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

Authors: Marwan Abukhaled; Ibrahim Sadek
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

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.


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## Body

## 1. 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].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 [9–11]. Typical examples are the applications of Walsh functions [12], block-pulse functions [13], Bessel functions [14], Harr function [15], Chebyshev polynomials [16], Laguerre polynomials [17], Fourier Series [18], Taylor series [19], Jacobi series [20, 21], Hermite polynomials [22], and wavelets [23–29].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.


## 2. Equation of Motion for a Piezoelectric Beam

Consider a microbeam of lengthl, width b, height hs and covered by layers of piezoelectric materials of thickness hp. The dynamical equilibrium of the Euler-Bernoulli beam is defined by [7, 32](2.1)ρ¯wtt+EcoIcowxxxx=0,0<x<l,0<t<tf,
where ρ¯ is the mass per unit length of the layered beam, EcoIco 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(2.2)w(0,t)=w(l,t)=0,wxx(0,t)=wxx(l,t)=GpEcoIcof(t),(2.3)w(x,0)=w0(x),wt(x,0)=wl(x),
where(2.4)EcoIco=D11-B112A11
in which(2.5)A11=(Eshs+Ephp)b,B11=Ep-Es2hphsb,D11=b12(Eshs(hs2+3hp2)+Ephp(hp2+3hs2)),Gp=(hs2-B11A11)be31
in which e31=d31Ep, where Ep and Es are Young's modulus of the piezoelectric layer and the microbeam, respectively. d31 is the actuator piezoelectric constant, and f(t) is the applied voltage.For convenience, we introduce the nondimensional variables(2.6)W(X,T)=w(x,t)l,X=xl,T=tl2EcoIcoρ¯,Tf=tfl2EcoIcoρ,
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(2.7)WTT+WXXXX=0,0<X<1,0<T<Tf
with boundary conditions(2.8)W(0,T)=W(1,T)=0,WXX(0,T)=WXX(1,T)=F(T)
and initial conditions(2.9)W(X,0)=W0(X),WT(X,0)=W1(X).


## 3. Optimal Control Problem

### 3.1. Problem Statement

Consider the set of all admissible control functionsUad={F:F∈L2(0,Tf)}. The performance index is given by(3.1)J(F)=μ1∫01W2(X,Tf)dX+μ2∫01WT2(X,Tf)dX+μ3∫0TfF2(T)dT,
where μ1,μ2, and μ3 are all nonnegative constants and μ1+μ2>0 and μ3≠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:(3.2)FindF*∈UadsothatJ(F*)=minF∈UadJ(F),
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.


### 3.2. Equivalent Optimal Control Problem

Let(3.3)W¯(X,T)=W(X,T)-α(X)F(T),
where(3.4)α(X)=12(X2-X).Substituting (3.3) into (2.1) gives(3.5)W¯TT(X,T)+W¯XXXX(X,T)=-α(X)F′′(T),0<X<L,0<T<Tf,
and the new boundary and initial conditions are, respectively,(3.6)W¯(0,T)=W¯(1,T)=0,W¯XX(0,T)=W¯XX(1,T)=0,(3.7)W¯(X,0)=W0(X)-α(X)F(0),W¯T(X,0)=W1(X)-α(X)F′(0).
The optimal control problem, that is equivalent to (3.2), is(3.8)FindF*∈UadsothatJ(F*)=minF∈UadJ(F),
and such that W¯(X,T) satisfies (3.5)–(3.7).


## 3.1. Problem Statement

Consider the set of all admissible control functionsUad={F:F∈L2(0,Tf)}. The performance index is given by(3.1)J(F)=μ1∫01W2(X,Tf)dX+μ2∫01WT2(X,Tf)dX+μ3∫0TfF2(T)dT,
where μ1,μ2, and μ3 are all nonnegative constants and μ1+μ2>0 and μ3≠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:(3.2)FindF*∈UadsothatJ(F*)=minF∈UadJ(F),
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.


