Design of Virtual Reference Feedforward Controller for an Active Suspension System

This paper presents a method to design a virtual reference feedforward controller (VRFC) for an active suspension system. Generally, it is not easy to apply a feedforward control to an active suspension system because a reference or disturbance is difficult to measure or estimate. Instead of measuring references or disturbances, a virtual reference on heave motion of a sprung mass representing a bump is defined and used for feedforward control in this paper. Feedforward controller with the virtual reference is combined with feedback controllers such as linear quadratic regulator (LQR) and linear quadratic (LQ) static output feedback (SOF) controller. To fully take advantages of the virtual reference for an active suspension system, it is necessary to find optimal parameters of the virtual reference which maximizes control performance. For the purpose, a simulation-based optimization is formulated and solved by a heuristic optimization method. A simulation with a simulation package shows that the proposed VRFC is quite effective in improving the ride comfort with an active suspension system.


I. INTRODUCTION
Generally, it is considered that the goal of a suspension design for a vehicle is to achieve both ride comfort and road holding. [1,2]. Typically, ride comfort has been evaluated with a vertical acceleration of a sprung mass. For better ride comfort, the vertical acceleration of a sprung mass should be reduced [3,4]. According to ISO2631-1, the target frequency ranges of the vertical acceleration of a sprung mass span between 4 and 10Hz [4]. This paper will focus on improving ride comfort with an active suspension system. An active suspension system has been used to reduce the road-induced vertical acceleration and improve the ride comfort by exerting a control force with some actuators. To date, so many papers have been published to design and implement a controller for active suspension [5][6][7]. Most of these papers have adopted only feedback control. On the contrary, there have been fewer papers on feedforward control for an active suspension system. A typical disturbance used for feedforward control in an active suspension system is a road profile acting on an unsprung mass [7]. LQ optimal control was adopted to design a feedforward controller with a measured or estimated road profile [8,9]. However, it is difficult to measure or estimate a road profile in real-time. For the reason, there have been few papers on feedforward control with a road profile. On the other hand, a road profile can be measured a priori by a look-ahead sensor. With the previewed road profile, a preview control has been applied to date [7,8]. In fact, most of the papers on feedforward control for an active suspension system have adopted a preview control [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Among them, LQ optimal preview control has been widely adopted for designing a preview controller [10][11][12][13][14][15][16]19,21,23]. H optimal preview control, sliding mode control and model predictive control (MPC) were also adopted for preview controller design [13,17,18,20,22,24].
The drawback of the preview control for an active suspension system is that it is difficult to measure a road profile a priori with a sensor. There are several methods to measure a road profile with look-ahead sensors. The most popular method is to use computer vision with a camera or a laser scanner. For example, a radar and laser scanner were used as a preview sensor for active suspension control [25,26]. eActive3 developed by Toyota has laser displacement sensors which are located at the front bumper and are angled downward the front of the vehicle. The laser contact point of this sensor is set to a position 1.4 m forward of the center of the front axle [27]. Active Body Control (ABC) developed by Mercedes-Benz in 2007 has laser scanners used for look-ahead scan on road surface [6,28]. Moreover, Magic Body Control (MBC) developed by the same company in 2013 has the Road Surface Scan function with a stereo camera. It was known that this system can scan road surface up to 15 m ahead of a vehicle at speeds up to 130 km/h [29]. Recent advances in computer vision technology for autonomous driving makes it possible to detect a bump in front of a vehicle with a stereo camera [30][31][32]. The profile of a bump obtained by the stereo camera can be used for preview control. In spite that there are available look-ahead sensors for preview control, installation of these sensors on a mass-produced vehicle requires additional cost and high-performance processor to handle computational burdens. Hence, it is necessary to develop a new method to design a feedforward controller which does not need additional cost and highperformance processor for signal processing on preview sensors for an active suspension system. This paper presents a virtual reference feedforward control (VRFC) for an active suspension system. Fig. 1 shows the overall control structure for an active suspension system proposed in this paper. In Fig. 1, LQR and LQ SOF control are used as a feedback controller. As shown in Fig.  1, a 2-DOF quarter-car model is adopted as a vehicle model because a controller designed with it can be directly applied to the full-car model [33,34]. In this paper, a virtual reference on heave motion of a sprung mass representing a bump is defined with an exponential function or a normal distribution, regardless of a real bump profile. A method to generate a desired road height profile was proposed to be used for feedforward compensation in the previous study [20]. As another case, a virtual feedforward controller (VFC) was proposed for periodic disturbances [35,36]. The VFC proposed in the study was a transfer function with not a reference but a disturbance. Different from the previous studies, the virtual reference proposed in this paper is about not the road profile but the height of a sprung mass. Moreover, the virtual reference has the shape of normal distribution, which has nothing to do with that of a real bump. In other words, the virtual reference does not need information on the shape of a road profile obtained from preview sensors. Hence, it is easier to obtain the virtual reference than a road profile on actual vehicles. The feedforward controller, i.e., VRFC, is designed with the virtual reference. More specifically, the VRFC is just a proportional control (P-control) with the error between the height of a sprung mass and the virtual reference. The VRFC has a single gain for P-control. It is also easy to implement it on actual vehicles. The VRFC is combined with the feedback controllers such as LQR and LQ SOF controller, as shown in Fig. 1. To maximize the performance of the VRFC, it is necessary to optimize the parameters of the virtual reference and the proportional gain of the VRFC. For this purpose, a simulation-based optimization is formulated to find a virtual reference and control gain which give the optimal performance. A heuristic optimization method is employed for simulation-based optimization. For optimization, several objective functions including the vertical acceleration of a sprung mass, suspension stroke and tire deflection are defined. Several VRFCs designed with 3 objective functions are compared among one another through simulation. When applying the VRFC, the speed variation of a vehicle is important because the optimality of the parameters and gain will be broken if the vehicle speed varies over time. Hence, it is necessary to fix the virtual reference and to optimize the single gain of the P-controller over speed variation. This is also done with a simulationbased optimization. To check the performance of the proposed VRFC for active suspension control, a simulation on CarMaker, a vehicle simulation package, is conducted.
The proposed VRFC is compared with conventional approaches, i.e., LQR, LQ SOF and LQ preview controllers, via simulation in terms of ride comfort.
The main contributions of this study are summarized as follows: 1) For active suspension control, the virtual reference is defined and the feedforward controller, the VRFC, is designed with the virtual reference. This type of virtual reference has not been proposed to date. 2) A simulation-based optimization is formulated to find optimum parameters of the VRFC and solved by a heuristic optimization method. 3) To use the VRFC in actual vehicles, the speed variation of a vehicle is considered and VRFC is optimized over speed variation. This paper is composed of four sections. In Section II, LQR, LQ SOF and preview controllers for an active suspension system are designed with 2-DOF quarter-car model. VRFC is defined and optimized in Section III. In Section IV, A simulation with a vehicle dynamics simulation program is performed to evaluate the proposed controllers, and simulation results are analyzed and compared with one another. The conclusions are given in Section V.

