Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 3(8), 86-95, August (2014) Res.J.Recent Sci. International Science Congress Association 86 A Study of the Performance of the PID Controller and Nonlinear Controllers in Vehicle Suspension Systems Considering Practical ConstraintsS. H. Hashemipour, M. Rezaei lasboei and M. KhalijiDepartment of Electrical and Mechanical Engineering, Young Researcher Club, Roudsar and Amlash Branch, Islamic Azad University, Roudsar, IRAN Available online at: www.isca.in , www.isca.me Received 11th September 2013, revised 1st November 2013, accepted 7th December 2013Abstract In this paper first an applicable PID controller has been designed for the vehicle suspension system with considering the nonlinear dynamics of the hydraulic actuator. In this method, linear sky-hook model was chosen as the reference model. In order to improve the characteristics of the reference model in terms of ride comfort and stability and also considering the practical constraints of the suspension system, an optimal LQR controller has been designed for the sky-hook model. To improve the reference model behavior in terms of ride comfort with considering the limitations of the suspension system working space and create a tradeoff between ride comfort and handling, an optimum LQR controller with adjustable weight matrices for the sky-hook model was designed. Of course, this controller had no suitable performance with regards to the system non-linear dynamics. Then, a non-linear controller was designed based on Lyapunov method. Simulation results indicate that this controller is successful in reducing the vertical acceleration to improve the ride comfort, but it cannot control the stability and stability of the vehicle. Meanwhile, the designed controller is not robust enough to system parameter perturbations. Therefore, the sliding mode control as a robust nonlinear control method has been adopted as an alternative way for the controller design. In this method, the sliding surfaces are selected in a way that the nonlinear system tracks a sky-hook model which has desirable behavior. Simulation results revealed that the non-linear model of the suspension system with a sliding mode controller could satisfactorily track the behavior of the new improved sky-hook reference model. Also, the sliding mode controller showed a good behavior when parameters changed. Keywords: Suspension system, PID, Sliding Mode, Lyapunov method. IntroductionOne of the causes for vehicle vibrations and shakes is road roughness. Shocks due to bumps on the road are transferred to the vehicle body through the wheels and cause discomfort for passengers and reduce driving quality. The suspension system is responsible for absorbing these shocks and reducing vehicle shakes as much as possible thereby providing more comfort to passengers. In passive suspension systems, the reduction in vertical acceleration results in an increase in suspension travel. The suspension travel has to be limited because the movement of vehicle spring-damper system is constrained. Since the reduction in vertical acceleration results in an increase in suspension travel, there’s a limit on the maximum reduction of vertical acceleration. To overcome these limitations and increase ride comfort within system constraints, and also to increase the vehicle stability, active suspension systems have been considered1, 2. In these systems, by applying an extra force to the suspension system using a hydraulic actuator, the vertical acceleration exerted on passengers is reduced while also reducing suspension travel considerably and thus improving ride comfort and driving quality. The LQR control method and robust control have been used widely for designing controllers for active suspension systems2, 3. Sam et al designed a sliding mode PI controller for the quarter vehicle model and compared their results with those obtained by the LQR control method. The abovementioned methods are based on a linear model and neglect the dynamics of the hydraulic actuator which is highly nonlinear. However, the nonlinearity in the actuator dynamics has a large effect on the overall behavior of the system and cannot be neglected. Using nonlinear control methods, Kurimoto et al designed a sliding mode controller for active suspension. They used a 4-DOF quarter model of vehicle which included a simplified model of nonlinear actuator dynamics. Yokoyama designed a sliding mode controller and observer for a