## 3.2. Equivalent Optimal Control Problem

Let(3.3)W¯(X,T)=W(X,T)-α(X)F(T),
where(3.4)α(X)=12(X2-X).Substituting (3.3) into (2.1) gives(3.5)W¯TT(X,T)+W¯XXXX(X,T)=-α(X)F′′(T),0<X<L,0<T<Tf,
and the new boundary and initial conditions are, respectively,(3.6)W¯(0,T)=W¯(1,T)=0,W¯XX(0,T)=W¯XX(1,T)=0,(3.7)W¯(X,0)=W0(X)-α(X)F(0),W¯T(X,0)=W1(X)-α(X)F′(0).
The optimal control problem, that is equivalent to (3.2), is(3.8)FindF*∈UadsothatJ(F*)=minF∈UadJ(F),
and such that W¯(X,T) satisfies (3.5)–(3.7).


## 4. 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].


### 4.1. Control Problem in Modal Space

A finite orthonormal expansion ofW¯(X,T) in terms of a complete basis {φn(x)=2sinλnx}∈L2([0,1]) gives the representation(4.1)W¯(X,T)=∑n=1NZn(T)φn(X),
where λn=nπ. The orthogonality of the Fourier sine series converts (3.5) into(4.2)d2dT2Zn(T)+λn4Zn(T)=-2ηnd2dT2F(T),
where ηn=∫01α(X)sin(λnX)dX,n=1,2,…,N.The performance index takes on the form(4.3)JN(F(T))=12μ1∑n=1NZn2(Tf)+12μ2∑n=1NddtZn2(Tf)+μ3∫0TfF2(τ)dτ.Integrating (4.2) twice over the interval (0,t) gives(4.4)Zn(T)-TZn′(0)-Zn(0)+λn4∫0T∫0ηZn(τ)dτdη=-2ηn[F(T)-F(0)-TF′(0)]
or, in vector notation(4.5)Z⃗(T)-G(T)ddTZ⃗(0)-Z⃗(0)+Λ∫0T∫0ηZ⃗(τ)dτdη=E⃗1+E⃗2+E⃗3,
where(4.6)Z⃗(T)=[Z1(T),Z2(T),…,ZN(T)]tr,G(t)=TIN×N,Λ=diag[λi4]N×N,E⃗1=-2DF⃗(t),E⃗2=2DF⃗(0),E⃗1=2tDF′⃗(0),D=diag[ηi]N×N,
where “tr” stands for the matrix transpose. The new optimal control problem is now formulated(4.7)FindF*(T)∈UadsothatJN(F*(T))=minF(T)∈UadJN(F(T))
subject to the integral equation (4.5).


