DOCSLIB.ORG

Linearization of Nonlinear Dynamical Systems: a Comparative Study

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linearization of Nonlinear Dynamical Systems: a Comparative Study Volume 5, Number 6, Dec. 2011 ISSN 1995-6665 JJMIE Pages 567 - 571 Jordan Journal of Mechanical and Industrial Engineering Linearization of Nonlinear Dynamical Systems: A Comparative Study M. Ababneh*,a, M. Salaha, K. Alwidyana aMechatronics Engineering Department, Hashemite University, P.O. Box 330127, Zarqa 13133, Jordan Abstract Linearization of nonlinear dynamical systems is a main approach in the designing and analyzing of such systems. Optimal linear model is an online linearization technique for finding a local model that is linear in both the state and the control terms. In this paper, a comparison between the performance of both optimal linear model and Jacobian linearization technique is conducted. The performance of these two linearization methods are illustrated using two benchmark nonlinear systems, these are inverted pendulum system; and Duffing chaos system. These two systems where chosen because they are inherently nonlinear unstable systems. © 2011 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved Keywords: Nonlinear systems; linearization; inverted pendulum. Jacobian method performance. Finally, conclusions are presented in section 4. 1. Introduction Linearization of nonlinear dynamical systems makes 2. Optimal Linear Model of Non-linear Systems use of available literature in the linear system to design and analyze nonlinear systems [1]. Optimal linear model is Consider a nonlinear system model a promising linearization technique that continues to find wide acceptance in the areas of nonlinear and chaos x() t F (()) x t G (())(), x t u t (1) systems [2, 3, 4]. This method was first introduced by Teixeira and Zak [5]. where F :nn and G :n nxm are nonlinear In fact, typical approach to handle nonlinear systems is function, xt()n is the state vector, and ut()m is the to utilize linearization at their operating points, including control input. The optimal linear model objective is to find Jacobian analysis for local dynamics of control systems. linearized models of a nonlinear dynamical system around Optimal linear model is another method that generates operating points and it is described as follows. optimal local models; it is an online linearization technique Suppose that it is desired to have a local linear model at the i-th operation point of interest , for finding a local model that is linear in both the state and (,)ABop op (,)xuop op the control terms. This technique provides a new tool to which is not necessarily an equilibrium point of the control nonlinear systems and it is briefly described in the system. Let the linearized model be given as next section. Furthermore, the inverted pendulum problem has been x()()() t Aop x t B op u t (2) used as benchmark to motivate the study of nonlinear control systems and techniques [6, 7]. This system is where A and B are constant matrices of appropriate inherently nonlinear unstable non-minimum phase system op op dimensions. For this purpose, Taylor expansion method is and provides a challenging system to test different control commonly used in this case, however, the truncation used techniques [8]. in this method results in an affined rather than linear In addition, Duffing system is well-known chaos model. Suppose that the operating point (,)xu is an attractor and has used to address many practical op op equilibrium point, where n and m , that is, applications in Engineering and Science [9, 10]. In this xtop () utop () paper two Duffing systems are synchronized together. Where synchronization is when two systems come to F( xop ) G ( x op ) u op 0. (3) behave in accordance with each other as time passes. The rest of paper is organized as follows. In Section 2, The linear model then is expressed as: the optimal linear model is discussed and generation of linearized models around operating points is shown In d ()()()()()xxop Fx op Gxu op op Axx op op Buu op op Section 3, the effectiveness of the optimal linear model is dt (4) A( x x ) B ( u u ). demonstrated and its performance is compared with the op op op op * Corresponding author. e-mail: [email protected] 568 © 2011 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 5, Number 6 (ISSN 1995-6665) The model (4) can be represented in the following form: in which is arbitrary but should be close to so that x x op the approximation is good. To determine a constant xt( ) Axtop ( ) But op ( ) ( Ax op op Bu op op ). (5) vector, aT , such that it is as close as possible to [Fx ( )]T i i op and satisfies T consider the following The state equation (5) is an affined rather than a linear ai x op F i( x op ), model due to the non-vanishing constant term in (5). One constrained minimization problem: exception is the trivial case where the equilibrium point is zero, which, however, cannot be ensured throughout a 2 minE :1 F ( x ) a nonlinear control process. The goal is to construct a local 2 i op i 2 model, linear in state and also linear in control, that can well approximate the dynamical behavior of (1) around the subject to operating point In other words, two constant (xuop , op ). aT x F() x matrices, A and B , are to be found such that they are i op i op (15) op op located in a neighborhood of x op , Given that this is a constrained optimization problem; therefore, the first order necessary condition for a F()() x G x u Aop x B op u (6) minimum of E is also sufficient, which is E ( aT x F ( x )) 0 for any u, aii a i op i op (16) and T ai x op F i( x op ), (17) F()() x G x u A x B u op op op op op (7) where λ is the Lagrange multiplier and the subscript in ai indicates that the gradient is taken with respect to . ai for any u. ai Then Since the control input u is to be designed, it is arbitrary, and therefore ai F i( x op ) x op 0. (18) Bop G( x op ), (8) Recalling the case where x 0, so by solving (18), the op Furthermore, following can be obtained T xop F i()() x op F i x op F() x Aop x (9) . 2 (19) x op 2 And Substituting this λ into (18) gives F() xop A op x op (10) T Fi()() x op x op F i x op a F() x x j i op2 op (20) Now let aT denotes the i-th row of the matrix A . Then x op op op 2 T Fi (x) ai x, i 1,2,3,, n (11) where It is easily verified that when x op 0 x op 0 equation (18) yields and a F() x (21) F( x ) aT x , i 1,2,3, , n (12) i i op i op op op which is also a special case of (20). where n is the i-th component of F. Then, Fi : expanding the left-hand side of (1) about , and x op neglecting the second and higher order terms, the 3. Simulation Examples following equation can be obtained Most often, control algorithms are tested on standard TT Fi( x op ) [ F i ( x op )] ( x x op ) a i x , (13) nonlinear models, and the objective is to first find the linearized model then to design suitable controller based where nn is the gradient column vector of on our skills with linear systems. These linearized models Fxi(): op of the nonlinear model are valid only for small deviations F evaluated at x . Now, using (12), equation (13) can i op of the state values from their nominal value. Such a be written as nominal value is called the equilibrium point. Therefore, the linear models are acceptable around a small range of TT [Fi ( x op )] ( x x op ) a i ( x x op ), (14) the operating point. In this section, the effectiveness of the above method described in previous section will be presented and its performance is compared with Jacobian method performance. © 2011 Jordan Journal of Mechanical and Industrial Engineering. All rights reserved - Volume 5, Number 6 (ISSN 1995-6665) 569 3.1. Inverted Pendulum: And 0 1 0 0 The inverted pendulum, shown in Figure 1, is highly 2 b(sin x3 ) x 3 x 2 b (sin x 3 ) x 3 0a b (cos x3 ) b (sin x 3 ) x 3 0 nonlinear system which can be considered as an important xx22 A 22 benchmark system for controller testing. op 0 0 0 1 d(sin x ) x2 d (sin x ) x x 3 2 3 3 2 0d (cos x3 ) d (sin x 3 ) x 2 e 0 xx22 22 (25) 0 c B for x 2 0 op 0 2 hxcos 3 where 2 2 2 2 f() I ml m l g xxxxxxxxxa[ ][ ]T , r , b 2 1 2 3 4 1 2 3 4 ()()mMI mMl22 mMI mMl ()()I ml2 mlf m M mgl c,, d r e 2 2 2 Figure 1: Inverted pendulum system. ()()()mMImMl mIMIml mIMIml ml h 2 The nonlinear dynamical equations of motion is given mI M() I ml by [11]. The Jacobian model is gradient of the system is given by: 2 (m M ) x fr x ml cos ml sin F (22) (I ml2 ) mgl sin mlx cos 0 0 1 0 0 Fx() 0a b cos x b (sin x ) x 0 i 3 3 3 (26) where m is the pole mass, M is the cart mass, fr is the cart x 0 0 0 1 friction coefficient, x is the horizontal displacement, l is 0d cos x d (sin x ) x e 0 the pole length, θ is the angle of the pole from upright 3 3 2 position, F is the applied force on the cart, I is the pole moment of inertia, and g is the gravity.