semi-active suspension system. State feedback control, fuzzy control, and optimal stochastic control are some other control strategies which have been used for active suspension system design. In the present article, the studied model was a nonlinear one. First, a PID controller was designed for the nonlinear model. The controlled sky-hook linear model was chosen as the reference model here. In the method based on Lyapunov theory, a reverse procedure is applied where, in the first step, a candidate Lyapunov function is chosen considering the representation of system state equations and the direct relation between the decrease in vertical acceleration and the increase in ride comfort. Then, the controller is designed in a way that the Research Journal of Recent Sciences _____________________________________________________________ISSN 2277-2502Vol. 3(8), 86-95, August (2014) Res. J. Recent Sci. International Science Congress Association 87 candidate Lyapunov function be the true Lyapunov function of the controlled system. The sliding mode control method is then applied for the design of a nonlinear and robust controller for the active suspension system. A novel innovative reference model has been used in the present work for choosing sliding surfaces. In the literature on sliding mode control for active suspension systems, a linear sky-hook model with a fairly desirable behavior has been used alone as the reference model to be tracked by the controlled system . However, in the present work, in order to obtain greater reduction in vertical acceleration, while taking into account practical system constraints, a controlled sky-hook model is chosen as the reference model. Since reduction in vertical acceleration results in an increase in suspension travel, in order to make a trade-off, first the LQR optimal control method is used for designing a controller for the sky-hook model. In this method, by adjusting weight matrices, the sky-hook model is controlled in a way that maximum reduction in vertical acceleration is attained while remaining within allowable displacement limits of the suspension system workspace. The sliding surfaces are designed in a way that the controlled system follows the behavior of this optimally controlled sky-hook model. In the next section of the article, the vehicle suspension system model has been introduced along with the hydraulic actuator and the sky-hook reference model. Then, controlling theories are introduced with their simulation results presented. The last section of the article includes comparison and conclusion. Quarter Model including Hydraulic Actuator Dynamics One of the most widely used models in active suspension system design is the vehicle quarter model. This model encompasses the main components of the suspension system including sprung and unsprung masses, spring and the main damper. The actuator dynamics can also be included in this model with ease. Figure 1 shows the vehicle quarter model. Figure-1 Quarter- car Model for Active Suspension Control DesignThe hydraulic actuator in this model is a four-way valve-piston system. The actuator force is calculated using AP where is the area of the piston and is the pressure drop across the piston. The time derivative of can be expressed as in equation (1) 11. ()LtpLsu PCPAxxQ b =--+  (1) In the above equation, is the total actuator volume, is the pressure drop, is the effective bulk modulus, tp is the total leakage coefficient of the piston, and is the hydraulic load flow which can be calculated using Equation (2): [] sgn() dvsvL QCwxPxP =- (2) where is the discharge coefficient, is the spool valve displacement, is the spool valve area gradient, is the supply pressure and is the hydraulic fluid density. Using Newton’s second law and actuator equations, the following relations can be obtained: ()() sssussusa mxkxxCxxF =-+-+  (3) mx=-K(x-x)-C(x-x)+C(r-x)+k(r-x)-F uusussustutua   (4) aL FAP (5) 24 ()() LLvsLv PPAxxxPPsignx bag=-+-+-  (6) 1 () vv xxu t =-+ (7) Choosing the state variables as in Equation (8) the state space representation of the system can be written as Equations (9)-(14): ( ) ,,,,, uuussLv XrxxxxxPx =--  (8) 12 xrx =-  (9) () 2124325 ()tsst xKxCxxKxCrxAx =---+--    (10) 324 xxx =-  (11) () 42435 ss xCxxKxAx =-++   (12) ( ) 5524665 () s xxAxxxPsignxx bag=-+-+- (13) ()66 xxu t =-+ (14) Where 4,e tp t C aba==and 1 Cwga r . In equation (8) the state variables are, respectively, the unsprung mass displacement, the unsprung mass