Strojniški vestnik - Journal of Mechanical Engineering 61(2015)6, 374-382 © 2015 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2014.2224 Original Scientific Paper Received for review: 2014-10-08 Received revised form: 2015-03-23 Accepted for publication: 2015-04-08 Design of Anti-Roll Bar Systems Based on Hierarchical Control Balazs Varga1 - Balazs Németh2 - Péter Gaspar12* 1 Budapest University of Technology and Economics, Department of Control for Transportation and Vehicle Systems, Hungary 2 Hungarian Academy of Sciences, Institute for Computer Science and Control, Systems and Control Laboratory, Hungary This paper proposes the modelling and control design of active anti-roll bars. The aim is to design and generate active torque on the chassis in order to improve roll dynamics. The control system also satisfies the constraint of limited control current derived from electrical conditions. The dynamics of the electro-hydraulic anti-roll bar are formulated with fluid dynamical, electrical and mechanical equations. A linear model is derived for control-oriented purposes. Several different requirements and performances for the control influence the hierarchical handling of the control design. In the hierarchical architecture, a high level improves chassis roll dynamics via a gain-scheduling linear quadratic (LQ) control, while a low level guarantees the input limitation and produces the necessary actuator torque by a constrained LQ control. The operation of the designed anti-roll bar control system is illustrated through simulation examples. Keywords: anti-roll bar, hydraulic actuator, gain-scheduling, LQ, automotive control application Highlights • Electro-hydraulic actuator modelling of an anti-roll bar system. • Constrained LQ control design for the actuator control. • Development of a gain-scheduling strategy for the control of roll dynamics. • Handling slow and fast roll dynamics performances together in the control. 0 INTRODUCTION AND MOTIVATION The improvement of roll dynamics is a relevant problem in vehicles with a high centre of gravity. Several roll-control systems that enhance the protection of cargo and improve roll stability have been developed. One of the most preferred roll-control solutions is anti-roll bars, which increase the stiffness of the suspension system. In this control system, torsion bars connect the left- and right-side suspensions on an axle. Active anti-roll bars can adapt to the current road conditions and lateral effects, while roll stability is improved. Several papers propose methods to reduce the chassis roll motion of road vehicles. Three different active systems are applied: anti-roll bars, auxiliary steering angle and differential braking forces [1]. Active anti-roll bars commonly apply hydraulic actuators to achieve appropriate roll moment [2]. In [3], an active roll-control system based on a modified suspension system is developed with the distributed control architecture. Active steering uses an auxiliary steering angle to reduce the rollover risk of the vehicle. However, this method also influences the lateral motion of the vehicle significantly [4]. The advantages of the differential braking technique are its simple construction and low cost [5]. In this case, different braking forces are generated on the wheels to reduce the lateral force. Several papers deal with the integration of the systems mentioned above. In [6], the integration of the active anti-roll bar and active braking is presented, while [7] investigates the coordination of active control systems, which could be controlled to alter the vehicle rollover tendencies of the vehicle. The benefits of the integration of anti-roll bars and the lateral control are presented in [8]. Furthermore, the control design of anti-roll bars for the articulated vehicles is a significant and novel topic in [9]. An analysis of the snaking stability of a tractor-light trailer vehicle, in which the trailer contains anti-roll bars is presented in [10]. A special construction of semi-active anti-roll bars, which guarantees both ride and roll performances is shown in [11]. The ride and roll performances for an active anti-roll system using a PID control are analysed in [12]. The active system proposed in this paper integrates an electro-hydraulic actuator into an anti-roll bar. The system contains a high-level controller, which improves the roll dynamics of the chassis using active torque; thus, the roll motion of the chassis is influenced. The high-level control strategy is realized with a gain-scheduling Linear Quadratic (LQ) controller. The actuator of the anti-roll bar is an oscillating hydromotor with a servo valve on the low level. The actuator control guarantees the generation of the necessary active torque and satisfies the input constraint of the electric circuit. The control design is based on a constrained LQ method [13]. The goal of this paper is to demonstrate a multi-level control design of an anti-roll bar system. The paper is organized as follows. Section 1 presents the control-oriented formulation of chassis 374 *Corr. Author's Address: Budapest University of Technology and Economics, H-1111, 2. Stoczek utca, Budapest, Hungary, gaspar.peter@mail.bme.hu Strojniski vestnik - Journal of Mechanical Engineering 61(2015)6, 374-382 roll dynamics and the electro-hydraulic actuator using fluid dynamical, electrical and mechanical equations. Section 2 describes the architecture of the active anti-roll bar control system and details the design methods of the vehicle dynamics and actuator controllers with demonstration examples. The actuation of the control system is illustrated with a simulation example in Section 3. Finally, Section 4 summarizes the contributions of the paper. 