Open access peer-reviewed chapter

A Computational Approach to Vibration Control of Vehicle Engine-Body Systems

By Hamid Reza Karimi

Submitted: December 21st 2011Reviewed: May 30th 2012Published: September 5th 2012

DOI: 10.5772/50295

Downloaded: 1472

1. Introduction

In recent years, the noise and vibration of cars have become increasingly important [20, 23, 29, 30, 35]. A major comfort aspect is the transmission of engine-induced vibrations through powertrain mounts into the chassis (see Figure 1). Engine and powertrain mounts are usually designed according to criteria that incorporate a trade-off between the isolation of the engine from the chassis and the restriction of engine movements. The engine mount is an efficient passive means to isolate the car chassis structure from the engine vibration. However, the passive means for isolation is efficient only in the high frequency range. However the vibration disturbance generated by the engine occurs mainly in the low frequency range [8, 19, 23, 30]. These vibrations are result of the fuel explosion in the cylinder and the rotation of the different parts of the engine (see Figure 2). In order to attenuate the low frequency disturbances of the engine vibration while keeping the space and price constant, active vibration means are necessary.

A variety of control techniques, such as Proportional-Integral-Derivative (PID) or Lead-Lag compensation, Linear Quadratic Gaussian (LQG), H2, H, μ-synthesis and feedforward control have been used in active vibration systems [1, 3, 4, 10, 11, 15, 24, 26, 31, 32, 34, 35]. The main characteristic of feedforward control is that information about the disturbance source is available and is usually realised with the Filtered-X Least-Mean-Squares (Fx-LMS) algorithms. However, the disturbance source is assumed to be unknown in feedback control, then different strategies of feedback control for vibration attenuation of unknown disturbance exist ranging from classical methods to a more advanced methods. Recently, the performance result obtained by Hfeedback controller with the result obtained by feedforward controller using Fx-LMS algorithms for vehicle engine-body vibration system was compared in [30, 35].

On the other hand, wavelet theory is a relatively new and an emerging area in mathematical research [2]. It has been applied in a wide range of engineering disciplines such as signal processing, pattern recognition and computational graphics. Recently, some of the attempts are made in solving surface integral equations, improving the finite difference time domain method, solving linear differential equations and nonlinear partial differential equations and modelling nonlinear semiconductor devices [5, 6, 7, 13, 16, 17, 18, 21, 27].

Figure 1.

Front axis of AUDI A 8 from [22, 30] (Werkbild Audi AG).

Figure 2.

Chassis excited by the engine vibration.

Orthogonal functions like Haar wavelets (HWs) [13, 16], Walsh functions [7], block pulse functions [27], Laguerre polynomials [14], Legendre polynomials [5], Chebyshev functions [12] and Fourier series [28], often used to represent an arbitrary time functions, have received considerable attention in dealing with various problems of dynamic systems. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations for the solution of problems described by differential equations, such as analysis of linear time-invariant, time-varying systems, model reduction, optimal control and system identification. Thus, the solution, identification and optimisation procedure are either greatly reduced or much simplified accordingly. The available sets of orthogonal functions can be divided into three classes such as piecewise constant basis functions (PCBFs) like HWs, Walsh functions and block pulse functions; orthogonal polynomials like Laguerre, Legendre and Chebyshev as well as sine-cosine functions in Fourier series [21].

In the present paper, we, for the first time, introduce a computational solution to the finite-time robust optimal control problem of the vehicle engine-body vibration system based on HWs. To this aim, mathematical model of the engine-body vibration structure is presented such the actuators and sensors used to investigate the robust optimal control are selected to be collocated. Moreover, the properties of HWs, Haar wavelet integral operational matrix and Haar wavelet product operational matrix are given and are utilized to provide a systematic computational framework to find the approximated robust optimal trajectory and finite-time Hcontrol of the vehicle engine-body vibration system with respect to a Hperformance by solving only the linear algebraic equations instead of solving the differential equations. One of the main advantages is solving linear algebraic equations instead of solving nonlinear differential Riccati equation to optimize the control problem of the vehicle engine-body vibration system. We demonstrate the applicability of the technique by the simulation results.

The rest of this paper is organized as fallows. Section 2 introduces properties of the HWs. Mathematical model of the engine-body vibration structure is stated in Section 3. Algebraic solution of the engine-body system is given in Section 4 and Haar wavelet-based optimal trajectories and robust optimal control are presented in Sections 5 and 6, respectively. Simulation results of the robust optimal control of the vehicle engine-body vibration system are shown in Section 7 and finally the conclusion is discussed.