Load more

Recommended publications
  • 1 Probelms on Implicit Differentiation 2 Problems on Local Linearization
    Math-124 Calculus Chapter 3 Review 1 Probelms on Implicit Di®erentiation #1. A function is de¯ned implicitly by x3y ¡ 3xy3 = 3x + 4y + 5: Find y0 in terms of x and y. In problems 2-6, ¯nd the equation of the tangent line to the curve at the given point. x3 + 1 #2. + 2y2 = 1 ¡ 2x + 4y at the point (2; ¡1). y 1 #3. 4ey + 3x = + (y + 1)2 + 5x at the point (1; 0). x #4. (3x ¡ 2y)2 + x3 = y3 ¡ 2x ¡ 4 at the point (1; 2). p #5. xy + x3 = y3=2 ¡ y ¡ x at the point (1; 4). 2 #6. x sin(y ¡ 3) + 2y = 4x3 + at the point (1; 3). x #7. Find y00 for the curve xy + 2y3 = x3 ¡ 22y at the point (3; 1): #8. Find the points at which the curve x3y3 = x + y has a horizontal tangent. 2 Problems on Local Linearization #1. Let f(x) = (x + 2)ex. Find the value of f(0). Use this to approximate f(¡:2). #2. f(2) = 4 and f 0(2) = 7. Use linear approximation to approximate f(2:03). 6x4 #3. f(1) = 9 and f 0(x) = : Use a linear approximation to approximate f(1:02). x2 + 1 #4. A linear approximation is used to approximate y = f(x) at the point (3; 1). When ¢x = :06 and ¢y = :72. Find the equation of the tangent line. 3 Problems on Absolute Maxima and Minima 1 #1. For the function f(x) = x3 ¡ x2 ¡ 8x + 1, ¯nd the x-coordinates of the absolute max and 3 absolute min on the interval ² a) ¡3 · x · 5 ² b) 0 · x · 5 3 #2.
    [Show full text]
  • DYNAMICAL SYSTEMS Contents 1. Introduction 1 2. Linear Systems 5 3
    DYNAMICAL SYSTEMS WILLY HU Contents 1. Introduction 1 2. Linear Systems 5 3. Non-linear systems in the plane 8 3.1. The Linearization Theorem 11 3.2. Stability 11 4. Applications 13 4.1. A Model of Animal Conflict 13 4.2. Bifurcations 14 Acknowledgments 15 References 15 Abstract. This paper seeks to establish the foundation for examining dy- namical systems. Dynamical systems are, very broadly, systems that can be modelled by systems of differential equations. In this paper, we will see how to examine the qualitative structure of a system of differential equations and how to model it geometrically, and what information can be gained from such an analysis. We will see what it means for focal points to be stable and unstable, and how we can apply this to examining population growth and evolution, bifurcations, and other applications. 1. Introduction This paper is based on Arrowsmith and Place's book, Dynamical Systems.I have included corresponding references for propositions, theorems, and definitions. The images included in this paper are also from their book. Definition 1.1. (Arrowsmith and Place 1.1.1) Let X(t; x) be a real-valued function of the real variables t and x, with domain D ⊆ R2. A function x(t), with t in some open interval I ⊆ R, which satisfies dx (1.2) x0(t) = = X(t; x(t)) dt is said to be a solution satisfying x0. In other words, x(t) is only a solution if (t; x(t)) ⊆ D for each t 2 I. We take I to be the largest interval for which x(t) satisfies (1.2).
    [Show full text]
  • Linearization of Nonlinear Differential Equation by Taylor's Series
    International Journal of Theoretical and Applied Science 4(1): 36-38(2011) ISSN No. (Print) : 0975-1718 International Journal of Theoretical & Applied Sciences, 1(1): 25-31(2009) ISSN No. (Online) : 2249-3247 Linearization of Nonlinear Differential Equation by Taylor’s Series Expansion and Use of Jacobian Linearization Process M. Ravi Tailor* and P.H. Bhathawala** *Department of Mathematics, Vidhyadeep Institute of Management and Technology, Anita, Kim, India **S.S. Agrawal Institute of Management and Technology, Navsari, India (Received 11 March, 2012, Accepted 12 May, 2012) ABSTRACT : In this paper, we show how to perform linearization of systems described by nonlinear differential equations. The procedure introduced is based on the Taylor's series expansion and on knowledge of Jacobian linearization process. We develop linear differential equation by a specific point, called an equilibrium point. Keywords : Nonlinear differential equation, Equilibrium Points, Jacobian Linearization, Taylor's Series Expansion. I. INTRODUCTION δx = a δ x In order to linearize general nonlinear systems, we will This linear model is valid only near the equilibrium point. use the Taylor Series expansion of functions. Consider a function f(x) of a single variable x, and suppose that x is a II. EQUILIBRIUM POINTS point such that f( x ) = 0. In this case, the point x is called Consider a nonlinear differential equation an equilibrium point of the system x = f( x ), since we have x( t )= f [ x ( t ), u ( t )] ... (1) x = 0 when x= x (i.e., the system reaches an equilibrium n m n at x ). Recall that the Taylor Series expansion of f(x) around where f:.
    [Show full text]
  • Linearization Extreme Values
    Math 31A Discussion Notes Week 6 November 3 and 5, 2015 This week we'll review two of last week's lecture topics in preparation for the quiz. Linearization One immediate use we have for derivatives is local linear approximation. On small neighborhoods around a point, a differentiable function behaves linearly. That is, if we zoom in enough on a point on a curve, the curve will eventually look like a straight line. We can use this fact to approximate functions by their tangent lines. You've seen all of this in lecture, so we'll jump straight to the formula for local linear approximation. If f is differentiable at x = a and x is \close" to a, then f(x) ≈ L(x) = f(a) + f 0(a)(x − a): Example. Use local linear approximation to estimate the value of sin(47◦). (Solution) We know that f(x) := sin(x) is differentiable everywhere, and we know the value of sin(45◦). Since 47◦ is reasonably close to 45◦, this problem is ripe for local linear approximation. We know that f 0(x) = π cos(x), so f 0(45◦) = πp . Then 180 180 2 1 π 90 + π sin(47◦) ≈ sin(45◦) + f 0(45◦)(47 − 45) = p + 2 p = p ≈ 0:7318: 2 180 2 90 2 For comparison, Google says that sin(47◦) = 0:7314, so our estimate is pretty good. Considering the fact that most folks now have (extremely powerful) calculators in their pockets, the above example is a very inefficient way to compute sin(47◦). Local linear approximation is no longer especially useful for estimating particular values of functions, but it can still be a very useful tool.
