Session Keynote Lyapunov Function Candidates for Descriptor Systems Problems and Solutions

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英语语言学function of language知识分享

Referential: to convey message and information Poetic: to indulge in language for its own sake Emotive: to express attitudes, feelings and emotions Conative: to persuade and influence others through
Ideational metafunction
The ideational function is language concerned with building and maintaining a theory of experience. It includes the experiential function and the logical function.
Halliday’s view
Interpersonal metafunction
The interpersonal function refers to the grammatical s that enable speakers to enact their complex and diverse interpersonal relations.
元语言功能（弄清意图、词语和意义）。
Halliday’s view
From early on in his account of language, Halliday has argued that it is inherently functional. His early papers on the grammar of English make reference to the "functional components" of language, as "generalized uses of language, which, since they seem to determine the nature of the language system, require to be incorporated into our account of that system." Halliday argues that this functional organization of language "determines the form taken by grammatical structure".

有界噪声激励下典型非线性系统响应的最大Lyapunov指数估计

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否是混沌的。
关键词：声学；有界噪声；空间重构；小数据最法；最大Ｌａｕｏ指数；沌相ｙｐｎｖ混中图分类号：２；７０３４Ｏ１５文献标识码：ＡＤＯ编码：１９９ｓ０６１５ —０１２０３Ｉ０６／ｉｎ１０ —３５２１００３ｊｓ

经典的lyapunov函数方法

经典的lyapunov函数方法什么是经典的Lyapunov函数方法，并探讨其在动力系统和控制理论中的应用？一、引言中括号（[ ]）是数学中常用的符号之一，用来表示某种运算或者代表一类操作。

在这篇文章中，我们将探讨经典的Lyapunov函数方法，并研究其在动力系统和控制理论中的应用。

Lyapunov函数方法是一种基于Lyapunov稳定性理论的分析方法，通过构造合适的Lyapunov函数来判断系统的稳定性。

在这篇文章中，我们将按照以下步骤对经典的Lyapunov函数方法进行详细介绍和分析。

二、什么是Lyapunov函数？在进一步讨论Lyapunov函数方法之前，我们首先需要了解什么是Lyapunov 函数。

Lyapunov函数是数学中的一种特殊函数，被广泛应用于动力系统和控制理论中。

Lyapunov函数具有以下特点：1)它是一个实值函数，通常对于特定的系统状态，其值是实数；2)它能够量化系统的稳定性，即通过函数值的大小可以判断系统是否稳定；3)它具有非负性，即在所有系统状态下，函数值始终大于等于零。

三、Lyapunov稳定性理论Lyapunov函数方法基于Lyapunov稳定性理论，该理论由俄国数学家M.A. Lyapunov在19世纪末提出。

Lyapunov稳定性理论主要研究动力系统的稳定性。

在给定一个动力系统的演化方程之后，通过构造一个合适的Lyapunov函数来判断系统是否具有稳定性。

四、如何构造Lyapunov函数？Lyapunov函数的构造是Lyapunov函数方法的关键步骤。

在实际应用中，通常通过以下步骤构造Lyapunov函数：1)选择一个合适的函数形式，通常是系统状态的某种线性组合；2)确定函数的系数，通常通过经验或者结合实际问题的特点进行选择；3)验证函数的非负性和系统稳定性。

通过这些步骤，我们可以构造出一个合适的Lyapunov函数。

五、Lyapunov稳定性和系统稳定性的判定在Lyapunov函数方法中，通过对Lyapunov函数进行分析，可以判断系统是否稳定。

New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems

m m
x ˙(t ) = y(t );
y(t ) =
i=1
Di y(t − hi ) +
i=0
Ai x(t − hi ):
(2)
The latter can be represented in the form of descriptor system with discrete and distributed delay in the “fast variable” y:
i=1
x(t ) y(t )
+ V 1 + V2 ;
(4)
I 0 ; 0 0
m t t −hi
P=
P1 P2
0 ; P3
T P1 = P1 ¿0;() (6)yT (s)Qi y(s) d s;
Qi ¿0
and
m
V2 =
i=1
0 −hi
t t +Â
yT (s)Ri y(s) d s d Â;
Ri ¿0:
E. Fridman ∗
Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 69978, Israel Received 10 September 2000; received in revised form 14 February 2001
m m m
x ˙(t ) = y(t );
0 = − y(t ) +
i=1
D i y (t − h i ) +
i=0
Ai
x(t ) −
i=1
Ai
t t −hi
y(s) d s:

Observer-based adaptive fuzzy backstepping dynamic surface control for a class of non-linear ....

Published in IET Control Theory and Applications Received on 29th October 2010 Revised on 10th February 2011 doi: 10.1049/iet-cta.2010.0632
1426 & The Institution of Engineering and Technology 2011
fuzzy or neural backstepping control approaches suffer from the problem of ‘explosion of complexity’. The ‘explosion of complexity’ is caused by repeated differentiations of some non-linear functions at each step within the conventional backstepping technique. As a result, the complexity of a controller drastically grows as the order of the system increases. Recently, the dynamic surface control (DSC) technique has been proposed to avoid this problem by introducing a ﬁrst-order low-pass ﬁlter at each step of the conventional backstepping design procedure [17 – 20]. In [17], a robust control is studied for non-linear systems in strict feedback form. In [18], a simpliﬁed DSC algorithm is developed for non-linear systems in parametric strict feedback form. In [19, 20], adaptive DSC approaches are proposed for adaptive tracking control of a class of canonical-form SISO systems with and without time-delays, respectively. However, the above-mentioned DSC approaches have two limitations. One is that these approaches require that the controlled non-linear dynamics models be known exactly or the unknown non-linear functions can be linearly parameterised. If those kinds of knowledge are not available a priori, these adaptive backstepping controllers cannot be applied. The other is that they require that all the states be available for measurements. To cope with the problem of ‘explosion of complexity’ inherent in the existing adaptive fuzzy or neural backstepping control schemes, an adaptive neural backstepping control

非线性系统(第三版)(英文版)chapter4[2页][001]精选全文完整版

1Additional Exercises for Chapter 41.For each of the following systems,use a quadratic Lyapunov function candidate to show that the origin is asymptotically stable.Then,investigate whether the origin is globally asymptotically stable.(1)˙x 1=−x 1+x 22,˙x 2=−x 2(2)˙x 1=(x 1−x 2)(x 21+x 22−1),˙x 2=(x 1+x 2)(x 21+x 22−1)(3)˙x 1=−x 1+x 21x 2,˙x 2=−x 2+x 1ing V (x )=x 21+x 22,study stability of the origin of the system˙x 1=x 1(k 2−x 21−x 22)+x 2(x 21+x 22+k 2),˙x 2=−x 1(k 2+x 21+x 22)+x 2(k 2−x 21−x 22)when (a)k =0and (b)k =0.ing the variable gradient method,ﬁnd a Lyapunov function V (x )that shows asymptotic stability ofthe origin of the system˙x 1=x 2,˙x 2=−(x 1+x 2)−sin(x 1+x 2)4.Consider the system˙x 1=x 2,˙x 2=x 1−sat(2x 1+x 2)Show that the origin is asymptotically stable,but not globally asymptotically stable.5.Show that the origin of the following system is unstable.˙x 1=−x 1+x 62,˙x 2=x 32+x 616.Consider the system˙z =−m i =1a i y i ,˙y i =−h (z,y )y i +b i g (z ),i =1,2,...,mwhere z is a scalar,y T =(y 1,...,y m ).The functions h (·,·)and g (·)are continuously diﬀerentiable for all (z,y )and satisfy zg (z )>0,∀z =0,h (z,y )>0,∀(z,y )=0,and z0g (σ)dσ→∞as |z |→∞.The constants a i and b i satisfy b i =0and a i /b i >0,∀i =1,2,...,m .Show that the origin is an equilibrium point,and investigate its stability using a Lyapunov function candidate of the formV (z,y )=α z 0g (σ)dσ+mi =1βi y 2i7.Consider the system˙x 1=x 2,˙x 2=−x 1−x 2sat(x 22−x 23),˙x 3=x 3sat(x 22−x 23)where sat(·)is the saturation function.Show that the origin is the unique equilibrium point,and useV (x )=x T x to show that it is globally asymptotically stable.8.The origin x =0is an equilibrium point of the system˙x 1=−kh (x )x 1+x 2,˙x 2=−h (x )x 2−x 31Let D ={x ∈R 2| x 2<1}.Using V (x )=14x 41+12x 22,investigate stability of the origin in each ofthe following cases.(1)k >0,h (x )>0,∀x ∈D ;(2)k >0,h (x )>0,∀x ∈R 2;(3)k >0,h (x )<0,∀x ∈D ;(4)k >0,h (x )=0,∀x ∈D ;(5)k =0,h (x )>0,∀x ∈D ;(6)k =0,h (x )>0,∀x ∈R 2.29.Consider the system˙x 1=−x 1+g (x 3),˙x 2=−g (x 3),˙x 3=−ax 1+bx 2−cg (x 3)where a ,b ,and c are positive constants and g (·)is a locally Lipschitz function that satisﬁesg (0)=0and yg (y )>0,∀0<|y |<k,k >0(a)Show that the origin is an isolated equilibrium point.(b)With V (x )=12ax 21+12bx 22+ x 3g (y )dy as a Lyapunov function candidate,show that the origin is asymptotically stable.(c)Suppose yg (y )>0∀y =0.Is the origin globally asymptotically stable?10.Consider the system˙x 1=x 2,˙x 2=−a sin x 1−kx 1−dx 2−cx 3,˙x 3=−x 3+x 2where all coeﬃcients are positive and k >a .Using V (x )=2a x 10sin y dy +kx 21+x 22+px 23with some p >0,show that the origin is globally asymptotically stable.11.Show that the system˙x 1=11+x 3−x 1,˙x 2=x 1−2x 2,˙x 3=x 2−3x 3has a unique equilibrium point in the region x i ≥0,i =1,2,3,and investigate stability of this point using linearization.12.For each of the following systems,use linearization to show that the origin is asymptotically stable.Then,show that the origin is globally asymptotically stable.(1)˙x 1=−x 1+x 2˙x 2=(x 1+x 2)sin x 1−3x 2(2)˙x 1=−x 31+x 2˙x 2=−ax 1−bx 2,a,b >013.Consider the system˙x 1=−x 31+α(t )x 2,˙x 2=−α(t )x 1−x 32where α(t )is a continuous,bounded function.Show that the origin is globally uniformly asymptoticallystable.Is it exponentially stable?14.Consider the system˙x 1=x 2,˙x 2=−x 1−(1+b cos t )x 2Find b ∗>0such that the origin is exponentially stable for all |b |<b ∗.15.Consider the system˙x 1=x 2−g (t )x 1(x 21+x 22),˙x 2=−x 1−g (t )x 2(x 21+x 22)where g (t )is continuously diﬀerentiable,bounded,and g (t )≥k >0for all t ≥0.Is the originuniformly asymptotically stable?Is it exponentially stable?16.Consider two systems represented by˙x =f (x )(1)˙x =h (x )f (x )(2)where f :R n →R n and h :R n →R are continuously diﬀerentiable,f (0)=0,and h (0)>0.Show that the origin of (1)is exponentially stable if and only if the origin of (2)is exponentially stable.17.Investigate input-to-state stability of the system˙x 1=(x 1−x 2+u )(x 21+x 22−1),˙x 2=(x 1+x 2+u )(x 21+x 22−1)。

