A Galerkin-Parameterization Method for the Optimal Control of Smart Microbeams

A proposed computational method is applied to damp out the excess vibrations in smart microbeams, where the control action is implemented using piezoceramic actuators. From a mathematical point of view, we wish to determine the optimal boundary actuators that minimize a given energy-based performance measure. The minimization of the performance measure over the actuators is subjected to the full motion of the structural vibrations of the micro-beams. A direct state-control parametrization approach is proposed where the shifted Legendre polynomials are employed to solve the optimization problem. Legendre operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the lumped parameter systems with respect to the quadratic cost function by solving linear algebraic equations. Numerical examples are provided to demonstrate the applicability and efficiency of the proposed approach.


Introduction
Devices of Microelectromechanical systems MEMSs find wide applications as sensors and actuators.The analysis of methods of actuation and sensing has been a topic of interest over the past several years.Different actuation and sensing properties such as piezosensitive, piezoelectric, electrostatic, thermal, electromagnatic, and optical have been used 1 .Piezo materials can be integrated in various structural components as distributed sensors or actuators.
The use of the piezoelectric actuators has been proved to be effective control devices for the control of structural vibrations in a wide range of engineering applications.One of the most widely used piezo materials in active control is piezoceramic such as PZT.This due to their large bandwidth, mechanical simplicity, and their mechanical ability in producing controlling forces 2 .

Mathematical Problems in Engineering
In this contest, the microelectromechanical systems MEMSs seem to be attractive in improving the mechanical efficiency of structural active control.Borovic et al. 3 highlight some MEMS control issues and provide an overview of MEMS control.
Piezoelectrically actuated microbeams have recently received considerable attention.The behavior of electrically actuated microbeams has been studied using different models and approaches 4-8 .With the growth of using microbeams MEMS, it is necessary to study their dynamic behavior.In this paper, the studied microsystem is a microbeam totally covered by a piezoelectric, PZT, film.While it is possible to cover the entire structure with piezoelectric material, it is not possible to do so on large structures.The optimal boundary vibration control for microbeams is investigated.The control action is implemented using piezoceramic actuators to damp out the vibrations of microbeams where the control function appears in the boundary conditions in the form of a moment.In the control problem, we wish to determine the optimal boundary control actuators that minimize a given energy-based performance measure.The minimization of the performance measure over the actuators is subjected to the equation of motion governing the structural vibration, the imposed initial conditions, and the boundary conditions.The performance measure is specified as a quadratic functional of displacement and velocity along with a suitable penalty term involving the boundary control function.For the determination of the optimal boundary controls, it is necessary to convert the problem from one in which there is boundary control into one in which there are distributed controls.The Galerkin-based approach is used then to reduce the modified problem to the optimal control of a linear time-invariant lumped-parameter system.In contrast to standard optimal control or variational methods for lumped parameter systems, a direct state-control parameterization by orthogonal polynomial expansion is employed to solve the modal space optimization problem.
In general, the approach is based on converting differential equations into integral equations through integration, approximating state and control involved in the equation by finite term series, and using the operational matrix of integration to eliminate the integral operations.This method has been used in obtaining the continuous control of various distributed and lumped parameter system modals [9][10][11]  A computational method is proposed to sole the modal optimization problem with quadratic performance index.The method is based on parametrizing the state and control variables by finite-term Legendre series whose coefficient values will be determined.The Legendre operational matrix and the properties of Kronecker product are used to relate the unknown coefficients of control variables to coefficients of the state variables.The performance index, as a result, can be expressed in terms of the unknown coefficients of the state variables.The necessary condition for optimality is derived as a system of linear algebraic equations in terms of the unknown coefficients of the state variables.Of the attractive features of our proposed approach is that we can avoid difficult integral equations, which are produced by variational methods 30 .This is achieved by reducing the problem to the solution of algebraic system of equations.Moreover, solving a system of coupled initial boundary-terminal-value problems as a requirement for the maximum principle 31 can now be avoided.Numerical simulations are presented to assess the effectiveness and the capabilities of piezo actuation by means of moments to damp out the vibration of microbeams with a minimum level of voltage applied on the piezo actuators.

Equation of Motion for a Piezoelectric Beam
Consider a microbeam of length l, width b, height h s and covered by layers of piezoelectric materials of thickness h p .The dynamical equilibrium of the Euler-Bernoulli beam is defined by 7, 32 where ρ is the mass per unit length of the layered beam, E c o I c o is the bending stiffness of the beam including the piezoelectric layers, and w x, t is the transverse displacement of the beam.
The boundary and initial conditions, respectively, are w 0, t w l, t 0, where For convenience, we introduce the nondimensional variables where W X, T , X, and T are dimensionless transverse displacement, position, and time, respectively.Substituting 2.6 into 2.1 -2.3 leads to the nondimensional equation of motion with boundary conditions and initial conditions 2.9

Problem Statement
Consider the set of all admissible control functions U ad {F : F ∈ L 2 0, T f }.The performance index is given by where μ 1 , μ 2 , and μ 3 are all nonnegative constants and μ 1 μ 2 > 0 and μ 3 / 0. The last term on the right hand side of 3.1 is a penalty on control energy.The optimal control problem is now formulated: and such that W X, T satisfies all 2.7 -2.9 .As we will assume the existence of the optimal control 3.2 , it can be easily shown that this quadratic control problem has at most one solution.

