Infinite systems of non-colliding Brownian particles

朗道十卷英文名朗道十卷是世界著名物理学教材。

朗道十卷英文名如下所示。

一、《朗道理论物理学教程第一卷：力学》Course of Theoretical Physics Volume one: Mechanics二、《朗道理论物理学教程第二卷：场论》Course of Theoretical Physics Volume two: The Theory of Fields三、《朗道理论物理学教程第三卷：量子力学(非相对论理论) 》Course of Theoretical Physics Volume three: Quantum Mechanics (Non-relativistic Theory)四、《朗道理论物理学教程第四卷：量子电动力学》Course of Theoretical Physics Volume four: Quantum Electrodynamics 五、《朗道理论物理学教程第五卷：统计物理学》Course of Theoretical Physics Volume five: Statistical Physics六、《朗道理论物理学教程第六卷：流体动力学》Course of Theoretical Physics Volume six: Fluid Mechanics七、《朗道理论物理学教程第七卷：弹性理论》Course of Theoretical Physics Volume seven: Theory of Elasticity八、《朗道理论物理学教程第八卷：连续介质电动力学》Course of Theoretical Physics Volume eight: Electrodynamics of Continuous Media九、《朗道理论物理学教程第九卷：统计物理学》Course of Theoretical Physics Volume nine: Statistical Physics十、《朗道理论物理学教程第十卷：物理动理学》Course of Theoretical Physics Volume ten: Physical Kinetics。

我奇怪的想法英文作文The Curious Mind: A Journey Through Unusual Thoughts.In the vast expanse of the universe, our minds are tiny islands floating on a sea of infinity. They are the repositories of our thoughts, dreams, and imaginations, and sometimes, they are the birthplaces of strange and unusual ideas. These ideas, often labeled as "weird" or "strange" by society, are actually the most fascinating aspects of our existence. They push the boundaries of our understanding, challenge our perceptions, and force us to question the world we know.For me, one such strange idea has always fascinated me: the concept of parallel universes. The idea that there could be an infinite number of worlds, each with its own laws of physics, history, and culture, is mind-boggling. What if, somewhere out there, there is a universe where gravity works in reverse, or where the sun shines at night? Or perhaps a universe where history unfolded differently,and the outcomes of major events were entirely different?The concept of parallel universes is not just a figment of my imagination; it has been explored by physicists, philosophers, and writers alike. The idea gained popularity in the 20th century with the development of quantum physics, which suggested that the universe might be made up of multiple realities that coexist simultaneously. This theory, known as the Many-Worlds Interpretation, proposed thatevery possible outcome of a quantum event occurs in a separate universe.While the scientific community is still debating the validity of this theory, it has sparked a wave ofcreativity among writers and artists. It has given us a platform to explore the limitless possibilities ofexistence and to imagine worlds that are entirely different from our own. Novels, movies, and TV shows have beeninspired by the concept of parallel universes, allowing usto escape the confines of our reality and immerse ourselves in exciting new worlds.Another strange idea that intrigues me is the concept of time travel. The idea that we could travel through time, visit the past or future, has fascinated humans for centuries. From the time-traveling heroes of sciencefiction novels to the philosophical debates about the nature of time, this concept has always captivated our imaginations.The possibility of time travel raises a number of fascinating questions. Could we change the course ofhistory by interfering with past events? Would we even be able to recognize the future if we saw it? And what wouldit mean to travel through time and find ourselves in a world that is entirely different from the one we left?These are questions that science has yet to answer, but they are questions that continue to inspire us to push the boundaries of our understanding. The concept of time travel may never become a reality, but it remains a powerful tool for exploring our understanding of the universe and our place within it.In conclusion, strange ideas are not just figments of our imaginations; they are windows to a world beyond our comprehension. They challenge our perceptions, push the boundaries of our understanding, and inspire us to question everything we know. Whether it's the concept of parallel universes or the possibility of time travel, these ideas force us to reevaluate our understanding of the world and our place within it. As we continue to explore the vast expanse of the universe and the infinite possibilities of our minds, these strange ideas will continue to guide us on our journey through existence.。