velocity, the sprung mass displacement, the sprung mass velocity, the pressure drop across the piston, and the valve displacement. The parameters used for the simulation of this model are given in table I 11: Research Journal of Recent Sciences _____________________________________________________________ISSN 2277-2502Vol. 3(8), 86-95, August (2014) Res. J. Recent Sci. International Science Congress Association 88 Table-1 Parameters of the quarter car model 51 22 1.5459[/] eNmKg g= 1000[/] s CNsm = 290[] s MKg = 2 3.354[] Aem =- 500[/] t CNsm = 59[] u MKg = 5 4.51513[/] eNm a= 1/30[sec] t = 16812[/] s KNm = 1.00[1/] Sec b = 10342500[] s PPa = 190000[/] t KNm = In figure 2.a, the vertical acceleration response of the linear suspension system without the nonlinear hydraulic actuator dynamics has been plotted. The same input has been applied to the nonlinear system including the actuator dynamics and the response has been plotted. It can be observed that nonlinearity, arising from actuator dynamics, drastically affects the behavior of the system and is not negligible. It should be noted that in the present work, the controller has been designed for the suspension system while taking the ill-behaved actuator dynamics as depicted in figure 2.b into account. a) Vertical acceleration without considering actuator dynamic b) Vertical acceleration with considering actuator dynamic Figure-2 Impact of hydraulic actuator dynamic on system behavior The Optimally Controlled Sky-Hook Model System Equations: The sky-hook model is a virtual model that can be used as a reference model in suspension system controller design for increasing ride comfort 10. In this paper, a controlled sky-hook model is used as the reference model. The purpose is to improve the reference model so that the trade-off between the reduction in vertical acceleration and the increase in suspension travel is improved. The model is shown in Figure 3 and its state equations can be written as:Figure-3 The sky-hook model [] []122132324443()tstxrx xKxKxCrxu xxxxCxKxu=-=-+--=-=-++   (15) In Equation (24), the state variables are the unsprung mass displacement, the unsprung mass velocity, the sprung mass displacement, and the sprung mass velocity, respectively. Optimal Controller Design for the Sky-Hook Model: In order to control and improve the behavior of the sky-hook reference model, state-space equations for the linear system were written as follows and an optimum LQR controller was designed for it. (16) Where 111 xAxBuDr =++   Research Journal of Recent Sciences _____________________________________________________________ISSN 2277-2502Vol. 3(8), 86-95, August (2014) Res. J. Recent Sci. International Science Congress Association 89 0-100 1 KCKtts--0- t MMMMuuuuA=,B=,D= u 111010-10 0 K1-C00 0 MMss  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \nIf the cost functional is chosen in the form of () TT JxQxuRudt =+ (17) The control law to minimize the above functional can be obtained as: ukx =- (18) kRBP (19) Where is obtained from the following algebraic Ricatti equation: 1111 0 TTPAAPPBRBPQ +-+= (20) By adjusting weight coefficients in this method, maximum reduction of vertical accelerations for admissible displacements in the suspension system’s working space was achieved and it resulted in a good compromise between the vehicle’s ride comfort and handling. In the present work, these matrices were selected as in equation (30) by trial and error so that both a desirable decrease in vertical acceleration be obtained and suspension travel remain within limits: } { 0.1,13,100,15,56 RQdiageee ==Through the above-mentioned choice, not only vertical acceleration was adequately reduced, but also suspension displacement remained in its admissible range. Thus, by tracking the above-said reference model’s behavior, a suitable ride comfort and handling could be simultaneously achieved. PID Controller PID controller is one of the widely applicable linear controllers in science and industry16. Many complicated non-linear systems can be controlled by this kind of controller. In this article, the sky-hook controlled linear model has been chosen as the reference model and the difference between the system vertical acceleration and its desired vertical acceleration is applied to the PID controller as an error. PDIde uKeKKedt dt=++ (30) Simulation results in figure 4 show that the controller has a relatively good tracking ability, but the obtained control signal cannot be applied