Intelligent semi-active vibration control of eleven degrees of freedom suspension system using magnetorheological dampers

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1 Journal of Mechanical Science and echnology 26 (2) (2012) 323~334 DOI /s Intelligent semi-active vibration control of eleven degrees of freedom suspension system using magnetorheological dampers Seiyed Hamid Zareh *, Atabak Sarrafan, Amir Ali Akbar Khayyat and Abolghassem Zabihollah School of Science and Engineering, Sharif University of echnology, Iran (Manuscript Received February 1, 2011; Revised August 8, 2011; Accepted September 13, 2011) Abstract A novel intelligent semi-active control system for an eleven degrees of freedom passenger car s suspension system using magnetorheological (MR) damper with neuro-fuzzy (NF) control strategy to enhance desired suspension performance is proposed. In comparison with earlier studies, an improvement in problem modeling is made. he proposed method consists of two parts: a fuzzy control strategy to establish an efficient controller to improve ride comfort and road handling (RCH) and an inverse mapping model to estimate the force needed for a semi-active damper. he fuzzy logic rules are extracted based on Sugeno inference engine. he inverse mapping model is based on an artificial neural network and incorporated into the fuzzy controller to enhance RCH. o verify the performance of the NF controller (NFC), comparisons with existing semi-active techniques are made. he typical control strategy are linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controllers with clipped optimal control algorithm, while inherent time-delay and non-linear properties of MR damper lie in these strategies. Simulation results demonstrated that the NFC has better control performance and less control effort than the optimal in improving the service life of the suspension system and the ride comfort of a car. Keywords: Clipped optimal control algorithm; Full car model; Linear quadratic Gaussian; MR damper; Neuro-fuzzy strategy; Suspension system Introduction Suspension systems have long been of great concern for car industries. It performs multiple tasks such as maintaining contact between vehicle tires and the road, addressing the stability of the vehicle, and isolating the frame of the vehicle from road-induced vibration and shocks. In general, ride comfort, road handling, and stability are the most important factors in evaluating suspension performance. In present work, passenger car suspension system is modified to reduce the amplitude of the car vibration caused by applied road profile. In the passive suspension system, the stiffness and damping parameters are fixed and effective over a certain range of frequencies. o overcome this problem, the use of semi-active suspension systems which have the capability of adapting to changing road conditions by the use of an actuator has been considered; therefore an MR damper is added to the usual suspension systems while the other parts of suspension system are intact. he significance of MR damper is that its viscosity changes as the magnetic field is changed. A schematic model his paper was recommended for publication in revised form by Associate Editor Hyoun Jin Kim * Corresponding author. el.: , Fax.: address: zareh_hamid@alum.sharif.edu KSME & Springer 2012 Fig. 1. A schematic model of MR damper. of MR damper is shown in Fig. 1. Considered suspension model is controlled by applied Linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG). By using employed controller results, the amount of viscosity of MR damper can be calculated incorporated with clipped optimal strategy. Unfortunately, due to the inherent nonlinear nature of the MR damper to generate a force, a model like that for its inverse dynamics is difficult to obtain mathematically. Because of this reason, a neural network with fuzzy logic controller is constructed to copy the inverse dynamics of the MR damper. Neuro-fuzzy controller is an artificial neural network, which is used to aggregate rules and provides control result for the