Nonlinear Systems and Control Nonlinear systems and control are essential topics in the field of engineering and mathematics. These systems are characterized by their complex behavior, which cannot be fully described by linear equations. Nonlinear systems can exhibit awide range of behaviors, including chaos, bifurcation, and instability, making them challenging to analyze and control. One of the key challenges in dealingwith nonlinear systems is the lack of a general theory that can be applied to all such systems. Unlike linear systems, which can be analyzed using well-established techniques such as Laplace transforms and transfer functions, nonlinear systems require more sophisticated methods, such as Lyapunov stability analysis, phase plane analysis, and numerical simulations. These methods often require a deep understanding of the underlying mathematics and can be computationally intensive, making the analysis and control of nonlinear systems a daunting task. Another challenge in dealing with nonlinear systems is the presence of uncertainties and disturbances. In real-world applications, nonlinear systems are often subject to external disturbances and uncertainties in the system parameters, which can significantly affect their behavior. This makes it difficult to design controllers that can effectively stabilize and control the system in the presence of such uncertainties. Robust control techniques, such as H-infinity control and sliding mode control, have been developed to address these challenges, but they often require a detailed knowledge of the system dynamics and uncertainties, which may not always be available. Nonlinear systems also pose challenges in terms of their control and optimization. Unlike linear systems, where the optimal control and optimization problems can often be solved analytically, nonlinear systems require the use of numerical optimization techniques, such as gradient descent and genetic algorithms. These methods can be computationally expensive and may not always guarantee convergence to the global optimum, especially for highly nonlinear and complex systems. Despite these challenges, the study of nonlinear systems and control is of great importance in many engineering and scientific disciplines. Nonlinear systems are ubiquitous in nature, appearing in fields such as physics, biology, and economics. Understanding and controlling these systems is essentialfor developing advanced technologies, such as autonomous vehicles, robotic systems,and renewable energy systems, which often exhibit highly nonlinear behavior. In conclusion, the analysis and control of nonlinear systems pose significant challenges due to their complex behavior, uncertainties, and the lack of a general theory. However, the study of nonlinear systems is crucial for advancing technology and understanding natural phenomena. Researchers and engineers continue to develop new methods and techniques to address these challenges, with the goal of effectively analyzing and controlling nonlinear systems in a wide range of applications.。

二叠纪-三叠纪灭绝事件二叠纪-三叠纪灭绝事件（Permian–Triassic extinction event）是一个大规模物种灭绝事件，发生于古生代二叠纪与中生代三叠纪之间，距今大约2亿5140万年[1][2]。

若以消失的物种来计算，当时地球上70%的陆生脊椎动物，以及高达96%的海中生物消失[3]；这次灭绝事件也造成昆虫的唯一一次大量灭绝。

计有57%的科与83%的属消失[4][5]。

在灭绝事件之后，陆地与海洋的生态圈花了数百万年才完全恢复，比其他大型灭绝事件的恢复时间更长久[3]。

此次灭绝事件是地质年代的五次大型灭绝事件中，规模最庞大的一次，因此又非正式称为大灭绝（Great Dying）[6]，或是大规模灭绝之母（Mother of all mass extinctions）[7]。

二叠纪-三叠纪灭绝事件的过程与成因仍在争议中[8]。

根据不同的研究，这次灭绝事件可分为一[1]到三[9]个阶段。

第一个小型高峰可能因为环境的逐渐改变，原因可能是海平面改变、海洋缺氧、盘古大陆形成引起的干旱气候；而后来的高峰则是迅速、剧烈的，原因可能是撞击事件、火山爆发[10]、或是海平面骤变，引起甲烷水合物的大量释放[11]。

目录? 1 年代测定? 2 灭绝模式o 2.1 海中生物o 2.2 陆地无脊椎动物o 2.3 陆地植物? 2.3.1 植物生态系统? 2.3.2 煤层缺口o 2.4 陆地脊椎动物o 2.5 灭绝模式的可能解释? 3 生态系统的复原o 3.1 海洋生态系统的改变o 3.2 陆地脊椎动物? 4 灭绝原因o 4.1 撞击事件o 4.2 火山爆发o 4.3 甲烷水合物的气化o 4.4 海平面改变o 4.5 海洋缺氧o 4.6 硫化氢o 4.7 盘古大陆的形成o 4.8 多重原因? 5 注释? 6 延伸阅读? 7 外部链接年代测定在西元二十世纪之前，二叠纪与三叠纪交界的地层很少被发现，因此科学家们很难准确地估算灭绝事件的年代与经历时间，以及影响的地理范围[12]。

