Design of a Modal Controller with Simple Models for an Active Suspension System

This paper presents a method to design a modal controller with simple 1-DOF models for an active suspension system. Full-state feedback controller, especially, linear quadratic regulator (LQR) and H∞ controller, designed with 7-DOF full-car model is hard to implement in actual vehicles because there are so many state variables and gain elements needed to be precisely measured and finely tuned, respectively. To overcome the problem, it is required to design a simple controller with a smaller number of gain elements and sensor signals. For the purpose, a modal controller is designed from controllers designed with three 1-DOF models describing heave, roll and pitch motions of a sprung mass. With these 1-DOF models, discrete-time LQR and sliding mode control (SMC) are adopted to design three feedback controllers which generate vertical force, roll and pitch moments for controlling the heave, roll and pitch motions of a sprung mass, respectively. In the modal controller, three control inputs are converted into active forces at four corners with input decoupling transformation. The modal controller is a type of static output feedback (SOF) one. By LQ SOF control methodology, the modal controller itself is designed with a heuristic optimization method. A frequency domain analysis and a simulation on vehicle simulation software, CarMaker®, show that the proposed modal controllers are effective in controlling the active suspension system for ride comfort.


I. INTRODUCTION
Generally, ride comfort and road holding are two key goals for vehicle suspension design. These are directly related to two physical measures: vertical acceleration of a sprung mass and tire deflection, respectively [1]. According to ISO2631-1, ride comfort is evaluated with vertical acceleration of a sprung mass [2,3]. On the other hand, Road holding depends on tire deflection. It has been well known that there is conflict between the two goals, i.e., ride comfort and road holding [1,4,5]. In other words, the one is improved while the other is deteriorated. An active suspension system has been developed for the purpose of improving ride comfort by reducing the vertical acceleration of a sprung mass with an active actuator. Up to now, lots of studies on controller design methods for an active suspension system have been done [6][7][8]. Since 2015, LQR, LQ SOF control, H control, fuzzy control, adaptive control, sliding mode control, back-stepping control, neural network based control and model predictive control (MPC) have been adopted as controller design methodology [9][10][11][12][13][14][15][16][17][18][19][20][21][22].
Most of these papers have taken actuator limitations and state constraints such as actuator saturation, actuator bandwidth, dynamic tire load and suspension space limits into account in controller design stage. However, no actuator dynamics was considered nor experimental investigation was done in those studies except the reference [12].
Among the controller design methodologies applied to an active suspension system, LQR and H control have been adopted because those can systematically design a full-state feedback controller [1,4,5,13,21,22]. Those controllers have been designed with the 2-DOF quarter-car, 4-DOF half-car and 7-DOF full-car models. Hereafter, the terms 2-DOF and 7-DOF are omitted. Most of studies [9][10][11][12][13][14][15][16][17][18][19][20][21][22] have used the quarter-car model except two studies [13,15] because it is simple to be used for controller design and performance analysis on active suspension system. However, this model cannot handle the roll and pitch motions of a sprung mass. Whereas, the full-car model is most pertinent to controller design for an active suspension system because it can handle the roll and pitch motions of a sprung mass [5]. Meanwhile, a LQR designed with the quarter-car model has not been used for the full-car case except in two studies [21,22]. In the previous studies, the LQR designed with the quarter-car model was adopted for the full-car model and it was verified that it can provide equivalent performance to LQR designed with the full-car model [21,22].
The number of state variables in the state-space equation derived from the full-car model is fourteen. Hence, these state variables must be measured if the full-car model is used to design a full-state feedback controller, e.g., LQR and H controller, for an active suspension system. However, it is difficult to measure the state variables in the model using sensors on actual vehicles. To overcome this problem, a state observer or an output feedback controller such as Kalman filter or LQG has been designed with the model [13,23]. Moreover, it is also difficult to implement a full-state feedback controller designed with the full-car model on an actual vehicle because the size of gain matrix in the full-state feedback controller is large. For example, there are four control inputs in the full-car model. Hence, the dimension of the gain matrix of the full-state feedback controller is 414, which is too large to be implemented in actual vehicles. For this reason, it is challenging to implement the full-state feedback controller on actual vehicles or on vehicle simulation software such as CarMaker or CarSim. Therefore, it is required to design a controller with fewer sensor signals for feedback control and fewer elements in the gain matrix.