2 324 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~334 Fig. 2. A full-car model with 11-DOFs. designed fuzzy logic controller. Application of fuzzy inference systems as a fuzzy logic controller (FLC) has gradually been recognized as the most significant and fruitful application for fuzzy logic and fuzzy set theory. Finally, the results of applied control strategies are compared. here are three models for modeling passenger car s suspension systems: quarter-car, half-car and full-car model. In quarter-car model, it is assumed that each of four-wheel has independent suspension to simulate the actions of an active vehicle suspension system, and therefore, quarter-car models are using for many simulations of suspension system. he model of quarter-car for vehicle suspension system has been used by Wang et al. [1]. Narayanan et al. [2] applied half-car model for simulating semi-active suspension system. hey modeled MR damper parameters by the modified Bouc Wen model and determined them to fit the hysteretic behavior. Vibration control of passenger car utilizing half-car was discussed by Yahaya et al. [3]. o improve the accuracy of car model Zabihollah et al. [4] modeled the vehicle suspension system by a full car model; in which the accuracy of model improved compared to quarter-car and half-car models. Semi-active and active control methods have been developed using different actuators such as electrorhrological (ER) and MR dampers. Salem et al. [5] controlled an active quartercar suspension system by fuzzy logic controller. Hyun et al. [6] utilized an adaptive LQG control for semi-active suspension system for quarter-car model. Semi-active control is used by Chen et al. [7] where the MR damper utilized as actuator of suspension system. Jialin et al. [8] presented and designed a full-state LQR controller for a half-car suspension system composed of actuators in parallel with conventional spring-damper passive suspension. Yang et al. [9] modeled and controlled an intelligent active suspension system using fuzzy controller by adaptive filter method. Zhou and Sun [10] have done a semi-active vibration control on five degrees of freedom suspension system using adaptive fuzzy PID control. hey combined the advantages of PID and fuzzy controllers and applied them to the vibration control of engineering vehicle. he parameters of PID tuned on line by fuzzy controller. Golnaraghi et al. [11] controlled a semi-active quarter car suspension system intelligently. hey utilized an inverse mapping model to estimate the current based on an artificial Neural Network and incorporated into the fuzzy logic controller. Sadati et al. [12] designed a neuro-fuzzy controller for a vehicle suspension system. hey controlled a half car model with four degrees of freedom using feedback error learning. he previous studies made full use of the advantages of the neural-network and the fuzzy logic controller and solved the different problems in suspension systems. Few researches involved combination of the two techniques to solve the timedelay and the inherent nonlinear nature of the MR dampers in semi-active strategy for full car model with high degrees of freedom. In this paper, four MR dampers are added in a suspension system between body and wheels parallel with passive dampers. For the intelligent system, fuzzy controller which inputs are relative velocities across MR dampers that are excited by road profile for predicting the force of MR damper to receive a desired passenger s displacement and velocity is applied. When predicting the displacement and velocity of MR dampers, a four-layer feed forward neural network, trained on-line under the Levenberg Marquardt (LM) algorithm, is adopted. In order to verify the effectiveness of the proposed neuro-fuzzy control strategy, the uncontrolled system and the clipped optimal controlled suspension system are compared with the neuro-fuzzy controlled system. hrough a numerical example under actual road profile excitation, it can be concluded that the control strategy is very important for semi-active control, the neuro-fuzzy control strategy can determine currents of the MR damper quickly and accurately, and the control effect of the neuro-fuzzy control strategy is better than that of the other control strategy. 2. Full car model In the full-car model, 11-DOFs are assumed; all wheels and passengers are dependent on each other and on the car s body. It is assumed that each wheel has an effect on the spring and damper of other wheels, and two axes of vehicle are relevant. MR actuator is utilized to damp the effect of road profile on the passengers. Note that MR shock absorber is added to the axel and car body. In a full-car model, the effect of body rotations around roll and yaw axis is simulated. he suspension system using a full-car model has 11-DOFs, four of them for the four wheels, three for body displacement and its rotations and the last four for passengers. A schematic of a full-car model with 11-DOFs and addition MR damper is shown in Fig. 2. he dynamic equations of each DOF are given as in Eq. (1)- (11). As a result, the state space form of the equation is shown in Eq. (12). he state space form and the corresponding matrices are observed in Eq. (13)-(19). E is a location matrix of actuators. Matrices K, S and are defined as stiffness, damp-