【国家自然科学基金】_共同lyapunov函数_基金支持热词逐年推荐_【万方软件创新助手】_20140731

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数学英文论文

070451 Controlling chaos based on an adaptive nonlinear compensatingmechanism*Corresponding author,Xu Shu ,email:123456789@Abstract The control problems of chaotic systems are investigated in the presence of parametric u ncertainty and persistent external distu rbances based on nonlinear control theory. B y designing a nonlinear compensating mechanism, the system deterministic nonlinearity, parametric uncertainty and disturbance effect can be compensated effectively. The renowned chaotic Lorenz system subject to parametric variations and external disturbances is studied as an illustrative example. From Lyapu nov stability theory, sufficient conditions for the choice of control parameters are derived to guarantee chaos control. Several groups of experiments are carried out, including parameter change experiments, set-point change experiments and disturbance experiments. Simulation results indicate that the chaotic motion can be regulated not only to stead y states but also to any desired periodic orbits with great immunity to parametric variations and external distu rbances.Keywords: chaotic system, nonlinear compensating mechanism, Lorenz chaotic systemPACC: 05451. IntroductionChaotic motion, as the peculiar behavior in deterministic systems, may be undesirable in many cases, so suppressing such a phenomenon has been intensively studied in recent years. Generally speaking chaos suppression and chaos synchronization[1-4 ]are two active research fields in chaos control and are both crucial in application of chaos. In the following letters we only deal with the problem of chaos suppression and will not discuss the chaos synchronization problem.Since the early 1990s, the small time-dependent parameter perturbation was introduced by Ott,Grebogi, and Y orke to eliminate chaos,[5]many effective control methods have been reported in various scientific literatures.[1-4,6-36,38-44,46] There are two lines in these methods. One is to introduce parameter perturbations to an accessible system parameter, [5-6,8-13] the other is to introduce an additive external force to the original uncontrolled chaotic system. [14-37,39-43,47] Along the first line, when system parameters are not accessible or can not be changed easily, or the environment perturbations are not avoided, these methods fail. Recently, using additive external force to achieve chaos suppression purpose is in the ascendant. Referring to the second line of the approaches, various techniques and methods have been proposed to achieve chaos elimination, to mention only a few:(ⅰ) linear state feedback controlIn Ref.[14] a conventional feedback controller was designed to drive the chaotic Duffing equation to one of its inherent multiperiodic orbits.Recently a linear feedback control law based upon the Lyapunov–Krasovskii (LK) method was developed for the suppression of chaotic oscillations.[15]A linear state feedback controller was designed to solve the chaos control problem of a class of new chaotic system in Ref.[16].(ⅱ) structure variation control [12-16]Since Y u X proposed structure variation method for controlling chaos of Lorenz system,[17]some improved sliding-mode control strategies were*Project supported by the National Natural Science Foundation of C hina (Grant No 50376029). †Corresponding au thor. E-mail:zibotll@introduced in chaos control. In Ref.[18] the author used a newly developed sliding mode controller with a time-varying manifold dynamic to compensate the external excitation in chaotic systems. In Ref.[19] the design schemes of integration fuzzy sliding-mode control were addressed, in which the reaching law was proposed by a set of linguistic rules. A radial basis function sliding mode controller was introduced in Ref.[20] for chaos control.(ⅲ) nonlinear geometric controlNonlinear geometric control theory was introduced for chaos control in Ref.[22], in which a Lorenz system model slightly different from the original Lorenz system was studied considering only the Prandtl number variation and process noise. In Ref.[23] the state space exact linearization method was also used to stabilize the equilibrium of the Lorenz system with a controllable Rayleigh number. (ⅳ)intelligence control[24-27 ]An intelligent control method based on RBF neural network was proposed for chaos control in Ref.[24]. Liu H, Liu D and Ren H P suggested in Ref.[25] to use Least-Square Support V ector Machines to drive the chaotic system to desirable points. A switching static output-feedback fuzzy-model-based controller was studied in Ref.[27], which was capable of handling chaos.Other methods are also attentively studied such as entrainment and migration control, impulsive control method, optimal control method, stochastic control method, robust control method, adaptive control method, backstepping design method and so on. A detailed survey of recent publications on control of chaos can be referenced in Refs.[28-34] and the references therein.Among most of the existing control strategies, it is considered essentially to know the model parameters for the derivation of a controller and the control goal is often to stabilize the embedded unstable period orbits of chaotic systems or to control the system to its equilibrium points. In case of controlling the system to its equilibrium point, one general approach is to linearize the system in the given equilibrium point, then design a controller with local stability, which limits the use of the control scheme. Based on Machine Learning methods, such as neural network method[24]or support vector machine method,[25]the control performance often depends largely on the training samples, and sometimes better generalization capability can not be guaranteed.Chaos, as the special phenomenon of deterministic nonlinear system, nonlinearity is the essence. So if a nonlinear real-time compensator can eliminate the effect of the system nonlinearities, chaotic motion is expected to be suppressed. Consequently the chaotic system can be controlled to a desired state. Under the guidance of nonlinear control theory, the objective of this paper is to design a control system to drive the chaotic systems not only to steady states but also to periodic trajectories. In the next section the controller architecture is introduced. In section 3, a Lorenz system considering parametric uncertainties and external disturbances is studied as an illustrative example. Two control schemes are designed for the studied chaotic system. By constructing appropriate L yapunov functions, after rigorous analysis from L yapunov stability theory sufficient conditions for the choice of control parameters are deduced for each scheme. Then in section 4 we present the numerical simulation results to illustrate the effectiveness of the design techniques. Finally some conclusions are provided to close the text.2. Controller architectureSystem differential equation is only an approximate description of the actual plant due to various uncertainties and disturbances. Without loss of generality let us consider a nonlinear continuous dynamic system, which appears strange attractors under certain parameter conditions. With the relative degree r n(n is the dimension of the system), it can be directly described or transformed to the following normal form:121(,,)((,,)1)(,,,)(,,)r r r z z z z za z v wb z v u u d z v u u vc z v θθθθθθθθ-=⎧⎪⎪⎪=⎪=+∆+⎨⎪ ++∆-+⎪⎪ =+∆+⎪=+∆⎩ (1) 1y z =where θ is the parameter vector, θ∆ denotes parameter uncertainty, and w stands for the external disturbance, such that w M ≤with Mbeingpositive.In Eq.(1)1(,,)T r z z z = can be called external state variable vector,1(,,)T r n v v v += called internal state variable vector. As we can see from Eq.(1)(,,,,)(,,)((,,)1)d z v w u a z v w b z v uθθθθθθ+∆=+∆+ ++∆- (2)includes system nonlinearities, uncertainties, external disturbances and so on.According to the chaotic system (1), the following assumptions are introduced in order to establish the results concerned to the controller design (see more details in Ref.[38]).Assumption 1 The relative degree r of the chaotic system is finite and known.Assumption 2 The output variable y and its time derivatives i y up to order 1r -are measurable. Assumption 3 The zero dynamics of the systemis asymptotically stable, i.e.,(0,,)v c v θθ=+∆ is asymptotically stable.Assumption 4 The sign of function(,,)b z v θθ+∆is known such that it is always positive or negative.Since maybe not all the state vector is measurable, also (,,)a z v θθ+∆and (,,)b z v θθ+∆are not known, a controller with integral action is introduced to compensate theinfluenceof (,,,,)d z v w u θθ+∆. Namely,01121ˆr r u h z h z h z d------ (3) where110121112100ˆr i i i r r r r i i ii r i i d k z k k k z kz k uξξξ-+=----++-==⎧=+⎪⎪⎨⎪=----⎪⎩∑∑∑ (4)ˆdis the estimation to (,,,,)d z v w u θθ+∆. The controller parameters include ,0,,1i h i r =- and ,0,,1i k i r =- . Here011[,,,]Tr H h h h -= is Hurwitz vector, such that alleigenvalues of the polynomial121210()rr r P s s h sh s h s h --=+++++ (5)have negative real parts. The suitable positive constants ,0,,1i h i r =- can be chosen according to the expected dynamic characteristic. In most cases they are determined according to different designed requirements.Define 1((,,))r k sign b z v θμ-=, here μstands for a suitable positive constant, and the other parameters ,0,,2i k i r =- can be selected arbitrarily. After011[,,,]Tr H h h h -= is decided, we can tune ,0,,1i k i r =- toachievesatisfyingstaticperformances.Remark 1 In this section, we consider a n-dimensional nonlinear continuous dynamic system with strange attractors. By proper coordinate transformation, it can be represented to a normal form. Then a control system with a nonlinear compensator can be designed easily. In particular, the control parameters can be divided into two parts, which correspond to the dynamic characteristic and the static performance respectively (The theoretic analysis and more details about the controller can be referenced to Ref.