### 4.2. State-Control Parametrization

In this section, a direct method for solving the modal control problem (4.7) is developed by parametrizing the state variables Zi(t) and the control variable F(t).Let(4.8)Zi(T)=∑j=0m-1aijSj(T),F(T)=∑j=0m-1bjSj(T),
where(4.9)aij=2j+1Tf∫0TfZi(τ)Sj(τ)dτ,bj=2j+1Tf∫0TfF(τ)Sj(τ)dτ0≤j≤m-1,1≤i≤N, and Sj(T) are shifted Legendre polynomials on the interval [0,Tf] (see [33]). Using the expansions (4.8), we write(4.10)Z⃗(T)=AN×mS⃗m×1orZ⃗(T)=SN×NmA⃗Nm×1,F⃗(T)=BN×m1S⃗m×1orF⃗(T)=SN×NmB1⃗Nm×1,F⃗(0)=BN×m2S⃗m×1orF⃗(0)=SN×NmB2⃗Nm×1,F′⃗(0)=BN×m3S⃗m×1orF′⃗(0)=SN×NmB3⃗Nm×1,G(T)ddtZ⃗(0)=CN×mS⃗m×1orG(T)ddtZ⃗(0)=SN×NmCNm×1,Z⃗(0)=WN×mS⃗m×1orZ⃗(0)=SN×NmW⃗Nm×1,
where
(4.11)A=[a10a11⋯a1,m-1a20a21⋯a2,m-1⋮⋮⋮⋮aN0aN1⋯aN,m-1],A⃗=[A1⃗A2⃗⋮AN⃗],C=[cc0⋯0cc0⋯0⋮⋮⋮⋮⋮cc0⋯0],C⃗=[C1⃗C2⃗⋮CN⃗],S⃗=[S0(T)S1(T)⋮Sm-1(T)],S=[S⃗tr(T)S⃗tr(T)⋱S⃗tr(T)],W=[Z1(0)0⋯0Z1(0)0⋯0⋮⋮⋮⋮Z1(0)0⋯0],W⃗=[W1⃗W2⃗⋮WN⃗],B1=[b0b1⋯bm-1b0b1⋯bm-1⋮⋮⋮⋮b0b1⋯bm-1],B2=[b0-b1b2-b3⋯(-1)m-1bm-1b0-b1b2-b3⋯(-1)m-1bm-1⋮⋮⋮⋮⋮⋮b0-b1b2-b3⋯(-1)m-1bm-1],B3=[02b1-6b212b3-20b4⋯(-1)m(m-1)mbm-102b1-6b212b3-20b4⋯(-1)m(m-1)mbm-1⋮⋮⋮⋮⋮⋮⋮02b1-6b212b3-20b4⋯(-1)m(m-1)mbm-1],
in which c=(Tf/2)(d/dt)Z1(0),Ai⃗=[ai0,ai1,…,ai,m-1]tr,Ci⃗=[c,c,0,…,0]tr, and Wi⃗=[Z1(0),0,…,0]tr.The double integral on the left-hand side of (4.5) is simplified as(4.12)∫0T∫0ηZ⃗(τ)dτdη=A∫0T∫0ηS⃗(τ)dτdη=AH2S⃗(T),
where H is the shifted Legendre operational matrix and is given by(4.13)H=T2[-1100⋯000-130130⋯0000-15015⋯000⋮⋮⋮⋮⋮⋮⋮⋮0000⋯-12m-3012m-30000⋯0-12m-10].Using (4.10) and (4.12), (4.5) takes on the form(4.14)AS⃗-CS⃗-WS⃗+ΛAH2S⃗=D[-2B1S⃗︸F(t)+2B2S⃗(0)︸F(0)βS⃗︸1+2B3S⃗′(0)︸F′(0)αS⃗︸t],
where(4.15)β1×m=[β0,β1,…,βm-1],α1×m=[α0,α1,…,αm-1]
are the coefficients of shifted Legendre polynomials resulting from the expansions of 1 and T, respectively [33].By lettingD¯=2D and using the orthogonality of S⃗(T), (4.14) becomes(4.16)A+ΛAH2=C+W-D¯B1+D¯B2S⃗(0)β+D¯B3S⃗′(0)α.
Now using the vec notation and kronecker product [34], (4.16) takes on the form(4.17)A⃗+(Λ⊗Γ)A⃗=C⃗+W⃗-(D¯⊗Im)B1⃗+(D¯⊗Ω1)B2⃗+(D¯⊗Ω2)B3⃗,
where A⃗,C⃗,W⃗,B1⃗,B2⃗, and B3⃗ are Nm×1 vectors, Γ,Ω1,Ω2, and Im are m×m matrices in which Γ=(H2)tr,Ω1=(S⃗(0)β)tr,Ω2=(S⃗′(0)α)tr, and Im is the m×m identity matrix, and D¯ and Λ are N×N matrices.Using the substitutions(4.18)X=INm+(Λ⊗Γ),U1=D¯⊗Im,U2=D¯⊗Ω1,U3=D¯⊗Ω2,Y⃗=C⃗+W⃗,
equation (4.17) becomes(4.19)XA⃗=Y⃗+U1B1⃗+U2B2⃗+U3B3⃗=Y⃗+KB1⃗,
where(4.20)K=U1+U2I1+U3I2
in which(4.21)I1=diag[(-1)i+1]i=1m,I2=diag[(-1)i(i-1)i]i=1m.Solving (4.19) for B1⃗, we obtain(4.22)B1⃗=MA⃗+N⃗,
where M=K-1X and N⃗=-K-1Y⃗.


### 4.3. Approximation of Modal Performance Index

The cost functional (4.3) can be transformed into modal space(4.23)JN(F⃗)=12∑n=1NZn⃗tr(Tf)R1Zn⃗(Tf)+12∑n=1NddTZn⃗tr(Tf)R2ddTZn⃗(Tf)+1N∫0TfF⃗tr(τ)R3F⃗(τ)dτ,
where Ri=diag [μi], i=1, 2, 3. Inserting (4.10) into (4.23) gives
(4.24)JN(F⃗)=12[A⃗trStr(Tf)R1S(Tf)A⃗+A⃗trddTStr(Tf)R2ddTS(Tf)A⃗]+1N∫0TfB⃗1trStr(τ)R3S(τ)B1⃗dτ,
or by using (4.22),(4.25)JN(A)=12[A⃗trP1(Tf)A⃗+A⃗trP2(Tf)A⃗]+1N∫0Tf(MA⃗+N⃗)trP3(τ)(MA⃗+N⃗)dτ,
where(4.26)P1(T)=Str(T)R1S(T),P2(T)=ddTStr(T)R2ddTS(T)P3(T)=Str(T)R3S(T).,
Expanding (MA⃗+N⃗)tr and letting P3*=∫0TfP3(t)dt, the performance index (4.25) takes on the form
(4.27)JN(A⃗)=1N[A⃗trMTP3*MA⃗+N⃗trP3*MA⃗+A⃗trMTP3*N⃗+N⃗trP3*N⃗]+12[A⃗tr(P1(Tf)+P2(Tf))A⃗].
To this end, the optimal control problem (3.8) is converted into the mathematical programming problem:(4.28)FindA*⃗∈ℝNsothatJN(A*⃗)=minA⃗∈ℝNJN(A⃗).


### 4.4. Optimal Control Characterization

LetP4=P1+P2 and P5=MtrP3*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 A⃗, we obtain
(4.29)∂JN(A⃗)∂A⃗=1N[(P5(Tf)+P5tr(Tf))A⃗+(N⃗trP3*M)tr+(MtrP3*N⃗)]+12(P4(Tf)+P4tr(Tf))A⃗.Let(4.30)P=12(P4(Tf)+P4tr(Tf))+1n(P5(Tf)+P5tr(Tf)),Q=-1N(MTP3*tr+MtrP3*).
Now we find A*⃗ so that ∂JN(A*⃗)/∂A⃗=0, we obtain(4.31)PA*⃗-QN⃗=0
or(4.32)A*⃗=P-1QN⃗
and from (4.22), we have(4.33)B*⃗=M(P-1QN⃗)+N⃗=(MP-1Q+I)N⃗.The optimal state variableW*(X,T) is obtained from (3.3), so we have(4.34)W*(X,T)=W¯*(X,T)+α(X)F*(T),
where(4.35)W¯*(X,T)=∑n=1N(∑j=0m-1aij*Sj(T))φn(X),F*(T)=∑j=0m-1bj*Sj(T)
in which aij* and bj* are the components of A*⃗ and B*⃗, respectively.


## 4.1. Control Problem in Modal Space

A finite orthonormal expansion ofW¯(X,T) in terms of a complete basis {φn(x)=2sinλnx}∈L2([0,1]) gives the representation(4.1)W¯(X,T)=∑n=1NZn(T)φn(X),
where λn=nπ. The orthogonality of the Fourier sine series converts (3.5) into(4.2)d2dT2Zn(T)+λn4Zn(T)=-2ηnd2dT2F(T),
where ηn=∫01α(X)sin(λnX)dX,n=1,2,…,N.The performance index takes on the form(4.3)JN(F(T))=12μ1∑n=1NZn2(Tf)+12μ2∑n=1NddtZn2(Tf)+μ3∫0TfF2(τ)dτ.Integrating (4.2) twice over the interval (0,t) gives(4.4)Zn(T)-TZn′(0)-Zn(0)+λn4∫0T∫0ηZn(τ)dτdη=-2ηn[F(T)-F(0)-TF′(0)]
or, in vector notation(4.5)Z⃗(T)-G(T)ddTZ⃗(0)-Z⃗(0)+Λ∫0T∫0ηZ⃗(τ)dτdη=E⃗1+E⃗2+E⃗3,
where(4.6)Z⃗(T)=[Z1(T),Z2(T),…,ZN(T)]tr,G(t)=TIN×N,Λ=diag[λi4]N×N,E⃗1=-2DF⃗(t),E⃗2=2DF⃗(0),E⃗1=2tDF′⃗(0),D=diag[ηi]N×N,
where “tr” stands for the matrix transpose. The new optimal control problem is now formulated(4.7)FindF*(T)∈UadsothatJN(F*(T))=minF(T)∈UadJN(F(T))
subject to the integral equation (4.5).