A. LQR DESIGN WITH A QUARTER-CAR MODEL
The configuration of a 2-DOF quarter-car model is depicted in Fig. 2. The vehicle body is modeled as sprung and unsprung masses to describe the vertical motions. This model assumes that the spring and damper are linear. For spring, it is assumed that a helical spring is used in the suspension. The force caused by the spring is linear with respect to the suspension stroke because it is associated not with the bending but with the torsion in the wire [37]. For damper, a viscous damping, i.e., linear damping is assumed. Viscous damping is valid when the fluid flow is relatively slow, i.e., laminar [38]. So, the suspension force f with control input u is calculated as (1). Based on the suspension force f, the equations of motion of the sprung and unsprung masses are derived as (2). By replacing f of (2) with that of (1) and rearranging it, the vector-matrix form of (2) is obtained as (3). After defining the vectors and matrices as given in (4), (3) is converted into (5). By defining the state vector as (6), the state-space equation for the 2-DOF quarter-car model is derived as (7) [33]. In (7), the system and input matrices are obtained as (8) from (4).
Based on the state vector of the quarter-car model, the LQ objective function J for LQR is defined as (9). The weight i is used to adjust the effect of each term in J to control gain. The value of i is determined by Bryson's rule given in (10) [39]. In (10), the maximum allowable value  is defined to determine i on the corresponding term in J. Under the condition that all weights are fixed, 1 for the first term of J, the vertical acceleration of the sprung mass, should be set to a lower value. Meanwhile, to improve the road holding, 3 for the tire deflection should have a higher value. After reorganizing J with the state vector (6), the weighting matrices, Q, N, and R, can be derived as given (9). LQR, a full-state feedback controller, is used with the gain matrix K, which is determined by the Riccati equation. Riccati equation is defined with A, B2, Q, N, and R. In this paper, the controller with K is denoted as LQRq.
In the previous literature, LQRq was designed for a full-car model under the assumption that a full-car model can be composed of four quarter-car models [33,34]. The controller derived from LQRq for the full-car model was called LQRfq. As shown in (6) and (11), LQRq and LQRfq do not need roll and pitch angles of a sprung mass for feedback control. However, it was shown that LQRq and LQRfq improved the roll and pitch motions of a sprung mass without measuring the roll and pitch angles of a sprung mass [34].