[38]).3. Illustrative example-the Lorenz systemThe Lorenz system captures many of the features of chaotic dynamics, and many control methods have been tested on it.[17,20,22-23,27,30,32-35,42] However most of the existing methods is model-based and has not considered the influence ofpersistent external disturbances.The uncontrolled original Lorenz system can be described by112121132231233()()()()x P P x P P x w x R R x x x x w xx x b b x w =-+∆++∆+⎧⎪=+∆--+⎨⎪=-+∆+⎩ (6) where P and R are related to the Prendtl number and Rayleigh number respectively, and b is a geometric factor. P ∆, R ∆and b ∆denote the parametric variations respectively. The state variables, 1x ,2x and 3x represent measures of fluid velocity and the spatial temperature distribution in the fluid layer under gravity ， and ,1,2,3i w i =represent external disturbance. In Lorenz system the desired response state variable is 1x . It is desired that 1x is regulated to 1r x , where 1r x is a given constant. In this section we consider two control schemes for system (6).3.1 Control schemes for Lorenz chaotic system3.1.1 Control scheme 1The control is acting at the right-side of the firstequation (1x), thus the controlled Lorenz system without disturbance can be depicted as1122113231231x Px Px u xRx x x x x x x bx y x =-++⎧⎪=--⎨⎪=-⎩= (7) By simple computation we know system (7) has relative degree 1 (i.e., the lowest ordertime-derivative of the output y which is directly related to the control u is 1), and can be rewritten as1122113231231z Pz Pv u vRz z v v v z v bv y z =-++⎧⎪=--⎨⎪=-⎩= (8) According to section 2, the following control strategy is introduced:01ˆu h z d=-- (9) 0120010ˆ-d k z k k z k uξξξ⎧=+⎪⎨=--⎪⎩ (10) Theorem 1 Under Assumptions 1 toAssumptions 4 there exists a constant value *0μ>, such that if *μμ>, then the closed-loop system (8), (9) and (10) is asymptotically stable.Proof Define 12d Pz Pv =-+, Eq.(8) can be easily rewritten as1211323123z d u v Rz z v v vz v bv =+⎧⎪=--⎨⎪=-⎩ (11) Substituting Eq.(9) into Eq.(11) yields101211323123ˆz h z d dv R z z v v v z v bv ⎧=-+-⎪=--⎨⎪=-⎩ (12) Computing the time derivative of d and ˆdand considering Eq.(12) yields12011132ˆ()()dPz Pv P h z d d P Rz z v v =-+ =--+- +-- (13) 0120010000100ˆ-()()ˆ=()d k z k k z k u k d u k d k z k d d k dξξξ=+ =--++ =-- - = (14)Defining ˆdd d =- , we have 011320ˆ()()dd d P h P R z P z v P v P k d=- =+- --+ (15) Then, we can obtain the following closed-loop system101211323123011320()()z h z dvRz z v v v z v bv d Ph PR z Pz v Pv P k d⎧=-+⎪=--⎪⎨=-⎪⎪=+---+⎩ (16) To stabilize the closed-loop system (16), a L yapunovfunction is defined by21()2V ςς=(17)where, ςdenotes state vector ()123,,,Tz v v d, isthe Euclidean norm. i.e.,22221231()()2V z v v dς=+++ (18) We define the following compact domain, which is constituted by all the points internal to the superball with radius .(){}2222123123,,,2U z v v d zv v dM +++≤(19)By taking the time derivative of ()V ςand replacing the system expressions, we have11223322*********01213()()(1)V z z v v v v dd h z v bv k P d R z v P R P h z d P v d P z v d ς=+++ =----++ +++-- (20) For any ()123,,,z v v d U ∈, we have: 222201230120123()()(1)V h z v b v k P dR z v PR Ph z d P v d d ς≤----+ ++++ ++ (21)Namely,12300()(1)22020V z v v dPR Ph R h R P ς⎡⎤≤- ⎣⎦++ - 0 - - 1 - 2⨯00123(1)()2Tb PR Ph P k P z v v d ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥0 ⎢⎥2⎢⎥++⎢⎥- - - +⎢⎥⎣22⎦⎡⎤⨯ ⎣⎦(22) So if the above symmetrical parameter matrix in Eq.(22) is positive definite, then V is negative and definite, which implies that system (16) is asymptotically stable based on L yapunov stability theory.By defining the principal minor determinants of symmetrical matrix in Eq.(22) as ,1,2,3,4i D i =, from the well-known Sylvester theorem it is straightforward to get the following inequations:100D h => (23)22004RD h =-> (24)23004R b D bh =-> (25)240302001()(1)(2)821[2(1)]08P M D k P D b PR Ph PR D Pb Ph R PR Ph =+-+++--+++>(26)After 0h is determined by solving Inequalities (23) to (25), undoubtedly, the Inequalities (26) can serve effectively as the constraints for the choice of 0k , i.e.20200031(1)(2)821[2(1)]8P M b PR Ph PR D Pb Ph R PR Ph k P D ++++ ++++>- (27)Here,20200*31(1)(2)821[2(1)]8P M b PR Ph PR D Pb Ph R PR Ph P D μ++++ ++++=-.Then the proof of the theorem 1 is completed. 3.1.2 Control scheme 2Adding the control signal on the secondequation (2x ), the system under control can be derived as112211323123x P x P x x R x x x x u xx x bx =-+⎧⎪=--+⎨⎪=-⎩ (28) From Eq.(28), for a target constant 11()r x t x =,then 1()0xt = , by solving the above differential equation, we get 21r r x x =. Moreover whent →∞,3r x converges to 12r x b . Since 1x and 2x havethe same equilibrium, then the measured state can also be chosen as 2x .To determine u , consider the coordinate transform:122133z x v x v x=⎧⎪=⎨⎪=⎩ and reformulate Eq.(28) into the following normal form:1223121231231zRv v v z u vPz Pv v z v bv y z =--+⎧⎪=-⎨⎪=-⎩= (29) thus the controller can be derived, which has the same expression as scheme 1.Theorem 2 Under Assumptions 1, 2, 3 and 4, there exists a constant value *0μ>, such that if *μμ>, then the closed-loop system (9), (10) and (29) is asymptotically stable.Proof In order to get compact analysis, Eq.(29) can be rewritten as12123123z d u v P z P v vz v bv =+⎧⎪=-⎨⎪=-⎩ (30) where 2231d Rv v v z =--Substituting Eq.(9) into Eq.(30),we obtain:1012123123ˆz h z d dv P z P v v z v bv ⎧=-+-⎪=-⎨⎪=-⎩ (31) Giving the following definition:ˆdd d =- (32) then we can get22323112123212301()()()()dRv v v v v z R Pz Pv Pz Pv v v z v bv h z d =--- =--- ----+ (33) 012001000ˆ-()d k z k k z k u k d u k dξξ=+ =--++ = (34) 121232123010ˆ()()()(1)dd d R Pz Pv Pz Pv v v z v bv h z k d=- =--- --+-+ (35)Thus the closed-loop system can be represented as the following compact form:1012123123121232123010()()()(1)zh z d v Pz Pv v z v bv d R Pz Pv Pz Pv v v z v bv h z k d⎧=-+⎪⎪=-⎪=-⎨⎪=---⎪⎪ --+-+⎩(36) The following quadratic L yapunov function is chosen:21()2V ςς=(37)where, ςdenotes state vector ()123,,,Tz v v d , is the Euclidean norm. i.e.,22221231()()2V z v v dς=+++ (38) We can also define the following compact domain, which is constituted by all the points internalto the super ball with radius .(){}2222123123,,,2U z v v d zv v dM =+++≤ (39)Differentiating V with respect to t and using Eq.(36) yields112233222201230011212322321312()(1)(1)()V z z v v v v dd h z P v bv k dP R h z d P z v z v v P b v v d P v d P z v d z v d ς=+++ =----+ +++++ ++--- (40)Similarly, for any ()123,,,z v v d U ∈, we have: 2222012300112133231()(1)(1)(2V h z P v b v k dPR h z d P z v v P b d P v d d M z dς≤----+ +++++ ++++ + (41)i.e.,12300()(12)22V z v v dPR M h P h P Pς⎡⎤≤- ⎣⎦+++ - -2 - 0 ⨯ 001230(12)(1)2TP b PR M h P k z v v d ⎡⎤⎢⎥⎢⎥⎢⎥ - ⎢⎥⎢⎥⎢⎥ ⎢⎥22⎢⎥⎢⎥ +++ - - -+⎢⎥⎣22⎦⎡⎤⨯ ⎣⎦(42) For brevity, Let1001(12)[(222)82(23)]P PR M h b PR P h M P b α=++++++ ++(43) 2201[(231)(13)]8P M P b b PR h α=+-+++ (44)230201(2)[2(12)8(2)(4)]PM P b P P PR M h P b Ph P α=++ +++ ++- (45)Based on Sylvester theorem the following inequations are obtained:100D h => (46)22004PD h P =-> (47)3202PMD bD =-> (48)403123(1)0D k D ααα=+---> (49)where,1,2,3,4i D i =are the principal minordeterminants of the symmetrical matrix in Eq.(42).