practically. Also, the more time passes after simulation, the higher the tracking error value would be. Also to study the performance of the designed controller in the presence of parameter uncertainties, it is assumed that the vehicle mass and tire stiffness vary as large as 30% and 40%, respectively. Figure 4.g shows the vehicle acceleration responses with and without model uncertainties. The road input for the analysis is the one in figure 4.a. The said results show that the controller is highly dependent on system parameters. Of course, these results are not unexpected. Due to the non-linear dynamic of the system, one should not expect a linear controller to cope with it. Thus, a non-linear controller is designed for the system in the next section of the article. Then, the controller’s performance was studied with the purpose of reducing maximum vertical acceleration irrespective of vehicle handling and stability. Lyapunov Method In general, two approaches exist to nonlinear controller design using Lyapunov direct method, both of which have a trial and error nature. In the first approach, after choosing a control law, a Lyapunov function is sought which justifies the law. Conversely, the second approach requires that a candidate Lyapunov function be chosen and then a suitable law be sought which turns the candidate Lyapunov function into a true Lyapunov function. The second method has been adopted in the present paper. The main objective in controller design is the minimization of the vertical acceleration transferred to passengers which has a direct relation with ride comfort. Considering the state space equations in equation (12) where  denotes the sprung mass and passengers’ acceleration, () 2 4 x  has been selected as the candidate Lyapunov function: PDIde uKeKKedt dt=++ (31) where ( ) 2435 ss BCxxKxAx =-++   (32) Constraining the derivative of the candidate function in Equation (31) to be negative, we have: 44 00 VxxBBBB ==£®£  (33) Substituting equation (32) into Equation (33) and using equations (9)-(13) one can write: () kCBBttC-+x+r-x-+K(x-x)-ss1224MMMM BB=B£0 uuusx+A(x-x)+AxP-xsign(x)552466 \r         (34) Research Journal of Recent Sciences _____________________________________________________________ISSN 2277-2502Vol. 3(8), 86-95, August (2014) Res. J. Recent Sci. International Science Congress Association 90 b) The second road input a) The first road input d) Vertical acceleration c) Vertical acceleration f) Suspension travel e) Suspension Travel h) Control Signal g) Robustness analyze Figure-4 PID controller for suspension system 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Time(s)Height(m) 0 0.5 1 1.5 0 0.005 0.01 0.015 0.02 0.025 0.03 Time(s)Bump Height(m) 0 0.5 1 1.5 2 2.5 3 -0.1 -0.05 0 0.05 0.1 0.15 Time(s)Acceleration(m/s PID Sky Hook 0 0.5 1 1.5 2 2.5 3 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Time(s)Vertical Acceleration (m/s PID SkyHook 0 0.5 1 1.5 2 2.5 3 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Time(s)Suspension Travel (m) 0 0.5 1 1.5 2 2.5 3 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 Time(s)Suspension Travel (m) 0.8116 0.8117 0.8118 0.8119 0.812 0.8121 0.8122 0.8123 1695.5 1696 1696.5 1697 1697.5 1698 Time(s)Control Signal 0 0.5 1 1.5 2 2.5 3 -0.01 0 0.01 0.02 0.03 0.04 0.05 Acceleration(m/sTime(s) With Uncertainty Without Uncertainty Research Journal of Recent Sciences _____________________________________________________________ISSN 2277-2502Vol. 3(8), 86-95, August (2014) Res. J. Recent Sci. International Science Congress Association 91 The above equation can be written in the simplified form: (35) If 2 2 VB l =-, the above equation will always be negative. Using this equation, substituting Equation (34) into the left hand side of equation (35) and simplifying the resulting equation, we have: ()() 1224524656()()tsstuuKCCCxrxkxxAxMMBBAxxAxPxsignxag  +-+--+ \n -= \n \n -+-  (36) Substituting from Equation (32) into Equation (36), the following equation results: ()()656123524()()()()tsstsuussKCCCAxPxsignxxrxkxMMAAxKACxxllbal -=----- --++-(37) If 20 l ³- is chosen, 0 V £  will result. According to equation (14), the control signal, , has a linear relation with the state variable, 6 x . Thus, obtaining a control law for 6 x can be generalized to . 