The notations used throughout the paper are fairly standard. The matricesIr, 0rand 0r×sare the identity matrix with dimension r×rand the zero matrices with dimensions r×randr×s, respectively. The symbol and tr(A)denote Kronecker product and trace of the matrixA, respectively. Also, operator vec(X)denotes the vector obtained by putting matrix Xinto one column. Finally, given a signalx(t), x(t)2denotes the L2norm ofx(t); i.e.,

x(t)22=0x(t)Tx(t)dtE1
.

2. Properties of Haar Wavelets

Properties of HWs, which will be used in the next sections, are introduced in this section.

2.1. Haar Wavelets (HWs)

The oldest and most basic of the wavelet systems is named Haar wavelet that is a group of square waves with magnitude of.±1. in the interval [0,1)[6]. In other words, the HWs are defined on the interval [0,1)as

ψ0(t)=1,t[0,1),ψ1(t)={1,fort[0,12),1,fort[12,1),E2

and ψi(t)=ψ1(2jtk)for i1and we write i=2j+kfor j0and0k<2j. We can easily see that the ψ0(t)and ψ1(t)are compactly supported, they give a local description, at different scalesj, of the considered function.

2.2. Function approximation

The finite series representation of any square integrable function y(t)in terms of an orthogonal basis in the interval[0,1), namelyy^(t), is given by

y^(t)=i=0m1aiψi(t):=aTΨm(t)E3

where a:=[a0a1am1]Tand Ψm(t):=[ψ0(t)ψ1(t)ψm1(t)]Tfor m=2jand the Haar coefficients aiare determined to minimize the mean integral square error ε=01(y(t)aTΨm(t))2dtand are given by

ai=2j01y(t)ψi(t)dtE4

Remark 1. The approximation error, Ξy(m):=y(t)y^(t), is depending on the resolution mand is approaching zero by increasing parameter of the resolution.

The matrix Hmcan be defined as

Hm=[Ψm(t0),Ψm(t1),,Ψm(tm1)]E5

where imti<i+1mand using (2), we get

[y^(t0)y^(t1)y^(tm1)]=aTHmE6
.

The integration of the vector Ψm(t)can be approximated by

0tΨm(t)dt=PmΨm(t)E7

where the matrix

Pm=<0tΨm(τ)dτ,Ψm(t)>=010tΨm(r)drΨmT(t)dtE8

represents the integral operator matrix for PCBFs on the interval [0,1)at the resolutionm. For HWs, the square matrix Pmsatisfies the following recursive formula [13]:

Pm=12m[2mPm2Hm2Hm210m2]E9

with P1=12and Hm1=1mHmTdiag(r)where the matrix Hmdefined in (4) and also the vector ris represented by

r:=(1,1,2,2,4,4,4,4,,(m2),(m2),,(m2)(m2)elements)TE10

form>2. For example, at resolution scalej=3, the matrices H8and P8are represented as

H8=[ψ0(t0)ψ1(t0)ψ2(t0)ψ3(t0)ψ4(t0)ψ5(t0)ψ6(t0)ψ7(t0)ψ0(t1)ψ1(t1)ψ2(t1)ψ3(t1)ψ4(t1)ψ5(t1)ψ6(t1)ψ7(t1)ψ0(t7)ψ1(t7)ψ2(t7)ψ3(t7)ψ4(t7)ψ5(t7)ψ6(t7)ψ7(t7)]=[1111111111111111111100000000111111000000001100000000110000000011],E11

and

P8=116[84H14H1102H24H210H4H410]=116[16P4H4H410]E12
=164[3216884444160884444440044004400004411200000112000001102000011020000],E13

for further information see [13, 25].

2.3. The product operational matrix

In the study of time-varying state-delayed systems, it is usually necessary to evaluate the product of two Haar function vectors [13]. Let us define

Rm(t):=Ψm(t)ΨmT(t)E14

where Rm(t)satisfies the following recursive formula

Rm(t)=12m[Rm2(t)Hm2diag(Ψb(t))(Hm2diag(Ψb(t)))Tdiag(Hm21Ψa(t))]E15

with

R1(t)=ψ0(t)ψ0T(t)E16

and

{Ψa(t):=[ψ0(t),ψ1(t),,ψm21(t)]T=Ψm2(t)Ψb(t):=[ψm2(t),ψm2+1(t),,ψm1(t)]T.E17

Moreover, the following relation is important for solving optimal control problem of time-varying state-delayed system:

Rm(t)am=a˜mΨm(t)E18

where a˜1=a0and

a˜m=[a˜m2Hm2diag(ab)diag(ab)Hm21diag(aaTHm2)]E19

with

{aa:=[a0,a1,,am21]T=am2(t)ab:=[am2(t),am2+1(t),,am1(t)]T.E20

Figure 3.