1 CONTROL-ORIENTED SYSTEM MODELLING In this section, the mechanical and hydraulic equations expressing the operation of the actuator are presented. The linear vehicle model, describing the roll dynamics of the chassis is modelled, which is enhanced with the active anti-roll bar system. The actuator for this system consists of a hydromotor and a valve. The four degrees-of-freedom vehicle dynamic model is illustrated in Fig. 1. 1.1 Modelling of Chassis Roll Dynamics Concerning the rolling motion of the chassis (sprung mass), an anti-roll bar is required so as to reduce the effect of load transfer and roll angle. The intervention of the anti-roll bar system is a force couple on the unsprung masses, which is provided by an active torque of the electro-hydraulic actuator Mact. Lateral force Fat on the vehicle chassis and road excitations on the wheels g01, g02 are disturbances active in the system. In the model, the masses, spring stiffness, damping ratios and geometrical parameters are constants. h is the distance between the roll centre of the chassis, and its centre of gravity and r is the half-track of the vehicle. The length of the anti-roll bar arm in the longitudinal direction is denoted by aarm. In the model, the effects of the side-slip angle and under-/oversteering are ignored. The vehicle dynamics are derived from the Euler-Lagrange formalism in four second-order differential equations: mz = —(dJ + d2)z — (d2r - dlr)
and their derivatives. The control input uveh =Mact of the system is the active torque generated by the electro-hydraulic actuator. The disturbances of the system Wveh =[ go2' F,ali are road excitations on the wheels and lateral forces. 1.2 Electro-Hydraulic Actuator Model of Anti-Roll Bar System The active torque Mact is generated by the electro-hydraulic actuator. The actuator that realizes the torque is an oscillating hydromotor, see Fig. 2. An oscillating hydromotor is a rotary actuator with two cells, separated by vanes. The pressure difference between the vanes generates a torque on the central shaft, which has a limited rotation angle. The anti-roll bar is split in two halves, and the motor connects them. The shaft of the motor is connected to one side of the roll bar, and the housing is to the other. When the vehicle chassis rolls, a torque appears in the house, Design of Anti-Roll Bar Systems Based on Hierarchical Control 375 Strojniski vestnik - Journal of Mechanical Engineering 61(2015)6, 374-382 which can be countered by the pressure difference in the two chambers provided by a pump. The hydromotor is connected to a symmetric 4/2 four-way valve, and the spool displacement of this valve is realized by a permanent magnet flapper motor. Since the presented system has high energy density, it requires little space and has low mass. Furthermore, the actuator has a simple construction, but it requires an external high-pressure pump [14]. Fig. 2. Electro-hydraulic actuator The physical input of the actuator is the valve current i, the output is the active torque Mact . The flapper motor and the spool can be modelled as a second order linear system, which creates a linear dependence between the valve current and the spool displacement. The motion of valve is modelled as: 1 .. 2D . —7X„ +--- x,, + x, = K„l, (3) where kv valve gain equals kv = - Qn 1 /2 uvmax where QN is the rated flow at rated pressure and maximum input current, pN is the pressure drop at rated flow and uvmax is the maximum rated current. Dv is the valve damping coefficient, which can be calculated from the apparent damping ratio. Dv stands for the natural frequency of the valve [15]. Note that the modelling of the valve motion poses several difficulties. Although Eq. (3) results in a suitable form for control-oriented purposes, the null positioning of the valve is a crucial problem. The pressures in the chambers depend on the flows of the circuits Qj, Q2. pL is the load pressure difference between the two chambers. The average flow of the system, assuming supply pressure ps is constant: Ql (x, Pl ) = CdA(xv )lp- ]Pl (4) This equation can be linearized around (xv0; pL,o) see [14] Ql = Kxv - Kp, (5) where Kq is the valve flow gain coefficient and Kc is the valve pressure coefficient. In this modeling principle, the hydromotor