Backstepping Control Design for a Semiactive Suspension System with MR Rotary Brake

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1 Backstepping Control Design for a Semiactive Suspension System with Rotary Brake K.M.I.U.Ranaweera, K.A.C.Senevirathne,M.K. Weldeab, H.R.Karimi Department of Engineering, Faculty of Engineering and Science University of Agder, 4879 Grimstad, Norway Hamid.r.karimi@uia.no Abstract The purpose of this paper to design a semicative controller for a vehicle suspension system that employs a magnetorheological ( damper as the actuator. The controller was developed using back stepping methodology for the laboratory model Semi Active Suspention System (SAS which represents a physical system for a quarter car model, which can be used to analyze the vertical dynamics of a car body due to the road disturbances. First the SAS system was modeled by the mathematical equations and then the backstepping controller was designed to suppress the vibrations of the car body. The dhal model was used to model the nonlinear behavior of the damper. The mathematical model of the SAS system and the control system were designed in Matlab/Simulink and system performance was evaluated. Finally the controller was connected to the experimental model and the vibration responses were obtained for different disturbances. wheel,rotational damper and a spring. It is driven by a DCmotor with gear coupled to an eccentric small wheel. The suspended car wheel rolls due to the small wheel rotation and oscillates up and down due to the small wheel eccentricity. I. INTRODUCTION One of the most common applications of suspension system is the automobile industry to suppress the vibrations of the vehicle body, created by the irregularities of the road surface. A good vibration suspension system will provide comfortable ride for the passengers. Many researchers and academicians nowadays made the research and development regarding the suspension systems in order to achieve a good car ride and handling quality[7]. One of the better suspension systems currently using for the vibration isolation is the Semi Active Suspension (SAS System. It consists of a magnetorheological damper which creates braking torque by changing the viscosity of the fluid inside the brake according to the applied current[5]. By controlling the input current, output torque of the damper can be controlled to make the system stable quickly. In this paper a control mechanism for controlling the torque of the damper is developed for the laboratory model Semi Active Suspension System which represents the quarter car model by using Backstepping control mechanism. II. SEMIACTIVE SUSPENSION SYSTEM The study of vehicle vibration reduction is made by analyzing the dynamics of the Semi-Active Suspension (SAS System developed by the Polish Company Inteco Limited. SAS laboratory model represents a physical system of a quarter car model, and can be used to analyze the vertical dynamics of the car wheel. As shown in Figure, the SAS systemconsists of an upper beam which represents the car body, a Figure : Semiactive Suspension System (SAS Figure provides the geometrical view of the SAS system. Using this figure the mathematical modeling equations for the upper lever and the lower lever can be derived as follows[4]. Figure : The geometrical view of the SAS system The equations of motion of the upper beam are; J ω d α dα + k + M cosα r k ( l l T ( i s os s = ISBN:

2 J ωɺ = kω M cosα + rk s ( los ls + T ( i The equations of motion of the lower beam are; J ω d α + M cos( β α + r k k R cos( β α ( l g dα R cos( β α = T og + R sin( β α + r D ( i s ( l os dα ls + k x + u ex f g d( Dx u ( Where; = ( r cosα r cosα + ( r sinα r α l s sin J, J - Moment of inertia of the lower beam and the upper beam with respect to its axis rotation M, M -Gravitational moment of the upper and lower beam k, k - Viscous friction coefficients of the upper beam and lower beam k -The elasticity coefficient of the spring s los -The length of the no-load spring ls -The length of the loaded spring -The elasticity coefficient of the tire k g fg - The absorption coefficient of the tire uex -The kinetic sinusoidal excitation - damper torque T III. BACKSTEPPING CONTROLLER DESIGN The control methodology must be implemented to decrease the vibrations of the body when the tire is subjected to the external excitation force such as bumps or vibrations. Therefore in backstepping control method we mainly focuson the vibration suppression of the body[] so that we use only the mathematical model of the body. Let s define; The viscous friction damping torque, M d = kω The gravitational force torque, M g = M cosα Spring torque, M = r k ( l l s s os s Then the equation of the upper beam can be written as, ω ωɺ = J ( M d M g + M s+ T ( i The equilibrium point of the system can be found when, ( ɺ α,ωɺ = (, It was found as, ( α, eq,ω, eq = (.55 rad, when T ( =. Defining the new coordinates z and z by shifting the origin to the equilibrium point. ( z, z = ( α α, eq, ω ω, eq = ( α α, eq,ω In the new coordinate system equation of the upper beam becomes, z ɺ = z ex = J ( M d M g + M s + J T In order to apply the backstepping technique let s define standard backstepping variables. e = z e ɺ = z e = z + he, h > eɺ = + heɺ e = z+ h z e ɺ =ɺ z + h z Lyapunov s direct method of stability is used to determine a rule for the stability of the system. Candidate Lyapunov candidate function[]; V = V + V = e + e Derivative of V, Vɺ = eeɺ + eeɺ V ɺ = e z+ e ( + h z V ɺ = e ( e h e + e + h e z V ɺ = he + ee + e + he z Vɺ = he + ee + e + he z + he he ; h > V ɺ = he he + e( z+ + h z + h ( z + he V ɺ = he he + e( z+ + h z + h z + h h z V ɺ = h e he + e( z(+ hh + + z( h + h + J T + ω ( h + h Vɺ = he he + e (( α α, eq (+ hh + J ( M d M g + M s V ɺ h e h e <, given that h h, > If, ( ( ( e α α + h h + J M M + M + J T + ω ( h + h (, eq d g s Then, V ɺ < Also V > and V (, = Therefore according to the Lyapunov second method of stability the system is stable. Then, e, e (, ( ( z, z (, ( α, ω ( αeq This implies that,, In order to be the system stable let s define the control law, ( α α (+ h h + J ( M M + M + J T + ω ( h + h e (, eq d g s = ( α α, eq (+ hh + J ( M d M g + M s + J T + ω( h + h = J T = ( α α, eq (+ hh J ( M d + M g M s + ω( h + h T T ( α = α, eq (+ hh J ( M d + M g M s + ω( h + h J = J ( α α, eq(+ hh + ( M d + M g M s ωj( h h + The controller must command the damper to produce the torque derived above to suppress the vibrations of the body as soon as possible. The damper can be controlled by the input current or voltage. Therefore the appropriate current or voltage must be found to give as the input to the damper to produce the required torque. ISBN:

3 IV. DAMPER The damper is a type of semi-active damper where the flow of fluid is controlled by varying the amount of current supplied and hence changes the level of damping [5]. The torque- current relationship of the damper is highly nonlinear which is called as hysteresis behavior. Therefore developing a mathematical model for a vibration suspension system is a big challenge if damper is used as the vibration suppression device. To describe the hysteretic torque/velocity and torque/displacement response of damper, several models have been proposed in the literature, including the Bingham model, Dhal model, the Bouc-Wen phenomenological model, Lugre Model, neural networks, etc[6],[9]. In this paper Dhal model is selected to model the damper. The governing equations of the damper are; T = K ( i ɺ θ+ K ( i z with = α ( ɺ θ ɺ θ z x y where θ is the angle, K x is the damping coefficients which depend linearly on the current (i and the z is the hysteretic variable. Parameters and α control the shape of the hysteresis curve. K y The current dependent parameters K x and K y described by following equations. Kx = Ka + Kb i K = K K i y + Figure 3: SAS system with the backstepping controller The following damper parameters for the dhal model are selected for the Simulation. K = 5, K =.5, K a =., K b =., α = 5 Backstepping control was applied to the SAS model with the parameters h = and h =. Vibration response of the body was obtained for different and the results are illustrated in the following figures (a Damper must produce the damping torque of; T = J α α (+ h h + ( M + M M ω J ( h, C (, eq d g s + h Damper Torque given from the Dhal model; T = K + K i ɺ θ+ ( K K i z, C ( a b + Therefore the control current that should be sent to the damper is, T, C K ɺ aθ Kz i = or K ɺ θ + K z b J ( α α, eq(+ h h + ( Md + M g Ms ωj( h + h K ɺ aθ Kz i = K ɺ bθ + Kz V. SIMULATION RESULTS The SAS system with the backstepping controller is modeled in the Matlab/Simulink and the Simulink diagram is given in Figure (b (c ISBN:

