Modeling and Control Active Suspension System for a Full Car Model

Modeling and Control Active Suspension System fora Full Car Model Rosheila Darus 1 , Yahaya Md. Sam 2 1 Faculty of Electrical Engineering, Universiti Teknologi Mara (UiTM), Cawangan Terengganu, 23000 Dungun, Terengganu, MALAYSIA 2 Faculty of Electrical Engineering, Universiti Teknologi Malaysia (UTM), 81310 Skudai, Johor, MALAYSIA Abstract-The purpose of this paper is to investigate the performance of a full car model active suspension system using LQR controller. Dynamic model used in this study is a linear model. A linear model can capture basic performances of vehicle suspension such as body displacement, body acceleration, wheel displacement, wheel deflection, suspension travels, pitch and yawn. Performance of suspension system is determined by the ride comfort and vehicle handling. It can be measured by car body displacement and wheel displacement performance. Two types of road profiles are used as input for the system. Simulation is based on the mathematical model by using MATLAB/SIMULINK software. Results show that the performance of body displacement and wheel displacement can be improved by using Linear Quadratic Regulator control (LQR). I. INTRODUCTION Suspension is a common property to all automobiles. It’s isolated the vehicle body from road disturbances for comfortable ride. Conventional suspension system consists of a spring and damper, also known as passive suspension. In active suspension system control strategy is a very important. With correct control strategy, it will give better compromise between ride comfort and vehicle handling. Nowadays there a lot of researches have been done to get more efficient and better performance of active suspension by having suitable control strategy. In most of the research has been done, a linearized model is used. One of the reason is it can be derived easily and it can capture basic features of a real vehicle problem. Most of the studies are concerning on controlling the forces created by the suspension damper and springs. Application of the LQR method to the active suspension system has been proposed [1]. In this study, the LQR method is used to improve the vehicle handling and the ride comfort for a quarter car model. II. DYNAMIC MODELING Many ongoing research studies about designing a control system have used the well know 7-DOF as shown in Fig. [1]. Mathematical modeling of the system obtained from [2]. Tf Tr roll axis pitch axis a b mu1 mu4 mu2 mu3 zr1 zr2 zr4 zr3 Zs ms Zu2 Zu3 Zu4 Zu2 Fig. 1. Suspension System for Full Car Model The dynamic model derived by [2] divided into 7 main parts: Bouncing of the sprung mass 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1 2 3 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s s s u s u s u f s u s u s u s u s u m Z b Z Z bf Z Z br Z Z br Z Z kf Z Z kf Z Z kr Z Z kr Z Z u u u u = ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ + + + +          (1) Pitching of the sprung mass 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1 2 3 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p s r s u s u s u f f r s u s u s u f f r r s u s u I b a Z Z b a Z Z b b Z Z b b Z Z k a Z Z k a Z Z k b Z Z k b Z Z au au bu bu  = ÷ ÷ ÷ ÷ ÷ ÷ + ÷ ÷ ÷ + ÷ + ÷ + ÷ + + ÷ ÷          (2) (2) Rolling of the sprung mass 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1 2 3 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r s r r s u s u s u f f f f r r s u s u s u f f f f r r r r r r s u s u f f I b T Z Z b T Z Z b T Z Z b T Z Z k T Z Z k T Z Z k T Z Z k T Z Z T u T u T u T u  = ÷ ÷ ÷ ÷ ÷ ÷ + ÷ ÷ ÷ + ÷ ÷ ÷ + ÷ + ÷ + ÷         (3) Vertical Direction for each wheel 1 1 1 1 1 1 1 1 ( ) ( ) u s u s u u r uf f f tf tf m Z b Z Z k Z Z k Z u k Z = ÷ + ÷ ÷ ÷ +    (4) 2 2 2 2 2 2 2 2 ( ) ( ) u s u s u u r uf f f tf tf m Z b Z Z k Z Z k Z u k Z = ÷ + ÷ ÷ ÷ +    (5) 3 3 3 3 3 3 3 3 ( ) ( ) ur r r tr tr u s u s u u r m Z b Z Z k Z Z k Z u k Z = ÷ + ÷ ÷ ÷ +    (6) 4 4 4 4 4 4 4 4 ( ) ( ) ur r r tr tr u s u s u u r m Z b Z Z k Z Z k Z u k Z = ÷ + ÷ ÷ ÷ +    (7) 13 2009 5th International Colloquium on Signal Processing & Its Applications (CSPA) 978-1-4244-4152-5/09/$25.00 ©2009 IEEE where; 1 s s s s f Z T a Z   = + + 1 s s s s f Z T a Z   = + +     2 s s s s f Z T a Z   = ÷ + + 2 s s s s f Z T a Z   = ÷ + +     3 r s s s s Z T b Z   = ÷ ÷ + 3 r s s s s Z T b Z   = ÷ ÷ +     4 r s s s s Z T b Z   = ÷ ÷ + 4 r s s s s Z T b Z   = ÷ ÷ +     Fig. 1 is an accurate 7-DOF where linearization and model reduction technique are used to find a simple version of a complex full car model. There are roll, pitch and the vertical displacement of sprung mass and four unsprung mass are included. TABLE 1 DEFINITIONS OF VARIABLES Variables Definitions s Z Vertical displacement s Z  Vertical velocity 1 u Z Vertical displacement of front right wheel 1 u Z  Vertical velocity of front right wheel 2 u Z Vertical displacement of front left wheel 2 u Z  Vertical velocity of front left wheel 3 u Z Vertical displacement of rear right wheel 3 u Z  Vertical velocity of rear right wheel 4 u Z Vertical displacement of rear left wheel 4 u Z  Vertical velocity of rear left wheel s  Pitch angle s   Pitch rate s  Roll angle s  Roll rate III. CONTROLLER DESIGN Assume quadratic performance index in the form of 1 ( ) 2 0 t T T J x Q x u R u d t = + } (8) Matrix Q is symmetric positive semi-definite and R is positive symmetric definite. Matrix K is presented as; 1 ' K R B P ÷ = (9) Linear feedback control law is obtained as; ( ) ( ) u t Kx t = ÷ (10) Designed matrix K is determined by using MATLAB by setting suitable matrix Q and R. IV. SIMULATION Input disturbance or irregular excitation from the road surface divided into 2 types; single bump and two bumps. It can show the ability of LQR to adapt different types of road disturbance. Case 1: A single bump input shows in (11) for front wheel and (12) for rear wheel. Assume that front right and left wheel reached bump at the same time. ( ) (1 cos 8 ), 0.5 0.75 0 otherwise d t a t s t s  ¦ ¦ ´ ¦ ¹ = ÷ s s (11) ( ) (1 cos 8 ), 3.0 3.25 0 otherwise d t a t s t s  ¦ ¦ ´ ¦ ¹ = ÷ s s (12) Fig. 1. Input Disturbance of a Single Bump for Front Right and Left Wheel. Fig. 2. Input Disturbance of a Single Bump for Rear Right and Left Case 2: Two bumps input shows in (13) for front wheel and (14) for rear wheel. Assume that front right and left wheel reached bump at the same time 14 ( ) (1 cos 8 ), 0.5 0.75 (1 cos 8 ) /2 6.5 6.75 0 otherwise d t a t s t s a t s t s   = ÷ s s ÷ s s ¦ ¦ ´ ¦ ¹ (13) ( ) (1 cos 8 ), 3.0 3.25 (1 cos 8 ) /2 9.0 9.255 0 otherwise d t a t s t s a t s t s   = ÷ s s ÷ s s ¦ ¦ ´ ¦ ¹ (14) Fig. 3. Input Disturbance of a Two Bumps for Front Right and Left Fig. 4. Input Disturbance of a Two Bumps for Rear Right and Left Parameters for the full car model of active suspension systems shows in table 1 [3]. TABLE 2 CAR PARAMETER mass of the car body or sprung mass, m s 1500 kg front mass of the wheel or unsprung mass, m uf 59 kg rear mass of the wheel or unsprung mass, m ur 59 kg Pitch of moment of inertia, I p 2160 kg m 2 roll of moment of inertia, I r 460 kg m 2 stiffness of car body spring for front, k f 35000 N/m stiffness of car body spring for rear, k r 38000 N/m front treat, T f 0.505 m rear treat, T r 0.557m font and rear tire stiffness, k tf and ktr 190000 N/m front damping , b f 1000 N/m rear damping, b r 1100 N/m c.g to front wheel, a 1.4m c.g to rear wheel, b 1.7 m V. RESULTS For comparison purpose, the performance of active suspension system using LQR technique is compared with passive suspension system. Fig. 5. Force Generate for Front Right and Left Actuator by Using LQR controller (Case 1) 15 Fig. 6. Force Generate for Rear Right and Left Actuator by Using LQR controller (case 1) Fig. 5 and Fig. 6 show force generated by actuators for each suspension sets for road profile with single bump. Front right and front left give same amount of force due to input disturbance is the same. Rear right and rear left also have same types of disturbance; therefore force generated at both actuators for front wheels are same and same condition happen to rear actuators. Fig. 7. Body Displacement (case 1) Fig. 8. Wheel Displacement for Front Right and Left (case 1) Fig. 9. Wheel Displacement for Rear Right and Left (case 1) Fig. 10. Suspension Travel Front Right