*0k μ>*12331D αααμ++=- (50)The theorem 2 is then proved.Remark 2 In this section we give two control schemes for controlling chaos in Lorenz system. For each scheme the control depends on the observed variable only, and two control parameters are neededto be tuned, viz. 0h and 0k . According to L yapunov stability theory, after 0h is fixed, the sufficient condition for the choice of parameter 0k is also obtained.4. Simulation resultsChoosing 10P =,28R =, and 8/3b =, the uncontrolled Lorenz system exhibits chaotic behavior, as plotted in Fig.1. In simulation let the initial values of the state of thesystembe 123(0)10,(0)10,(0)10x x x ===.x1x 2x1x 3Fig.1. C haotic trajectories of Lorenz system (a) projected on12x x -plane, (b) projected on 13x x -plane4.1 Simulation results of control the trajectory to steady stateIn this section only the simulation results of control scheme 2 are depicted. The simulation results of control scheme 1 will be given in Appendix. For the first five seconds the control input is not active, at5t s =, control signal is input and the systemtrajectory is steered to set point2121(,,)(8.5,8.5,27.1)T Tr r r x x x b =, as can be seen inFig.2(a). The time history of the L yapunov function is illustrated in Fig.2(b).t/sx 1,x 2,x 3t/sL y a p u n o v f u n c t i o n VFig.2. (a) State responses under control, (b) Time history of the Lyapunov functionA. Simulation results in the presence ofparameters ’ changeAt 9t s =, system parameters are abruptly changed to 15P =,35R =, and 12/3b =. Accordingly the new equilibrium is changedto 2121(,,)(8.5,8.5,18.1)T Tr r r x x x b =. Obviously, aftervery short transient duration, system state converges to the new point, as shown in Fig.3(a). Fig.4(a) represents the evolution in time of the L yapunov function.B. Simulation results in the presence of set pointchangeAt 9t s =, the target is abruptly changedto 2121(,,)(12,12,54)T Tr r r x x x b =, then the responsesof the system state are shown in Fig.3(b). In Fig.4(b) the time history of the L yapunov function is expressed.t/sx 1,x 2,x 3t/sx 1,x 2,x 3Fig.3. State responses (a) in the presence of parameter variations, (b) in the presence of set point changet/sL y a p u n o v f u n c t i o n Vt/sL y a p u n o v f u n c t i o n VFig.4. Time history of the Lyapunov fu nction (a) in the presence of parameter variations, (b) in the presence of set point changeC. Simulation results in the presence ofdisturbanceIn Eq.(5)external periodic disturbance3cos(5),1,2,3i w t i π==is considered. The time responses of the system states are given in Fig.5. After control the steady-state phase plane trajectory describes a limit cycle, as shown in Fig.6.t/sx 1,x 2,x 3Fig.5. State responses in the presence of periodic disturbancex1x 3Fig.6. The state space trajectory at [10,12]t ∈in the presence ofperiodic disturbanceD. Simulation results in the presence of randomnoiseUnder the influence of random noise,112121132231233xPx Px x Rx x x x u xx x bx εδεδεδ=-++⎧⎪=--++⎨⎪=-+⎩ (51) where ,1,2,3i i δ= are normally distributed withmean value 0 and variance 0.5, and 5ε=. The results of the numerical simulation are depicted in Fig.7,which show that the steady responses are hardly affected by the perturbations.t/sx 1,x 2,x 3t/se 1,e 2,e 3Fig.7. Time responses in the presence of random noise (a) state responses, (b) state tracking error responses4.2 Simulation results of control the trajectory to periodic orbitIf the reference signal is periodic, then the system output will also track this signal. Figs.8(a) to (d) show time responses of 1()x t and the tracking trajectories for 3-Period and 4-period respectively.t/sx 1x1x 2t/sx 1x1x 2Fig.8. State responses and the tracking periodic orbits (a)&( b)3-period, (c)&(d) 4-periodRemark 3 The two controllers designed above solved the chaos control problems of Lorenz chaoticsystem, and the controller design method can also beextended to solve the chaos suppression problems of the whole Lorenz system family, namely the unified chaotic system.[44-46] The detail design process and close-loop system analysis can reference to the author ’s another paper.[47] In Ref.[47] according to different positions the scalar control input added,three controllers are designed to reject the chaotic behaviors of the unified chaotic system. Taking the first state 1x as the system output, by transforming system equation into the normal form firstly, the relative degree r (3r ≤) of the controlled systems i s known. Then we can design the controller with the expression as Eq.(3) and Eq.(4). Three effective adaptive nonlinear compensating mechanisms are derived to compensate the chaotic system nonlinearities and external disturbances. According toL yapunov stability theory sufficient conditions for the choice of control parameters are deduced so that designers can tune the design parameters in an explicit way to obtain the required closed loop behavior. By numeric simulation, it has been shown that the designed three controllers can successfully regulate the chaotic motion of the whole family of the system to a given point or make the output state to track a given bounded signal with great robustness.5. ConclusionsIn this letter we introduce a promising tool to design control system for chaotic system subject to persistent disturbances, whose entire dynamics is assumed unknown and the state variables are not completely measurable. By integral action the nonlinearities, including system structure nonlinearity, various disturbances, are compensated successfully. It can handle, therefore, a large class of chaotic systems, which satisfy four assumptions. Taking chaotic Lorenz system as an example, it has been shown that the designed control scheme is robust in the sense that the unmeasured states, parameter uncertainties and external disturbance effects are all compensated and chaos suppression is achieved. Some advantages of this control strategy can be summarized as follows: (1) It is not limited to stabilizing the embeddedperiodic orbits and can be any desired set points and multiperiodic orbits even when the desired trajectories are not located on the embedded orbits of the chaotic system.(2) The existence of parameter uncertainty andexternal disturbance are allowed. The controller can be designed according to the nominal system.(3) The dynamic characteristics of the controlledsystems are approximately linear and the transient responses can be regulated by the designer through controllerparameters ,0,,1i h i r =- .(4) From L yapunov stability theory sufficientconditions for the choice of control parameters can be derived easily.(5) The error converging speed is very fast evenwhen the initial state is far from the target one without waiting for the actual state to reach the neighborhood of the target state.AppendixSimulation results of control scheme 1.t/sx 1,x 2,x 3t/sL y a p u n o v f u n c t i o n VFig.A1. (a) State responses u nder control, (b) Time history of the Lyapunov functiont/sx 1,x 2,x 3t/sx 1,x 2,x 3Fig.A2. State responses (a) in the presence of parameter variations, (b) in the presence of set point changet/sL y a p u n o v f u n c t i o n Vt/sL y a p u n o v f u n c t i o n VFig.A3. Time history of the L yapu nov fu nction (a) in the presence of parameter variations, (b) in the presence of set point changet/sx 1,x 2,x 3Fig.A4. State responses in the presence of periodic disturbanceresponsest/sx 1,x 2,x 3Fig.A5. State responses in the presence of rand om noiset/sx 1x1x 2Fig.A6. State response and the tracking periodic orbits (4-period)References[1] Lü J H, Zhou T S, Zhang S C 2002 C haos Solitons Fractals 14 529[2] Yoshihiko Nagai, Hua X D, Lai Y C 2002 C haos Solitons Fractals 14 643[3] Li R H, Xu W , Li S 2007 C hin.phys.16 1591 [4]Xiao Y Z, Xu W 2007 C hin.phys.16 1597[5] Ott E ，Greb ogi C and Yorke J A 1990 Phys.Rev .Lett. 64 1196 [6]Yoshihiko Nagai, Hua X D, Lai Y C 1996 Phys.Rev.E 54 1190 [7] K.Pyragas, 1992 Phys. Lett. A 170 421 [8] Lima,R and Pettini,M 1990 Phys.Rev.A 41 726[9] Zhou Y F, Tse C K, Qiu S S and Chen J N 2005 C hin.phys. 14 0061[10] G .Cicog na, L.Fronzoni 1993 Phys.Rew .E 30 709 [11] Rakasekar,S. 1993 Pramana-J.Phys.41 295 [12] Gong L H 2005 Acta Phys.Sin.54 3502 (in C hinese) [13] Chen L,Wang D S 2007 Acta Phys.Sin.56 0091 (in C hinese) [14] C hen G R and Dong X N 1993 IEEE Trans.on Circuits andSystem-Ⅰ:Fundamental Theory and Applications 40 9 [15] J.L. Kuang, P.A. Meehan, A.Y.T. Leung 2006 C haos SolitonsFractals 27 1408[16] Li R H, Xu W, Li S 2006 Acta Phys.Sin.55 0598 (in C hinese) [17] Yu X 1996 Int.J.of Systems Science 27 355[18] Hsun-Heng Tsai, C hyu n-C hau Fuh and Chiang-Nan Chang2002 C haos,Solitons Fractals 14 627[19] Her-Terng Yau and C hieh-Li C hen 2006 C hao ,SolitonsFractal 30 709[20] Guo H J, Liu J H, 2004 Acta Phys.Sin.53 4080 (in C hinese) [21] Yu D C, Wu A G , Yang C P 2005 Chin.phys.14 0914 [22] C hyu n-C hau Fuh and Pi-Cheng Tu ng 1995 Phys.Rev .Lett.752952[23] Chen L Q, Liu Y Z 1998 Applied Math.Mech. 19 63[24] Liu D, R en H P, Kong Z Q 2003 Acta Phys.Sin.52 0531 (inChinese)[25] Liu H, Liu D and Ren H P 2005 Acta Phys.Sin.54 4019 (inChinese)[26] C hang W , Park JB, Joo YH, C hen GR 2002 Inform Sci 151227[27] Gao X, Liu X W 2007 Acta Phys.Sin. 56 0084 (in C hinese) [28] Chen S H, Liu J, Lu J 2002 C hin.phys.10 233 [29] Lu J H, Zhang S. 2001 Phys. Lett. A 286 145[30] Liu J, Chen S H, Xie J. 2003 C haos Solitons Fractals 15 643 [31] Wang J, Wang J, Li H Y 2005 C haos Solitons Fractals 251057[32] Wu X Q, Lu JA, C hi K. Tse, Wang J J, Liu J 2007 ChaoSolitons Fractals 31 631[33] A.L.Fradkov , R .J.Evans, 2002 Preprints of 15th IF AC W orldCongress on Automatic Control 143[34] Zhang H G 2003 C ontrol theory of chaotic systems (Shenyang:Northeastern University) P38 (in C hinese)[35] Yu-Chu Tian, Moses O.Tadé, David Levy 2002Phys.Lett.A.296 87[36] Jose A R , Gilberto E P, Hector P, 2003 Phys. Lett. A 316 196 [37] Liao X X, Yu P 2006 Chaos Solitons Fractals 29 91[38] Tornambe A, V aligi P.A 1994 Measurement, and C ontrol 116293[39] Andrew Y.T.Leung, Liu Z R 2004 Int.J.Bifurc.C haos 14 2955 [40] Qu Z L, Hu,G .,Yang,G J, Qin,G R 1995 Phys.Rev .Lett.74 1736 [41] Y ang J Z, Qu Z L, Hu G 1996 Phys.Rev.E.53 4402[42] Shyi-Kae Yang, C hieh-Li Chen, Her-Terng Yau 2002 C haosSolitons Fractals 13 767。