The sketch of engine-body vibration system

3. Mathematical model description

A schematic of the vehicle engine-body vibration structure is shown in Figure 3. The actuator and sensor used to this control framework are selected to be collocated, since this arrangement is ideal to ensure the stability of the closed loop system for a slightly damped structure [26]. In our study, only the bounce and pitch vibrations in the engine and body are considered [35]. The engine with mass Meand inertia moment Ieis mounted in the body by the engine mounts keandce. The front mount is the active mount, the output force of which can be controlled by an electric signal. The active mount consists of a main chamber where an oscillating mass (inertia mass) is moving up and down. The inertia mass is driven by an electro-magnetic force generated by a magnetic coil which is controlled by the input current.

The vehicle body with mass Mband inertia moment Ibis supported by front and rear tires, each of which is modeled as a system consisting of a spring kband a damping devicecb. Therefore, a four degree-of-freedom vibration suspension model shown in Figure 3 can be described by the following equations

{Mex¨1+2cex˙1+2kex12cex˙22kex22(Ll)cex˙42(Ll)kex4=f(t)+de(t)Mbx¨2+2(ce+cb)x˙2+2(ke+kb)x22cex˙12kex1+2(Ll)cex˙4+2(Ll)kex4=f(t)Iex¨3+2l2cex˙3+2l2kex32l2cex˙42l2kex4=lf(t)Ibx¨4+((L2+(L2l)2)ce+2L2cb)x˙4+((L2+(L2l)2)ke+2L2kb)x42l2cex˙32l2kex32lcex˙12lkex1+2(Ll)cex˙2+2(Ll)kex2=Lf(t)E21

where the states x1(t),x2(t),x3(t)and x4(t)are the bounces and pitches of the engine and body, respectively, where displacement of the chassis (x2(t))is usually taken as an output. Input force, f(t), is used as the active force to compensate the vibration transmitted to vehicle body. Moreover, engine disturbance de(t)can be the excitation, generated by the motion up/down of the different parts inside the engine;

The system Eq. (14) can be represented in the following state-space form

{Mx¨(t)+Cx˙(t)+Kx(t)=Bff(t)+Bdde(t),t[0,Tf]z(t)=[C1x(t)C2x˙(t)C3f(t)]E22

where x(t)4is the state; f(t)is the control input; de(t)is the disturbance input which belongs toL2[0,); and z(t)3is the controlled output withC11×4, C21×4and C3is a positive scalar. The state-space matrices are also defined as

M=[Me0000Mb0000Ie0000Ib],
C=[2ce2ce02(Ll)ce2ce2(ce+cb)02(Ll)ce002l2ce2l2ce2lce2(Ll)ce2l2ce0]E23
,Bf=[11lL],
Bd=[1000]E24
,
K=[2ke2ke02(Ll)ke2ke2(ke+kb)02(Ll)ke002l2ke2l2ke2lke2(Ll)ke2l2ke(L2+(L2l)2)ke+2L2kb]E25
.

Taking displacement of the chassis (x2(t))as an output then a comparison of the displacement response respect to the input force f(t)and the external disturbance de(t)in the frequency range up to 1 KHz is depicted in Figure 4a) and 4b). Three relevant modes occur around the frequencies 1, 5 and 9 Hz, respectively, which represent the dynamics of the main degrees of freedom (DOFs) of the system.

4. Algebraic solution of system equations

In this section, we study the problem of solving the second-order differential equations of the engine-body system (14) in terms of the input control and exogenous disturbance using HWs and develop appropriate algebraic equations.