3 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~ ing coefficient and input matrix due to wheels stiffness, respectively. mx 11 && + ( k1+ kt1) x1 kx 15 kr 111ϕ kr 112θ + bx& bx& br & ϕ br & θ k x = t1i1 (1) br 8 82) x& 5 br 5 52x& 6+ br 6 62x& 7 br 7 72x& 8+ br 8 82x& 9 + ( br 1 12r11 b2r22 r21 br 3 32r31 + b4r42r41 + br 5 52r b66261 r r b77271 r r + b88281 r r )& ϕ + ( br 112+ b222 r br br br br br br 8 82) & θ = 0 mx 2&& 2 + ( k2 + kt2) x2 kx 2 5 kr 2 21ϕ + kr 2 22θ + bx& bx& br & ϕ + br & θ k x = t2 i2 (2) mx 5&& 6 kx 5 5+ kx 5 6 kr 5 51ϕ kr 5 52θ bx& bx& br & ϕ br & θ = (8) mx 3&& 3+ ( k3+ kt3) x3 kx 3 5+ kr 3 31ϕ kr 3 32θ + bx& bx& + br & ϕ br & θ k x = t3 i3 (3) mx 6&& 7 kx 6 5+ kx 6 7 kr 6 61ϕ + kr 6 62θ bx& + bx& br & ϕ + br & θ = (9) mx 4&& 4 + ( k4 + kt4) x4 kx kr 4 41ϕ + kr 4 42θ + bx& bx& + br & ϕ + br & θ k x = t4 i3 (4) mx 7&& 8 kx 7 5+ kx 7 8+ kr 7 71ϕ kr 7 72θ bx& + bx& + br & ϕ br & θ = (10) M && x k x k x k x k x + ( k + k b k + k + k + k + k + k ) x k x k x kx kx + ( kr + kr kr kr + kr kr kr kr ) ϕ + ( kr kr + kr kr 442+ kr 552 kr 662+ kr 772 kr 882) θ bx 11 & bx 2& 2 bx 3& 3 bx 4& 4 + ( b1+ b2 + b3 + b4 + b5 + b + b + b ) x& b x& b x& b x& b x& ( br br br 3 31 br br br 6 61 br br )& ϕ + ( br br + br br + br br br br )& θ = I && ϕ k r x k r x + k r x + k r x ( kr + k r kr k r + kr + k r kr kr ) x kr x kr x + kr x ( ) ϕ ( kr x + kr + kr + kr + kr + kr + kr + kr + kr + kr r kr r kr r + kr r + kr r + kr r kr r + kr r ) θ br x& br x& + br x& + br x& ( br + br br br + br + br br br ) x& br x& br x& + br x& + br x& br 111 b2r21 br br br br ) & ϕ ( ( br + br + br r br r br r + br r + br r + br r br r + br r )& θ = I && θ k r x + k r x k r x + k r x + ( k r kr + kr kr + kr kr + kr kr ) x kr x + kr x kr x + kr x ( k r r k r r k r r + k r r + k r r ) ϕ ( 1r12 + k2r ) θ kr r kr r + kr r + k + kr + kr + kr + kr + kr + kr br x& + b r x& b r x& + b r x& + ( br br + br br + br br + br (5) (6) (7) mx && kx + kx + krϕ + kr θ = bx 8& 5 bx 8& 9 br 8 81& ϕ br & 8 82θ 0 x1 = x1, x2 = x2, x3 = x3, x4 = x4, x5 = x5, x6 = x6, x7 = x7, x8 = x8, x9 = x9, ϕ = x10, θ = x11, x& 1 = x12 x& 2 = x13, x& 3 = x14, x& 4 = x15, x& 5 = x16, x& 6 = x17, x& 7 = x18, x& 8 = x19, x& 9 = x20, & ϕ = x21, & θ = x22 x& = Ax + Bu + Gw y = Cx 0 I A = 1 1 M K M S 0 B = 1 M 0 G = 1 M E m m m m M M diag b L = m m m m I I t1 t2 t3 t = [ k,0,0,0;0, k,0,0;0,0, k,0;0,0,0, k ; zeros(7,4)] E = [ eye(4,4);1,1,1,1; zeros(4,4); r, r, r, r ; r, r, r, r ] (11) (12) (13) (14) (15) (16) (17) (18) (19) where M b, m 1, m 2, m 3, m 4, m 5, m 6, m 7 and m 8 stand for the mass of the car body, masses of four wheels and mass of passengers, respectively. I 1 and I 2 are the moments of inertia of the car body around two axes, respectively. he terms k 1, k 2, k 3, k 4, k 5, k 6, k 7 and k 8 are stiffnesses of the springs of the suspension

4 326 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~334 able 1. Numerical values of model. Symbol Quantity Value M b Mass of car body (kg) 670 I 1 Inertia around yaw (kg/m^2) 800 I 2 Inertia around roll (kg/m^2) 1100 m 1-4 Masses of Wheels (kg) 30 m 5-8 Masses of passengers (kg) 120 k 1-4 Stiffnesses of car springs (N/m) k 5-8 Stiffnesses of seats (N/m) 1750 b 1-4 Viscosity of car dampers (Ns/m) 1460 b 5-8 Viscosity of seats damper (Ns/m) 700 k t1-4 Stiffnesses of wheels (N/m) r 11 Length distance between the front left r 12 r 21 r 22 r 31 r 32 r 41 r 42 r 51 r 52 r 61 r 62 r 71 r 72 r 81 r 82 Width distance between the front left Length distance between the front right Width distance between the front right Length distance between the back right Width distance between the back right Length distance between the back left Width distance between the back left Length distance between the driver seat and center of mass (m) Width distance between the driver seat and center of mass (m) Length distance between the front right seat and center of mass (m) Width distance between the front right seat and center of mass (m) Length distance between the back left seat and center of mass (m) Width distance between the back left seat and center of mass (m) Length distance between the back right seat and center of mass (m) Width distance between the back right seat and center of mass (m) system and stiffnesses of the springs of passengers seat, respectively. he terms k t1, k t2, k t3 and k t4 are stiffnesses of the tires. he terms b 1, b 2, b 3, b 4, b 5, b 6, b 7 and b 8 are coefficients of car and passengers seat dampers, respectively. hen, b r1, b r2, b r3 and b r4 are coefficients of the MR dampers, respectively. x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9, φ and θ indicate the DOFs of the suspension system model, respectively. he terms x i1, x i2, x i3 and x i4 indicate load profile disturbance, respectively. he numerical values of model s dimensions for obtaining the responses are shown in able 1 [4]. 