functional 详解

functional 详解一、什么是functional编程Functional编程是一种编程范式，它将计算机程序视为一系列数学函数的计算。

在functional编程中，函数是一等公民，即函数可以被作为参数传递给其他函数，也可以作为返回值返回。

这种编程范式强调函数之间的独立性和纯粹性，即函数的输出只依赖于输入，不受外部环境的影响。

二、functional编程的特点1. Immutable Data（不可变数据）：在functional编程中，数据是不可变的，即一旦定义就不能被修改。

这样可以避免数据被意外修改，减少了程序中的bug，并且方便进行并发编程。

2. Pure Functions（纯函数）：纯函数是指没有副作用的函数，即函数的输出只依赖于输入，不会对外部环境产生任何影响。

纯函数易于测试和调试，并且可以进行函数的组合和重用。

3. High-order Functions（高阶函数）：高阶函数是指可以接受其他函数作为参数或返回函数的函数。

高阶函数可以实现函数的组合和抽象，提高代码的复用性和可读性。

4. Recursion（递归）：functional编程中常用递归替代循环，递归可以简化代码逻辑，并且方便进行尾递归优化，减少内存消耗。

5. Declarative Style（声明式编程）：functional编程强调定义程序的逻辑而不是控制流程，通过声明式的方式来描述问题，提高了代码的可读性和可维护性。

三、functional编程的应用1. 并发编程：由于functional编程强调不可变数据和纯函数，因此在并发编程中表现出色。

并发编程通常需要处理共享数据的同步和竞态条件，而functional编程通过不可变数据避免了这些问题，提高了程序的并发性能。

2. 数据处理：functional编程适用于对大量数据进行处理和转换的场景。

通过函数的组合和管道操作，可以方便地对数据进行处理和转换，提高了代码的可读性和可维护性。

非线性时滞切换系统的稳定性研究

切换系统的研究第一主要是研究系统的稳定性以及切换信号，第二是系统适用于何种切换序列，第三是研究系统的镇定问题。本文主要研究切换系统的信号，并利用时间和状态驱动切换策略设计切换信号。状态依赖的研究主要有李雅普诺夫函数、抽象方法等。［２］在对时滞系统稳定性分析时李雅普诺夫函数是比较有效的方法。本文利用李雅普诺夫函数，引入自由权衡矩阵与积分不等式２种方法相互结合，设计出新的切换信号。
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lyapunov-krasovskii泛函方法

lyapunov-krasovskii泛函方法Lyapunov-Krasovskii 泛函方法是一种在控制理论中广泛应用的分析工具，其主要用于描述和分析非线性动态系统的稳定性。

该方法基于Lyapunov 稳定性理论的基本原理，通过构建适当的 Lyapunov 函数来评估系统的稳定性。

本文将介绍 Lyapunov-Krasovskii 泛函方法的原理和应用。

在控制系统中，稳定性是一个至关重要的性质，它决定了系统是否能够在给定的条件下稳定地运行。

Lyapunov 稳定性理论提供了一种评估系统稳定性的方法。

根据 Lyapunov 稳定性理论，一个连续时间系统在给定平衡点附近稳定，如果存在一个正定函数 V(x) （x 为系统状态），满足以下条件：1.V(x)>0，当且仅当x≠0；2. V(x) 的导数沿着系统的轨迹为负定，即 dV/dt < 0。

Lyapunov-Krasovskii 泛函方法是基于这个原理进行扩展和应用的。

它的主要思想是通过构建适当的 Lyapunov 函数来设计稳定性指标，并进一步采用泛函分析方法对系统进行分析。

具体来说，Lyapunov-Krasovskii 泛函方法提供了一种通过稳定性矩阵不等式来描述 Lyapunov 函数的方法，并基于该矩阵不等式设计控制器。

Lyapunov-Krasovskii 泛函方法的一个典型应用是在网络控制系统中。

网络控制系统是一种由传感器、执行器和通信网络组成的控制系统。

由于通信延迟和不确定性等因素的存在，网络控制系统容易受到时延和数据包丢失等问题的影响，从而导致系统的不稳定性。

为了解决这些问题，Lyapunov-Krasovskii 泛函方法被广泛应用于网络控制系统的设计和分析中。

在网络控制系统中，主要的问题是如何通过设计适当的控制器来保证系统的稳定性。

Lyapunov-Krasovskii 泛函方法通过构建合适的Lyapunov 函数来评估系统的稳定性，并设计基于 Lyapunov 函数的控制器，从而解决这个问题。