B. DESIGN OF LQR AND LQ CONTROLLER WITH A QUARTER-CAR MODEL
When implementing LQR based suspension controller with the quarter-car model, the precise measurement of the state variables of (6) is required to achieve the intended control performance. However, it is difficult to directly measure the height and velocity of the sprung and unsprung masses. To cope with the problem, it is desirable to use SOF control with available sensor signals [40,41]. Therefore, SOF control has been adopted for the controller design of the active suspension system [42,43]. In real vehicles, the vertical acceleration of the sprung mass and the suspension stroke are typically available. These signals have been used for semiactive suspension control [44][45][46][47][48]. In the previous work, it is assumed that the suspension stroke and its derivative are available to determine the control inputs [34]. This paper follows the assumption.
The SOF controller is defined as (12). Since the suspension stroke and its derivative are available for SOF control, the vector of sensor outputs is defined as y as given in (13) from the output matrix C and the state vector of (6). Because there are two sensor outputs, the dimension of the gain matrix, KSOF, is two by one. By replacing y in (12) with Cx, the control input of SOF controller, uSOF, is obtained as the full-state feedback form of (14). From (14), Z is the fullstate feedback gain matrix of uSOF.
Optimization problem is formulated to design LQ SOF controller. The optimization problem, (15), is solved to find KSOF by minimizing the J in (9). In (15), the optimization variables are two elements of KSOF. Since the optimization problem for LQ SOF is non-convex and Z is structured, it has been known that there have been no methods, which guarantee to find a global optimum or a stable initial solution [40,41].
In this paper, the heuristic optimization method, CMA-ES, is applied to determine the optimum gain matrix KSOF [49]. A detailed description of the optimization procedure with CMA-ES can be found in the previous work [34]. Let denote the LQ SOF controller, KSOF, which is obtained by solving (15), as LQSOFq. As mentioned earlier, LQSOFq was applied to a full-car model under the assumption that a full-car model can be represented by four quarter-car models [33,34].

C. DESIGN OF LQ OPTIMAL PREVIEW CONTROLLER WITH A QUARTER-CAR MODEL
In general, LQ optimal preview control has been designed in the discrete-time domain. For the reason, the continuous-time state-space equation (7) is converted to the discrete-time one (16) by the discretization (17). In (17), Ts is the sampling time or rate of the discrete-time system.
Let the preview period Tp be a multiple of the sampling period Ts, i.e., pTs. Let w(k) be the vector containing all the preview signals within the preview period at instant k, as given in (18). With the definition of (18), the state-space equation of the vector w(k) is expressed as (19).
With the vector and matrices of (21), (20) is rewritten as the state-space equation of (22).
With the matrices Q, N and R in (9), the new matrices for the augmented system are defined as (23). With those matrices and the augmented system (22), the LQ objective function (9) is also augmented with the previewed disturbances as (24).
The discrete-time LQ optimal preview controller is obtained as (25) from LQR for the augmented system (22) with LQ objective function (24). As shown in (25), LQR for the augmented system has the form of the full-state feedback control, which consists of the feedback and feedforward parts, KFB and KFF, corresponding to the vectors of state and disturbance, respectively. In (25), KFB is identical to K in (11).
The future road profile should be measured for preview control, as shown in (18). Moreover, those signals should be interpolated according to speed variation of a vehicle [50]. This interpolation should be done even though the signals are given via V2V communication from other vehicles. The virtual reference proposed in this paper does not need the future road profile or the shape of a bump because it requires only the center position of a real bump.