3. LQR controller strategy LQR controller responds to changes in the location of the poles of the system to the optimal place. ime response, overshoot and steady state errors depend on the location of the system s poles. LQR controller controls the system by a matrix gain Eq. (20). his gain is achieved from Eq. (21). o solve this energy equation of the system, Riccati equation is used, and this relation is given in Eq. (22), in which Q is a symmetric positive semi-definite matrix and R is a symmetric positive definite matrix. he result of solving Riccati equation is matrix S. Gain of pole placement is achieved from Eq. (23) by using S matrix. x& = ( A BK) x+ Gw y = Cx 0 (20) 1 ( ( ) ( ) J = x t Qx t + u ( t ) Ru ( t )) dt (21) 2 1 A S + SA SBR B S + Q = 0 (22) 1 K = R B S (23) Gain K by B matrix made a square matrix, subtracted from A matrix, and changes dynamic properties and the pole of the control system. he A matrix presents the dynamic properties of the system. New A matrix shows the system with new position pole. herefore by finding optimal gain, pole of the system is shifted to the optimal position. Here, Q 22*22 is an identity matrix (I 22*22 ) because all states are important and the best responses are obtained by this value. R 4*4 is 45*I 4*4, which is obtained by trial and error to receive the desired response. 4. LQG controller strategy In order to control a system with disturbance and noise by modern control method a controller with noise filter must be used. Optimal control, linear quadratic Gaussian, is the most appropriate to control the full car model. he LQG controller is simply the combination of a Kalman filter (i.e., a linearquadratic-estimator (LQE) with a linear-quadratic-regulator). o eliminate the effect of disturbance on the suspension system, the LQE part of the LQG controller is utilized. Perhaps, the main step to design the LQE is to identify the amplitude and position of disturbance. As a result, disturbance vector and position of disturbance entrance should be identified. he disturbance of the system is entered to the system as the input part. In the present work, it is assumed that some states of the system can be detected by sensors attached to the parts of the suspension system (seven states of twenty two states that are denoted as C lqg ). Consequently, the controller of the suspension system is not a full state control. he state space of the suspension system is controllable and observable. o design the LQG controller, first, the disturbance vector needs to be identified. hen, the matrices W and V are produced; the process noise (disturbance) is simulated by road

5 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~ profile that is explained in next subsection; the measurement noise is simulated by fraction of road profile. Kalman filter is a state estimator given a state-space model of the plant and the process and measurement noise covariance data. he Kalman estimator provides the optimal solution to the following continuous or discrete estimation problems. x& = Ax + Bu + Gw (24) y = C x+ v (25) lqg he observer structure with known inputs u, white process noise w, and white measurement noise v, is in the form of: xˆ & = Axˆ+ Bu + L( y C xˆ) lqg (26) where ˆx is an LQG optimal estimate of x. In the next step, the energy equation of the LQE must be solved, as expressed in Eq. (27). In order to solve the energy equation, Riccati equation is produced. Riccati equation is given in Eq. (28). Solving Riccati equation, the P matrix will be achieved. he P matrix is utilized to find the observer gain, L, in Eq. (29). J = [ xvx+ uwudt ] (27) 0 lqg 1 lqg AP + PA PC V C P + W = 0 (28) 1 L = PC V (29) lqg where W = E[ ww ], V = E[ vv ] are the plant disturbance and measurement noise covariances. he input u itself is generated by a state feedback law as: u = Kxˆ. (30) Finally, using LQG controller in close-loop system, it leads the original system convert to following equations: x& A BK x xˆ & = LClqg A BK LC lqg xˆ G 0 w + 0 L v zeros(11 11) zeros(11 11) Clqg = zeros(11 11) C11 I(5 5) zeros(5 4) zeros(5 2) C11 = zeros(4 5) zeros(4 4) zeros(4 2). zeros(2 5) zeros(2 4) I (2 2) (31) (32) (33) Here, I is a identity matrix. he optimal force vector f c regulated only by the state vector x that is calculated by Eq. (34) [13]. f = K x (34) c LQR Fig. 3. Clipped optimal algorithm block diagram. 