keynote048研究解读

keynote048研究解读问题并解析研究。

[研究解读]：如何通过交换策略来改善决策者的决策质量和满意度？在现实生活中，决策是我们经常面临的任务。

无论是个人还是组织，在面临决策时，我们都希望能够做出最优的选择。

然而，在实际情况中，我们经常会受到信息和资源的限制，以及复杂的环境和不确定性的影响，这会导致我们的决策质量和满意度不尽如人意。

因此，研究者们开始探索通过交换策略来改善决策者的决策质量和满意度的方法。

首先，我们需要了解什么是交换策略。

交换策略是指在面临决策时，通过与他人进行资源和信息的交换，来获得更全面、准确的信息或者更好的决策解决方案的一种策略。

换句话说，交换策略是通过与他人合作、沟通和协商，来改进我们的决策过程和结果。

那么，为什么交换策略能够改善决策质量和满意度呢？研究表明，交换策略可以增加决策者的信息获取渠道和信息质量。

通过与他人的沟通和交流，我们可以获得他们的观点、经验和知识，从而更全面、准确地了解问题和解决方案。

此外，交换策略还可以增加决策者的选择集。

通过与他人交换资源和信息，我们可以获得更多的选择，并且可以更好地衡量不同选择之间的利弊，从而做出更优的决策。

接下来，我们要探讨交换策略的具体方法和应用场景。

研究者们提出了一些有效的交换策略，例如：信息共享、经验分享、协同决策等。

在信息共享方面，决策者可以与他人分享他们的信息，例如：报告、分析结果或者调查数据。

通过信息共享，决策者可以获得他人的反馈和意见，从而更好地评估不同选择的风险和回报。

在经验分享方面，决策者可以与他人分享他们的经验和教训。

通过经验分享，决策者可以借鉴他人的成功经验和教训，避免犯同样的错误或者学习到更好的解决方案。

在协同决策方面，决策者可以与他人合作，共同制定决策解决方案。

通过协同决策，决策者可以集思广益，整合各方意见和建议，从而制定更好的决策方案。

此外，交换策略适用于许多不同的决策场景。

它可以应用于个人决策、团队决策和组织决策等各个层面。

keynote048研究解读

keynote048研究解读一、研究背景随着科技的飞速发展，我国在经济、教育、医疗等多个领域取得了显著的成果。

在这一背景下，各行各业对人才的需求日益增加，人们对于个人能力和素质的提升也越来越重视。

在这样的背景下，研究解读成为一门热门课题。

本文将通过对keynote048的研究解读，探讨人才培养、团队协作等方面的问题。

二、研究方法本研究采用多种方法对keynote048进行深入剖析。

首先，通过搜集并整理相关文献资料，对研究对象的理论基础、现状、发展趋势等方面进行全面了解。

其次，运用案例分析法，选取具有代表性的实例进行分析，以揭示研究现象背后的规律。

此外，还将运用实证研究方法，对研究对象的实际情况进行调查、访谈等，以获得第一手数据。

三、研究结果经过深入研究，本文得出以下几个主要结论：1.人才培养方面，研究对象在选拔、培养、使用等方面具有较高的水平，为我国经济社会发展提供了有力的人才支持。

2.团队协作方面，研究对象注重团队建设，形成了良好的沟通协作氛围，提高了工作效率。

3.创新发展方面，研究对象紧跟行业发展趋势，不断探索创新，为行业注入新活力。

4.企业文化方面，研究对象积极传承和弘扬企业文化，形成了独具特色的价值观和核心竞争力。

四、结论与启示综上所述，keynote048在人才培养、团队协作、创新发展等方面具有较强的优势。

为此，本文提出以下建议：1.企业应继续加大对人才的培养力度，注重人才队伍建设，为可持续发展奠定基础。

2.加强团队建设，提高团队协作能力，促进各项业务高效开展。

3.紧跟行业发展趋势，加大创新投入，培育企业核心竞争力。

4.弘扬企业文化，激发员工归属感，为实现企业愿景共同努力。

Lyapunov函数的一些构造方法

* *
dL 并且 1 dt
y v 0 x x* , * * , w w* . y v M
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Lyapunov 函数的构造
三、Lyapunov函数形式的推广
* y u y u x* x*v* v u v* 其中 F * du * du du. * * x y v u u ky u x
推广形式：
u w* w* u du,
w

w w*
u w* u w du, * du, w f u
*

w( w* ) du, f u

w
w*
f (u ) f ( w* ) du. u
Lyapunov 函数的构造
四、一类SI模型的全局稳定性
Lyapunov 函数的一些构造方法
李建全
空军工程大学(西安)
E-mail: jianq_li@
Lyapunov 函数的构造
一、引言二、带有免疫的病毒动力学模型三、Lyapunov函数形式的推广四、一类SI模型的全局稳定性
Lyapunov 函数的构造
一、引言
全局渐近稳定性 (GAS)是动力系统研究的重要问题。对于高维自治微分系统，证明GAS最常用的方法是 Lyapunov - LaSalle定理，即若自治系统 * x' f x, x Rn
函数 2、情形f w 1. 定义Lyapunov

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* x x p 0 * * x 2 w w * x x M c

2
x* xy*v y v* x v 3 x x* yv* y *v 0,

李雅普诺夫函数

李雅普诺夫函数李雅普诺夫函数（Lyapunov function）是⽤来证明⼀动⼒系统或⾃治微分⽅程稳定性的函李雅普诺夫函数数。

其名称来⾃俄罗斯数学家亚历⼭⼤·李雅普诺夫（Aleksandr Mikhailovich Lyapunov）。

李雅普诺夫函数在稳定性理论及控制理论中相当重要。

李雅普诺夫候选函若⼀函数可能可以证明系统在某平衡点的稳定性，此函数称为李雅普诺夫候选函数（Lyapunov-candidate-function）。

不过⽬前还找不到⼀般性的⽅式可建构（或找到）⼀个系统的李雅普诺夫候选函数，⽽找不到李雅普诺夫函数也不代表此系统不稳定。

在动态系统中，有时会利⽤守恒律来建构李雅普诺夫候选函数。

针对⾃治系统的李雅普诺夫定理，直接使⽤李雅普诺夫候选函数的特性。

在寻找⼀个系统平衡点附近的稳定性时，此定理是很有效的⼯具。

不过此定理只是⼀个证明平衡点稳定性的充分条件，不是必要条件。

⽽寻找李雅普诺夫函数也需要碰运⽓，通常会⽤试误法（trial and error）来寻找李雅普诺夫函数。

⽬录[隐藏]1 李雅普诺夫候选函数的定义2 系统平衡点的转换3 ⾃治系统的基本李雅普诺夫定理3.1 稳定平衡点3.2 局部渐近稳定平衡点3.3 全域渐近稳定平衡点4 参见5 参考资料6 外部链接李雅普诺夫候选函数的定义[编辑]令为标量函数。