A. DESIGN OF A VIRTUAL REFERENCE FEEDFORWARD CONTROLLER
A virtual reference feedforward controller (VRFC) is defined as (26). In (26), zs,ref is a virtual reference on the vertical height of the sprung mass, used for VRFC and kff is the feedforward gain. This is virtual because zs,ref has no physical meaning. In other words, it is assumed that a real bump has the shape of a normal distribution regardless of a true shape of a bump, in this paper. The virtual reference zs,ref is defined as (27). This is an exponential function used to represent a normal distribution in statistics, as shown in Fig. 3. In Fig. 3, the legend Bump represents a real bump through where a vehicle passes. As shown in In (27), the virtual reference is a function of travel distance, x. In (27) and Fig. 3, h is a parameter used to tune the height of the virtual reference.  and  are the mean and standard deviation of the normal distribution, which are used to tune the center position and width of the virtual reference, respectively.
In this paper, the center position of the virtual reference, , is set to that of the real bump. In other words, this is set to the contact point of the tire and the road surface in the quartercar model. This needs information on the location and height of a real bump. Hence, it is assumed that the location and height of a real bump are known a priori. Three parameters, i.e., kff, h, and , are needed to describe the virtual reference.
These parameters should be determined to maximize the control performance of the VRFC. In (27), the parameter h is bounded as (28) in such a way that it is limited to the height of the bump, i.e., 0.1 in Fig. 3. By summing (11), (14) (26), the control inputs of LQRq and LQSOFq are calculated as (29) and (30), respectively.

FIGURE 3. Actual and virtual reference bump profiles
When optimizing (27), the first thing to do is to define an objective function. Generally, the objective of active suspension control is to improve ride comfort or reduce the vertical acceleration of the sprung mass. For the purpose, there are several types of objective functions. The first type This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3184417 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ is to use the LQ objective function, (9). This can be approximated as (31) by discretizing (7). The second type is to use (32). The third type is to use the maximum of the absolute vertical acceleration, i.e., (33). The next step for the optimization is to set a feedback controller used with the VRFC. There are two options: LQRq and LQSOFq as given in (29) and (30). After the objective function and the feedback controller are selected, the optimization is started.
The optimization variables are kff, h, and  in (27). This problem is non-linear and non-convex. Hence, there are no analytic methods to find an optimum [51]. For the reason, a heuristic optimization method is adopted. In this paper, MATLAB built-in function, fminsearch(), is used for optimization. This command is an implementation of Nelder-Mead simplex algorithm. The quarter-car model, (7), with the controller (29) or (30) is implemented with MATLAB/Simulink. In the Simulink code, the spring stiffness and the damping coefficient are replaced with the nonlinear curves, as given by Figs. 4 and 5, respectively. These were referred from Demo_Lexus_NX300h, which is the built-in vehicle model in CarMaker. Once the optimization variables are set to particular values, MATLAB/Simulink code is run by MATLAB command sim() and the variables in the objective functions are obtained from the Simulink code. Then, the objective functions, (31), (32) and (33), are calculated with the variables. Fig. 6 illustrates this procedure. Let denote the controllers of (29) with the VRFC optimized for (31), (32) and (33) as LQRqv1, LQRqv2, and LQRqv3, respectively. Let denote the controllers of (30) with the VRFC optimized for (31), (32) and (33) as LQSOFqv1, LQSOFqv2, and LQSOFqv3, respectively.

B. DESIGN OF A VELOCITY-DEPENDENT VIRTUAL REFERENCE FEEDFORWARD CONTROLLER
The VRFC presented in the previous subsection was designed under the assumption that the vehicle speed is constant. However, this is not valid in actual driving conditions. To cope with the speed change, a new type VRFC is proposed.
New VRFC has constant h, and . The values of h and  are set to 0.05 and 3.9279, respectively. Hence, there is only one parameter, kff, in (29) and (30), to be optimized. The vehicle speed is discretized from 10 km/h to 80 km/h with an interval of 10 km/h. For each speed, the feedforward gain, kff in VRFC, is optimized with the simulation-based optimization method. The parameters, h and , can be optimized over speed change. However, the virtual reference itself varies over speed change. This can distort the virtual reference profile. For the reason, it is assumed that h, and  are constant.
The optimized feedforward gains are interpolated with respect to vehicle speed. With the data, the feedforward gain varies according to the vehicle speed. Let denote this VRFC as VRFFC.