## 4.2. State-Control Parametrization

In this section, a direct method for solving the modal control problem (4.7) is developed by parametrizing the state variables Zi(t) and the control variable F(t).Let(4.8)Zi(T)=∑j=0m-1aijSj(T),F(T)=∑j=0m-1bjSj(T),
where(4.9)aij=2j+1Tf∫0TfZi(τ)Sj(τ)dτ,bj=2j+1Tf∫0TfF(τ)Sj(τ)dτ0≤j≤m-1,1≤i≤N, and Sj(T) are shifted Legendre polynomials on the interval [0,Tf] (see [33]). Using the expansions (4.8), we write(4.10)Z⃗(T)=AN×mS⃗m×1orZ⃗(T)=SN×NmA⃗Nm×1,F⃗(T)=BN×m1S⃗m×1orF⃗(T)=SN×NmB1⃗Nm×1,F⃗(0)=BN×m2S⃗m×1orF⃗(0)=SN×NmB2⃗Nm×1,F′⃗(0)=BN×m3S⃗m×1orF′⃗(0)=SN×NmB3⃗Nm×1,G(T)ddtZ⃗(0)=CN×mS⃗m×1orG(T)ddtZ⃗(0)=SN×NmCNm×1,Z⃗(0)=WN×mS⃗m×1orZ⃗(0)=SN×NmW⃗Nm×1,
where
(4.11)A=[a10a11⋯a1,m-1a20a21⋯a2,m-1⋮⋮⋮⋮aN0aN1⋯aN,m-1],A⃗=[A1⃗A2⃗⋮AN⃗],C=[cc0⋯0cc0⋯0⋮⋮⋮⋮⋮cc0⋯0],C⃗=[C1⃗C2⃗⋮CN⃗],S⃗=[S0(T)S1(T)⋮Sm-1(T)],S=[S⃗tr(T)S⃗tr(T)⋱S⃗tr(T)],W=[Z1(0)0⋯0Z1(0)0⋯0⋮⋮⋮⋮Z1(0)0⋯0],W⃗=[W1⃗W2⃗⋮WN⃗],B1=[b0b1⋯bm-1b0b1⋯bm-1⋮⋮⋮⋮b0b1⋯bm-1],B2=[b0-b1b2-b3⋯(-1)m-1bm-1b0-b1b2-b3⋯(-1)m-1bm-1⋮⋮⋮⋮⋮⋮b0-b1b2-b3⋯(-1)m-1bm-1],B3=[02b1-6b212b3-20b4⋯(-1)m(m-1)mbm-102b1-6b212b3-20b4⋯(-1)m(m-1)mbm-1⋮⋮⋮⋮⋮⋮⋮02b1-6b212b3-20b4⋯(-1)m(m-1)mbm-1],
in which c=(Tf/2)(d/dt)Z1(0),Ai⃗=[ai0,ai1,…,ai,m-1]tr,Ci⃗=[c,c,0,…,0]tr, and Wi⃗=[Z1(0),0,…,0]tr.The double integral on the left-hand side of (4.5) is simplified as(4.12)∫0T∫0ηZ⃗(τ)dτdη=A∫0T∫0ηS⃗(τ)dτdη=AH2S⃗(T),
where H is the shifted Legendre operational matrix and is given by(4.13)H=T2[-1100⋯000-130130⋯0000-15015⋯000⋮⋮⋮⋮⋮⋮⋮⋮0000⋯-12m-3012m-30000⋯0-12m-10].Using (4.10) and (4.12), (4.5) takes on the form(4.14)AS⃗-CS⃗-WS⃗+ΛAH2S⃗=D[-2B1S⃗︸F(t)+2B2S⃗(0)︸F(0)βS⃗︸1+2B3S⃗′(0)︸F′(0)αS⃗︸t],
where(4.15)β1×m=[β0,β1,…,βm-1],α1×m=[α0,α1,…,αm-1]
are the coefficients of shifted Legendre polynomials resulting from the expansions of 1 and T, respectively [33].By lettingD¯=2D and using the orthogonality of S⃗(T), (4.14) becomes(4.16)A+ΛAH2=C+W-D¯B1+D¯B2S⃗(0)β+D¯B3S⃗′(0)α.
Now using the vec notation and kronecker product [34], (4.16) takes on the form(4.17)A⃗+(Λ⊗Γ)A⃗=C⃗+W⃗-(D¯⊗Im)B1⃗+(D¯⊗Ω1)B2⃗+(D¯⊗Ω2)B3⃗,
where A⃗,C⃗,W⃗,B1⃗,B2⃗, and B3⃗ are Nm×1 vectors, Γ,Ω1,Ω2, and Im are m×m matrices in which Γ=(H2)tr,Ω1=(S⃗(0)β)tr,Ω2=(S⃗′(0)α)tr, and Im is the m×m identity matrix, and D¯ and Λ are N×N matrices.Using the substitutions(4.18)X=INm+(Λ⊗Γ),U1=D¯⊗Im,U2=D¯⊗Ω1,U3=D¯⊗Ω2,Y⃗=C⃗+W⃗,
equation (4.17) becomes(4.19)XA⃗=Y⃗+U1B1⃗+U2B2⃗+U3B3⃗=Y⃗+KB1⃗,
where(4.20)K=U1+U2I1+U3I2
in which(4.21)I1=diag[(-1)i+1]i=1m,I2=diag[(-1)i(i-1)i]i=1m.Solving (4.19) for B1⃗, we obtain(4.22)B1⃗=MA⃗+N⃗,
where M=K-1X and N⃗=-K-1Y⃗.