4 Figure 4: Simulation results of the beam vibrations for different According to Figure 4, it can be clearly seen that, when the backstepping controller is present the body stabilizes more quickly by reducing the amplitude of the vibration. VI. LABORATORY EXPERIMENT Vibration analysis was done for the experimental set up for the Semi Active Suspension system. The SAS system has been designed to operate with an external, PC-based digital controller. The control computer communicates with the level sensors, values and pump by a dedicated I/O board and the power interface. The I/O board is controlled by the real software which operates in Matlab/Simulink/RTWT rapid prototyping environment. The ler for vibration suppression was added to the simulink model that communicates with the SAS system, as illustrated in Figure (b (c Figure 6: Angle of the upper beam for different Experimental results Figure 5: Real experimental Simulink model of the SAS with back stepping controller Different signals were given as the reference rotatingfrequency of the eccentric wheel to introduce vibrations to the Wheel. Then the vibration response of the upper beam was obtained with and without connecting the ler. The results are illustrated in Figure 6. According to the results obtained from the experiment we can see that, the amplitudes of the vibrations were decreased when the back stepping controller is connected and the body stabilizes more quickly with the controller. In the experimental set up maximum input current for the damper is.5a. Therefore optimal control cannot be achieved due to the current limitation in the experimental setup as the results obtained with the simulations in Matlab VII. CONCLUSION In this paper we have developed a controller based on the backstepping method for a semi active suspension system. The semi active device is an damper and the proposed controller has been tested in Matlab/Simulink and with the real experimental model. The obtained results have shown a good performance of the backstepping controller in the simulation as well as the laboratory experiment. REFERNCES [] H.R. Karimi A Semiactive Vibration Control Design forsuspension Systems with Dampers, Vibration Analysis and Control - New Trends and Developments, InTech, DOI:.577/. [] M.Zapateiro, N.Luo, H.R.Karimi, J.Vehı Vibration control of a class of Semiactive suspension systemusing neural network and backstepping techniques, Mechanical Systems and Signal Processing, 3, 6, 9, [3] K. Hassan Nonlinear Systems, Third Edition Prentice Hall,. [4] INTECO Limited, Semiactive Suspension System (SAS: User s Manual, 7. [5] J. Poynor Innovative Designs for Magneto-Rheological Dampers, Master Thesis, Virginia Polytechnic Institute and State University,. [6] B.F. Spencer, S.J. Dyke, M.K. Sain and J.D. Carlson Phenomenological Model of a Magnetorheological Damper Journal of Engineering Mechanics, Vol. 3, No. 3, March 997, pp [7] Sara S. Sedeh, Reza S. Sedeh, Keivan Navi Using car semi-active suspension systems to decrease undesirable effects of road excitations on human health Proc. 6 International Conference on Bioinformatics & Computational Biology, Las Vegas, USA, 6. [8] Ningsu Luo, Mauricio Zapateiro, Hamid Reza Karimi Heuristic and Backstepping Control Strategies for Semiactive Suspension in Automotive Systems Equipped with Dampers IFAC Syposium on Advanced Automotive Control, Munich, Germany, (a ISBN:

5 [9]Jinung An and Dong-Soo Kwon Modeling of a Magnetorheological Actuator Including Magnetic Hysteresis Journal of Intelligent Material Systems and Structures, September 3 vol. 4 no [] C. Kaddissi, J.P. Kenné, M. Saad Drive by Wire Control of an Electro- Hydraulic Active Suspensiona Backstepping Approach IEEE Conference on Control Applications, Toronto, , 5. ISBN: