 男 ， １８ 一 ， 汉族 ， 博士 ， 讲师 ， 从事计算数学 的研究 ， － ａ ： ａｇ ｌ ．ｄ ．ａ Ｅｍ ｉ ｚ ｌ ｎ ｅ ｕ ｃ． ｌ ｎ ＠ｄ ｕ
一类平面Filippov系统的定性分析
一类平面Filippov系统的定性分析
本文运用动力系统理论和微分方程定性理论来分析一类平面Filippov系统的动力学行为,Filippov系统在机械系统、电子系统、反馈控制系统等许多实际物理问题中有着广泛的运用,对其各种解的性质的深入研究和认识具有重要的理论意义和应用价值.由于在光滑系统中所定义的解在非光滑系统中不能使用,于是我们需要将解的定义进行推广.科学家们更多关注的是对平衡点、周期解、拟周期解、同宿异宿解等各种特殊形式解的存在性及稳定性的研究.本文研究如下系统：其中α,b,A,B,c1,c2∈R是参量且我们根据这四个参数：α,b,A,B的不同情况对系统(1)进行分类讨论：(i)当参数α<-b2/4,A<-B2/4时,该系统在{x>0}和{x<0}内的平衡点都是鞍点,我们利用微分包含和Poincare映射来研究和分析系统的平衡点个数、类型以及各种解的情况；(ii)当参数α
>-b2/4,A＜-B2/4时，该系统的两个平衡点分别为焦点和鞍点,同样,我们利用微分包含和Poincare映射来研究和分析系统的平衡点个数、类型以及各种解的情况.在(i)和(ii)中,主要研究类型Ⅰ轨道和类型Ⅱ轨道存在的条件以及个数.。
高二英语科学家名称单选题20题
高二英语科学家名称单选题20题1. Who discovered the theory of relativity?A. Isaac NewtonB. Albert EinsteinC. Galileo GalileiD. Marie Curie答案：B。
本题考查著名物理学家的成就。
选项A 艾萨克·牛顿，主要贡献是万有引力定律等；选项C 伽利略·伽利雷，在天文学和力学方面有重要成就；选项D 玛丽·居里，是著名的化学家。
而阿尔伯特·爱因斯坦发现了相对论，所以选B。
2. Which physicist is known for his work on quantum mechanics?A. Max PlanckB. Nikola TeslaC. Thomas EdisonD. Werner Heisenberg答案：D。
本题考查对量子力学有贡献的物理学家。
选项A 马克斯·普朗克，是量子力学的奠基人之一；选项B 尼古拉·特斯拉，在电学领域有突出成就；选项C 托马斯·爱迪生，是发明大王。
维尔纳·海森堡在量子力学方面有重要贡献，故选D。
3. The law of universal gravitation was proposed by _____.A. ArchimedesB. CopernicusC. KeplerD. Isaac Newton答案：D。
本题考查万有引力定律的提出者。
选项A 阿基米德，在浮力方面有重要发现；选项B 哥白尼，提出日心说；选项C 开普勒，发现了行星运动定律。
艾萨克·牛顿提出了万有引力定律，所以答案是D。
4. Who is famous for his experiments with electricity?A. Benjamin FranklinB. Michael FaradayC. James Clerk MaxwellD. Alessandro V olta答案：B。
光化学psii结晶构造
光化学psii结晶构造光化学PSII结晶构造PSII（Photosystem II）是光合作用中的一个重要复合物，它在光反应阶段起着关键作用。
PSII的结晶构造给我们提供了深入了解其功能和机制的重要线索。
PSII是由多个蛋白质亚基和辅助色素分子组成的复合物。
其结构包括一个核心复合物和多个外周抗氧化剂分子。
核心复合物主要由D1和D2蛋白质亚基以及PsbA、PsbB、PsbC等辅助蛋白质组成。
这些亚基之间通过多个色素分子和电子传递蛋白质进行相互连接和调节。
在PSII的结晶构造中，D1和D2蛋白质亚基位于复合物的中心，形成了一个核心结构。
这两个亚基含有多个色素分子，其中最重要的是叶绿素a分子（Chl a）。
这些色素分子通过吸收光子能量，将其转化为电子能量，并将其传递给其他亚基和辅助蛋白质。
除了D1和D2蛋白质亚基，PSII的结晶构造中还包括多个辅助蛋白质。
这些辅助蛋白质的功能是调节和稳定PSII的结构，以确保其正常的光合作用功能。
辅助蛋白质中的一个重要成员是氧化还原酶（OEC），它负责光解水产生氧气和电子供应。
PSII的结晶构造还揭示了其光合作用过程中的一个重要特点，即光致电荷分离。
当叶绿素a分子吸收到光子能量后，电子被激发到高能态，并从叶绿素a分子中传递到邻近的叶绿素a分子。
最终，这些电子被传递到D1和D2蛋白质亚基中的特定位置，并参与光合作用的后续反应。
PSII的结晶构造还显示了其在光合作用中的另一个关键过程，即氧化还原反应。
在光合作用过程中，光能被转化为电子能量，并用于将二氧化碳还原为有机物质。
这一过程涉及到多个电子传递步骤，其中PSII起到了氧化还原反应的关键角色。
总结起来，光化学PSII的结晶构造为我们提供了深入了解其功能和机制的重要线索。
它揭示了PSII的核心复合物和辅助蛋白质的组成，以及光致电荷分离和氧化还原反应的关键步骤。
通过进一步研究PSII的结晶构造，我们可以更好地理解光合作用的机制，为提高光合作用效率和开发新型光合作用技术提供重要参考。