Based on HWs definition on the interval time[0,1], we need to rescale the finite time interval [0,Tf]into [0,1]by consideringt=Τfσ; normalizing the system Eq. (15) with the time scale would be as follows

Mx¨(σ)+Cx˙(σ)+Kx(σ)=Bff(σ)+Bdde(σ)E26

Now by integrating the system above in an interval[0,σ], we obtain

M(x(σ)x(0))+ΤfC0σx(τ)dτ+Τf2K0σ0ξx(τ)dτdξ=Τf2Bf0σ0ξf(τ)dτdξ+Τf2Bd0σ0ξde(τ)dτdξ+0σ(Mx˙(0)+ΤfCx(0))dξ.E27

By using the Haar wavelet expansion (2), we express the solution of Eq. (15), input force f(σ)and engine disturbance de(σ)in terms of HWs in the forms

x(σ)=XΨm(σ)E28
,
f(σ)=FΨm(σ)E29
,
de(σ)=DeΨm(σ)E30
,

whereX4×m,F1×mand De1×mdenote the wavelet coefficients ofx(σ), f(σ)andde(σ), respectively. The initial conditions of x(0)and x˙(0)are also represented by x(0)=X0Ψm(σ)andx˙(0)=X¯0Ψm(σ), where the matrices {X0,X¯0}4×mare defined, respectively, as

X0:=[x(0)04×104×1(m1)]E31
X¯0:=[x˙(0)04×104×1(m1)]E32

Therefore, using the wavelet expansions (18)-(20), the relation (17) becomes

M(XX0)+ΤfCXPm+Τf2KXPm2=Τf2BfFPm2+Τf2BdDePm2+(MX¯0+ΤfCX0)PmE33

For calculating the matrixX, we apply the operator vec(.)to Eq. (23) and according to the property of the Kronecker product, i.e.vec(ABC)=(CTA)vec(B), we have:

(ImM)(vec(X)vec(X0))+Τf(PmTC)vec(X)+Τf2(Pm2TK)vec(X)=Τf2(Pm2TBf)vec(F)+Τf2(Pm2TBd)vec(De)+Τf(PmTC)vec(X0)+(PmTM)vec(X¯0).E34

Solving Eq. (24) for vec(X)leads to

vec(X)=Δ1vec(F)+Δ2vec(De)+Δ3vec(X0)+Δ4vec(X¯0)E35
(25)

where the matrices {Δ1,Δ2}4m×mand{Δ3,Δ4}4m×4mare defined as

{Δ1=Τf2(Τf(PmTC)+Τf2(Pm2TK)+ImM)1(Pm2TBf)Δ2=Τf2(Τf(PmTC)+Τf2(Pm2TK)+ImM)1(Pm2TBd)Δ3=(Τf(PmTC)+Τf2(Pm2TK)+ImM)1(ImM+ΤfPmTC)Δ4=(Τf(PmTC)+Τf2(Pm2TK)+ImM)1(PmTM).E36

Consequently, using (25) and (26) and the properties of the Kronecker product, the solution of system (15) is

x(σ)=(ΨmT(σ)I4)vec(X)E37

and it is also clear that to find the approximated solution of the system, we have to calculate the inverse of the matrix Τf(PmTC)+Τf2(Pm2TK)+ImMwith dimension 4m×4monly once.

Figure 4.

Displacement of the chassis respect to f(t) (a) and de(t) (b).

5. Optimal control design

The control objective is to find the optimal control f(t)with respect to a quadratic cost functional approximately such acts as the active force to compensate the vibration transmitted to vehicle body. The quadratic cost functional weights the states and their derivatives with respect to time in the cost function as follows:

J=12xT(Τf)S1x(Τf)+12x˙T(Τf)S2x˙(Τf)+120Τf(xT(t)Q1x(t)+x˙T(t)Q2x˙(t)+Rf(t)2)dtE38

whereS1:4×4, S2:4×4, Q1:4×4and Q2:4×4are positive-definite matrices and Ris a positive scalar. We can rewrite the cost function (28) as follows:

J=12[xT(1)Τf1x˙T(1)]S˜[x(1)Τf1x˙(1)]+Τf201([xT(σ)Τf1x˙T(σ)]Q˜[x(σ)Τf1x˙(σ)]+Rf(σ)2)dσE39
.

where S˜=diag(S1,S2)and Q˜=diag(Q1,Q2)with the time scalet=Τfσ.