若要为李雅普诺夫候选函数，函数需为局部正定函数，亦即其中是的邻域。

系统平衡点的转换[编辑]令为⼀个⾃治（autonomous）的动态系统，其平衡点为:可利⽤的坐标转换，使得在新的系统中，其平衡点为原点。

若系统的平衡点不是原点，可⽤上述的⽅式，转换为另⼀个平衡点为原点的系统，因此以下的说明中，均假设原点是系统的平衡点。

⾃治系统的基本李雅普诺夫定理[编辑]主条⽬：李雅普诺夫稳定性令为以下⾃治系统的平衡点且令为李雅普诺夫候选函数的时间导数。

稳定平衡点[编辑]若在平衡点的邻域，李雅普诺夫候选函数为正定，且其时间导数半负定：则此平衡点为⼀稳定的平衡点。

分段lyapunov-krasovkii函数

分段lyapunov-krasovkii函数
分段Lyapunov-Krasovkii函数是用于描述分段线性或混沌系统的一种非常有用的方法。

这种函数结合了两种分析方法：李雅普诺夫函数分析和Krasovkii分析。

通过将这两种分
析方法结合起来，这种函数可以帮助我们确定系统稳定性的条件，并有效地控制系统。

为了更好地理解分段李雅普诺夫-克拉索夫基函数，我们需要了解一些基本概念。

首先，李雅普诺夫函数是一种用于描述动态系统稳定性的函数。

对于一个稳定的动态
系统，其李雅普诺夫函数在系统所有状态变量的值都趋近于零。

这意味着李雅普诺夫函数
可以用于确定系统是否稳定，并确定稳定性的条件。

Krasovkii分析是一种方法，用于控制分段线性或混沌系统。

这种方法可以将系统控
制为稳定状态，并且可以控制系统在所需的状态范围内保持稳定。

分段Lyapunov-Krasovkii函数将这两个分析方法结合起来，可以有效地控制系统。

具体来说，这种函数将系统分为若干个区域，并在每个区域内使用不同的Lyapunov函数进行分析。

然后，通过将这些Lyapunov函数相加，可以获得整个系统的Lyapunov函数。

此外，该函数还使用Krasovkii分析来确定分段函数的分界点，并确定控制策略以保持系统稳
定。

在实践中，分段Lyapunov-Krasovkii函数已经被广泛应用于机器人控制、智能交通系统、航空航天控制、医学图像处理等领域。

通过使用这种函数，我们可以更好地理解系统
稳定性的条件，并为系统提供有效的控制策略。

基于Lyapunov方法和快速终端滑模的轨迹跟踪控制

基于Lyapunov方法和快速终端滑模的轨迹跟踪控制张扬名;刘国荣;刘洞波;刘欢【期刊名称】《计算机应用》【年(卷),期】2012(32)11【摘要】针对移动机器人的运动学模型,提出一种具有全局渐近稳定性的跟踪控制器.该跟踪控制器的设计分为两部分:第一部分是采用全局快速终端滑动模态的思想设计了角速度的控制律,用来渐近镇定移动机器人跟踪的前向角误差；第二部分是采用Lyapunov方法设计了线速度的控制律,用来渐近镇定移动机器人跟踪的平面坐标误差.采用Lyapunov稳定性定理,证明了移动机器人在满足这些控制律条件下,实现了对参考轨迹的全局渐近跟踪.实验结果表明移动机器人能够有效地跟踪期望轨迹,有利于在实际应用中推广.%In view of the kinematic model of mobile robot, a tracking controller of global asymptotic stability was proposed. The design of tracking controller was divided into two parts: The first part designed the control law of angular velocity by using global fast terminal sliding mode in order to asymptotically stabilize the tracking error of the heading angle;the second part designed the control law of linear velocity by using the Lyapunov method in order to asymptotically stabilize the tracking error of the planar coordinate. By combining Lyapunov stability theorem and two control laws, the mobile robot can track the desired trajectory in a global asymptotic sense when the angular velocity and the linear velocity satisfy these control laws. The experimental results showthat the mobile robot can track desired trajectory effectively. It is helpful for promoting the practical application.【总页数】4页(P3243-3246)【作者】张扬名;刘国荣;刘洞波;刘欢【作者单位】湘潭大学信息工程学院,湖南湘潭411105;湘潭大学信息工程学院,湖南湘潭411105;湖南大学电气与信息工程学院,长沙410082;湖南工程学院电气信息学院,湖南湘潭411101;湖南大学电气与信息工程学院,长沙410082;湖南工程学院电气信息学院,湖南湘潭411101;湘潭大学信息工程学院,湖南湘潭411105【正文语种】中文【中图分类】TP242.6【相关文献】1.基于Lyapunov方法的轮式移动机器人全局轨迹跟踪控制 [J], 赵涛;刘明雍;周良荣2.移动机器人的快速终端滑模轨迹跟踪控制 [J], 吴青云;闫茂德;贺昱曜3.对带有死区的机械臂系统的快速非奇异终端滑模轨迹跟踪控制 [J], 徐贵4.非奇异快速终端滑模及动态面控制的轨迹跟踪制导律 [J], 陈琦; 王旭刚5.基于快速终端滑模的机器人轨迹跟踪避障方法 [J], 曹志斌; 杨卫; 邵星灵; 刘宁因版权原因，仅展示原文概要，查看原文内容请购买。