and Left (case 1) Fig. 11. Suspension Travel Rear Right and Left (case 1) Fig. 7 shows the comparison between active and passive suspension for body displacement. Body displacement is used to measure passenger ride comfort. Even it shows that the amplitude of active suspension using LQR technique is higher than passive suspension but it improved displacement settling time. Fig. 8 and Fig. 9 show the performance of wheel deflection for each wheel. It clearly shows that the amplitude for active suspension lower than passive suspension. In the simulation results also shows that the output performances for rear wheels give slightly higher output compare to front wheels it may cause by output from rear 16 actuators force which is slightly higher compare with the front actuators. Fig. 12. Force Generate for Front Right and Left Actuator by Using LQR controller (case 2) Fig. 13. Force Generate for Rear Right and Left Actuator by Using LQR controller (case 2) By using same parameter setting for Q and R output performance is captured to investigate either LQR can adapt in changes of road disturbance. Fig. 14. Body Displacement (case 2) Fig. 15. Wheel Displacement for Front Right and Left (case 2) Fig. 16. Wheel Displacement for Rear Right and Left (case 2) Fig. 17. Suspension Travel Front Right and Left (case 2) Fig. 18. Suspension Travel Rear Right and Left (case 2) 17 By comparing the performance of the passive and active suspension system using LQR control technique for full car model, it is clearly shows that there is a problem with robustness. The output performances for case 2 happen to have slightly higher amplitude compare with case 1 active suspensions performances for certain parameters such as Body Acceleration and Wheel Deflection for each wheel. Body Displacement improved even the amplitude is slightly higher compare with passive suspension system but the settling time is very fast. Body Displacement is used to represent ride quality. It is shows that LQR does not have capability to adapt variations in road profiles. VI. CONCLUSION AND FUTURE WORK LQR control technique has successfully implemented to the linear model active suspension system for full car model. Active suspension system that has been proposed in [1] is for a quarter car model. Input disturbance using in [1] is step input and random input. In this study by using dynamic model in [1] simulation has been done by using same road profile with full car model to compare the performance of LQR between this two model. Results show that there is a limitation using LQR control technique. It cannot perform in rough road disturbances especially for full car model. Performance shows that LQR control technique for a quarter car model can provide better ride comfort compare LQR control technique that implement in a full car model active suspension. However the LQR control technique still can improve ride comfort and car handling compare to the passive suspension. In this study also does not include dynamic models of the actuators. Force actuator generates needed force to achieve desired objectives. Thus the study of actuator must be including in future work so it can give real time performance. Different types of controller will be implementing to observe the output performance of the suspension. REFERENCES [1] Y.M. Sam, M.R.A. Ghani and N. Ahmad, “LQR controller for active car suspension,” IEEE Proceedings of TENCON 2000, 2000. I441-I444. [2] Kim C., Ro P.I, “An Accurate Full Car Ride Model Using Model Reducing Techniques,” Journal of Mechanical Design, vol. 124, pp. 697-705, December 2002. [3] S. Ikenaga, F. L. Lewis, J. Campos and L. Davis, “Active Suspension Control of Ground Vehicle based on a Full-Vehicle Model,” Proceedings of the American Control Conferenc, Chicago, Illinois June 2000. [4] Sam Y.M., Osman H.S.O., Ghani M.R.A., “A Class of Proportional- Integral sliding Mode Control with Application to Active Suspension System” System and Control Letters. 2004. 51:217-223 [5] Sam, Y.M. Proportional Integral Sliding Mode Control of an Active Suspension System. Malaysia University of Technology. PHD Dissertation. Malaysia University of Technology; 2004 18