From (15) and the relationx˙(σ)=X¯Ψm(σ), where X¯:4×mdenotes the wavelet coefficients of x˙(σ)after its expansion in terms of HFs, we read

[x(σ)Τf1x˙(σ)]=[XΤf1X¯]Ψm(σ):=XaugΨm(σ)E40

where Xaug=[XΤf1X¯]and

vec(Xaug)=[vecT(X)Τf1vecT(X¯)]TE41

Remark 2. By substituting x˙(σ)=X¯Ψm(σ)intox(σ)x(0)=0σx˙(t)dt, we have:

XΨm(σ)X0Ψm(σ)=0σX¯Ψm(τ)dτE42
,

and using (4), we readXX0=X¯Pm. Then, by applying the operator of vec(.)and according to the properties of Kronecker product in Appendix A1, we obtain

vec(X)vec(X0)=(PmTIn)vec(X¯)E43

By substituting the definition (31) in (33) and using the properties of the operator tr(.)in Appendix A1, the cost function (28) is given by

J=12(vecT(Xaug)Πm1vec(Xaug)+vecT(F)Πm2vec(F))E44

where the matrices Πm1:8m×8mand Πm2:m×mare defined as

Πm1=MfTS˜+Τf(MTQ˜)andΠm2=RΤfMm, respectively,

and the matrices Mm:m×mand Mmf:m×mare defined as

Mm:=01Ψm(σ)ΨmT(σ)dσandMmf:=Ψm(1)ΨmT(1), respectively.

It is clear that the cost function of J(.)is a function ofimσi<i+1m, then for finding the optimal control law, which minimizes the cost functionalJ(.), the following necessary condition should be satisfied

Jvec(F)=0E45

By consideringvec(Xaug), which is a function ofvec(F), and using the properties of derivatives of inner product of Kronecker product in Appendix A2, we find

Jvec(F)=[Δ1TΤf1Δ1T(Pm1I4)]Πm1vec(Xaug)+Πm2vec(F)E46

Then the wavelet coefficients of the optimal control law will be in vector form as

vec(F)=Πm21[Δ1TΤf1Δ1T(Pm1I4)]Πm1vec(Xaug)E47

Consequently, the optimal vectors of vec(X)and vec(F)are found, respectively, in the following forms

vec(X)=(I4m+Δ1(Πm21[Δ1TΤf1Δ1T(Pm1I4)]Πm1[I4mΤf1(PmTI4)1])1(Δ2vec(De)+(Δ1Πm21×[Δ1TΤf1Δ1T(Pm1I4)]Πm1[04mΤf1(PmTI4)1]+Δ3)vec(X0)+Δ4vec(X¯0)),E48

and

vec(F)=Πm21[Δ1TΤf1Δ1T(Pm1I4)]Πm1{[I4mΤf1(PmTI4)1]×(I4m+Δ1Πm21[Δ1TΤf1Δ1T(Pm1I4)]Πm1[I4mΤf1(PmTI4)1])1E49
×(Δ2vec(De)+(Δ1Πm21[Δ1TΤf1Δ1T(Pm1I4)]Πm1[04mΤf1(PmTI4)1]+Δ3)vec(X0)+Δ4vec(X¯0))[04mΤf1(PmTI4)1]vec(X0)}.E50

Finally, the Haar function-based optimal trajectories and optimal control are obtained approximately from Eq. (27) andf(t)=ΨmT(t)vec(F).

6. Robust optimal control design

In this section, an optimal state feedback controller is to be determined computationally such that the following requirements are satisfied:

  1. the closed-loop system is asymptotically stable;

  2. under zero initial condition, the closed-loop system satisfies z(t)2<γde(t)2for any non-zero de(t)[0,)where γ>0is a prescribed scalar.

The control objective is to find the approximated robust optimal control f(t)with Hperformance such f(t)acts as the active force to compensate the vibration transmitted to vehicle body, i.e. guarantees desired L2gain performance. Next, we shall establish the Hperformance of the system (15) under zero initial condition. To this end, we introduce

J=12xT(Τf)S1x(Τf)+12x˙T(Τf)S2x˙(Τf)+120Tf(zT(t)z(t)γ2de2(t))dt.E51

It is well known that a sufficient condition for achieving robust disturbance attenuation is that the inequality J<0for every de(t)L2[0,)[33, 36]. Therefore, we will establish conditions under which

Infvec(F)Supvec(De)J(vec(F),vec(De))0E52

From (15), the Eq. (40) can be represented as

J=12(xT(1)1Τfx˙T(1))S˜(x(1)Τf1x˙(1))+Τf201((xT(σ)Τf1x˙T(σ))C˜(x(σ)Τf1x˙(σ))+C32f2(σ)γ2de2(σ))dσE53

wheret=Τfσ, S˜=diag(S1,S2)andC˜=diag(C1TC1,C2TC2).

Using the relationx˙(σ)=X¯Ψm(σ), we read

[x(σ)Τf1x˙(σ)]=[XΤf1X¯]Ψm(σ):=XaugΨm(σ)E54

where Xaug=[XΤf1X¯]and

vec(Xaug)=[vecT(X)Τf1vecT(X¯)]TE55

Moreover, according to Remark 2 in [18], the following relation is already satisfied between vec(X)and vec(X¯)

vec(X)vec(X0)=(PmTI4)vec(X¯)E56

By using the definition (44) in Eq. (45), we have

J=12(tr(MmfXaugTS˜Xaug))+Τf2(tr(MmXaugTC˜Xaug)+tr(C32MmFTF)γ2tr(MmDeTDe))E57

Using the property of the Kronecker product, i.e.tr(ABC)=vecT(AT)(IpB)vec(C), (AC)(DB)=ADCBandvec(ABC)=(CTA)vec(B), we can write (42) as

J=12(vecT(Xaug)Πm1vec(Xaug)+C32vecT(F)Πm2vec(F)γ2vecT(De)Πm2vec(De))E58

where the matricesΠm18m×8m, Πm2m×mare defined as Πm1=MmfS˜+Τf2(MmC˜)andΠm2=Τf2Mm, respectively.

It is easy to show that the worst-case disturbance in Eq. (47) occurs when

vec(De)=γ2Πm21[Δ2TΤf1Δ2T(Pm1I4)]Πm1vec(Xaug):=γ2Πmdvec(Xaug)E59

By substituting Eq. (48) into Eq. (47) we obtain

Infvec(F)Supvec(De)J(vec(F),vec(De))=Infvec(F)J(vec(F),vec(De))E60

Minimizing the right-hand side of Eq. (49) results in the algebraic relation between wavelet coefficients of the robust optimal control and of the optimal state trajectories in the following closed form

vec(F)=C32Πm21[Δ1TΤf1Δ1T(Pm1I4)](Πm1γ2ΠmdTΠm2Πmd)vec(Xaug):=Πmfvec(Xaug).E61

As a result we have

Infvec(F)Supvec(De)J(vec(F),vec(De))vecT(Xaug)(Πm1+RΠmfTΠm2Πmfγ2ΠmdTΠm2Πmd)vec(Xaug)E62

Consequently, if there exists positive scalar γto the matrix inequality

Πm1+C32ΠmfTΠm2Πmfγ2ΠmdTΠm2Πmd0E63

then inequality (41) is concluded.

From the relations above we obtain the robust optimal vectors of vec(X)and vec(F)after some matrix calculations, respectively, in the following forms

vec(X)=(I4m(Δ1Πmf+γ2Δ2Πmd)[I4mΤf1(PmTI4)1])1((Δ3(Δ1Πmf+γ2Δ2Πmd)×[04mΤf1(PmTI4)1])vec(X0)+Δ4vec(X¯0)),E64

and

vec(F)=Πmf{([I4mΤf1(PmTI4)1]((I4m(Δ1Πmf+γ2Δ2Πmd)[I4mΤf1(PmTI4)1])1×(Δ3(Δ1Πmf+γ2Δ2Πmd)[04mΤf1(PmTI4)1])[04mΤf1(PmTI4)1])vec(X0)+[I4mΤf1(PmTI4)1](I4m(Δ1Πmf+γ2Δ2Πmd)[I4mΤf1(PmTI4)1])1Δ4vec(X¯0))}E65

Finally, the Haar wavelet-based robust optimal trajectories and robust optimal control are obtained approximately from Eq. (27) andf(t)=ΨmT(t)vec(F), respectively.

7. Numerical results

In this section the proposed computational methodology is applied to the vehicle engine-body vibration system (15) such the exogenous disturbance de(t)is assumed to be a Sin(.)function at the frequency of10Hz. The system parameters, used for the design and simulation are given in Tables 1 and 2 in the Appendix B. Table 3 in the Appendix gives the pole-zero locations of 8th –order model of the vehicle engine-body vibration system. It is clear that the vehicle engine-body vibration system is unstable and has the nonminimum phase property. The objective is to find the approximated robust optimal displacement of the chassis and robust optimal input force with Hperformance using HWs at the finite time interval[0,1]. Moreover, the matrices {S1,S1}4×4and the vectorsC1, C2and the scalar C3in the controlled output z(t)in Eq. (15) are chosen asS1=S2=04,C1=[0,1,1,2], C2=[3,1,0,1]andC3=1.

