Linearization of Nonlinear Dynamical Systems: a Comparative Study

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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.

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 ()()()()()xxop  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

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()() xop G x op u  A op x op  B op u (7) where λ is the Lagrange multiplier and the subscript a in i 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 2 d(sin x3 ) x 2 d (sin x 3 ) x 3 x 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. The state space The inverted pendulum parameters are assigned the model can be presented as follows: following values [11], m = 0.23 kg, M = 2.4 kg, l = 0.38 m. fr = 0.05 Ns/m, I = 0.099 kg.m2, and g = 9.81 m/s2. Furthermore, the simulation for the controllers, with pole 0 1 0 0 2 22 placement at poles -1±j1 and -2±j1, was carried out in xx11 f() I ml m l gcos x    00r 3   SimuLink of MatLab using a fifth-order Dormand-Prince xx()()m MI  mMl22 m  MI  mMl 22    algorithm with a fixed integration step of 0.005 and initial xx 0 0 0 1   33condition of [0, 0, 0, 1]T. The performances for both     xx44 mlfr cos x 3 ()m M mgl   0022 optimal linear model and Jacobian method are displayed in mI MI()()  ml mI  MI  ml (21) Figure 2. It is obvious that first method has less overshoot 0 2 and better performance than the second method. ()I ml ()m M I mMl 2 u 1.5 0  Optimal Linear Model  mlcos x 3 Jacobian 2 mI M() I ml 1 Where

0.5 TT [][].x1 x 2 x 3 x 4  x x 

Given equations (20) and (21) the optimal model Angle, Theta 0

0 1 0 0 0a b (cos x ) b (sin x ) x 0 -0.5 A  3 3 3 op 0 0 0 1  0d (cos x ) d (sin x ) x e 0 3 3 2 -1 (24) 0 1 2 3 4 5 6 7 8 9 10 Seconds 0 Figure 2: Comparison between Optimal linear model and Jacobian c B for x2 0cos x method. op 0 2 3  hxcos 3

3.2. Synchronization of Duffing System: the two systems are said to be synchronized if

The chaotic Duffing system is a popular benchmark lim x(t)  xˆ(t)  0 (31) t example in the study of nonlinear system. The Duffing system can be expressed as [4, 12]: For the optimal linearized model xx12 (27) x( t ) Aop x ( t ) B op u ( t ), xx(0)  0 x  p x  p x  p x3  qcos( wt ) (32) 2 1 2 2 1 3 1 y(t)  Cx(t)

with parameters p1 = -1.1, p2 = 0.4, p3 = 1, q = 1.8 and w with A and B are the matrices from optimal linear =1.8, the system is chaotic and the attractor for op op model, then an observer can be designed as uncontrolled system is shown Figure 3 below.

2.5 xtˆ() Axt ˆ ()  But ()  Kyt (()  yt ˆ ()), xˆ(0)  xˆ o 0 (33) 2 yˆ(t)  Cxˆ(t) 1.5

1 where Ko is the observer gain matrix.

0.5 The simulation for this synchronization was carried out 0 in SimuLink of MatLab using a fifth-order Dormand-

-0.5 Prince algorithm with a fixed integration step of 0.005 and initial condition of [0, 5]T. Note that state x2 is the state -1 that not measurable. The performance of synchronization -1.5 for Duffing system is shown in Figure 4 for the optimal -2 linear model and in Figure 5 for the Jacobian method. It is

-2.5 obvious that first method is superior to the second method, -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 and it was able to synchronize completely with non- X2 Figure 3: Duffing system chaotic attractor. measurable state x2 but the Jacobian method fail to do so and the error was huge for most of the time. Applying equations (20) and (21), the optimal linear model 4 is derived as: Transmitter 2 Receiver 0 01 2 X1 A for x 0 op 2 2 p2 3 p 3 xk 1  p 1 -2 01 -4 2 0 1 2 3 4 5 6 7 8 9 10 A22p x43 p x x for x 0 op 2 3 1 3 1 2 2 p2 3 p 3 x 1   p 1  xx22 22 (28) 10 B  0 1T Transmitter k Receiver and 5 C  1 0 X2 k 0

-5 0 1 2 3 4 5 6 7 8 9 10 where and x2 [,][,] x xT x x is the square magnitude Seconds 2 1 2 1 2 Figure 4: Synchronization of Duffing system using optimal linear of the operating point. model. In general, synchronization is when two systems come to behave in accordance with each other as time passes; an example is the transmitter / receiver unit in communication 4 Transmitter system. The goal here is to consider the synchronization 2 Receiver problem from the point of view of control theory [3]. 0 Mathematically speaking, consider two dynamical X1 systems, -2

-4 0 1 2 3 4 5 6 7 8 9 10 x(t)  f (x(t)) x(0)  x0 , (29) y(t)  h(x(t)) 20 Transmitter 10 Receiver and 0 X2 ˆ ˆ ˆ xˆ(0)  xˆ -10 x(t)  f (x(t), y(t)), 0 (30) -20 yˆ(t)  h(xˆ(t)) 0 1 2 3 4 5 6 7 8 9 10 Seconds

Figure 5: Synchronization of Duffing system using Jacobian method.

4. Conclusions synchronization of Duffing system was performed, as shown by the convincing simulation results in Figure 4 and A linearization technique of optimal linear model was 5, the optimal linear model was able to perfectly reproduce briefly presented in this paper, and its performance was the non-measurable state x2, but the Jacobian failed to do compared with a popular Jacobian method. Two typical so with large error. It is obvious from these results that nonlinear benchmark examples were used to compare the optimal linear model has better performance than the two linearization methods; these are inverted pendulum Jacobian method, especially when the system is highly and chaotic Duffing system. In the inverted pendulum nonlinear. Future work might be in conducting example, the controllers were designed with pole performance comparison between optimal linear model placement for both cases, and as shown in Figure 2 the and feedback linearization method. performance of first method was superior to the second method with much less overshoot. Furthermore, the

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