Multibody SystemsMultibody Systems Multibody systems are a crucial concept in engineering and physics, playing a significant role in various fields such as robotics, biomechanics, and automotive engineering. In this essay, we will delve into the fundamentals of multibody systems, their applications, and the challenges associated with their analysis and design. To begin with, a multibody system consists of multiple interconnected bodies or links, each with its own motion and interaction with other bodies. These systems are used to model and analyze complex mechanical systems, such as vehicles, machinery, and even biological organisms. The dynamics of multibody systems are governed by Newton's laws of motion andEuler's equations, which describe the motion of each body and the forces and torques acting upon them. One of the key applications of multibody systems is in robotics, where they are used to model the motion and interaction of robot manipulators, humanoid robots, and other mechanical systems. By simulating the dynamics of these systems, engineers can optimize their design, control their motion, and ensure their stability and performance. In addition, multibody simulations are also used in virtual prototyping and testing of automotive and aerospace systems, allowing engineers to evaluate the behavior of complex mechanical systems under various operating conditions. Despite their wide-ranging applications, the analysis and design of multibody systems pose several challenges. One of the primary challenges is the computational complexity associated with simulating the dynamics of interconnected bodies. As the number of bodies andtheir interactions increase, the computational cost of simulating the system also increases significantly. This necessitates the use of advanced numerical methods and high-performance computing techniques to efficiently analyze and design multibody systems. Furthermore, the accurate modeling of contact and friction between bodies in multibody systems is another challenging aspect. Contact forces and frictional effects play a crucial role in the behavior of mechanical systems, especially in applications such as robotics and biomechanics. Therefore,developing accurate contact models and frictional algorithms is essential for the realistic simulation and analysis of multibody systems. In conclusion, multibody systems are a fundamental concept in engineering and physics, with diverseapplications ranging from robotics to biomechanics. The analysis and design of multibody systems present challenges in terms of computational complexity and accurate modeling of contact and friction. However, advancements in computational methods and simulation techniques continue to drive the development of innovative multibody systems with improved performance and reliability.。

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.。

三体中的名词术语英文刘宇昆The Three-Body Problem by Liu Cixin has captivated readers around the world with its intricate plot and rich scientific concepts. One of the unique aspects of this acclaimed science fiction novel is the author's introduction of various specialized terms and jargon related to the fictional "Three-Body" universe. These terms not only add depth and authenticity to the narrative but also challenge readers to delve deeper into the complex ideas presented. In this essay we will explore some of the key English terminology used in The Three-Body Problem and examine their significance within the context of the story.One of the most prominent terms in the novel is "Trisolaris" which refers to the fictional three-star planetary system that serves as the primary setting for much of the narrative. The word "Trisolaris" is a combination of the prefix "tri-" meaning three and the word "solaris" meaning related to the sun. This term encapsulates the core premise of the story where a planet orbits not one but three suns, leading to extreme climatic instability and unpredictability. The chaotic nature of the Trisolaris system is a key driver of the plot as it sets the stagefor the extraterrestrial civilization's attempts to flee their doomed homeworld and seek refuge on Earth.Another important term is "sophon" which describes the mysterious and highly advanced microscopic probes sent by the Trisolarians to infiltrate and monitor human civilization. The word "sophon" is derived from the Greek word "sophos" meaning wise or intelligent. This term effectively conveys the incredible technological capabilities of the Trisolarian civilization, as these sophons are able to manipulate the fundamental particles of the universe to gather information and even influence human affairs. The sophons serve as a constant threat and source of tension throughout the narrative, highlighting the asymmetry of power between the Trisolarians and humanity.A crucial concept introduced in the novel is that of "the Sphere" which represents the boundary of the Trisolaris system's influence. This term refers to an imaginary three-dimensional sphere within which the Trisolarian civilization can effectively exert its control and disrupt the normal laws of physics. The Sphere is a crucial plot device as it defines the limits of the Trisolarians' technological superiority and the extent to which they can intervene in human affairs. The constant struggle to expand or maintain the Sphere is a central theme in the novel, as both the Trisolarians and humanity attempt to gain the upper hand in this cosmic chess match.Another notable term is "dark forest" which describes the author's vision of a hostile and unforgiving universe where civilizations must remain hidden and silent to avoid annihilation. This metaphor is used to convey the idea that in a universe teeming with intelligent life, the safest strategy for survival is to avoid drawing attention to oneself, much like a hunter hiding in a dark forest. The "dark forest" concept is a key driver of the Trisolarians' decision to seek out and destroy humanity, as they believe that by remaining hidden, they can avoid the fate that has befallen countless other civilizations.The term "cosmic sociology" is also introduced in the novel to describe the study of the relationships and interactions between different civilizations on a cosmic scale. This field of study is central to the Trisolarians' understanding of the universe and their attempts to navigate the treacherous landscape of interstellar politics. The term "cosmic sociology" highlights the novel's exploration of the broader implications of humanity's first contact with an alien civilization, and the potential challenges and consequences that such an encounter may bring.Another intriguing term is "supra-quantum computer" which refers to the Trisolarians' most advanced computing technology. This term suggests a level of computational power that transcends the limitations of traditional quantum computers, allowing theTrisolarians to perform feats of calculation and simulation that are beyond the reach of human technology. The supra-quantum computer is a crucial plot device, as it enables the Trisolarians to model and manipulate the behavior of human civilization with uncanny precision.Finally, the term "Wallfacer" is used to describe a select group of individuals tasked with devising strategies to defend Earth against the Trisolarian invasion. The word "Wallfacer" evokes the image of a defensive wall or barrier, suggesting the critical role these individuals play in protecting humanity from the extraterrestrial threat. The Wallfacers are imbued with immense power and resources, but their true plans and motivations remain shrouded in mystery, adding to the suspense and intrigue of the narrative.These are just a few of the many specialized terms and concepts that are integral to the world-building and storytelling in The Three-Body Problem. Each of these terms serves to deepen the reader's understanding of the complex scientific and philosophical ideas that underpin the novel's narrative. By exploring these terms and their significance, we gain a deeper appreciation for the richness and depth of Liu Cixin's imaginative vision, and the ways in which he has seamlessly blended scientific speculation with captivating storytelling.。