## 4.3. Approximation of Modal Performance Index

The cost functional (4.3) can be transformed into modal space(4.23)JN(F⃗)=12∑n=1NZn⃗tr(Tf)R1Zn⃗(Tf)+12∑n=1NddTZn⃗tr(Tf)R2ddTZn⃗(Tf)+1N∫0TfF⃗tr(τ)R3F⃗(τ)dτ,
where Ri=diag [μi], i=1, 2, 3. Inserting (4.10) into (4.23) gives
(4.24)JN(F⃗)=12[A⃗trStr(Tf)R1S(Tf)A⃗+A⃗trddTStr(Tf)R2ddTS(Tf)A⃗]+1N∫0TfB⃗1trStr(τ)R3S(τ)B1⃗dτ,
or by using (4.22),(4.25)JN(A)=12[A⃗trP1(Tf)A⃗+A⃗trP2(Tf)A⃗]+1N∫0Tf(MA⃗+N⃗)trP3(τ)(MA⃗+N⃗)dτ,
where(4.26)P1(T)=Str(T)R1S(T),P2(T)=ddTStr(T)R2ddTS(T)P3(T)=Str(T)R3S(T).,
Expanding (MA⃗+N⃗)tr and letting P3*=∫0TfP3(t)dt, the performance index (4.25) takes on the form
(4.27)JN(A⃗)=1N[A⃗trMTP3*MA⃗+N⃗trP3*MA⃗+A⃗trMTP3*N⃗+N⃗trP3*N⃗]+12[A⃗tr(P1(Tf)+P2(Tf))A⃗].
To this end, the optimal control problem (3.8) is converted into the mathematical programming problem:(4.28)FindA*⃗∈ℝNsothatJN(A*⃗)=minA⃗∈ℝNJN(A⃗).