66 () xxu t =-+ (38) Thus, a control rule of the following form is obtained: 66 () xxu t =-+ (39) In figure 5 the vertical accelerations and suspension travels of an active suspension system with a Lyapunov controller and a passive controller are presented and compared. In both cases, the two suspension systems are subject to two road inputs, a bump figure 4.a and a random roughness figure 4.b. It can be seen from figure 5.a and 5.b that in the active control system, the amplitude of vertical acceleration decreases significantly which results in an increase in suspension travel, especially for the second road input. It should be noted that the increase in suspension travel must not exceed the allowed workspace limits. The allowed workspace limit in this study is in the range of 8 ± cm10-12. It is seen in figure 5.d that the suspension travel is close to its limits. It can be predicted that the suspension travel can exceed its limits for other road inputs. As it can be seen, with the Lyapunov controller, vertical accelerations are drastically decreased but the displacements increase to a level that is close to or exceeds the limits of the suspension system workspace. Generally, the objective of this method is only the reduction of vertical accelerations; the method doesn’t take the practical constraints of the workspace into account. Hence, a trade-off between the reduction of vertical accelerations and the increase in suspension travel is not possible in this method 15. a) Vertical acceleration b) Vertical acceleration c) Suspension Travel d) Suspension Travel 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time(s)Acceleration(m/s Lyapunov Passive 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time(s)Acceleration(m/s Lyapunov Passive 0 0.5 1 1.5 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 Time(s)Suspension Travel(m) Lyapunov Passive 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Time(s)Suspension Travel(m) Lyapunov Passive Research Journal of Recent Sciences _____________________________________________________________ISSN 2277-2502Vol. 3(8), 86-95, August (2014) Res. J. Recent Sci. International Science Congress Association 92 e) Robustness analyze Figure-5 Lyapunov Controller for suspension system On the other hand, in the controller design, nominal parameter values were assumed. Such an assumption cannot be made in practice and deviations from nominal values are inevitable. For example, the vehicle mass varies based on factors such as its load, number of passengers, and the amount of fuel. Also, the elasticity of vehicle tires changes based on vehicle speed, temperature increase, and road quality. Also robustness analyze perform for this controller and that presented in figure 5.e. It can be observed that the Lyapunov controller is very sensitive to changes in system parameter values and the amplitude of vertical acceleration with the new parameter values is a multiple of that with nominal parameter values. On the whole, the results of studying the Lyapunov controller indicate that although this controller acts successfully to improve ride comfort by reducing vertical accelerations, it is not robust against model uncertainties. In addition, for various road inputs, suspension travel is increased and cannot be controlled. These results show the necessity of designing a nonlinear controller with which a trade-off is possible between vertical acceleration and suspension travel for various road inputs. To solve the problems of the designed Lyapunov controller with respect to the practical aspects of the suspension system, the sliding mode method, being a non-linear and robust one, was used for designing the controller. Sliding Mode Controller Design By designing such a controller, achieving both ride comfort and suitable handling would be possible. Therefore, the sliding mode control method with the nonlinear and robust nature taken into account was used and its sliding surfaces were obtained according to the sky-hook model which was selected as a reference. However, to have a tradeoff between vertical acceleration and suspension displacements that was the requirement for optimization, an optimized linear controller was first designed for the sky-hook reference model in which by choosing suitable weight coefficients, a compromise could be created between acceleration reduction and increased suspension displacement, something that has been done in the previous section related to PID controller. The sliding mode controller tracked the behavior of the adjusted reference model. This controller will be introduced in the following sections of the article with its relevant simulation results. Comparing equation (12) and the fourth equation in (4) which represents the acceleration of the sprung mass in the nonlinear model and in the sky-hook model, respectively, one can write: 52 AxuCx =- (40) Using the feedback law, ukx =- (41) the actuator force can be obtained from: 2 5d kxCx x A =- (42) Here the error vector is defined as 155 d exx =-where 5 x is the hydraulic pressure drop across the piston and 5 d x is the desired pressure drop. Since equation (13) is of first order, the first sliding surface is the error vector and is defined as below: 155 d sxx =- (43) Selecting the sliding condition as the following first-order differential equation, 111 sks =-  (44) The term 1 s will tend to zero as time passes by. A value of zero for 1 s means that the system has reached the desired state. Satisfying this condition results in 6 d x , the spool valve displacement, is obtained as: 22 4215156 ()()()() ()ss kxCxkxCx AxxkxkAAPxsignxab ++ -+---  (45) To track the desired behavior of the controlled linear reference model, two sliding surfaces are needed to be designed. The first sliding surface is defined with the purpose of tracking the force of the linear actuator and the second is defined for tracking the desired spool valve displacement and obtaining the system input actuator voltage. The actuator voltage is defined in a way that the spool valve displacement coincides with its desired value, which is obtained from satisfying the sliding condition. The second error vector is defined in equation (46). This equation, similar to equation (14), is of first order. The sliding surface is defined in equation (47): 266 d exx =- (46) 266 d sxx =- (47) 222 sks =-  (48) Research Journal of Recent Sciences _____________________________________________________________ISSN 2277-2502Vol. 3(8), 86-95, August (2014) Res. J. Recent Sci. International Science Congress Association 93 The actuator voltage is obtained from the following equation: 26266 (1) dd ukxkxx ttt=-++  (49) Where 12 90 kk == and 1000 C = have been chosen . To simulate the system behavior, in Figure 6, the vertical acceleration and suspension travel have been shown for the actively controlled system with the sliding mode controller and the passively controlled system have been presented for the road inputs in figures 3.a and 3.b. Figure 6 shows that the nonlinear model tracks the behavior of the controlled linear model well and that the vertical acceleration of the active suspension system is smaller than that of the passive system and suspension travel is within bounds for both road inputs. In comparison to the Lyapunov method, the sliding mode controller, not only compensates for the nonlinearities in the model, but it also uses optimization in making a trade-off between ride comfort and practical constraints. This is because the sliding mode controller tracks the behavior of the controlled sky-hook model. The behavior of the controlled sky-hook model can be modified using the weighting matrices and based on design specifications. Figure 7 presents the robustness analysis of the sliding mode controller. It can be seen that the sliding mode controller is more robust against parameter variations. Figure 8 shows the actuator forces of the sliding mode and Lyapunov controllers and table 2 gives their respective control energy obtained from the following equation: 2 0t Eudt = (50) a) Vertical acceleration of the active and passive systems and the reference model b) Vertical acceleration of the active and passive systems and the reference model c) Suspension travel of the active and passive systems d) Suspension travel of the active and passive systems Figure-6 Sliding mode controller for suspension system 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time(s)Acceleration(m/s Sliding Passive Sky Hook 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time(s)Acceleration(m/s Sliding Passive 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 Time(s)Suspension Travel(m) Sliding Passive 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Time(s)Suspension Travel(m) Sliding Passive Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(8), 1-7, August (2014) Res. J. Recent Sci. International Science Congress Association 94 Figure-7 Robustness analysis for the sliding mode controllerFigure-8 Actuator forces for sliding mode and Lyapunov controllers Table-2 Control energy Lyapunov Controller Sliding Mode Controller 26590 J 13740 J Conclusion In this article, a PID linear controller and