For the purpose in this paper, a modal control can be adopted as a controller design methodology [24][25][26][27][28][29][30]. The first step of the modal control is to design three controllers separately for heave, roll and pitch motions of a sprung mass [24,30]. Let these controllers denote heave, roll and pitch controllers, respectively. The second step of the modal control is to convert three control inputs calculated from heave, roll and pitch controllers to four ones at each corner with input decoupling transformation [25,26,29]. The controllers designed in the first step have a simpler structure and require a smaller number of states for feedback.
Following the idea of the modal control, this paper uses three 1-DOF simple models describing heave, roll and pitch motions of a sprung mass. Three controllers, i.e., heave, roll and pitch controllers, are designed with corresponding three 1-DOF models using discrete-time LQR and SMC in order to generate vertical force, roll and pitch moments, respectively. Each controller requires two state variables for feedback. So, six state variables are needed for active suspension control. Three control force and moments are converted into four vertical control forces at each corner. Compared to LQR designed with the full-car model, this controller requires six state variables and has six corresponding gain elements. Fig. 1 shows the schematic diagram of the modal controller to be designed in this paper [25,29]. As shown in Fig. 1, the heave, roll and pitch controllers require two sensor signals, and generate the vertical force and roll and pitch moments. These three control inputs are converted into control forces at four corners with input decoupling transformation.  The modal controller designed in this paper is a type of structured SOF controller [21,25,29]. For example, the dimension of the gain matrix of the modal controller adopted in this paper is 36, which means that there are six sensor signals and three control inputs. However, there are six non-zero gain elements in the gain matrix. Hence, this is a structured controller, as given in [22,25,29]. For this reason, the gain matrix of the modal controller can be determined by LQ SOF control methodology. In this paper, a heuristic optimization method is adopted to find an optimum gain matrix of the modal controller.
The contributions of this paper are condensed as follows: 1) For active suspension control, the modal controller is designed with 1-DOF simple models describing the heave, roll and pitch motions of a sprung mass. 2) Discrete-time LQR and sliding mode control are adopted to design controllers with 1-DOF simple models. Among them, sliding mode control is quite simple to design.
3) The modal controller is designed with LQ SOF control methodology. For the purpose, an optimization problem is formulated and the solution of the problem is obtained by a heuristic optimization method. To evaluate the performance of the proposed modal controllers for an active suspension system, an analysis on frequency responses and a simulation on the vehicle simulation software, CarMaker, are done. The designed modal controllers are compared with one another using the analysis and simulation from the viewpoint of ride comfort.
This paper is organized with four sections. In Section II, discrete-time state-space equations are derived from quarter-car, full-car and three 1-DOF models. With those equations obtained from 1-DOF models, the discrete-time LQR and sliding mode controller are designed. In Section III, for the designed controllers, singular value plots are drawn and simulations on a vehicle simulation software are done. Section IV concludes this paper.

II. CONTROLLER DESIGN FOR ACTIVE SUSPENSION
A. CONTROLLER DESIGN WITH QUARTER-CAR AND FULL-CAR MODELS Fig. 2 shows a quarter-car model, which describes the vertical motions of the sprung and unsprung masses. The control input uq is generated by an actuator. The disturbance is the road profile. The force fq acting on the sprung and unsprung masses is calculated as (1) with the suspension displacement and its velocity. With fq, the equations of motion of the quarter-car model are derived as (2). The state vector of the quarter-car model is defined as (3). With the definition and some algebraic manipulations on (2), the continuous-time statespace equation for the quarter-car model is obtained as (4). the system and input matrices, Aq, B1q and B2q, are given in (5). The procedure of how to obtain these matrices can be found in the previous study [21]. Those matrices are discretized into q, q and q by (6) with the sampling time Ts. After discretization, the discrete-time state-space equation is obtained as (7).