## 4.4. Optimal Control Characterization

LetP4=P1+P2 and P5=MtrP3*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 A⃗, we obtain
(4.29)∂JN(A⃗)∂A⃗=1N[(P5(Tf)+P5tr(Tf))A⃗+(N⃗trP3*M)tr+(MtrP3*N⃗)]+12(P4(Tf)+P4tr(Tf))A⃗.Let(4.30)P=12(P4(Tf)+P4tr(Tf))+1n(P5(Tf)+P5tr(Tf)),Q=-1N(MTP3*tr+MtrP3*).
Now we find A*⃗ so that ∂JN(A*⃗)/∂A⃗=0, we obtain(4.31)PA*⃗-QN⃗=0
or(4.32)A*⃗=P-1QN⃗
and from (4.22), we have(4.33)B*⃗=M(P-1QN⃗)+N⃗=(MP-1Q+I)N⃗.The optimal state variableW*(X,T) is obtained from (3.3), so we have(4.34)W*(X,T)=W¯*(X,T)+α(X)F*(T),
where(4.35)W¯*(X,T)=∑n=1N(∑j=0m-1aij*Sj(T))φn(X),F*(T)=∑j=0m-1bj*Sj(T)
in which aij* and bj* are the components of A*⃗ and B*⃗, respectively.


## 5. 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,Tf, subject to the initial impact conditions:(5.1)W(x,0)=sinλ1x+sinλ2x,WT(x,0)=0,
where λi=iπ,i=1,2.For the numerical simulations, we letN=2 and m=10(the first 10 shifted Legendre polynomials). The measure of the total force used in the control process is given by(5.2)Jc(F0)=∫0Tf(F0(τ))2dτ,
and the controlled energy of the system is defined as(5.3)Ec(F0)=∫01{μ1[W0(x,Tf)]2+μ2[WT0(x,Tf)]2}dx,
where W0(x,T) corresponds to the optimal displacement of the microbeam. The controlled and uncontrolled energies are denoted by Ec and Eu, respectively, and the force used is denoted by Jc.In Tables 1–3, we study the effect of each two consecutive weighting factors μi and μi+1 for i=1,3,and5. 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.Table 1
Effect ofμ1 and μ2.
μ1=μ2Eu(F=0)Ec(F0)Jc(F0)0.06250.0147870.0000060.0000010.12500.0295740.0000100.0648680.25000.0591490.0000190.6915370.50000.1182970.0000330.0060491.00000.2365940.0000430.0647522.00000.4731880.0000560.013062Table 2
Effect ofμ3 and μ4.
μ3=μ4Eu(F=0)Ec(F0)Jc(F0)0.06250.2365940.0000420.0105270.12500.2365940.0000420.0002510.25000.2365940.0000420.0406560.50000.2365940.0000460.2635301.00000.2365940.0000430.0647522.00000.2365940.0000440.128104Table 3
Effect ofμ5 and μ6.
μ5=μ6Eu(F=0)Ec(F0)Jc(F0)0.06250.2365940.0000030.0033650.12500.2365940.0000060.0054340.25000.2365940.0000150.0000760.50000.2365940.0000320.1593321.00000.2365940.0000430.0647522.00000.2365940.0000560.023356Table 4
Effect of terminal time(Tf).
TfEu(F=0)Ec(F0)Jc(F0)10.2365940.00000430.06475250.0291060.0020170.005514100.3854000.0000030.006740150.255480.0000010.004564The following observations are made.(1)
The system achieves substantial energy reduction when the weighting factorsμ1 and μ2 are smaller. This implies that the smaller μ1 and μ2, the more emphasis on kinetic energies; see Table 1.(2)
The system energies remain unchanged as the weighting factorsμ3 and μ4 assume different values; see Table 2.(3)
The weighting factorsμ5 and μ6 are effective in regulating the system energies, see Table 3.(4)
At any given terminal time, the system is damping the energy out to a desired value; see Table4.


## 6. 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 the unknown parameters. The optimal value of these parameters is obtained by using quadratic programming results. The numerical examples presented support the theoretical study and reveal the usefulness of the proposed approach.Figure 1
Microbeam with distributed actuator layers.



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*Source: 530287-2009-12-16.xml*