    [Show full text]
  • Linearization Via the Lie Derivative ∗
    Electron. J. Diff. Eqns., Monograph 02, 2000 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp) Linearization via the Lie Derivative ∗ Carmen Chicone & Richard Swanson Abstract The standard proof of the Grobman–Hartman linearization theorem for a flow at a hyperbolic rest point proceeds by first establishing the analogous result for hyperbolic fixed points of local diffeomorphisms. In this exposition we present a simple direct proof that avoids the discrete case altogether. We give new proofs for Hartman’s smoothness results: A 2 flow is 1 linearizable at a hyperbolic sink, and a 2 flow in the C C C plane is 1 linearizable at a hyperbolic rest point. Also, we formulate C and prove some new results on smooth linearization for special classes of quasi-linear vector fields where either the nonlinear part is restricted or additional conditions on the spectrum of the linear part (not related to resonance conditions) are imposed. Contents 1 Introduction 2 2 Continuous Conjugacy 4 3 Smooth Conjugacy 7 3.1 Hyperbolic Sinks . 10 3.1.1 Smooth Linearization on the Line . 32 3.2 Hyperbolic Saddles . 34 4 Linearization of Special Vector Fields 45 4.1 Special Vector Fields . 46 4.2 Saddles . 50 4.3 Infinitesimal Conjugacy and Fiber Contractions . 50 4.4 Sources and Sinks . 51 ∗Mathematics Subject Classifications: 34-02, 34C20, 37D05, 37G10. Key words: Smooth linearization, Lie derivative, Hartman, Grobman, hyperbolic rest point, fiber contraction, Dorroh smoothing. c 2000 Southwest Texas State University. Submitted November 14, 2000.
    [Show full text]
  • Linearization and Stability Analysis of Nonlinear Problems
    Rose-Hulman Undergraduate Mathematics Journal Volume 16 Issue 2 Article 5 Linearization and Stability Analysis of Nonlinear Problems Robert Morgan Wayne State University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Morgan, Robert (2015) "Linearization and Stability Analysis of Nonlinear Problems," Rose-Hulman Undergraduate Mathematics Journal: Vol. 16 : Iss. 2 , Article 5. Available at: https://scholar.rose-hulman.edu/rhumj/vol16/iss2/5 Rose- Hulman Undergraduate Mathematics Journal Linearization and Stability Analysis of Nonlinear Problems Robert Morgana Volume 16, No. 2, Fall 2015 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Wayne State University, Detroit, MI Rose-Hulman Undergraduate Mathematics Journal Volume 16, No. 2, Fall 2015 Linearization and Stability Analysis of Nonlinear Problems Robert Morgan Abstract. The focus of this paper is on the use of linearization techniques and lin- ear differential equation theory to analyze nonlinear differential equations. Often, mathematical models of real-world phenomena are formulated in terms of systems of nonlinear differential equations, which can be difficult to solve explicitly. To overcome this barrier, we take a qualitative approach to the analysis of solutions to nonlinear systems by making phase portraits and using stability analysis. We demonstrate these techniques in the analysis of two systems of nonlinear differential equations. Both of these models are originally motivated by population models in biology when solutions are required to be non-negative, but the ODEs can be un- derstood outside of this traditional scope of population models.
    [Show full text]
  • 1 Introduction 2 Linearization
    I.2 Quadratic Eigenvalue Problems 1 Introduction The quadratic eigenvalue problem (QEP) is to find scalars λ and nonzero vectors u satisfying Q(λ)x = 0, (1.1) where Q(λ)= λ2M + λD + K, M, D and K are given n × n matrices. Sometimes, we are also interested in finding the left eigenvectors y: yH Q(λ) = 0. Note that Q(λ) has 2n eigenvalues λ. They are the roots of det[Q(λ)] = 0. 2 Linearization A common way to solve the QEP is to first linearize it to a linear eigenvalue problem. For example, let λu z = , u Then the QEP (1.1) is equivalent to the generalized eigenvalue problem Lc(λ)z = 0 (2.2) where M 0 D K L (λ)= λ + ≡ λG + C. c 0 I −I 0 Lc(λ) is called a companion form or a linearization of Q(λ). Definition 2.1. A matrix pencil L(λ)= λG + C is called a linearization of Q(λ) if Q(λ) 0 E(λ)L(λ)F (λ)= (2.3) 0 I for some unimodular matrices E(λ) and F (λ).1 For the pencil Lc(λ) in (2.2), the identity (2.3) holds with I λM + D λI I E(λ)= , F (λ)= . 0 −I I 0 There are various ways to linearize a quadratic eigenvalue problem. Some are preferred than others. For example if M, D and K are symmetric and K is nonsingular, then we can preserve the symmetry property and use the following linearization: M 0 D K L (λ)= λ + .