Figure 5.

Comparison of displacement of the chassis found by HWs at resolution level j=5 (solid) and by analytical solution (dashed).

To compare the approximate solutionsx2(t)andf(t), found by HWs, to the analytical solution found by Theorem 1 in the Appendix C, we choose the performance bound and the resolution level equal to3.15and5, respectively, i.e. γ=3.15andj=5. The time curves found are plotted in Figures 5 and 6. It is clear that the effect of the engine disturbance is attenuated onto the displacement of the chassis as the output as well. In other words, f(t)compensates the vibration transmitted to the chassis. Compare the Haar wavelet based solutions to the continuous solutions using the differential Riccati equation, the approximate solutions (53) and (54) deliver both, robust control f(t)and state trajectory x(t)in one step by solving linear algebraic equations instead of solving nonlinear differential Riccati equation, while accuracy can easily be improved by increasing the resolution levelj.

Figure 6.

Comparison of input force found by HWs at resolution level j=5 (solid) and by analytical solution (dashed).

8. Conclusion

This paper presented the modelling of engine-body vibration structure to control of bounce and pitch vibrations using HWs. To this aim, the Haar wavelet-based optimal control for vibration reduction of the engine-body system was developed computationally. The Haar wavelet properties were introduced and utilized to find the approximate solutions of optimal trajectories and robust optimal control by solving only algebraic equations instead of solving the Riccati differential equation. Numerical results were presented to illustrate the advantage of the approach.

Appendix

Appendix A

A1. Some properties of Kronecker product

LetA:p×q, B:q×r,C:r×sand D:q×tbe fixed matrices, then we have:

vec(ABC)=(CTA)vec(B),tr(ABC)=vecT(AT)(IpB)vec(C),tr(ABC)=vecT(AT)(IpB)vec(C),(AC)(DB)=ADCB.E66

A2. Derivatives of inner products of Kronecker product

Let A:n×nbe fixed constants and x:n×1be a vector of variables. Then, the following results can be established:

(Ax)x=vec(A),(Ax)xT=A,(xTAx)x=Ax+ATxE67

A3. Chain rule for matrix derivatives using Kronecker product

Let Ζbe a p×qmatrix whose entries are a matrix function of the elements ofY:s×t, where Yis a function of matrixΧ:m×n. That is, Ζ=Η1(Y), whereY=Η2(X). The matrix of derivatives of Ζwith respect to Χis given by

ΖΧ={vecT(Y)ΧIp}{InΖvec(Y)}E68
.

Appendix B

ParametersValues
Mb1000 [kg]
Ib810 [kg m2]
kb20000 [N/m]
cb300 [N/m/s]
Lb2.5 [m]

Table 1.

The vehicle body parameter.

ParametersValues
Me250 [kg]
Ie8.10 [kg m2]
ke200000 [N/m]
ce200 [N/m/s]
Le0.5 [m]

Table 2.

The engine parameters.

ZerosPoles
-6.23±i111.69-6.2313±i111.62
-0.97±i62.59-1.09±i58.17
0.03±i26.860.14±i29.48
-1.10-0.29±i6.19

Table 3.

Pole-zero locations of the 8th -order model.

Appendix C

Theorem 1 (State Feedback) [9]. Consider dynamical system

{x˙(t)=Ax(t)+B1u(t)+B2w(t)z(t)=Cx(t)+Du(t)E69

under assumption (A,B1,C)is stabilizable. For a givenγ>0, the differential Riccati equation

X˙=ATX+XA+X(γ2B2B2TB1B1T)X+CTCE70

has a positive semi-definite solution X(t)such that A(B1B1Tγ2B2B2T)X(t)is asymptotically stable. Then the control law u(t)=B1TX(t)x(t):=K(t)X(t)is stabilizing and satisfiesz(t)2<γw(t)2.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Hamid Reza Karimi (September 5th 2012). A Computational Approach to Vibration Control of Vehicle Engine-Body Systems, Advances on Analysis and Control of Vibrations - Theory and Applications, Mauricio Zapateiro de la Hoz and Francesc Pozo, IntechOpen, DOI: 10.5772/50295. Available from:

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