IV. SIMULATION
In this section, the parameters of the virtual reference are optimized, and a simulation is conducted to verify the control performance of the designed VRFF on the vehicle simulation software, IPG CarMaker. IPG CarMaker has been widely used for validation on vehicle stability and active/semi-active suspension control over the last decade [52][53][54][55]. Through the simulation, the objective functions for the virtual reference are compared to one another. Table I shows the descriptions and values of parameters for the quarter-car model. These values were referred from Demo_Lexus_NX300h, which is the built-in vehicle model in CarMaker and has characteristics of both internal combustion engine vehicles and electric vehicles. The weights in the LQ objective functions, (9), are calculated by (10) with the maximum allowable values given in Table II. The weights used in this study focus on the improvement of ride comfort. In other words, the reduction of the vertical acceleration of the sprung mass is the primary goal of the proposed controller. As a consequence, the road adhesion will deteriorate. In this paper, it is assumed that the bandwidth of the actuator is infinite in generating an active control force and that there are no limits on the maximum force of the actuator.

A. OPTIMIZATION ON VIRTUAL REFERENCE FEEDFORWARD CONTROLLER
Three parameters of the virtual reference are optimized for three objective functions, (31), (32) and (33). As mentioned earlier, this is done with MATLAB built-in function, fminsearch().
Figs. 7 and 8 show the virtual references optimized for the three objective functions with the feedback controllers, LQRq and LQSOFq, respectively. As mentioned earlier, the virtual reference is generated for the height of a sprung mass. Hence, it can be expected that the virtual reference near a real bump can give better performance than that far from one. As shown in Fig. 7, LQRqv2 nearly approaches the real bump. On the contrary, LQRqv1 is far from the real bump. Hence, it is expected that LQRqv2 and LQRqv1 give the best and worst performances in terms of ride comfort, respectively. The heights of the virtual reference optimized with LQSOFq as given in Fig. 8 are smaller than those optimized with LQRq as given in Fig. 7. Therefore, VRFC with LQRq is superior to one with LQSOFq.