With the state variables and control input of the quartercar model, LQ objective function for an active suspension system is defined as (8). The weights 1/i 2 in (8) are set by Bryson's rule, where i is the maximum allowable value on the corresponding term [31]. If a particular i is reduced while maintaining the other weights constant, then the corresponding term will decrease. For the purpose of improving ride comfort, 1 on the vertical acceleration of the sprung mass must be set to a lower value while maintaining the other i constant. Whereas, for the purpose of enhancing road holding or cornering, 2 and 3 on the suspension displacement and the tire deflection must be set to a higher one, respectively. The LQ objective function, (8), is converted into the vector-matrix form, (9), with the weighting matrices, Qq, Nq and Rq. LQR is a full-state feedback controller, as given in (11), which minimizes Jq. The controller gain matrix Kq is obtained by solving Riccati equation comprising q, q, Qq, Nq and Rq. As shown in (11), the number of elements in the gain matrix is 4, which is the same as the number of state variables given in (3). Fig. 3 shows a full-car model, which describes the heave, roll and pitch motions of a sprung mass, and the vertical motions of 4 unsprung masses. In Fig. 3, the four corners, i.e., front left (FL), front right (FR), rear left (RL) and rear right (RR) ones are indexed as , ,  and  for convenience, respectively. In the model, there are 4 external disturbances, i.e., the road profiles, zr1, zr2, zr3 and zr4, on the unsprung mass. The longitudinal and lateral accelerations caused by braking and cornering are not considered in this paper. The suspension forces at each corner in the model are calculated in (12). In (12), ui is the control input or active force acting on the i-th suspension or corner. The equations of motion for the sprung and unsprung masses are derived as (13) and (14), respectively. The equations of motion for the sprung mass in (13) can be rewritten into the vectormatrix form of (15). In (15), the matrix G stands for the geometric relationship between the suspension forces fi at each corner and the vertical force, roll and pitch moments acting on the sprung mass. This is called input decoupling transformation matrix or modal transform matrix or modal decomposition matrix [24,25,30].
f c 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.
To derive the state-space equation, new vectors and matrices are defined in (16) and (17), respectively. With those vectors and geometric relationship of a vehicle, the vertical displacements of four corners are calculated as (18) [21]. With (16), (17) and (18), the suspension forces are converted into the vector-matrix form of (19). The equations of motions of the sprung and unsprung masses are converted into the vector-matrix form of (20). By replacing f(t) in (20) with (19) and converting (20) into the vector-matrix form, (21) is obtained. With the new definitions of matrices as given in (22), (21) is converted into (23). From the definition of the state vector xf and (23), the state-space equation of the full-car model is derived as (24) [21]. The matrices Af, B1f and B2f in (24) are given in (25). By using the formula of (26), the continuous-time state-space equation is discretized into the discrete-time state-space one (27) LQ objective function for the full-car model is defined as (28). The weights 1/i 2 can be set by Bryson's rule, just as (8) [31]. The LQ objective function, (28), is converted into the vector-matrix form, (29), with the weighting matrices, Qf, Nf and Rf. LQR is a full-state feedback controller, as given in (30), which minimizes Jf. The gain matrix Kf of LQR is obtained by solving Riccati equation comprising f, f, Qf, Nf and Rf. Let denote this controller as LQRf1.
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.
By setting 5 ~ 10 to infinity, the LQ objective function (28) is reduced into (31), which has the same terms as (8).
The LQ objective function, (31), is converted into the vector-matrix form, (32), with the weighting matrices, Qfq, Nfq and Rfq. It is easy to obtain Kf by solving Riccati equation comprising Qfq, Nfq and Rfq. Let denote this controller as LQRf2.