5. Clipped optimal algorithm he clipped optimal control strategy for an MR damper usually involves two steps. he first step is to assume an ideal actively controlled device and construct an optimal controller for this active device. In the second step, a secondary controller finally determines the input voltage of the MR damper. hat is, the secondary controller clips the optimal force in a manner consistent with the dissipative nature of the device. he block diagram of clipped optimal algorithm is shown in Fig. 3. he clipped optimal control approach is to append a force feedback loop to induce the MR damper to produce approximately a desired control force f c. he linear quadratic regulator algorithm has been employed both for active control and for semi-active control. Using this algorithm, the optimal control force f c for f, which is force generated by a MR damper, may be obtained by minimizing the following scalar performance index t f J = ( x Qx+ F RF) dt. (35) t0 Q and R are weighting matrices and their values are selected depending on the relative importance given to different terms in their contributions to the performance index J. Solving the optimal control problem with J defined by Eq. (35), results in a optimal force vector f c regulated only by the state vector x, such that 1 fc ( R B P) x Gx = = (36) where matrix G represents the gain matrix; and the matrix P is the solution of the classical Riccati equation given by Eq. (37) 1 PA + A P PBR B P + Q = 0. (37) he force generated by the MR damper cannot be commanded. When the MR damper is providing the desired optimal force (i.e., f = f c ), the voltage applied to the damper

6 328 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~334 able 2. Numerical values of MR damper model. α a (N cm -1 ) α b (N cm -1 V -1 ) γ (cm -2 ) β (cm -2 ) able 3. Numerical values of MR damper model. Fig. 4. Mechanical model of a shear mode type MR damper. c 0a (Nscm -1 ) c 0b (Nscm -1 V -1 ) n η (s -1 ) A m should remain at the present level. If the magnitude of the force produced by the damper is smaller than the magnitude of the desired optimal force and the two forces have the same sign, the voltage applied to the current driver [14] varies continuously in the range of [0-V max ]. he secondary controller for continuously varying the command voltage can be stated as { } vi = VciH( fci fi fi) (38) μ ifci, for fci fmax Vci = (39) Vmax, for fci > fmax where V max is 12v and f max is maximum force produced by the damper (=3,000 N); μ is coefficient relating the voltage to the force (= V max /f max ); H (.) is the heaviside step function expressed as 0 or 1 [14]; f i is the force produced by i th MR dampers which is applied to the structure. In this paper, a simple mechanical model consisting of a Bouc-Wen element in parallel with a viscous damper is used, as shown in Fig. 4. his model has been verified to accurately predict the behavior of a prototype shear-mode MR damper over a wide range of inputs in a set of experiments, and is also expected to be appropriate for modeling a full-scale MR damper. he equations governing the force f i exerted by this model are as follows: & (40) fi = cx 0 +α z n 1 n z& = γ x& z z& βx& z + A x& (41) where x is the displacement of the device and z is the evolutionary variable that accounts for the history dependence of the response. he parameters γ, β, n and A m adjusted to determine the linearity in the unloading and the smoothness of the transition from the pre-yield region to the post-yield region. Device model parameters α and c 0 are determined by the dependency on the control voltage u, as follows: m α = α( u) = αa + αbu (42) c0 = c0( u) = c0a + c0bu. (43) Moreover, to account for a time-lag in the response of the Fig. 5. Hysteretic behavior of an MR damper. device to the changes in the command input, the first-order filter dynamics are introduced to the system as follows: u& = η ( u υ) (44) where v is a command voltage applied to the control circuit and η is the time constant of the first-order filter. he numerical values of parameters are shown in ables 2 and 3 [23]. he hysteretic behavior of the MR damper model according to the input voltage is shown in Fig Road profile simulation he random road excitation is generated using white noise as a road irregularity as a disturbance. he power spectral density (PSD) function of road irregularity is assumed to be in the form of Eq. (45). S h 2 σ α V = π ( ω ( α ) ) r rv (45) where σ 2 is the variance of the road profile, ω is the excitation frequency of the road input V is the vehicle forward constant velocity and α r is a coefficient depending on the type of road surface. he typical properties of unpaved road profile are shown in able 4 [2]. he PSD of the road surface is obtained by using MALAB.