two nonlinear controllers were designed using the sliding mode and Lyapunov methods for a vehicle active suspension system. Due to the linear nature of the PID controller, it was not successful in controlling the system. Though it had a relatively good tracking behavior, its control signal could not be applied and the system was highly dependent on parameters and did not prove to be robust at all. In Lyapunov method, by choosing the vertical acceleration square as the candidate Lyapunov function, the controller was designed in a way that the chosen function would be a real Lyapunov function. This controller did not control the effective factors of a vehicle’s handling and stability and excessive reduction of acceleration led to increased suspension displacement close to the set boundaries. Also, Lyapunov controller did not have a good robustness against parameter variation. However, by choosing a suitable sliding surface in the sliding mode method, the nonlinear model followed the improved desired linear model’s behavior. Since the LQR optimal controlling method was chosen for controlling the sky-hook reference model, system behavior could be desirably changed by adjusting R and Q weight matrices. Moreover, robustness analysis results indicated that the sliding mode controller showed a good robustness against parameter changes. Reference 1.H. Hashemipour, M. Amiri, M. Mirzaei and A. Maghol, Nonlinear control of vehicle active suspension considering actuator dynamics, Proceedings of the second International Conference on Computer and Electrical Engineering, IEEE, 362-366, (2009)2.Haiping Du, Nong Zhang, H control of active vehicle suspension with actuator time delay, Journal of Sound And Vibration, 19(4), 412-518, (2006)3.M. Eslamian, M. Mirzaei and G. Alizadeh, Enhancement of Vehicle Lateral Stability by Non-linear Optimal Control of Yaw Dynamics, Mech. and Aerospace Eng. J.,2, 3, (2007)4.H.D. Taghirad and E. Esmailzadeh, Automobile Passenger Comfort Assumed through LQG/LQR active Suspension, Vibration and control, (1997)5.Y. Sam, J. Osman and M. Ghani, A class of proportional-integral sliding mode Control with application to active suspension system, Journal system and control,51(3-4),217- 223, (2004)6.M. Kurimoto and T. Yashimura1, Active Suspension of Passenger Cars Using Sliding Mode Controllers (Based On Reduced Models), Int.J. of Vehicle Design, 19(4), 402-414, (1998)7.M. Yokoyama and J. arl. Hedrick , A Model Following Sliding Mode Controller for Semi-Active Suspension System with MR Dampers, Proceedings of the American Control Conference, (2001)8.E. Kose and F. Jabbari, Scheduled Controllers for Linear Systems with Bounded Actuators, Automatica,39, 1377-1387, (2003)9.Strfbrsky, K. Hyniova, J. Honcu and A. Kruczek, Using Fuzzy Logic to Control Active Suspension System for One Half-Car-Model, Acta Montanistica Slovaca, Rocnl'k, , csilo 4, (2003)10.S. Marzbanrad, G. Ahmadi, H. Zohoor and Y. Hojjat, Stochastic Optimal Preview Control of a Vehicle Suspension, Journal of sound and vibration, 275, 973-990, (2004) 0 0.5 1 1.5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Time(s)Acceleration(m/s Ms=290,Kt=190000 Ms=190,Kt=114000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -700 -600 -500 -400 -300 -200 -100 0 100 200 Time(s)Force(N) Lyapunov Sliding Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(8), 1-7, August (2014) Res. J. Recent Sci. International Science Congress Association 95 11.R. Kazemi, A.R. Ohadi and M. Sleymani, Nonlinear Sliding Mode Control OF Vehicle Active Suspension Considering Hydraulic Actuator Dynamics", Mashhad University, ISME, (2003)12.Z. Liu, C. Luo and D. Hu, Active Suspension Control Design Using a Combination of LQR and Backstepping, Proceedings of the 25th Chinese control Conference, 123-125, (2006)13.Ian Fialho and Gary J. Balas, Road Adaptive Active Suspension Design Using Linear Parameter-Varying Gain Scheduling, IEEE Transactions on control systems technology, 10(1), (2002)14.J. Lin, I. kanellakopoulos, Nonlinear Design of Active Suspension, IEEE Control System Magazine, 45-48, (1997) 15.H. Hashemipour, M. Mirzaei and G. Alizadeh, Nonlinear Controller Design for Vehicle Suspension System Considering Practical Constraints, Aerospace MechanicsJournal, 1-13, (2011)16.S.H. Hashemipour, A. ghoreishi, M. Mahdavinasab and M.N. Moghaddasi, PID Controller for Robotic Manipulator Nonlinear Model and compare with Sliding Mode Controller, Research Journal of Recent Sciences, (2013)