    [Show full text]
  • Graduate Macro Theory II: Notes on Log-Linearization
    Graduate Macro Theory II: Notes on Log-Linearization Eric Sims University of Notre Dame Spring 2017 The solutions to many discrete time dynamic economic problems take the form of a system of non-linear difference equations. There generally exists no closed-form solution for such problems. As such, we must result to numerical and/or approximation techniques. One particularly easy and very common approximation technique is that of log linearization. We first take natural logs of the system of non-linear difference equations. We then linearize the logged difference equations about a particular point (usually a steady state), and simplify until we have a system of linear difference equations where the variables of interest are percentage deviations about a point (again, usually a steady state). Linearization is nice because we know how to work with linear difference equations. Putting things in percentage terms (that's the \log" part) is nice because it provides natural interpretations of the units (i.e. everything is in percentage terms). First consider some arbitrary univariate function, f(x). Taylor's theorem tells us that this can be expressed as a power series about a particular point x∗, where x∗ belongs to the set of possible x values: f 0(x∗) f 00 (x∗) f (3)(x∗) f(x) = f(x∗) + (x − x∗) + (x − x∗)2 + (x − x∗)3 + ::: 1! 2! 3! Here f 0(x∗) is the first derivative of f with respect to x evaluated at the point x∗, f 00 (x∗) is the second derivative evaluated at the same point, f (3) is the third derivative, and so on.
    [Show full text]
  • Chapter 3. Linearization and Gradient Equation Fx(X, Y) = Fxx(X, Y)
    Oliver Knill, Harvard Summer School, 2010 An equation for an unknown function f(x, y) which involves partial derivatives with respect to at least two variables is called a partial differential equation. If only the derivative with respect to one variable appears, it is called an ordinary differential equation. Examples of partial differential equations are the wave equation fxx(x, y) = fyy(x, y) and the heat Chapter 3. Linearization and Gradient equation fx(x, y) = fxx(x, y). An other example is the Laplace equation fxx + fyy = 0 or the advection equation ft = fx. Paul Dirac once said: ”A great deal of my work is just playing with equations and see- Section 3.1: Partial derivatives and partial differential equations ing what they give. I don’t suppose that applies so much to other physicists; I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful ∂ mathematical relations If f(x, y) is a function of two variables, then ∂x f(x, y) is defined as the derivative of the function which maybe don’t have any physical meaning at all. Sometimes g(x) = f(x, y), where y is considered a constant. It is called partial derivative of f with they do.” Dirac discovered a PDE describing the electron which is consistent both with quan- respect to x. The partial derivative with respect to y is defined similarly. tum theory and special relativity. This won him the Nobel Prize in 1933. Dirac’s equation could have two solutions, one for an electron with positive energy, and one for an electron with ∂ antiparticle One also uses the short hand notation fx(x, y)= ∂x f(x, y).