model does not take into account the friction force and the external leakage flow. The compressibility of the fluid is considered constant [14]. The volumetric flow in the chambers is formed as: pl = ( - VP& + - c2PL ), (6) where pE is the effective bulk modulus, Vt is the total volume under pressure and Vp is proportional to the areas of vane cross-sections. cn and cl2 are parameters of the leakage flow. The motion equation of the shaft rotation due to the pressure difference pL and the external load Mext is: J 9 = -da9 + VppL + M ex (7) where J is the mass of the hydromotor shaft and vanes, and da is the damping constant of the system. Mext is the effect of disturbances on the chassis roll dynamics. In the linear form, the nonlinearities of the friction are ignored. The active torque of the actuator is determined by pL. The relationship is written as follows: Mact = IPLAa (8) where Av is the area of the vanes, and is the arm of the stabilizer bar in the longitudinal direction. The control design of the actuator requires the transformation of the previous equations into a statespace form. Eqs. (3), (6) and (7) are the necessary differential equations, Eq. (5) is a part of Eq. (6): = ay act act - B, w l,act act B2,actUact, (9a) y act Cat (9b) The state vector of the actuator model xact = [ xv xv p 9 ] contains the spool displacement xv and its derivative xv, the load pressure p and the shaft angular velocity 9 . The output yact = Mact = uveh of the system is formulated v 376 Varga, B. - Németh, B. - Gâspâr, P. Strojniski vestnik - Journal of Mechanical Engineering 61(2015)6, 374-382 using Eq. (8). The control input is uact=i, while the disturbance is the external load wact=Mext. Finally, the model of the anti-roll bar, incorporating vehicle dynamics (Eq. (2)) and actuator dynamics (Eq. (9)) is formulated as: x = Ax + Blw + B2u, (10) X = [xveh, xact]T, disturbance where x = [x . x w = [wwfo Wact]T , the input is u are: vector is act and the matrices A = B = B2,vehCact A Bl,veh 0 B "0 " , B2 = b2 t 2, act 2 HIERARCHICAL DESIGN OF ANTI-ROLL BAR CONTROL 2.1 Performances of the Control Problem In the previous section, the roll dynamics and the electro-hydraulic actuator were modeled, and a control-oriented model for active anti-roll bar control design was built. This section proposes the architecture and the optimal design of the control system. The anti-roll bar control system must fulfil several requirements. The role of the system is to enhance the roll dynamics of the vehicle, which has two main components: the roll angle q> and the roll angular acceleration cp. First, the roll angle of the chassis influences the traveling comfort of the vehicle, and the high roll angle increases the risk of the rollover motion. Second, it is essential to take into account the roll angular acceleration, due to the impulse-like excitations. These road excitations lead to the intense angular acceleration of the chassis, while the roll angle remains small. With the minimization of cp , the risk of rollover caused by sudden effects can be reduced. The vehicle dynamic performances are formulated such as: zj =ç Zj ^ mm, z2 = ff |z2| ^ min. (11a) (11b) The performances zx, z2 are arranged in a vector form, such as: z = [ z2 ] . (12) Another requirement for the control system is the minimization of the current i, for which there are two main reasons. First, there is the applied control energy, which is an economy requirement. Since the valve has a frequent intervention, the minimization of actuation energy is necessary. Second, the current has technical limits, such as -iiimit < i < iiimit. Thus, the control input u = i must be minimized: u ^ min, u < L (13) Criteria in Eqs. (11) and (13) show that the anti-roll bar system must fulfil several requirements. In the following, a cost function J, which incorporates the previous requirements, is formulated. The goal of the control design is to find a controller which minimizes the cost function: 1 œ J=1 j[ zTQz+"TRu ]dt 2 n ^ min, (14) where Q and R are constant weights that influence the solution of the minimization problem. The role of the weights is to find a balance between the performances and the control input. Although the design criterion (Eq. (14)) provides an adequate description of the control problem, it is difficult to find an appropriate solution. The overall formulation of the system (Eq. (10)) contains two subsystems (Eqs. (2) and (9)), whose dynamics are different: the dynamics of the chassis are slower than that of the hydraulic actuator. Moreover, the consideration of the input constraint in Eq. (13) also poses difficulties in high-order systems. It is beneficial to reduce the states of the system, which is guaranteed by the separation of the two subsystems. Furthermore, it is not necessary to constantly guarantee both of the performances (Eq. (11)). Using a changeable balance between the performances a less conservative controller can be achieved. However, it requires the reduction of the system order, which