A LQR requires full-state feedback. Hence, it is required that all state variables should be available for feedback. As shown in (16), there are 14 state variables in the full-car model. The dimension of Kf, given in (30), is 414. Hence, 14 state variables must be measured or estimated for Kf. However, it is very difficult to measure or estimate those variables with sensors in actual vehicles. For this reason, it is required to design a controller with a smaller number of state variables and gain elements. In the previous research, the controller Kq of (11) is used to derive Kf [21,22]. The procedure of how to derive Kf from Kq can be found in the previous studies [21,22]. Let denote the full-state feedback controller for the full-car model derived from Kq as LQRfq.

B. CONTROLLER DESIGN WITH SIMPLE 1-DOF MODELS
In the previous study, 3-DOF reference model was adopted to describe three motions, i.e., heave, roll and pitch motions of a sprung mass [27]. In the model, three motions were coupled with one another. In other words, there are crosscoupling terms on stiffness and damping matrices in the equations of motions under the basic assumption that the sprung mass is a rigid body. On the other hand, three separated 1-DOF models describing heave, roll and pitch motions are adopted in this paper. Fig. 4 shows three 1-DOF models, which describe the heave, roll and pitch motions of the sprung mass. In Fig. 4, Fzc, M and M are the vertical force, roll and pitch moments needed to control the corresponding motion of the sprung mass, respectively. Different from the model used in the previous study, these models have coupling terms among them.  The equations of motion for 1-DOF models are given in (33). In (33), Fzc, M and M, are the control inputs of the 1-DOF heave, roll and pitch models, respectively. The parameters of three 1-DOF models are calculated as (34) from the geometry and those of springs and dampers in the full-car model in Fig. 3.
The procedure how to derive K, C, K and C is explained. Fig. 5 shows the free body diagram of 1-DOF roll model. In Fig. 5, xf and xr are the vertical displacement of the sprung mass at each corner caused by roll motion. The equation of roll motion is derived as (35). xf and xr are calculated as (36) from the geometry. The suspension forces acting on the sprung mass are calculated as (37) from (12) and (36). By replacing fi of (35) with those of (37), (35) is rearranged into (38). From (38), K and C are obtained as (34). With the same manner, K and C of 1-DOF pitch model are calculated as (34). This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and   (35) sin For controller design, the state-space equations of these models are needed. The state-space equation of the 1-DOF heave model is derived as an example. The state variables of 1-DOF heave model are defined as (39). With the definition, the state-space equation is derived as (40). Under the assumption that the sampling time is Ts, the discrete-time state-space equation of 1-DOF heave model is obtained as (41). In (41), the definitions of the matrices z, z, and z are calculated by the identical way to (6). For the 1-DOF roll and pitch models, the identical procedure can be applied with the definitions of state variables, given in (42), in order to obtain the discrete-time state-space equations as (43).
With the discrete-time state-space equations of (41) and (43), LQR is designed. The LQ objective functions are defined as (44), (45) and (46) for the 1-DOF vertical, roll and pitch models, respectively. The full-state feedback controllers or the control inputs of these models are defined as (47). The controller gain matrices, KLQR,z, KLQR, and KLQR,, in (47) are calculated by solving Riccati equations with the matrices in the state-space equations and LQ objective functions.
For sliding mode controller design, the sliding surface for the 1-DOF heave model is defined as (48). To make the sliding surface be zero, the convergence condition is given in (49). However, the condition (49) is so severe that the control input obtained from it becomes quite large. As a result, responses of a vehicle with the controller designed with (49) show severe chattering phenomena [32]. To avoid the case, the new relaxed convergence condition (50) is used. Generally, a necessary and sufficient condition for the existence and the sliding motion and the convergence of sliding surface onto hyperplane is given as (51). Evidently, the new condition, (50), satisfies (51) [33]. In (50), z is the parameter used to tune the convergence speed. If z becomes smaller, the convergence speed gets faster and the control input does larger. According to the previous study, there is chattering in the responses of a vehicle and the control input of the controller if z is set to 0.3 or less in spite of better control performance [32]. By combining (41), (48) and (50), the equation (52) is obtained. From (52), the vertical force Fzc is calculated as (53). In (53), () + represents the pseudoinverse. With this manner, the control roll moment M and the control pitch moment M can be easily derived as (54).