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Session Keynote:Lyapunov Function Candidates for Descriptor Systems:Problems and SolutionsVladimir B.Baji´cCentre for Engineering Research,T echnikon Natal,P.O.Box953,Durban4000,Republic of SouthAfricae-mail:bajic.v@umfolozi.ntech.ac.zaKeywords:Lyapunov’s direct method,Lyapunov functions,descriptor(singular)systems,stability, qualitative analysisABSTRACTThis paper presents in a systematic way problems encountered in the construction of Lyapunov func-tion candidates for descriptor systems.The solutions to some of the major difficulties in the applica-tion of Lyapunov’s direct method to descriptor systems are presented.Some new results regarding the extension of Lyapunov’s direct method tied to the construction of the Lyapunov functions are given. Also,the algebraic necessary and sufficient conditions for some specific properties of motions of descriptor systems are developed.INTRODUCTION AND BACKGROUNDIt is a tradition to consider the equations describing internal system dynamics of continuous time systems in the normal form of the so-called state equations(%'s E|c%c 5U c(1) where('_*_|is the ordinary time derivative operator,and where|5U, 5U o and%5U c denote the time,the input vector and the state vector,respectively.Systems governed by the model (1)we usually call the state variable systems.However,there are physical systems for which the state models do not exist.It was recognized that the more natural models for these cases go beyond the classical state-variable description(1).These models are given bys E|c%c(%c 'f5U (2) where R'?may be allowed.Systems with such models we denote as implicit.In the special case the models(2)contain the canonical form of models linear in(%,and they are of the formE|c%c (%'s E|c%c (3) It should be pointed out that the matrix can even be a rectangular one.Systems governed by(3)are known as descriptor,as well as singular,semi-state,generalized state-space or differential-algebraic systems.ST ABILITY ANAL YSIS OF DESCRIPTOR SYSTEMSThe surveys of some of the results concerning both continuous and discrete descriptor and general implicit systems can be found,for example,in the books[1]-[3],[6]-[7]and[10]and in the special issues of the journal Circuits,Systems and Signal Processing[8]-[9].The systematic introduction and presentation of part of the results relating to the general application of Lyapunov’s direct method (LDM)for the analysis of DS is given in[2],[3].For some other results on stability of DS,particularly with regard to mechanical DS,see[12]-[16].It has been shown in[2],[4],that specific structure of DS may lead to several problems regarding the construction of Lyapunov function candidates(LFCs) for the intended qualitative analysis.These problems do not appear in the application of the LDM to systems in the normal forms(1).This paper aims to contribute to the general methodology relating to the construction of the LFCs for descriptor systems.W e present in a systematic way problemsencountered in the construction of LFCs for continuous-time DS,as well as the solutions to some of the major difficulties in the application of the LDM.Some new results regarding the extension of the LDM tied to the construction of LFCs are given.PROBLEM ST A TEMENT AND SIGNIFICANCE:LDM AND THE ROLE OF LFC There are essentially two problems that relate to the preliminary construction of the LFC:the problems that appear in the process of evaluation of the total time derivative(TTD)of an LFC along the motions of(2)and(3).the structure and properties of the matrix in(3)The application of the LDM requires the selection of a concrete LFC.When the exact analytical form of an LFC is known and when the necessary and sufficient conditions that ensure the appropriate qualitative concept of motion are expressed directly in terms of system and LFC parameters,then we say that the construction of the LFC is given.Unfortunately,in the general case the main drawback of the LDM is the conceptual nonexistence of a systematic procedure for the practical construction of an LFC for the qualitative concept of interest(one of possible general solutions for this is proposed in this paper).There are only a few results of the general nature on the construction of the LFC for DS(see references in[2]).So far there are no global answers on how to select an LFC for DS.Thus any general hint that concerns the construction of an LFC is of particular importance.The aim of this presentation is to provide some suitable solutions of this problem in a very specific way by the evaluation of the derivative of an LFC along the solutions of DS.This poses practical problems in finding a TTD of an LFC for DS,and,also,directly restricts the classes of useful LFCs.T o make this more evident let the matrix and the functions s and in(3)be sufficiently smooth and such that the systems considered possess the continuous and differentiable in|solutions in some domain. For simplicity,let T E|c% be an arbitrary function of|and%,which is continuously differentiable, and let it be an LFC for systems(2)or(3).In any qualitative analysis of the solutions of systems considered by means of the LDM,specific properties of the LFC are required.In what follows the TTD of T along the solutions of any of the systems(1)to(3)we will denote by(P T In all cases,the calculation of the TTD(P T d|c E| o of the LFC T,along the system motion ,has to be found.In general,this is not possible in a direct way for systems(2)or(3),unless T E|c% depends on%in a specific manner.As is well known,(P T d|c E| o can be found without the knowledge of solutions if the system analyzed is in the normal form(1)[17,p.12].Using the same argument, the(P T d|c E| o of the LFC T E|c% along the motion is given by(P T d|c E| o'YT d|c E| onYT d|c E| oe( E| (4)may be computed along the solutions of(2)or(3)if( is obtained from these equations and substi-tuted in(4).It is obvious that this can be done directly for systems in the normal form(1),in which case along the solutions of(1)we have(P T E|c% 'YT E|c%Y|nYT E|c%Y%es E|c%c (5)However,systems(2)or(3)need not be solvable in(%.Thus,in the cases when(2)or(3)are inher-ently singular(non-solvable in(%),(4)cannot be used directly for the evaluation of(P T d|c E| o. This is one of the main difficulties relating to successful construction of the LFC for DS.In this exposition will provide some general solutions to this problem.LFC FOR IMPLICIT SYSTEMS:ALGEBRAIC NECESSARY AND SUFFICIENTCONDITIONW e consider a system P governed by implicit differential equations(2).Let W'i|5U G|5o|r c|s d j, |r c|s5U,|r |s.The solutions of(2)are functions G W W U c6E|c|f c%f $ E|c|f c%f 5U c c where |f and %f are regarded as the initial moment and the initial value of a solution c re-spectively .The notation E |c |f c %f represents the value of the solution of P at the moment |c that at the moment |f had the value %f W e consider the case when (2)has only unique solutions that are continuous functions of their arguments and differentiable in |.Also,we allow that the E |f c %f 5W V 8E |f c |s c where V 8is the set of the consistent initial values %f at the moment |f such that from each %f 5V 8E |f c |s at least one solution that exists on W E |f 'd |f c |s d W c is generated.Further,we will consider properties of solutions of P only in the domain G 'W V 8E |r c |s 6E |c % .Obviously ,G is invariant w .r.t.solutions c i.e.if E |f c %f 5G c then E |c E |c |f c %f 5G for any | |f c |5W .W e will provide the algebraic necessary and sufficient conditions for a function T to be a L yapunov function for (2),by which the following useful qualitative concepts can be verified.The concepts considered are expressed byS d |c %E |c |f c %f o d |c |f c S E |f c %f o c S E |f c %f ' d |f c |f c S E |f c %f o c (6)where S and are scalar real functions.When all solutions of P in the domain G satisfy (6)we say that P has the property in G W e can give different meaning to the functions S andTheorem 1.Let an auxiliary function T G W U c 6E |c % $T E |c % 5U c T E |c % 5 E W U c c be chosen and let C G W U $U define a scalar differential equation ( 'C E |c that ;E |f c f 5W U has unique solutions E |c |f c f that are continuously differentiable in |.Let the Eulerian derivative (P T along the solutions of the model of (2)be given by (P T E |c % '}E |c % If (2)has unique solutions c then for every solution that exists on W E |f the property given by (6)will hold if and only if }E |c % C d |c T E |c % o in W E |f V 8E |r c |s The last conditions are also necessary and sufficient that T E |c % S E |c % be the Lyapunov function for EP cThis result is based on [5]and is quite opposite to Gruji´c ’s method [11].It gives the algebraic nec-essary and sufficient conditions for the verification of the examined property as well as the direct construction of T Some partial results of this approach have been already successfully applied to the problems of nonlinear uncertain and fuzzy logic systems.GENERAL PROCEDURE FOR ELIMINA TION OF (%FROM THE TTD OF AN LFC Consider the model (3)and assume that possibly all entries of the matrix are dependent on |and %.The construction of an LFC in such a situation is a complicated task.The method that follows is a general method for the calculation of the TTD of an LFC for DS (3)with nonlinear and possibly rectangular matrix .Let T be a scalar LFC,T G U U c 6E |c % $T E |c % 5U and let T be differentiable in the ordinary sense.If the matrix equation .'.7 c where .' YT E |c % Y%e (7)has any solution 7,then the formula for the TTD of the LFC T along the solutions of DS (3)is (P T E |c % 'YT E |c % Y|n YT E |c % Y%e 7s E |c %c This procedure is general and can be used directly for stability problem analysis as given in the following results [2].Theorem 2.Let W be a continuous open time interval,W U c and let l U .Let T G W l 6E |c % 5U n c T 5 E W l c and let T be a differentiable function of its arguments.Assume that >G W $U is an integrable function.If there is any solution 7of the matrix equation (7),and if ;E |c % 5W l YT E |c % Y|n YT E |c % Y% e 7s E |c %c >E | Tc (8)then any solution of (3)which starts in l at |'|f ,satisfies as long as it remains in l theestimateT d |c E | o T d |f c E |f o i T ]||f>E r _r(9)T o analyze systems (3)with inputs we considered the following.Theorem 3.Let W be a continuous open time interval,W U c and let l U c Assume that the function s in (3)is of the form s E |c %c ' E |c % %n (E |c % ,and that is a positive defi-nite symmetric matrix .Let 7be any solution of the matrix equation .'.7 in W l c where .'Y E % *Y%e Let 8' e n Y *Y|n .7 and '(n .7( Let 8f be any positive definite constant matrix and let a solution of (3)passes through l Assume that >G W $U is an integrable function,and m E |c % m Rc m E |c % m v .Let K' %5U c G m %m Rvb 3 T E 8f .If the matrix ~f '8n 8f >E | is negative semi definite for all values of its arguments,the estimate (9)holds with T '%e %c as long as E | 5K _l c providing %f 5K _l ,and if K _l is not empty.If E |c % f ,(9)holds in lNONDIFFERENTIABLE LFCThere are some classes of physical systems for which the ’’natural ’’LFCs are not very regular and may lead to nondifferentiable LFC.The extension of the LDM to the nondifferentiable LFC was provided for systems in the normal form by Y oshizawa [18,pp.3-4].Let (n P T denote the upper right-hand Dini derivative of an LFC T E |c % ,for the system (1).Calculation of (n P T requires a knowledge of the so-lutions of (1).However,the formula for the calculation of the (n P T ,which requires no knowledge of the system solution c is given in [18].This formula is valid under the mild assumptions that T E |c % should be continuous in the domain of interest and should locally satisfy the Lipschitz condition with respect to % It states that (n P T d |c %E | o G '* 4Å<f n d 3E T d |n c %n s E |c % o T E |c % o c where s is the function on the right side of the equality sign in (1),and where it is assumed that the input variable does not appear in s .Unfortunately ,this result cannot be directly applied to systems(2)or (3).T o overcome this,we introduce a new variable 5U r ,and consider the problem in terms of E |c instead of E |c % Let W be an open time interval of interest and let K %be an open set,K % U r Denote the LFC T as T E |c G W K %$U c T 5 E W K % c T 5Lip %E W K % c where Lip %E W K % denotes the class of functions that locally satisfy the Lipschitz condition with respect to when E |c 5W K %.Let (2)has solutions %E | on W .At an arbitrary moment |5W the values of all solutions %E | of (2)which exist on W determine the points in the system descriptor space U c which form the set V 8E | .Define also the set V %E | 'i 5U r G 'C E |c % c %5V 8E | j c ;|5W .Then the following hold [2].Theorem 4.Let W be an open continuous time interval and let the function C G W U c $U r define the variable 'C E |c % and is such that:(a)for a given function G W U c ,the TTD ( d |c %c E | o along the solutions of the (2)is known,and is a bounded function when E |c belongs to a compact set in W U r ;(b)when E |c % continuously change in the set W V 8E | then E |c continuously change in the set W V %E | .Let K %be an open set,V %E | K % U r c ;|5W c and let and LFC T G W K %6E |c $T E |c 5U c T 5 E W K % c T 5Lip %E W K % If (2)has continuous solutions %E | on W ,such that E | 'C d |c %E | o satisfies E |c E | 5W V %E | c then the upper right-hand Dini derivative of T ,along the solutions %E | at the moment |5W c is given by (n P T d |c E | o '* 4t T E 3 E T E |n c n ( T E |c G $f nTIME DISCONTINUOUS LFCConsider the situation where and s in (3)are time-discontinuous for a fixed %.This is motivated,for example,by the usage of time-discontinuous components like switching capacitors and resistive switches in singular electrical circuits.Our interest will be in treating the case when and s are continuous in %for a fixed |,and piecewise continuous in |for a fixed %,while for a fixed %the matrix is differentiable for all |where both and s are continuous.W e will develop sufficient conditions for the upper bounds on the system response.From these bounds the conditions for theasymptotic behavior of response will be derived.W e will utilize the LDM and a time-discontinuous Lyapunov function.Consider the singular model (3)of the formd |c %E | o (%E | 's d |c %E | o c %c s E |c % 5U c c E |c % 5U c f c (10)where |5W U .Let %5U c be fixed and denote by W 7 the set consisting of all points of W where either or s are discontinuous in |.W e will assume that the set W 'V i W 7 G %5U c j is such that W _E where E W and where E is bounded,has finite number of elements | i.e.W 'i | j .Denote by W the set W q W .Without loss of generality we will assume that the elements of W are ordered,i.e.; : ,|&:| .W e will assume that all discontinuities in |for and s are of the first kind,and that is continuously differentiable on A for a fixed %5U c .W e assume that and s are such as to ensure the existence and uniqueness of solutions of (10)which are continuous on W and continuously differentiable on W .Such solutions will satisfy (10)for all |5W .The set of such solutions we denote by 7 .Let W E |f 'W _i |5W G | |f j for |f 5W .In what follows U n 'i @5U G @ f j and P denotes the complement of the set P .W e can state the following results based on [4]Theorem 5.Consider the model (10)for which V '>.Let W be a continuous open time interval,W U ,and let l U c .Assume that there exist a function T G W l 6E |c % $T E |c % 5U n ,such that:(i)T is continuous in E |c % on W l ,T E | c % 'T E |n c % for all E | c % 5W l ,and T has only discontinuities in |of the first kind;(ii)YT E |c % *Y|exists and is continuous in E |c % on W l ;(iii)d YT E |c % *Y%o e exists and is continuous in E |c % on W l ;(iv)f q E | 4E % T E |c % k E | 4E % where k G W $U ,q G W $U ,k E | :f ,q E | :f ,and where k and q are continuous functions on A and bounded for finite |5W ,and 4G l 6%$4E % 5U n ,where 4is continuous on W and such that q E | 4E % :f ,;E |c % 5P ,where P 'i E |c % 5W l G T E |c % 'f j .Let >G W $U be a function continuous on W .If there is any solution 7of the matrix equation .'.7 ,where .'d YT E |c % *Y%o e ,such that ;E |c % 5W l equation (8)holds,then for any solution 5V of (10),which starts from %f 5l at |'|f 5A ,the estimate T d |c E |c |f c %f o S T E |f c %f i T ]||f >E r _r c S '3C 8E | \ '&E |fk E |n q E |3 4D (11)holds as long as remains in l ,where is the subscript of the first moment |&of discontinuity of or s in W _d |f c n 4d .Theorem 6Let l U c and in addition to the other conditions of Theorem 3let there exist :f c 5U ,such that E ;|W 5W for which E W _d |W c |W n d '>we have 3C 8E | \ '&E |Wk E |n q E |3 4D i T ]||W >E r _r c Z c ;|5W _d |W c |W n d where is the subscript of the first moment |&of discontinuity of or s in d |W c |W n d .Let Å'|f n ,where is a nonnegative integer .Then for any solution 5V of (10),which starts from %f 5l at |'|f 5A ,the estimate T d |c E |c |f c %f o Z ÅS T E |f c %f i T ]| Å>E r _r c ;|5W _d Åc Ån d c S '3C 8E | \ '&E Å k E |n 3 4D holds.Moreover ,if W 'd @c n 4d c @5U c and q E | 6:f on W ,then * 4|<n "4d E |c |f c %f o 'fCONCLUSIONSThis paper gives a systematic approach to the elimination of some of the crucial problems in the construction of the LFC for time-continuous descriptor systems.The results obtained can be used as a basis for further development of stability and boundedness problems of descriptor 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