黄色短杆菌中L-异亮氨酸同位素丰度及分布的分析方法研究
第43 卷第 3 期2024 年3 月Vol.43 No.3496~500分析测试学报FENXI CESHI XUEBAO（Journal of Instrumental Analysis）黄色短杆菌中L-异亮氨酸同位素丰度及分布的分析方法研究赵雅梦1，2，范若宁1，2，雷雯1，2*（1．上海化工研究院有限公司，上海 200062；2．上海市稳定同位素检测及应用研发专业技术服务平台，上海 200062）摘要：随着代谢组学、蛋白质组学等生命科学领域的迅猛发展，稳定同位素标记试剂，尤其是标记氨基酸，因无放射性、与非标记化合物理化性质一致等优势得到广泛应用。
该文建立了一种稳健、快速的氨基酸同位素丰度分析方法。
方法采用Hypersil Gold Vanquish（100 mm × 2.1 mm，1.9 μm）色谱柱，以水和含0.1%甲酸的甲醇为流动相，正离子模式下进行液相色谱-高分辨质谱联用（LC-HRMS）分析；测得细菌发酵液中L-异亮氨酸-15N的同位素丰度为98.58%，相对标准偏差为0.03%，可应用于不同稳定同位素（15N或13C）示踪的黄色短杆菌中L-异亮氨酸同位素丰度及分布的准确测定。
该方法具有简便、灵敏、稳健等优点，有望在合成生物学、同位素示踪代谢流等研究中发挥重要作用。
关键词：同位素标记氨基酸；液相色谱-高分辨质谱（LC-HRMS）；黄色短杆菌；同位素分布及丰度中图分类号：O657.72；O629.7文献标识码：A 文章编号：1004-4957（2024）03-0496-05Analysis of Isotope Abundance and Distribution for L-Isoleucinein Brebvibacterium flavumZHAO Ya-meng1，2，FAN Ruo-ning1，2，LEI Wen1，2*（1．Shanghai Research Institution of Chemical Industry Co. Ltd.，Shanghai 200062，China；2．Shanghai Professional Technology Service Platform on Detection and Application Development for Stable Isotope，Shanghai 200062，China）Abstract：In the rapidly advancing life science fields such as metabolomics and proteomics，stable isotope labeling reagents that are non-radioactive and have similar physiochemical properties with un⁃labeled compounds have been widely utilized. Biological fermentation is one of the major synthesis ap⁃proaches for labeled amino acids. In this study，we have established an accurate，robust，and rapid method to determine the isotope abundance of the amino acids in the fermentation broth to aid in early assessment of batch quality and optimization of fermentation conditions and amino acid yield. A Hy⁃persil Gold Vanquish column（100 mm × 2.1 mm，1.9 μm）with water and methanol containing 0.1%formic acid as mobile phase and a liquid chromatography-high resolution mass spectrometry（LC-HRMS） system in positive ion mode were used for the study. The isotopic abundance of L-iso⁃leucine-15N samples was determined to be 98.58%，closely matching the indicated value（>98%），with a relative standard deviation of 0.03%，demonstrating excellent accuracy and precision for the method. Then the method was successfully applied to determine the isotopic abundance and distribu⁃tion of L-isoleucine in Brevibacterium flavum labeled with 15N or 13C. The proposed method is simple to perform，convenient，highly