For closed-loop stability, the tuning parameters, z,  and , should be less than 1.
Different from MDLQR and MDSMC, the six gain 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.  (31). In other words, the six gain elements in KFB should be determined in order to design the LQ SOF controller. If these elements can be found to minimize the LQ objective function (28) using some heuristic optimization methods, then it is the optimal LQ SOF controller. This problem is formulated as the optimization one, (60). In (60), eig(X) and mag(y) are the operators used to calculate the eigenvalues of the matrix X and the magnitudes of the complex numbers y, respectively. In this paper, the evolutionary strategy, CMA-ES, is used to find the optimum KFBS [38]. The stability of the closed-loop system with KFBS is guaranteed by solving (60) because the objective function Jfs cannot be calculated if the closedsystem is unstable. Let denote this structured LQ SOF controller as SLQSOF.
As shown in (44)

III. SIMULATION
In this section, analysis on frequency responses and simulation are performed to evaluate the performance of the designed controllers, MDLQR, MDSMC and SLQSOF. Through frequency response analysis, five controllers, LQRf2, LQRfq, MDLQR, MDSMC and SLQSOF, are compared to one another. In the previous work, it was shown that LQRf2 and LQRfq are equivalent to each other [22]. So, these controllers are used as a base one for comparison. Table I shows the parameters of 7-DOF full-car and 1-DOF simple models, which were referred from Demo_Lexus_NX300h given in CarMaker. LQRf2 and LQRfq are designed with the weights given in Table II. To avoid the chattering in control responses, the values of the tuning parameters, z,  and , in MDSMC are set to 0.6.  Table III shows the largest magnitudes of the eigenvalues of the closed-loop system with the designed controllers. For stability, the largest magnitude of the eigenvalues of the discrete-time closed-loop system should be less than 1. As shown in Table III, all the designed controllers are stable because the largest magnitudes of the eigenvalues are smaller than 1.

A. ANALYSIS ON REQUENCY RESPONSES
To check the effects of MDLQR, MDSMC and SLQSOF on ride comfort, singular value plots were drawn from the road profile into several outputs. To draw the plots for these controllers, the state-space equation of the full-car model, (27), with the parameters given in Table 1 was used. Fig. 6 shows singular value plots from the road profile into several outputs. In these plots, the outputs are those of the sprung mass, the suspension displacement and the tire deflection. As shown in Fig. 6-(a), MDSMC show the best performance in controlling the vertical acceleration of the sprung mass. This means that MDSMC can provide good ride comfort. Moreover, MDLQR is also effective for ride comfort. Notable feature of MDLQR and MDSMC, i.e., a modal controller, is that those make ride comfort near 10Hz worse than the passive system, and that the resonant frequency of the tire was move to slightly lower. These are typical phenomena when using a modal controller [39]. On the other hand, SLQSOF shows worse performance in terms of ride comfort, compared to MDLQR and MDSMC. This tendency is also valid to the height of the sprung mass, as shown in Fig. 6-(b). As shown in Fig. 6-(c) and -(d), five controllers are nearly equivalent to one another in controlling the roll and pitch motions of the sprung mass. As a counter effect to good ride comfort, the suspension displacements were increased and the tire deflections decreased. This means that road holding was deteriorated by those controllers.
Among those controllers, MDSMC shows the largest suspension displacement and the least tire deflection within the range from 2 to 10Hz as a result of the minimum vertical acceleration and height of the sprung mass. This is wellknown trade-off between ride comfort and road holding. From the frequency response analysis, it can be concluded that MDSMC is the best controller in terms of ride comfort.