B. SIMULATION OF VRFC ON CARMAKER
The simulation for the designed VRFCs is conducted on the co-simulation environment with MATLAB/Simulink and the vehicle simulation package, IPG CarMaker. The simulation scenario is a single bump. Three sets of controllers are used for simulation. The first set consists of the passive system, LQRq, LQRqv1, LQRqv2 and LQRqv3. The second set consists of the passive system, LQSOFq, LQSOFqv1, LQSOFqv2 and LQSOFqv3. The third set consists of the passive system, LQRqv2, LQSOFqv2 and the preview controller presented in the subsection III.C.
The vehicle model is Demo_Lexus_NX300h, which is the built-in model of CarMaker. Fig. 9 shows the bump profile used for simulation, which has a length of 3.6 m and a height of 0.1 m. The initial condition of the vehicle is a stand-still position. Then, the vehicle accelerates to 30 km/h by a builtin speed controller in CarMaker. After reaching 30 km/h, the vehicle passes the bump. In the simulation, the tire-road friction coefficient is set to 0.8. The simulation results of the first set of controllers are summarized in Fig. 10. As shown in Fig. 10, three controllers with VRFC are superior to LQRq in terms of ride comfort. This is expected from the fact that VRFCs are optimized after LQRq is designed. Among VRFCs, LQRqv2 shows the best performance in terms of ride comfort. Moreover, LQRqv2 and LQRqv3 show nearly identical performance in terms of ride comfort. This was expected from the results as given in Fig. 7. The difference between LQRqv2 and LQRqv3 is the magnitudes of the control inputs, as shown in Fig. 10-(e). The maximum control input of LQRqv3 is smaller than that of LQRqv2 while these shows the nearly same peaks of vertical accelerations. The most notable feature of VRFC is that the vertical acceleration and the suspension stroke were simultaneously reduced because these are conflicting with each other in general.   11 shows the simulation results for the second set of the controllers, i.e., VRFCs with LQSOFq. As shown in Fig.  11, three controllers with VRFC are superior to LQSOFq in terms of ride comfort. This is expected from the fact that VRFCs are optimized after LQSOFq is designed. As shown in Fig. 11, LQSOFqv2 shows the best performance in terms of ride comfort. Different from the results in Fig. 7, LQSOFqv3 is superior to the other VRFCs in terms of other measures. Hence, it is desirable to use LQSOFqv2, i.e., LQSOFq with VRFC, for active suspension control. Different from the controllers, LQRqv2 and LQRqv3, the peak values of the control inputs and the suspension strokes of VRFCs with LQSOFq are nearly same to one another. The simulation for the third set of controllers, i.e., the passive system, LQRqv2 and LQSOFv2, and LQ preview controller is conducted. For the LQ preview controller, the sampling time and the preview period were set to 0.001sec and 0.1sec, respectively. Fig. 12 shows the simulation results for the third set of the controllers. As shown in Fig. 12-(a) and -(b), the VRFC, i.e., LQRqv2, is superior to the LQ preview controller in terms of ride comfort. LQSOFqv2 is comparable to the LQ preview controller. As shown in Fig.  12-(c) and -(d), the LQ preview controller gives larger positive suspension strokes and smaller tire deflections than the others. This is typical for the LQ controller designed with the full-car model. Hence, it is desirable to use the VRFC, LQRqv2, instead of the LQ preview controller for active suspension control.  Tables III shows the peak-to-peak values of the responses of the front left corner in Figs. 10, 11 and 12 for each controller, respectively. Table IV shows the percentage  reduction of the responses of the front left corner with respect  to the passive case, calculated from Table III. In these tables, the suspension stroke and the tire deflection are abbreviated to SS and TD, respectively. As shown in Tables III and IV,  LQRqv2 and LQRqv3 give the smallest vertical acceleration than LQRq, LQRqv1 and the LQ preview controller. In other words, these controllers can provide the best performance in terms of ride comfort. For example as shown in Table IV, the vertical accelerations of LQRv2 and LQRv3 were reduced to 16% of the passive case and to 30% of LQRq. In terms of the pitch angle, LQRqv2 shows the best performance among controllers although it uses the largest control input. Compared to LQRq with VRFC, LQSOF controllers with VRFC shows poor performance in terms of all measures. In view of the feedforward control, LQRv2 is better than the LQ preview controller in terms of ride comfort. The vertical accelerations of LQRv2 and the LQ preview controller were reduced to 16% and to 29% of the passive suspension. Hence, it is not recommended that VRFC is combined with LQSOFq This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.   Fig. 13, these gains significantly increase over 60 km/h. This is natural because the vehicle speed has a large effect on the vertical acceleration of the sprung mass. In fact, it is very hard for the active suspension to reduce the vertical acceleration of the sprung mass over 60 km/h due to actuator limitations on the maximum force, bandwidth, and moving velocity. For the reason, the control performance of VRFFC deteriorates as the vehicle speed increases over 60 km/h. This fact can be checked in Fig. 14. As shown in Fig. 14-(c), the active suspension controllers, LQRq and LQRq+VRFFC, have little effects on controlling the vertical acceleration of the sprung mass if the vehicle speed is over 80 km/h. On the contrary, VRFFC shows good performance in controlling the vertical acceleration of the sprung mass near 30 km/h.

V. CONCLUSION
In this paper, the virtual reference on heave motion of a sprung mass representing a bump was proposed for an active suspension system under the assumption that a real bump has the shape of a normal distribution regardless of a true shape of a bump. This virtual reference has the shape of normal distribution, whose two parameters are used to describe it. The feedforward controller with the form of P-control, called VRFC, was designed with the virtual reference. To maximize the performance for active suspension control, the two parameters of the virtual reference and feedforward gain were optimized with MATLAB/Simulink model and fminsearch() given in MATLAB. To cope with the speed variation of a vehicle, the feedforward gain was optimized with respect to a particular vehicle speed while the other parameters were fixed. With this manner, the feedforward gain varies according to the change in vehicle speed. To check the performance of the VRFC, the simulation on the vehicle simulation package, CarMaker, was conducted. From the simulation results, it was confirmed that the vertical acceleration of the VRFC designed with the quadratic objective function on it reduced to 16% of the passive case and to 30% of LQRq. Hence, in terms of ride comfort, it can be concluded that the VRFC designed with the quadratic objective function on the vertical acceleration is quite effective. It was also confirmed that the VRFC has little effect on ride comfort as the vehicle speed increases over 80 km/h. Further research can include the design of a tracking controller to make the height of the sprung mass follow the virtual reference.