7 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~ able 4. ypical properties of unpaved road profile. σ 2 (m 2 ) ω (rad/s ) V (m/s) α r (rad/m) 3e-4 200π Neuro fuzzy strategy Unfortunately, due to the inherent nonlinear nature of the MR damper to generate a force, a model like that for its inverse dynamics is difficult to obtain mathematically. Because of this reason, a neural network with fuzzy logic controller is constructed to copy the inverse dynamics of the MR damper. Unlike conventional controllers, such controllers do not require mathematical model and they can easily deal with the nonlinearities and uncertainties of the controlled systems. Also, a Levenberg-Marquardt neural controller has been designed for variable geometry suspension systems with MR actuators. In the present research, an optimal controller (LQR) is designed for the control of a semi-active suspension system for a full-model vehicle, using a neuro-fuzzy along with Levenberg-Marquardt learning. he purpose in a vehicle suspension system is reduction of transmittance of vibrational effects from the road to the vehicle s passengers, hence providing ride comfort. o accomplish this, one can first design a LQR controller for the suspension system, using an optimal control method and use it to train a neuro-fuzzy controller. his controller can be trained using the LQR controller output error on an online manner. Once trained, the LQR controller is automatically removed from the control loop and the neuro-fuzzy controller takes on. In case of a change in the parameters of the system under control, the LQR controller enters the control loop again and the neural network gets trained again for the new condition [14]. An important characteristic of the proposed controller is that no mathematical model is needed for the system components, such as the non-linear actuator, spring, or shock absorbers. he basic idea of the proposed neuro-fuzzy control strategy is that the force of the MR dampers is determined by a fuzzy controller, whose inputs are the measured displacement and velocity response provided by a neural network. he architecture of this strategy is shown in Fig. 6, which consists of four parts to perform different tasks. he first part is the neural network to be trained on-line. he numbers of the sample data pairs are 3500, the training data pairs increase step by step during the entrance disturbance from road profile, predicted voltage by clipped method. he neural network is trained to generate the one step ahead prediction of the displacement x k+1 and the velocity x k+1. Inputs to this network are the delayed outputs (x k+3, x k+2, x k+1, x k, x k+3, x k+2, x k+1, x k), the delayed force which is predicted by fuzzy controller (f k+1 ), and the disturbance input (d k ). At the initial time, the inputs of the network will be taken to have the value of zero in accordance with the actual initial circumstance. Before online training, the network trained off-line to achieve update weights near to desired. Fig. 6. Architecture of the Neuro-Fuzzy control strategy. he second part is the fuzzy controller, whose inputs are the measured displacement and velocity across MR dampers. he disturbance can be calculated by road profile model. he output of the fuzzy controller is control force of the MR dampers. he main aim of this part is to determine control force of the MR dampers quickly in accordance with the input excitation. How to design the fuzzy controller will be explained in the following section. In order to reach this aim, it is required to predict the responses of passengers in accordance with the optimal responses. At the same time, the actual responses will feed back to the neural network and the weights and bias will be revised real time. In this research work, the calculated results by optimal control history analysis method are used to simulate the actual measured responses. he errors between the predicted responses and the actual responses are used to update the weights of the neural network on-line. 7.1 he neural network based on Levenberg-Marquardt (LM) algorithm he MR damper model discussed earlier in this research estimates damper forces based on the inputs of the reactive velocity. In such case, it is essential to develop an inverse dynamic model that predicts the corresponding control force to be sent from dampers so that an appropriate damper force can be generated [16]. Neural network is a simplified model of the biological structure found in human brains. his model consists of elementary processing units (also called neurons). It is the large amount of interconnections between these neurons and their capabilities to learn from data to enable neural network as a strong predicting and classification tool. In this study, hreelayer feed forward neural network, which consists of an input layer, one hidden layer, and an output layer is selected to predict the responses with MR dampers. he net input value net k of the neuron k in some layer and the output value O k of the same neuron can be calculated by the following Eqs. (46) and (47): netk w jko j = (46) O = f( net + θ ) (47) k k k where w jk is the weight between the j th neuron in the previous

8 330 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~334 layer and the k th neuron in the current layer, O j is the output of the j th neuron in the previous layer, f(.) is the neuron s activation function which can be a linear function, a radial basis function, and a sigmoid function, and y k is the bias of the k th neuron. Feed forward neural network often has one or more hidden layers of sigmoid neurons followed by an output layer of linear neurons. Multiple layers of neurons with nonlinear transfer functions allow the network to learn nonlinear and linear relationships between input and output vectors. In the neural network architecture as shown in Fig. 6, the logarithmic sigmoid transfer function is chosen as the activation function of the hidden layer, Eq. (48): 1 Ok = f( netk + θk) =. (48) ( net ) 1 + e k+ θk he linear transfer function is chosen as the activation function of the output layer, Eq. (49): Fig. 7. Membership function of front-left damper velocity (m/s). Fig. 8. Membership function of front-right damper velocity (m/s). Ok = f( netk + θk) = netk + θk. (49) We note that neural network needs to be trained before predicting responses. As the inputs are applied to the neural network, the network outputs (.) ˆ are compared with the targets (.). he difference or error between both is processed back through the network to update the weights and biases of the neural network so that the network outputs match closer with the targets. he input and output data are usually represented by vectors called training pairs. he process as mentioned above is repeated for all the training pairs in the data set, until the network error converged to a threshold minimum defined by a corresponding performance function. In this research, the mean square error (MSE) function is adopted (desired MSE is 1e-5). LM algorithm is adapted to train the neural network, which can be written as Eq. (50): 1 2 i+ 1 i E E w = w + μi 2 i i w w (50) i where i is the iteration index, E/ w is the gradient descent of the performance function E with respect to the parameter matrix w i, μ 0 is the learning factor, and I is the unity matrix. During the vibration process, the neural network updates the weights and bias of neurons real time in accordance with sampling pairs till the objective error is satisfied, i.e. the property of the system is acquired. As we know, the main aim of the neural network is to predict the dynamic responses of the system and to provide inputs of fuzzy controller and data of calculating control force of MR dampers. hus outputs of the neural network are predicted values of displacement x k+1 and velocity x k+1. In order to predict the dynamic responses of the system accurately, the most direct and important factors which affect the predicted dynamic responses are considered, i.e. the Fig. 9. Membership function of back-left damper velocity (m/s). Fig. 10. Membership function of back-right damper velocity (m/s). delayed outputs (x k+3, x k+2, x k+1, x k, x k+3, x k+2, x k+1, x k), the predicted force (f k+1 ), and the disturbance input (d k ). LM algorithm is encoded in neural networks toolbox in MALAB software. 7.2 Design of fuzzy controller he first step in designing a fuzzy controller is to determine the basic domains of inputs and outputs. he desired displacement and velocity responses are chosen as inputs of the fuzzy controller. he output of fuzzy controller is the control force of the MR damper, which basic domain is -700N 300N same as the working force of the MR damper calculated using LQR. he membership functions are usually chosen in accordance with the characteristics of the membership functions and designing experience. For simplifying the calculation, triangle or trapezoid form functions are usually adopted as the membership functions. he triangle membership function is more

9 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~ Fig. 11. Displacement of front-left seat from front left wheel excited with 60 km/h. Fig. 13. Displacement of front-left seat from front left wheel excited with 30 km/h. Fig. 12. Acceleration of front-left seat from front left wheel excited with 60 km/h. Fig. 14. Acceleration of front-left seat from front left wheel excited with 30 km/h. sensitive to inputs than the trapezoid form function, in expectation that the control forces of the MR dampers are sensitive to excitations and responses [13], but in this case are used Gaussian and triangle form because considered form had better response by trial and error. In this research, Gaussian and triangle functions are adopted as the membership functions of velocity. he membership function curves of velocity are shown in Figs Results he full-car model with MR damper and disturbance is modeled by the dynamic equations and state space matrices. One of the desired points of this study is to decrease the amplitude of passenger s displacements, when the suspension system excited from the road profile. herefore the effect of LQR and LQG controllers and neuro-fuzzy strategy are simulated for road excitation with calculated their amplitude, and then compared with each other. he random road surface are generated compatible with the power spectral density given in Eq. (45) using a sum of bumper with 7 cm height and 4 cm width. he displacement and acceleration trajectories for front-left passenger s seat that is excited by bumper with 7 cm height and 4 cm width with 60 and 30 km/h constant velocity in unpaved road under front left wheel are shown in Figs , respectively. Notice that, in all graphs, time duration is selected for the best resolution and critical responses are happened when car strikes with bumper. he graphs which are presented show that LQR controller designed cannot eliminate the effect of the distributed road, but the LQG controller can eliminate the effect of the distributed road. However, this modeling and controller which have been theoretically designed may be experimental model cannot work such as the theoretical model. Also, the trajectories of neuro-fuzzy strategy show that this strategy reduces the amplitude of vibration lower than the passive system and also to some extent as well as optimal controllers; because displacements and acceleration are predicted by feed forward neural networks. he primary oscillations are due to the less number of network input to train, on the other hand, there are not strong history in transient, therefore the transient part of response not as well as steady state part. o investigate, are there any primary unwanted oscillations due to use of neurofuzzy strategy at the other bumpiness after the first one, is