    [Show full text]
  • Note: Linearization @ 2020-01-21 01:01:05-08:00
    Note: Linearization @ 2020-01-21 01:01:05-08:00 EECS 16B Designing Information Devices and Systems II Fall 2019 Note: Linearization Overview So far, we have spent a great deal of time studying linear systems described by differential equations of the form: d ~x(t) = A~x(t) + B~u(t) +~w(t) dt or linear difference equations (aka linear recurrence relations) of the form: ~x(t + 1) = A~x(t) + B~u(t) +~w(t): Here, the linear difference equations came from either discretizing a continuous model or were learned from data. In the real world, however, it is very rare for physical systems to perfectly obey such simple linear differential equations. The real world is generally continuous, but our systems will obey nonlinear differential equations of the form: d ~x = ~f (~x(t);~u(t)) +~w(t); dt where ~f is some non-linear function. Through linearization, we will see how to approximate general systems as locally linear systems, which will in turn allow us to apply all the techniques we have developed so far. 1 Taylor Expansion First, we must recall how to approximate the simplest nonlinear functions — scalar functions of one variable — as linear functions. To do so, we will use the Taylor expansion we learned in calculus. This expansion tells us that for well-behaved functions f (x), we can write 2 d f 1 d f 2 f (x) = f (x0) + (x − x0) + (x − x0) + ··· dx 2 dx2 x=x0 x=x0 near any particular point x = x0, known as the expansion point.
    [Show full text]
  • 32-Linearization.Pdf
    Math S21a: Multivariable calculus Oliver Knill, Summer 2011 1 What is the linear approximation of the function f(x, y) = sin(πxy2) at the point (1, 1)? We 2 2 2 have (fx(x, y),yf (x, y)=(πy cos(πxy ), 2yπ cos(πxy )) which is at the point (1, 1) equal to f(1, 1) = π cos(π), 2π cos(π) = π, 2π . Lecture 10: Linearization ∇ h i h− i 2 Linearization can be used to estimate functions near a point. In the previous example, 0.00943 = f(1+0.01, 1+0.01) L(1+0.01, 1+0.01) = π0.01 2π0.01+3π = 0.00942 . In single variable calculus, you have seen the following definition: − ∼ − − − The linear approximation of f(x) at a point a is the linear function 3 Here is an example in three dimensions: find the linear approximation to f(x, y, z)= xy + L(x)= f(a)+ f ′(a)(x a) . yz + zx at the point (1, 1, 1). Since f(1, 1, 1) = 3, and f(x, y, z)=(y + z, x + z,y + ∇ − x), f(1, 1, 1) = (2, 2, 2). we have L(x, y, z) = f(1, 1, 1)+ (2, 2, 2) (x 1,y 1, z 1) = ∇ · − − − 3+2(x 1)+2(y 1)+2(z 1)=2x +2y +2z 3. − − − − 4 Estimate f(0.01, 24.8, 1.02) for f(x, y, z)= ex√yz. Solution: take (x0,y0, z0)=(0, 25, 1), where f(x0,y0, z0) = 5. The gradient is f(x, y, z)= y=LHxL x x x ∇ (e √yz,e z/(2√y), e √y).
    [Show full text]
  • 1.4 Stability and Linearization
    42 Introduction 1.4 Stability and Linearization Suppose we are studying a physical system whose states x are described by points in Rn. Assume that the dynamics of the system is governed by a given evolution equation dx = f(x). dt Let x0 be a stationary point of the dynamics, i.e., f(x0) = 0. Imagine that we perform an experiment on the system at time t = 0 and determine that the initial state is indeed x0. Are we justified in predicting that the system will remain at x0 for all future time? The mathematical answer to this question is yes, but unfortunately it is probably not the question we really wished to ask. Experiments in real life seldom yield exact answers to our idealized models, so in most cases we will have to ask whether the system will remain near x0 if it started near x0. The answer to the revised question is not always yes, but even so, by examining the evolution equation at hand more carefully, one can sometimes make predictions about the future behavior of a system starting near x0. A simple example will illustrate some of the problems involved. Consider the following two differential equations on the real line: x!(t)= x(t) (1.4.1) − and x!(t)=x(t). (1.4.2) The solutions are, respectively, t x(x0,t)=x0e− (1.4.3) and +t x(x0,t)=x0e . (1.4.4) Note that 0 is a stationary point of both equations. In the first case, for all x0 R, we have limt x(x0,t) = 0.
    [Show full text]