Advances in Mechanical Engineering, ISSN: 2160-0619 Volume 2, Number 3, September, 2012

On an Optimal Linear Control of a Chaotic Non-Ideal Duffing System

Fbio Roberto Chavarette Faculty of Engineering, UNESP Univ Estadual Paulista, Departament of Mathematics, Avenida

Brasil, 56, 15385-000, Ilha Solteira, SP, Brazil

fabioch@mat.feis.unesp.br

doi.10.5729/ame.vol2.issue3.1

Keywords: Chaos, Optimal linear control, Non-ideal duffing system.

Abstract. In this work, we use a nonlinear control based on Optimal Linear Control. We used as mathematical model a Duffing equation to model a supporting structure for an unbalanced rotating machine with limited power (non-ideal motor). Numerical simulations are performed for a set control parameter (depending on the voltage of the motor, that is, in the static and dynamic characteristic of the motor) The interaction of the non-ideal excitation with the structure may lead to the occurrence of interesting phenomena during the forward passage through the several resonance states of the system. Chaotic behavior is obtained for values of the parameters. Then, the proposed control strategy is applied in order to regulate the chaotic behavior, in order to obtain a periodic orbit and to decrease its amplitude. Both methodologies were used in complete agreement between them. The purpose of the paper is to give suggestions and recommendations to designers and engineers on how to drive this kind of system through resonance.

1. Introduction The non-ideal dynamics in its great part is found in the literature for mechanical systems [1, 2] and recently by [3]. The operation of mechanical systems depends, basically, of two effects: the dissipation (damping) and the excitation.

The dissipation is produced for the friction. It is the responsible for the reduction of the amount of kinetic and potential energy of the mechanical system.

The excitation will depend on the characteristics that constitute the dynamical system. When the excitation is not influenced by the response of the system, this is said ideal excitation that has origin in an ideal energy source. On the other hand, when the excitation is influenced by the response of the system, the excitation is said to be non-ideal [2].

As a first characteristic that differentiate them, the system non-ideal is that it presents an equation that describes the interaction of the energy source with the ideal dynamic system.

Therefore, the non-ideal system the dynamics depends directly on the properties of the excitation whose available potency in the energy source, generally, it is limited.

As a second characteristic, is observed that the dynamics of a non-ideal system approaches of the ideal case as the supplied potency becomes sufficiently big and vice-versa.

Prof. Duffing introduced a nonlinear ideal and oscillator, with a cubic stiffness term, to describe the hardening spring effect observed in many mechanical problems. Since then, this ideal equation has become one of the most popular models, in the classical studies of nonlinear oscillations, bifurcations and chaos.

In Duffing original formulation, the equation had the traditional form

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3 cos( )x x px x wt + + + = (1) where t is the time variable, , p, , , and w where assumed to be real constants, and,

particularly, the coefficient of the linear stiffness term, p, was chosen to be positive. Duffing equation was exhaustively studied in current literature by a number of authors.

We remarked that, in the last years, a significant interest in control theory of the nonlinear systems, exhibiting chaotic behavior, has been observed. Among strategies of chaos control, with feedback, the most popular is OGY (Ott-Grebogi-York) method [4]. This method uses the Poincar map of the considered dynamical system.

Recently, a methodology to linear system with periodic coefficients, based on the application of the Lyapunov-Floquet transformation, was proposed by Sinha et al. [5] in order to solve this kind of problem. This method allows directing the chaotic motion to any desired periodic orbit or to a fixed point. It is based on linearization of the equations, which described the error between the actual and desired trajectories.

Here, we use different approaches, which were proposed by Rafikov and Balthazar in [6]: the Dynamic Programming was used to solve the formulated optimal control problems [7, 8] and we will obtain functions that satisfied the Hamilton-Jacobi-Bellman, among the correspondent Lyapunov functions of the Duffing non-ideal equation.

Finally, we mention that this paper is organized, as follows: in section 2, we analyze the ideal behavior of the nonlinear dynamics of the Duffing equation. In section 3, we analyze the non-ideal behavior of the nonlinear dynamics of the Duffing equation. In section 4, we proposed an optimal control design problem to Duffing equation. In section 5, we do some concluding remarks of this work and in section 6, we list the main bibliographic references used and next we do some acknowledgements.

2. Dynamic ideal to duffing equation In this section, we consider ideal Duffing model. Varying the parameters in the equation (1) we found chaotic responses. Here taken the following values of the dimensionless parameters =0.3, p=1.0, =1.0 w=1.2 and a set of dimensionless values for excitation: =0.25, =0.27, =0.29 and =0.45.

