Optimal control of a distributed parameter system with applications to beam vibrations using piezoelectric actuators

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Abstract

This paper addresses the issue of the active vibration control of the transverse modes in a flexible elastic systems. The control is implemented by discrete sets of piezoelectric actuators that apply the optimal forces. The performance index is a time-dependent quadratic functional of state variables and their time derivatives, and control forces which are determined by minimizing the objective functional subject to a penalty term on the control functions. A combination of Galerkin and variational approaches are employed to determine the control forces in the time domain explicitly in terms of coupled amplitudes and velocities. The effectiveness of the proposed method is demonstrated by applying it to a physical problem controlled by piezoelectric patch actuators.

Introduction

The active control is aimed at damping out the vibrations of a mechanical system by applying optimal control forces and thereby modifying the system's response. A review of vibration suppression of smart structures with piezoelectric control actuator is presented in [1]. In the last decade, active vibration control has been implemented to damp out structural vibrations by incorporating active materials within the host structure. The piezoelectric materials have many outstanding advantages to produce actuators and sensors, especially in vibration control, because of their electromechanical coupling characteristics and frequency response [2]. A review for the state of the art of smart structures is given in [3].

In the present study, active vibration control of flexible elastic structures by using piezoelectric patch actuators is considered. Specifically, the structure to be controlled has transverse modes of vibration. The control is exercised by a set of actuators located over the surface of the structure and the control forces are determined on the basis of the minimization of the total energy of the system subject to a penalty on the time-dependent control force at any time. The total energy of the system is defined as the weighted sum of the kinetic energy and potential energy, and the time-dependent control forces constituting the penalty term. A combination of Galerkin and variational methods is used to determine the control forces in the time domain that are obtained explicitly in terms of amplitudes and velocities. In this paper, it is aimed to formulate a method of solution which is applicable to the control of structural elements using piezopatches as actuators and the proposed approach lies in using this combination which facilitates the computation of the solutions. It is also more efficient than solving a system of coupled initial-boundary-terminal-value problems which is the method of solution for the approach using the maximum principle [4]. The numerical simulations are given to assess the effectiveness and capabilities of piezoactuators for the vibration control.

The use of piezoelectric materials for active control of structures has been investigated by a number of researchers (see e.g. [5], [6], [7], [8], [9], [10], [11]). Active control of flexible structures by means of the classical variational techniques has been used in different types of control problems (see e.g., [12], [13]). The paper [14] gives an analysis and comparison of the classical and optimal feedback control strategies for the active control of vibrations of smart piezoelectric beams.

Section snippets

Problem formulation

Let Ω be an open connected and bounded region in the n-dimensional Euclidean space Rn with sufficiently smooth boundary Ω. The spatial vector and time co-ordinates are denoted by X=(x1,x2,,xn) and t[0,)R1, respectively. Consider a class of continuous controlled flexible elastic system that can be described by a generalized wave equationm(X)wtt(X,t)+Lw(X,t)=f(X,t)+i=1Nakivi(t)X[δ(Xγi)],(X,t)D:=Ω×(0,),where m(X) is the mass distribution, w(X,t) is the deflection of the structure at the

Method of solution

In this section, the problem introduced in Eqs. (1), (6) is converted to that of the optimal control problem of lumped parameter systems by using the Galerkin projections of the state and control functions. Necessary optimality conditions are derived in the form of integral equations by using variational approach. Finally, the optimum state variable is obtained as the solution of a second order temporal differential equation with variable coefficients.

Example: control of a simply supported beam with piezopatches

As an illustrative example of vibration suppression in the distributed parameter system, we consider the vibration control of a beam with piezoelectric actuators bonded on the surface, as shown in Fig. 1. The non-dimensional formulation of the controlled system is considered by expressing the equation of motion as [5]wtt+Lw(x,t)=i=1Naζivi(t)d2dx2[H(xxi)H(xxi+1)],where w=w(x,t) is the deflection of the beam, Lw(x,t):=wxxxx, ζi=(L3/EI)ki, H(x) is Heaviside step function, and vi is the ith

Numerical results

In this section, the simulation will be run for the first mode to show the robustness of the technique that is developed in the previous sections. For N=Na=1, Eq. (32) becomesw¨1(t)+g1(t)w˙1(t)+g2(t)w1(t)=0,where g1(t)=μ2r1β1[2γ1(ϕ1(x2)ϕ1(x1))]2sinβ1tandg2(t)=β12+μ1r1β12[2γ1(ϕ1(x2)ϕ1(x1))]2(1cos(β1t)).The simulations are conducted in MATLAB. In the simulations, x1=1/3, x2=2/3, k1=108,μ1=10,μ2=0.1, and r1=0.01(;100) are taken. Initial conditions for Eq. (33) are taken as w(x,0)=0 and wt(x,

Conclusion

The paper addresses the active control of an elastic system using piezoelectric actuators to control the transverse vibrations. The control objective is to minimize the total energy of the system and the control effort. We first apply Galerkin projection method to reduce the optimal control of the distributed parameter system into the optimal control of a lumped parameter system. A variational approach is presented to determine the control forces in the time domain. In general, the

Acknowledgements

The authors are very grateful to Professor Sarp Adali from the University of KwaZulu-Natal from South Africa for valuable discussions and improving the manuscript. The first author acknowledges the financial support of TUBITAK, Turkey.

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