B. SIMULATION IN VEHICLE SIMULATION SOFTWARE WITH THE DESIGNED CONTROLLERS
Simulation was conducted on the vehicle simulation software, CarMaker, connected with MATLAB/Simulink. IPG CarMaker has been widely used for validation on vehicle stability and active/semi-active suspension control over the last decade [40][41][42][43]. The vehicle model used for simulation was Demo_Lexus_NX300h provided in CarMaker. Fig. 7 shows the single bump profile used for simulation. There were no actuator models in the simulation. In other words, an actuator used to generate a control force has infinite bandwidth and has no limits on the magnitude of it. The vehicle speed was set to 30km/h and maintained as it by a speed controller given in CarMaker. Then, the vehicle passes the bump. The tire-road friction coefficient is set to 0.85. There are no roll motions since the bump is evenly applied to left and right wheels.  , MDLQR shows the best performance in controlling the vertical acceleration and the pitch angle of the sprung mass. This is natural because MDLQR generates the largest control inputs, as shown in Fig. 8-(e). As a result, MDLQR gives the largest suspension displacement and the smallest tire deflection, as shown in Fig. 8-(c) and -(d). MDSMC shows relatively good performance in terms of ride comfort. These results are different from those presented in the frequency response of Fig. 6. On the other hand, SLQSOF shows good performance in controlling the pitch angle. However, it gives poor performance in controlling the vertical acceleration. This is opposite to LQRfq. LQRfq shows relatively poor performance in controlling the pitch angle because the LQ objective function of LQRfq has no terms on it.  Tables IV and V show the peak-to-peak and root-meansquare values of the responses of Fig. 8 for each controller at front left corner, respectively. In Tables IV and V, SD and TD denotes the suspension displacement and the tire deflection at front left corner, respectively. As shown in Tables IV and V, MDLQR shows the best performance from the viewpoint of ride comfort. MDSMC can be also good choice for ride comfort. However, SLQSOF is not recommend because it cannot control the vertical acceleration of the sprung mass. These results are coincided with those given in Fig. 8. Three controllers, MDLQR, MDSMC and SLQSOF, have the identical structure of (57). The difference among them is controller design methodology. So, it is not valid that these controllers are compared with simulations under the identical condition. For example, the performance of MDSMC can be improved by decreasing the value of . Moreover, the performance of SLQSOF can be improved by reducing the value of 1 in (31).
The key point of this paper is to reduce the number of elements of the controller gain matrix, which is identical to that of signals used for feedback control. The number of the gain elements of LQRfq is 4, which means that it requires 4 signals for feedback control. On the contrary, the number of the gain elements of MDLQR, MDSMC and SLQSOF is 6. This is much smaller than that of LQRf2, i.e., 56. Moreover, MDLQR and MDSMC are much easier to design than LQRf2, LQRfq and SLQSOF.

IV. CONCLUSION
In this paper, the modal controller was derived from three controllers, i.e., heave, roll and pitch controllers, designed with three 1-DOF models describing the vertical, roll and pitch motions of the sprung mass. In the discrete-time domain, heave, roll and pitch controllers are designed with these 1-DOF models by applying discrete-time LQR and SMC. The control inputs of these controllers are converted into four ones at each corner in the full-car model with input decoupling transformation. The modal controller requires six state variables and has six gain elements. Hence, the number of gain elements of these controllers is much smaller than those of LQR for the full-car model. By the nature of structured and SOF control of the modal controller, the modal controller itself was designed by the heuristic optimization method, i.e., CMA-ES. Through a frequency response analysis and a simulation on the vehicle simulation package, the modal controllers designed with LQR and SMC show quite effective in controlling the motions of the sprung mass in terms of ride comfort. Further research will include the design of feedforward controller design with measured acceleration signals, which can improve the control performance against external disturbances.