10 332 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~334 Fig. 15. Displacement of front-left seat from front left wheel excited with 60km/h due to two bumpers. Fig. 18. Generated force by front-left MR damper from front left wheel excited with 60 km/h. Fig. 16. Road holding for front-left damper excited with 60 km/h. Fig. 17. Road holding for front-left damper excited with 30 km/h. added another after five seconds. he displacement of frontleft seat due to two bumpers is shown in Fig. 15. he graph which is presented is shown that the unwanted oscillations due to use of neuro-fuzzy strategy at the beginning of first excitation to some extent removed at the beginning of second bumper. he road holding for front-left damper that is excited by bumper with 7 cm height and 4 cm width with 60 km/h and 30 km/h constant velocity under front left wheel are shown in Figs. 16 and17, respectively. he stability of automobile due to use of neuro-fuzzy strategy is better than other two strategies, because the oscillation of front left wheel due to road excitation in neuro-fuzzy algorithm is less than other strategies. he trajectories of requirement forces to obtain the desire displacements and acceleration is shown in Fig. 18. he force is calculated by use of optimal controllers to some extent have the same trend. But the forces of neuro-fuzzy cannot follow them; because, optimal forces depend on twenty two state variables and the forces obtained by the fuzzy part of neuro-fuzzy strategy depend on one state variable (relative velocity across MR damper). One of the main advantages of using neuro-fuzzy, the control effort of dampers is less than LQR and LQG responses. he clipped optimal method responses to generate a requirement voltage for two different velocities are shown in Figs. 19 and 20, respectively. hese are obtained by another neuro-fuzzy strategy (the output of fuzzy part is voltage). he voltages are calculated by use of optimal controllers to some extent have the same trend. he voltages are calculated using neuro-fuzzy with less oscillations, therefore saving energy and cost. 9. Conclusions Usual suspension systems are utilized in the vehicle, and damped the vibration from road profile. However, passive suspension systems have long settling time. When cars are moved along bumpy road, the passive suspension system driver cannot react in effective time. As a result, the usual suspension system cannot damp the excitation with small time interval. In order to remove this problem the properties of the suspension system should be variable. his task is done by adding MR dampers as an actuator to the suspension system. In order to send commands to the actuator, LQR controller is utilized. It can decrease the amplitude of the vibration of passenger seats, but it cannot eliminate the effect of bumpy roads as a disturbance. herefore, LQG controller was designed. his controller by LQE estimator can estimate the

11 S. H. Zareh et al. / Journal of Mechanical Science and echnology 26 (2) (2012) 323~ Acknowledgment he authors wish to express their gratitude to the International campus of Sharif University of echnology for the support provided for this research. References Fig. 19. Requirement voltage to front-left MR damper from front left wheel excited with 60 km/h. Fig. 20. Requirement voltage to front-left MR damper from front left wheel excited with 30 km/h. state of the system without disturbance effect. hen the estimated system is controlled by LQR part of LQG controller. Consequently, the disturbance effect is eliminated, and the response of this controller in the theoretical model is much performed. It is seen that the control forces with LQG algorithm and LQR algorithm completely suitable for road excitation, meaning that the Kalman filter gives accurate estimations of structural states. As can be seen, LQG can eliminate the effect of disturbances better than LQR. Unfortunately, due to the inherent nonlinear nature of the MR damper to generate force, a model like that for its inverse dynamics is difficult to obtain mathematically. Because of this reason, a neural network with fuzzy logic controller is constructed to copy the inverse dynamics of the MR damper. According to the graphs that show above, the trajectories of neuro-fuzzy strategy can reduce the amplitude of vibration to some extent as well as optimal controllers with less control effort and oscillation. [1] Y. P. Wang, Y. Shio, J. Guo, M. H. Chiang and Y. K. Din, Semi-active control of vehicle suspension system using electrorheological dampers, 3rd Institution of Engineering and echnology Conference (2007). [2] R. S. Prabakar, C. Sujatha and S. Narayanan, Optimal semiactive preview control response of a half car vehicle model with magnetorheological damper, Journal of sound and vibration, 326 (2009) [3] Y. Md. Sam and J. H. S. B. Osman, Modeling and control of the active suspension system using proportional integral sliding mode approach, Asian Journal of Control, 7 (2) (2005) [4] A. F. Jahromi, A. Zabihollah and Linear Quadratic Regulator and fuzzy controller application in full-car model of suspension system with magnetorheological shock absorber, IEEE/ASME International Conference on Mechanical and Embedded Systems and Applications (2010) [5] M. M. M. Salem and Ayman A. Aly, Fuzzy control of a quarter-car suspension system, World Academy of Science Engineering and echnology, 53 (2009) [6] H. C. Sohn and K.. Hong, An adaptive LQG control for semi-active suspension systems, International Journal Vehicle Design, 34 (4) (2004) [7] Y. Chen, an, L. A. Bergman and. C. sao, Smart suspension systems for bridge-friendly vehicles, Annual International Symposium on Smart Structures and Materials (2002). [8] S. Jialin, A. Ordys and K. Panahi, LQR-based linear controller of active suspension on a half-body model for improved ride comfort, 3rd IAR Workshop on Advanced Control and Diagnosis (2008). [9] J. Sun and Q. Yang, Modeling and intelligent control of vehicle active suspension system, IEEE International conference on RAM (2008) [10] J. Sun, Q. Zhou, Vibration control of vehicle suspension with five degrees of freedom, Proc. 9 th IEEE International conference on cognitive informatics (2010) [11] M. Biglarbegian, W. Melek and F. Golnaraghi, Intelligent control of vehicle semi-active suspension system for improved ride comfort and road handling, IEEE International conference (2006) [12] S. H. Sadati, M. A. Shooredeli and A. D. Panah, Designing a neuro-fuzzy controller for a vehicle suspension system using feedback error learning, Journal of mechanics and aerospace, 4 (3) (2008) [13] Zh. D. Xu, Y. Q. Guo and Neuro-Fuzzy control strategy for earthquake-excited nonlinear magnetorheological structures, Soil Dynamics and Earthquake Engineering, 28 (2008) [14] O. Yoshida and S. J. Dyke, Seismic control of a nonlinear benchmark building using smart dampers, Journal of engi-

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