For =0.25, fig 1a, the Duffing system is 1-periodic. For : =0.27, fig. 1b, the system is 1periodic, but with two period . For =0.29, show multiples periods. Finally, for =0.45 we have the behavior chaotic.

Figure 1. (a) Phase portrait for =0.25. (b) FFT for =0.25.

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Figure 2. (a) Phase portrait for =0.27. (b) FFT for =0.27.

Figure 3. (a) Phase portrait for =0.29. (b) FFT for =0.29.

Figure 4. (a) Phase portrait for =0.45. (b) FFT for =0.45.

3. Dynamic non-ideal to duffing equation In this section, we consider an extended Duffing equation with a Non-ideal excitation, this way, we add one degree that original Duffing equation to describe the interaction of system and the motor. An Non-Ideal Duffing equation may defined as follows:

2 31 1 cos( )x x w x px f t T

+ + = + (2)

sinz b z r x z + + (3) 2

( cos sin )T q z z z z = + (4)

Where z is non-ideal excitation response, a, b, q and r are dimensionless constant positives. The term T is due interaction between the dynamical system and an energy source, for example, a DC motor with limited power supply. T is responsible to the Non-idealization of the Duffing oscillator.

Note also that in the equation (3) we may observe an interaction term sinr x z

.The parameter a is a constant depending on initial conditions and b is an internal damping in Dc Motor.

For numerical simulations we taken dimensionless values =0.3, p=1.0, =1.0 w=1.2, =0.45 (of the Duffing Equation in section 2) and a=2.3,b=1.5,r=0.5 and q=0.5

In the figure 5, show the chaotic behavior, being, (a) the time history for x(t), (b) the time history

for ( )x t

, (c) the Phases portrait and (d) the FFT.

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Figure 5.The chaotic behavior for Non-Ideal Duffing Equation

4. Optimal linear control design In this section, we also developed an optimal linear control design for the non-ideal Duffing equation reducing the oscillatory movement of the non-ideal Duffing equation to a periodic orbit

We consider the nonlinear system

( )x Ax g x = + (5)

where nx R is vector of states, n nx R is constant matrix, and g(x) is vector, whose elements are continuous functions. In several engineering problems, the objective is to choose the control law U that moves the system of the disturbed regime to a desired one, either an equilibrium fixed point or a periodic or not periodic orbit.

Lets consider a vector function~x that characterizes the desired trajectory. The controlled

system is

( )x Ax g x U = + + (6)

where mU R is a control vector that consists of two parts ~

tU u= + (7) The part u is the feed forward control that may be expressed as:

~ ~ ~ ~( )x A x g x

= (8) and the control u

f is a linear feedback and it may be expressed as:

tu B= (9) where n mB R is constant matrix.

Defining ~

y x x= (10) as variation of the trajectory of the system Eq. (6) of the trajectory desired, and admitting Eq.

(7)(9), arrive at the variation equation : ~

( ) ( )y Ay g x g x Bu = + + (11)

The non-linear part of the system Eq. (10) can be written as ~ ~ ~

( ) ( ) ( , )( )g x g x G x x x x = (12)

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where ~

( , )G x x is a limited matrix, whose elements depend on x and ~x . Admitting Eq. (12),

the system Eq. (10) has the following form: ~

( , )y Ay G x x Bu = + + (13)

Then we will use the theorem done by Rafikov and Balthazar [6-8]. In addition, with the feedback control (6), there exists a neighborhood 0 ,

n , of the origin such that if 0 0x , the solution ( ) 0, 0x t t= , of the controlled system (11) is locally asymptotically stable, and

0min 0(0)T xJ x P= Finally, if n = then the solution ( ) 0, 0y t t= > , of the

controlled system (11) is globally asymptotically stable. Using the theorem by Rafikov and Balthazar the dynamic error y can be minimized ( y 0 ) [6-8] Theorem. If there exist matrixes Q and R, positive definite, being Q symmetric, such that the matrix

~ ~ ~( , ) ( , )TQ Q G x x P PG x x= (14)

is positive definite for the limited matrix G, then the linear feedback control 1 Tu R B Py= (15)

is optimal, in order to transfer the non-linear system (13) of from any initial state to the terminal state

( ) 0y = (16) minimizing the functional

~ ~

0

( )T TJ y Q y u Ru dt

= + (17) where the symmetric matrix P is calculated from the algebraic nonlinear Riccati equation:

1 0T TPA A P PBR B P Q+ + = (18) Next, we will apply this methodology to Non-Ideal Duffing Equation mathematical model. The equations Eq. (19) describing the Non-Ideal Duffing Equation controlled, can be written in the following form

1

1

1 1

3 21 1 1

1.5 0.5 sin( ) 2.3 ;

0.29 31.1364 1 0.7cos(5.85) (0.5( cos( ) sin( )));

x x

y y

x y x y U

y x x x y y y y

=== + + += + + + + + +

(19)

Where the function of control U is determinate by Eq. (7). The matrix A is

2 21 1 1

1

0 1 0 01 1 11 3 31.1364 sin( ) cos( ) cos( ) sin( )2 2 2

0 0 0 11 1 30 sin( ) cos( )1 2 2

x y y y y y y yA

y x y

+ + =

(20)

and

5

1000

B

= (21)

Where

~

1 1~

2 2

x xy

x x

= ,

~

0.10.10.10.1

x

= and the martrix

1 1 0 00 1 0 00 0 1 00 0 0 1

Q

= is a definitive positive

matrix. Where

0 1 0 00.9724 1.0025 1.0805 0.0247

0 0 0 10.0485 0.05 0.1409 1.4888

A

= (22)

Obtaining

0 0 120 800 120 36400 234000 36400 128610 82570

120 23400 82570 53010

P

= (23)

Solving the algebraic equation of Riccati Eq. (18) the function of optimal control u has the following form:

u =3.8554 x1 3.2246 x2 124.0621 x3 78.2124 x4 (24)

In the figure 6, show in (a) the trajectories ( )x t of the system without control. (b) the trajectories ~( )x t of the system without control , (c) the control Phases portrait, (d) the FFT for control system,

(e) the trajectories ( )x t of the system without control and controlled system and (f) the trajectories

( )x t

of the system without control and controlled system

Figure 5. The control non-ideal duffing equation

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5. Conclusions remarks In this work, a non-ideal problem and a control strategy was proposed for Duffing Equation. The Non-Ideal Duffing Equation when subjected to a harmonic excitation producing chaotic behavior and a control strategy was applied to regulate the chaotic behavior.

We considered Optimal Linear Feedback Control strategy for Non-Ideal Duffing Equation. The theory allowed reducing the chaotic behavior of the system to a limit cycle, where the theory of the Optimal Linear Feedback Control uses constants coefficients and it is an analytical procedure.

This control strategy may be extending to other problems.

References

[1] A.H. Nayfeh, Nonlinear Interactions: Analytical, Computational, and Experimental Methods, Wiley Series in Nonlinear Science Wiley, New York, (1979), pp. 760.

[2] V.O. Kononenko, Vibrating System with a Limited Power Supply, Illife Books, London (1969).

[3] J.M. Balthazar, D.T. Mook, H.I. Weber, M.L.R.F. Reyolando , A. Fenili, D. Belato, J.L.P. Felix, and F.J. Garzeri, A review on new vibration issues due to non-ideal energy sources. In: Dynamics systems and Control, edited by Udwadia F.E., Weber H.I., Leitman, G, Stability and Control: theory. Methods and Applications, 22, Chapman & Hallick, (2004), pp. 237-258

[4] E. Ott, C. Grebogi, J. A. Yorque, Controlling Chaos, Phys. Rev. Lett. 66, (1990), pp. 1196.

[5] S.C. Sinha, J.T. Henrichs, B.A. Ravindra, A General Approach in the Design of active Controllers for Nonlinear Systems Exhibiting Chos. Int. J. Bifur. Chaos, 10:1, (2000), pp.165.

[6] M. Rafikov, J.M. Balthazar, On control and synchronization in chaotic and hyperchaotic systems. Communications in Nonlinear Science & Numerical Simulation, vol. 13, (2008) pp. 12461255.

[7] F.R.Chavarette , J.M. Balthazar, M. Rafikov, H.A. Hermini, On Non-Linear and Non-ideal Dynamics Behavior and Optimal Control Design of the Potential of Action Membrane, A.A.M. Applied Mechanics and Materials. Bath, vol.5, (2006), pp.47-54.

[8] F.R.Chavarette, N.J.Peruzzi, J.M.Balthazar, L. Barbanti, B.C. Damasceno. On an Optimal Control Applied to a Non-Ideal Load Transportation System, Modeled with Periodic Coefficients. Applied Mechanis and Materials,vol. 52, (2011), pp. 13-18.

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