is guaranteed by the separation. In practice, dividing the optimization problem (Eq. (14)) into two sub-problems is recommended. This results in two optimal solutions to the sub-problems; however, they are suboptimal, considering the original problem. In the following, the overall system (Eq. (10)) is divided into the vehicle (Eq. (2)), and actuator (Eq. (9)) subsystems. These are the high level and the low level in the hierarchy. The input of the high-level vehicle system is the actuator torque Mact, which is the output of the low-level actuator. The interconnection between the subsystems is created by Mact. During the separation, the requirements for the controllers must be redefined. The highlevel controller must fulfil the vehicle dynamic performances (Eq. (2)). The control input of the u 0 Design of Anti-Roll Bar Systems Based on Hierarchical Control 377 Strojniski vestnik - Journal of Mechanical Engineering 61(2015)6, 374-382 high level in anti-roll bar is the active torque Mact. Due to economy and technical aspects, Mact must be minimized: Uveh = Mac, > ^ mm. (15) Using the control input Mact , the roll dynamic performances (Eq. (2)) must be guaranteed. However, physically, it is the output of the actuator, see (Eq. (9)). The required control input is computed with the high-level controller and is denoted by Mact,ref. The purpose of the low-level control is to guarantee the minimum error between the required and the physical torque. Thus, the next performance is formed for the low-level control design: = Mac, ,rf - Mac ^ mm. (16) A further requirement for the control input of low-level i is defined in Eq. (13). Based on the separation of vehicle dynamics and actuator, the optimization problem of the cost function J is divided into two parts: min J < min Jveh + min Jac,, K Khigh Klow (17) where 1 Jveh = ~ f[ ZQvehZ + UTvehRvehUveh ] dt, (18a) Jact 2 i - zT o z act z-s act act + uTRaclu ] dt, (18b) where K is the optimal controller of the problem (Eq. (14)), Khigh is the vehicle dynamic controller and K[ow is the actuator controller. Note that the solution of the minimizations in Eq. (17) results in a suboptimal solution to the original minimization problem (Eq. (14)). However, in this way, a solution to the constrained optimization problem can be found. The architecture of the hierarchical control is illustrated in Fig. 3. 2.2 Vehicle Level Control Design In the following, the control design of the high level is presented. The roll dynamic performances of the system are the minimization of the roll angle and the roll angular acceleration, see Eq. (11). A further requirement for the control system is the minimization of the control input Mact in Eq. (15). Note that it is not necessary to simultaneously guarantee all of the requirements. There are priorities among them, which depend on the current vehicle dynamic status. The priority between the performances is represented with a scheduling variable pveh, which is chosen as a linear combination of q and ip : Pveh (
and ip . pveh is calculated during the measurements of the roll angle and angular acceleration signals. The scheduling variable is taken into consideration in the further design of the control architecture.
Fig. 3. Architecture of control system
Three criteria are defined in Section 2: the minimization of q>, ip and Mact. Using pveh, different weights are defined for these criteria:
_ (Pveh _mi )2
4 (Pveh) = e , Hi (Pveh 1, i = [1;2;3], (20)
where mi and oi are scale parameters of the curves belonging to the respective criteria. £ weights depend on pveh , and the functions have symmetric bell curve shapes, see Fig. 4. This is adequately chosen to express the importance of each criterion at a given pveh. Where pveh) has a high value, the consideration of the related criterion has a high priority.
Based on the Jveh cost function minimization problem, three different LQ controllers Khighi i= [1; 2; 3] are designed. The resulting Khighi are Lq controllers computed with different Qveh, Rveh weights.
• Khigh, j operates at low roll angles and low angular accelerations. In the absence of a critical situation, the actuator intervention is not necessary. As it saves energy, it is an economical mode of the anti-roll bar system. The weights of the LQ control design are Qveh=Rveh.
• Khigh, 2 controller is activated when q> and ip increase. It is essential to take into account both conditions, e.g. at impulse-like excitations
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Strojniski vestnik - Journal of Mechanical Engineering 61(2015)6, 374-382
angular acceleration of the chassis increases, while the roll angle is still small. With this approach, the risk of a rollover caused by sudden effects can be reduced. The weights of the LQ control design are , Qveh >Rveh which guarantees the appropriate actuation. Khigh,3 has an important role in the limitation of Mact, see Eq. (16). This controller prevents the actuator from being overload. The weights of the LQ control design are Qveh