Equivalent Optimal Control Problem
and the new boundary and initial conditions are, respectively,

3.7
The optimal control problem, that is equivalent to 3.2 , is Find F * ∈ U ad so that J F * min and such that W X, T satisfies 3.5 -3.7 .

Optimal Control of Lumped Parameter System
In this section, the distributed parameter system optimization problem 3.5 is transformed into a modal Lumped parameter problem by means of Galerkin approach 30 .

Control Problem in Modal Space
A finite orthonormal expansion of W X, T in terms of a complete basis where λ n nπ.The orthogonality of the Fourier sine series converts 3.5 into where η n 1 0 α X sin λ n X dX, n 1, 2, . . ., N. The performance index takes on the form Integrating 4.2 twice over the interval 0, t gives or, in vector notation where where "tr" stands for the matrix transpose.The new optimal control problem is now formulated subject to the integral equation 4.5 .

State-Control Parametrization
In this section, a direct method for solving the modal control problem 4.7 is developed by parametrizing the state variables Z i t and the control variable F t .Let where and S j T are shifted Legendre polynomials on the interval 0, T f see 33 .Using the expansions 4.8 , we write
The double integral on the left-hand side of 4.5 is simplified as where H is the shifted Legendre operational matrix and is given by Using 4.10 and 4.12 , 4.5 takes on the form where are the coefficients of shifted Legendre polynomials resulting from the expansions of 1 and T , respectively 33 .By letting D 2D and using the orthogonality of − → S T , 4.14 becomes Now using the vec notation and kronecker product 34 , 4.16 takes on the form Using the substitutions where in which

Approximation of Modal Performance Index
The cost functional 4.3 can be transformed into modal space where R i diag μ i , i 1, 2, 3. Inserting 4.10 into 4.23 gives or by using 4.22 , where

4.27
To this end, the optimal control problem 3.8 is converted into the mathematical programming problem:

Optimal Control Characterization
Let P 4 P 1 P 2 and P 5 M tr P * 3 M.By employing the properties of matrix differentiation 34 , we can obtain the necessary condition of the optimal control by differentiating the performance index 4.27 with respect to the unknown vector − → A, we obtain

4.30
Now we find and from 4.22 , we have

4.33
The optimal state variable W * X, T is obtained from 3.3 , so we have

Numerical Experiments
Numerical results are given to show the effectiveness of piezo actuators in controlling the system and damping out the vibrations of the microbeam with a minimal use of voltage applied to the piezo actuators at various terminal times, T f , subject to the initial impact conditions: where λ i iπ, i 1, 2.
For the numerical simulations, we let N 2 and m 10 the first 10 shifted Legendre polynomials .The measure of the total force used in the control process is given by and the controlled energy of the system is defined as where W 0 x, T corresponds to the optimal displacement of the microbeam.The controlled and uncontrolled energies are denoted by E c and E u , respectively, and the force used is denoted by J c .In Tables 1-3, we study the effect of each two consecutive weighting factors μ i and μ i 1 for i 1, 3, and 5.While we vary each two consecutive μ s, we set the other four as well as the terminal time at 1.In Table 4, we study the effect of the terminal time on the system energies.
The following observations are made.
1 The system achieves substantial energy reduction when the weighting factors μ 1 and μ 2 are smaller.This implies that the smaller μ 1 and μ 2 , the more emphasis on kinetic energies; see Table 1.

Conclusions
In this paper, a computational approach was presented to optimal boundary control of smart mirco-beams with quadratic performance index.The Galerkin method is first used to reduce the problem to optimal control of lumped parameter system.An algorithm based on parametrizing the state and control variables by shifted Legendre polynomials was employed to solve the lumped parameter optimization problem.The control parameters are obtained from the integrated system state equations as a function of the approximated state parameters, and the performance index was evaluated by an algorithm, which was also proposed in the current study.The optimal control problem in lumped parameter system was converted into a parameter optimization problem, which was quadratic in

and − → B 3 are,
Nm × 1 vectors, Γ, Ω 1 , Ω 2 , and I m are m × m matrices in which Γ H 2 tr , and I m is the m × m identity matrix, and D and Λ are N × N matrices.

, 3
t dt, the performance index 4.25 takes on the form a * ij and b * j are the components of − − → A * and − → B * , respectively.

Chebyshev polynomials 16 , Laguerre polynomials 17 , Fourier Series 18 , Taylor series 19 , Jacobi series 20, 21 , Hermite polynomials 22 , and wavelets 23- 29 .
31d 31 E p , where E p and E s are Young's modulus of the piezoelectric layer and the microbeam, respectively.d 31 is the actuator piezoelectric constant, and f t is the applied voltage.

Table 4 :
Effect of terminal time T f .