高中英语世界著名科学家单选题50题1. Albert Einstein was born in ____.A. the United StatesB. GermanyC. FranceD. England答案：B。

解析：Albert Einstein（阿尔伯特·爱因斯坦）出生于德国。

本题主要考查对著名科学家爱因斯坦国籍相关的词汇知识。

在这几个选项中，the United States是美国，France是法国，England是英国，而爱因斯坦出生于德国，所以选B。

2. Isaac Newton is famous for his discovery of ____.A. electricityB. gravityC. radioactivityD. relativity答案：B。

解析：Isaac Newton 艾萨克·牛顿）以发现万有引力gravity）而闻名。

electricity是电，radioactivity是放射性，relativity 是相对论，这些都不是牛顿的主要发现，所以根据对牛顿主要成就的了解，选择B。

3. Marie Curie was the first woman to win ____ Nobel Prizes.A. oneB. twoC. threeD. four答案：B。

解析：Marie Curie 居里夫人）是第一位获得两项诺贝尔奖的女性。

这题主要考查数字相关的词汇以及对居里夫人成就的了解，她在放射性研究等方面的贡献使她两次获得诺贝尔奖，所以选B。

4. Thomas Edison is well - known for his invention of ____.A. the telephoneB. the light bulbC. the steam engineD. the computer答案：B。

解析：Thomas Edison（ 托马斯·爱迪生）以发明电灯（the light bulb）而闻名。

The mysteries of the universeMultiversesThe mysteries of the multiverse have long captivated the minds of scientists, philosophers, and curious individuals alike. The concept of multiple universes existing simultaneously, each with its own set of physical laws and possibilities, opens up a realm of infinite possibilities and questions that challenge our understanding of reality. One of the most intriguing aspects of the multiverse theory is the idea that there could be an infinite number of universes, each branching off from our own in different ways. This idea stems from the concept of quantum mechanics, which suggests that every possible outcome of a given situation actually occurs in a separate universe. This means that there could be universes where dinosaurs still roam the Earth, where humans never evolved, or where gravity works in reverse. The implications of the multiverse theory are profound, as it raises questions about the nature of reality and our place within it. If there are indeed multiple universes, each with its own version of us, what does that meanfor our sense of self and identity? Are we truly unique individuals, or are we merely one version of ourselves among an infinite number of possibilities? Some argue that the multiverse theory offers a solution to the fine-tuning problem in physics, which suggests that the fundamental constants of the universe are so precisely balanced that even the slightest change would render life impossible. In a multiverse scenario, it is proposed that there are an infinite number of universes, each with its own set of constants, and we just happen to find ourselves in one that is conducive to life. This idea has sparked debate among scientists and theologians alike, with some seeing it as evidence of a grand design, while others view it as a purely natural phenomenon. From a philosophical standpoint, the multiverse theory challenges our notions of causality and determinism. If every possible outcome of a given situation occurs in a separate universe, does that mean that free will is an illusion? Are our choices predetermined by the laws of physics, or do we have the ability to shape our own destinies across multiple universes? The concept of the multiverse also raises questions about the nature of consciousness and existence. If there are infiniteversions of ourselves in different universes, are we all connected on some level? Could there be a way to communicate or travel between universes, and if so, what would that mean for our understanding of reality and the universe at large? In conclusion, the mysteries of the multiverse continue to intrigue and perplex us, challenging our understanding of reality and our place within it. The concept of multiple universes existing simultaneously opens up a realm of infinite possibilities and questions that push the boundaries of science, philosophy, and human imagination. As we continue to explore and contemplate the implications of the multiverse theory, we are faced with profound questions about the nature of existence, consciousness, and the very fabric of reality itself.。