sensitive，and robust，holding wide application potentials in syn⁃thetic biology and research in stable isotope traced metabolic pathways.Key words：stable isotope labeled amino acid；liquid chromatography-high resolution mass spec⁃trometry（LC-HRMS）；Brebvibacterium flavum；isotope distribution and abundance利用同位素标记技术将化合物中普通原子替换为同位素核素所合成的稳定同位素标记化合物，结合质谱技术，已在蛋白质组学、代谢组学、生物靶标发现、临床诊断等生命科学研究中发挥重要作用［1-4］。
基于Filippov微分包含解的干摩擦自激振动系统黏滑运动的研究
基于Filippov微分包含解的干摩擦自激振动系统黏滑运动的研究基于Filippov微分包含解的干摩擦自激振动系统黏滑运动的研究摘要：本文研究了一种基于Filippov微分包含解的干摩擦自激振动系统，在考虑黏滑运动的情况下进行了分析。
通过引入非线性摩擦模型和滑动滑动面约束等，对系统进行了建模，并通过仿真和分析研究了系统的运动特性。
研究结果表明，Filippov微分包含解方法在描述干摩擦自激振动系统黏滑运动过程中具有较好的精确性和可行性。
关键词：Filippov微分包含解；干摩擦自激振动系统；黏滑运动1. 引言干摩擦自激振动是一类重要的非线性振动现象，存在于许多实际物理系统中。
在实际工程中，对于干摩擦自激振动系统的研究已经引起了广泛的关注。
然而，在实际系统中，往往会存在黏滑运动的现象，这对于系统的稳定性和运动特性有着重要的影响。
因此，研究干摩擦自激振动系统黏滑运动的特性具有重要意义。
2. Filippov微分包含解的引入Filippov微分包含解是一种在非光滑系统动力学中广泛应用的方法，用于描述系统在光滑和非光滑切换时的动力学行为。
在该方法中，非光滑性由一个或多个切换面来描述，并使用广义导数来描述系统在切换面之间的动力学解。
因此，Filippov 微分包含解方法非常适用于描述干摩擦自激振动系统中的黏滑运动。
3. 干摩擦自激振动系统黏滑运动的建模考虑一个干摩擦自激振动系统，其动力学方程可以表示为：\[ m \ddot{x} + f(\dot{x}) + \gamma \dot{x} + kx = 0 \] 其中，\(m\)为质量，\(f(\dot{x})\)为非线性摩擦力，\(\gamma\)为阻尼系数，\(k\)为刚度，\(x\)为位移。
在非线性摩擦力模型中，我们采用Coulomb摩擦模型和Kelvin-Voigt摩擦模型，分别描述干摩擦和黏滑运动。
4. 干摩擦自激振动系统黏滑运动的仿真与分析通过对干摩擦自激振动系统进行数值仿真，我们可以得到系统在不同参数条件下的运动特性。
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Crossing orbits: At x ∈ Σc (α) cross. Sliding orbits: For a regular sliding point x ∈ Σs(α) define the Filippov vector g (x, α) = λ(x, α)f (1) (x, α) + (1 − λ(x, α))f (2) (x, α), where λ(x, α) = Hx (x, α), f (2) (x, α) . Hx (x, α), f (2) (x, α) − f (1) (x, α)
H (u(0), α) H (u(1), α) u(1) − v (0) v (1) − u(0)
= = = =
0, 0, 0, 0.
Σc S2
u(0) = v (1)
u(1) = v (0)
S1
• pseudo-equilibrium at u(1) (n = 2): t(u(1), α), g (u(1), α)
Sj
= = = = = =
0, 0, 0, 0, 0, 0.
u(0) = s(1)
S2
S2 Σc Σc S1 T Σs
u(1) = s(0)
Σc Si
T Σs
Σs
S1
Notice that the condition H (u(1), α) = 0 holds automatically due to the invariancy of Σs.
= = = = =
0, 0, 0, 0, 0,
= 0, = 0, = 0.
Tangent point to pseudo-equilibrium connection:
3. Introduction to AUTO97
f S2 Hx (x) g (x) x Σs S1
(1)
2. Supported n-dimensional cases 2.1. Local objects Pseudo-equilibrium:
H (x, α) = 0, (1) (x, α) + λ f (2) (x, α) = 0, λ f 1 2 λ1 + λ2 − 1 = 0.
Sj
u(0)
Σc
u(1)
Si
F : R n × R m × R m d → R nd , with m + md = nb − nd + 1.
