CHAOS CONTROL VIA STATE-DEPENDENT RICCATI EQUATION METHOD APPLIED A NON-IDEAL VIBRATION ABSORBER COUPLED TO A NONLINEAR OSCILLATOR

Fábio Roberto Chavarette1 , Nelson José Peruzzi2, Douglas da Costa Ferreira3, Mara Lucia Martins Lopes4 1,4Department of Mathematics Faculty of Engineering of Ilha Solteira UNESP – Univ.Estadual Paulista Brasil Avenue, 56, 15385-000, Ilha Solteira, SP, BRAZIL 2Department of Exact Sciences UNESP – Univ. Estadual Paulista Via de Acesso Prof. Paulo Donato Castellane s/n, 14884-900 Jaboticabal, SP, BRAZIL 3Mechanical Engineering Department UFMT – Federal University of Mato Grosso Rondonópolis, MT, BRAZIL


Introduction
In recent decades the control of structural vibration has become increasingly more interest from engineers and researchers because of the fact that these vibrations are unwanted and may cause noise emission, premature wear and fatigue failure of components.Thus there is great interest in reducing vibrations in the system through the use of control techniques [5,6,7].
The structural control is a technology for protection of structures which promotes a change in the stiffness and damping properties of the structure by adding external devices or by the action of external forces.Vibration absorbers have been applied to the control (reduction) of vibrations in structures, and the absorbers are simple devices that, when properly connected to a structure, are able to promote the reduction of its vibrations in an effective manner and consequently in many cases, reduction of noise levels, with the advantage of not requiring high costs for its implementation.
The tuned mass damper(TMD)is one popular device for minimize vibrations of mechanical structures.Frahm [1] was the first to introduce the TMD concept and considered a linear attachment composed of a mass and a spring coupled to a conservative linear oscillator(LO).Viguie and Kerschen proposed in [3] the tuning of a nonlinear vibration absorber coupled to an essentially nonlinear oscillator.The dynamic model of non-ideal power source can be defined for a beam excited by an unbalanced direct current (DC) motor was proposed by Kononenko [2].This paper proposes the control of the vibration absorber coupled to a nonlinear oscillator model,and for its development we organize the paper this way.In Section 2, we present the mathematical model.In Section 3, the control design control project is propose for the minimize vibration and in Section 4, the conclusions are presented.

Non-ideal Model
Figure 1 shows the non-ideal and nonlinear absorber coupled to nonlinear oscillator proposed, which consist of a nonlinear oscillator composed of a mass m 1 and a cubic stiffness k nl1 and the absorber with a mass m 2 and a nonlinear stiffness k nl2 .c 1 and c 2 are the weak damping to induce energy dissipation [3].
The angular position z is non-ideal excitation response, a, b are motor torque constants, r is the distance from unbalanced mass to rotation center of the DC motor and d is related to moment of inertia of the system.All parameters are dimensionless constant positives.The term z is due interaction between the dynamical system and an energy source.The parameter a is the applied torque constant and depends on initial conditions and b is resistive net torque and has no influence of initial conditions and is considered as internal damping of the DC Motor.
The equations that describe the device illustrate in Figure 1 are: Rewriting the equations of the dynamical system, in state form, making y 1 = x 1 , y 2 = ẋ1 ,y 3 = x 2 ,y 4 = ẋ2 , y 5 = z and y 6 = ż the governing equations may be written as being: .
Aiming to minimize vibrations and reduce the oscillatory motion caused in the system in the following section proposes the application of State-Dependent Riccati Equation (SDRE) to reduce this chaotic motion, see Figure 2b, to a

Optimal Control Design
The control objective is to stabilize the system with chaotic behavior in the small orbit periodic near the system origin.The Control Design via State-Dependent Riccati Equation (SDRE) approach for obtaining a suboptimal solution of the control problem has the following procedure [9] and [10]: 1. Represent the model in state-space form.Use direct parametrization to bring the nonlinear dynamics ẋ = f (x) + g(x) to the state-dependent coefficient (SDC) form, as follows: where, f (x) = A(x)x and g(x) = B(x), A(x) ∈ ℜ nxn is the dynamic matrix, B(x) ∈ ℜ nxm is the input matrix, C(x) ∈ ℜ sxn is the output matrix, x ∈ ℜ n is the state vector u ∈ ℜ m is the control law, y ∈ ℜ s is the output vector.
2. Define the initial conditions x(0) = x 0 , and choose the coefficients of positive definite weighting matrices Q(x) and R(x), which determine the relative importance of state x(t) and control effort u(t), respectively.
3. Solve the state-dependent Riccati equation given by: A T (x)P (x) + P (x)A(x) − P (x)B(x)R −1 (x)B T (x)P (x) + Q(x) = 0 (4) 4. Construct the nonlinear feedback control via: The control law ( 5) is calculated so that the performance index given by, In the multivariable case, there always exists an infinite number of SDC parameterizations.Therefore, the choice of the matrix A(x) isn't unique [9].
The pair {A(x), B(x)} is a controllable parametrization of the nonlinear system in a region Ω if {A(x), B(x)} is pointwise controllable in the linear sense for all x ∈ Ω.Therefore, the choice of A(x) must be such that the statedependent controllability matrix [B(x) A(x)B(x) ... A n−1 (x)B(x)] has full rank [10].
The SDRE technique has been used to control various systems [11,5].Details about the technique SDRE can be found in [9] and [10].

Figure 1 :
Figure 1: Non-ideal and nonlinear absorber coupled to nonlinear oscillator.