B : R nd × R nd × R m × R m d → R nb
Σs
exp.f
C---------------------------------------------------------------------C---------------------------------------------------------------------C exp : A boundary value problem (Bratu’s equation) C---------------------------------------------------------------------C---------------------------------------------------------------------C SUBROUTINE FUNC(NDIM,U,ICP,PAR,IJAC,F,DFDU,DFDP) C ---------- ---C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION U(NDIM),PAR(*),F(NDIM) C E=EXP(U(1)) F(1)=U(2) F(2)=-PAR(1)*E C RETURN END C SUBROUTINE STPNT(NDIM,U,PAR,T) C ---------- ----C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION U(NDIM),PAR(*) C PAR(1)=0 U(1)=0.0 U(2)=0.0 C
u ˙ − T f (i) (u, α) H (u(0), α) Hx (u(0), α), f (i) (u(0), α) H (u(1), α) λif (i) (u(1), α) + λj f (j ) (u(1), α) λi + λ j − 1
= = = = = =
0, 0, 0, 0, 0, 0.
• Algebraic continuation problems: F (x, α, β ) = 0, with n + m + md = nd + 1. • Boundary-value continuation problems: u ˙ (τ ) − F (u(τ ), α, β ) = 0, B (u(0), u(1), α, β ) = 0, where τ ∈ [0, 1], F : R n × R m × R m d → R nd
Sj
u(0) = u(1)
= = = =
0, 0, 0, 0.
Σs
Sj v (1)=s(0)
u(0)=s(1)
ΣcSiຫໍສະໝຸດ u(1)=v (0) Σc
Σs Si
Grazing in multidimensional (n ≥ 3) Filippov systems can create chaos.
Pseudo-equilibrium to boundary connection:
f (1) S2
(x)
H=0 S1 f (2)
f (2) (x)
Boundary equilibrium: f (i) (x, α) = 0, H (x, α) = 0.
This gives the sliding system: x ˙ = g (x, α), x ∈ Σs(α). (2)
• pseudo-equilibrium: g (P, α) = 0 and f (i)(P, α) are transversal to Σs(α) and anti-collinear; • boundary equilibrium: f (i) (X, α) = 0; • tangent point: Both f (i) (T, α) = 0 but Hx (T, α), f (i)(T, α) = 0.
Contents
Lecture 2. Filippov systems: Continuation of sliding bifurcations Yuri A. Kuznetsov
1. Reminder on Filippov systems
2. Supported n-dimensional cases
Σc
Sliding cycle with two standard segments:
u ˙ − Tif (i)(u, α) v ˙ − Tj f (j )(u, α) s ˙ − T0g (s, α) H (u(0), α) Hx (u(0), α), f (i)(u(0), α) H (u(1), α) v (0) − u(1) s(0) − v (1) s(1) − u(0)
= = = = = = = = =
Σc
0, 0, 0, 0, 0, 0, 0, 0, 0.
Grazing cycle:
u ˙ − T f (i) (u, α) H (u(0), α) (i)(u(0), α) H ( u (0) , α ) , f x u(0) − u(1)
2.2. Global objects Sliding cycle with one standard segment:
u ˙ − Tif (i)(u, α) s ˙ − T0g (s, α) H (u(0), α) Hx (u(0), α), f (i)(u(0), α) s(0) − u(1) s(1) − u(0)
3. Introduction to AUTO97
March 4, 2010
4. SlideCont 2.0
5. Examples
1. Reminder on Filippov systems References: x ˙= • Dercole, F. and Kuznetsov, Yu.A. User Guide to SclideCont 2.0, 2005. http://www.math.uu.nl/people/kuznet/cm/slidecont.pdf where x ∈ Rn, S1(α) = {x ∈ Rn : H (x, α) < 0}, S2 (α) = {x ∈ Rn : H (x, α) > 0}, H : Rn → R is smooth with Hx (x, α) = 0 on the discontinuity boundary • Dercole, F. and Kuznetsov, Yu.A. SlideCont: An AUTO97 driver for bifurcation analysis of Filippov systems. ACM TOMS 31 (2005), no.1, 95-119. Σ (α) = {x ∈ Rn : H (x, α) = 0} . For x ∈ Σ (α), define σ (x, α) = Hx (x, α), f (1)(x, α) Hx (x, α), f (2)(x, α) and introduce the crossing set: Σc (α) = {x ∈ Σ : σ (x, α) > 0}, • Kowalczyk, P., di Bernardo, M., Champneys, A. R., Hogan, S. J., Homer, M., Piiroinen, P. T., Kuznetsov, Yu. A., and Nordmark, A. Two-parameter discontinuity-induced bifurcations of limit cycles: Classification and open problems. Int. J. Bifurcation & Chaos 16 (2006), 601-629. and the sliding set: Σs(α) = {x ∈ Σ : σ (x, α) ≤ 0}. Points x ∈ Σs(α), where Hx (x, α), f (2) (x, α) − f (1) (x, α) = 0 are called singular sliding points. f (1)(x, α), x ∈ S1(α), f (2)(x, α), x ∈ S2(α), (1)