# Effect of Suspension Roll Angle Sensors on the Performance ofH∞ Active Antiroll Bar Control System of Trucks

Authors: Van Tan Vu; Ngoc Tuan Vu
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

The rollover phenomenon is quite familiar with trucks and often has severe consequences worldwide. This article focuses on developing a new control structure for the active antiroll bar system using the robust control method. The main objective is to improve the vehicle’s roll stability by using additional suspension roll angle sensors. First, a truck model equipped with an active antiroll bar system is introduced. Then, the basic robust controller structure diagram is described in detail. Based on this basic robust controller, the authors develop a new robust controller considering the suspension roll angle sensors. Finally, the response of a truck using an active antiroll bar system with two robust (H∞) controllers and a passive antiroll bar system is compared and evaluated in the time domain and in the frequency domain. The results showed that the use of two sensors of the suspension roll angle sensors increased the efficiency by about 20% compared to the basic robust controller. This has proven the effectiveness of a new approach in improving the roll stability of trucks.


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## Body

## 1. Introduction

The rollover of automobiles and trucks is a serious traffic safety problem worldwide. Although rollover accidents happen infrequently, they cause serious consequences for human life, the economy, and traffic infrastructure [1, 2]. In order to improve the roll stability of trucks, modern control solutions have been studied and applied, such as active braking system [3], active antiroll bar system [4–6], active steering system [7–9], and active suspension [10]. Among these systems, the active antiroll bar system directly meets the goal of preventing rollover and is gradually being perfected to be fitted to commercial vehicles soon.The active antiroll bar system for trucks is composed of two electro-hydraulic actuators on either side of the dependent suspension system as shown in Figure1. For each actuator, one end is linked to the vehicle body, and the other end is linked to the unsprung mass. When the actuators create forces equal in magnitude and opposite in direction, the system generates an active torque against the vehicle’s overturning. Antiroll torque control is achieved through the oil flow pressure regulation via a solenoid valve [12–14]. The main research directions related to the active antiroll bar system on trucks can be mentioned as follows:(i)
Building Vehicle Model. The models used include the roll model [11], the yaw-roll model [4–6], and the actuator model [14]. These models establish the stabilizing moment of the active antiroll bar system acting on the part of the sprung mass and unsprung mass in the roll plane. The main control signal is the torque of the active antiroll bar system or the input current supplied to the system when combined with the actuator model.(ii)
Designing Controller. Some of the control methods applied to the active antiroll bar control system on trucks are briefly recalled in the following: (1) Optimal control: Sampson and Cebon [13, 15, 16] proposed a state feedback controller which was designed by finding an optimal controller based on a linear quadratic regulator (LQR) for single-unit and heavy articulated vehicles. The LQR was also applied to an integrated model, including an electronic servo-valve hydraulic actuator model and a yaw-roll model of a single-unit heavy vehicle. The input current of the electronic servo-valve is the input control signal [17, 18]. (2) Neural network control: a reinforcement learning algorithm using neural networks is proposed to improve the roll stability for a single-unit heavy vehicle [1, 10, 19].(iii)
Robust control (LPV): Gaspar et al. [2, 4–6] applied linear parameter-varying techniques to control active antiroll bars combined with active brakes on the single-unit heavy vehicle. The forward velocity is considered the varying parameter.Figure 1
Active antiroll bar system on real trucks [11].The robust control method is applied to the active antiroll bar system on trucks to improve the vehicle’s roll stability. The sensors used are mainly the lateral acceleration sensor and the velocity of the sprung mass roll angle. When the truck approaches the lateral instability state, the lateral acceleration value rapidly increases. Furthermore, the lateral acceleration sensor is inexpensive and highly accurate [20]. Therefore, the lateral acceleration sensor is most often used to control the active antiroll bar system. The velocity of the sprung mass roll angle sensor is also used a lot for the above purpose. However, no studies have used the suspension roll angle sensor to control the active antiroll bar system. Meanwhile, the control goal is to maintain the magnitude of this angle not exceeding 7-8 degrees [2, 4, 21, 22].Based on the previous research using the genetic algorithm method to find the optimal weighting function of the robust controller for the active antiroll bar system in the 15th Mini Conference on Vehicle System Dynamics, Identification, Anomalies, Budapest, and Hungary [23], the main contributions of this paper can be listed as follows:(i)
Building a basicH∞ control structure for the active antiroll bar system (first H∞ active antiroll bar controller) using two common sensors: the lateral acceleration sensor and the sprung mass roll angle velocity sensor. This control structure is built on a truck model with an active antiroll bar system equipped at the two axles.(ii)
A newH∞ control structure (second H∞ active antiroll bar controller) is built using two suspension roll angle sensors. At the same time, these two signals’ magnitude values are considered the performance output that should be minimized.The structure of the paper is organized as follows: Section2 introduces the truck model, with the control signal being the torque of the active antiroll bar system. Section 3 introduces H∞ controller design with a basic structure and proposes a new structure. Evaluating the effectiveness of the proposed controller in the frequency domain is shown in Section 4, and the time domain is detailed in Section 5. The conclusions and further research directions are presented in Section 6.


## 2. Truck Modeling to Study Roll Stability

### 2.1. Yaw-Roll Model of a Truck

Figure2 illustrates the combined yaw-roll dynamics of a truck modeled by a three-body system, where ms is the sprung mass, muf the unsprung mass at the front, including the front wheels and axle, and mur the unsprung mass at the rear, with the rear wheels and axle. The model variables are given in Table 1 and the parameters are in [2].Figure 2
Yaw-roll model of a truck [2, 4].Table 1
Variables of the yaw-roll model [2, 4].
SymbolsDescriptionValuesmsSprung mass12487 kgmu;fUnsprung mass on the front axle706 kgmu;rUnsprung mass on the rear axle1000 kgmThe total vehicle mass14193 kghHeight of CG of sprung mass from the roll axis1.15 mhu;iHeight of CG of unsprung mass from the ground0.53 mrHeight of roll axis from the ground0.83 mCfTire cornering stiffness on the front axle582 × 103 kN/radCrTire cornering stiffness on the rear axle783 × 103 kN/radkfSuspension roll stiffness on the front axle380 × 103 kNm/radkrSuspension roll stiffness on the rear axle684 × 103 kNm/radbfSuspension roll damping on the front axle100 × 103 kN/radbrSuspension roll damping on the rear axle100 × 103 kN/radktfTire roll stiffness on the front axle2060 × 103 kNm/radktrTire roll stiffness on the rear axle3337 × 103 kNm/radIxxRoll moment of inertia of sprung mass24201 kgm2IxzYaw-roll product of inertia of sprung mass4200 kgm2IzzYaw moment of inertia of sprung mass34917 kgm2lfLength of the front axle from the CG1.95 mlrLength of the rear axle from the CG1.54 mlwHalf of the vehicle width0.93 mµRoad adhesion coefficient1vForward velocity—vwiComponents of the forward velocity—ayLateral acceleration—βSide-slip angle at the center of mass—ψHeading angle—ψ·Yaw rate—αSide slip angle—ϕSprung mass roll angle—ϕu,iUnsprung mass roll angle—δfSteering angle—In the vehicle modeling, the differential equations of motion of the yaw-roll dynamics of the truck, i.e., the lateral dynamics (1), the yaw moment (2), the roll moment of the sprung mass (3), the roll moment of the front (4), and the rear and unsprung masses (5). They are formalized in the equations (6).(1)mvβ·+ψ·−mshϕ··=Fyf+Fyr,(2)−Ixzϕ··+Izzψ··=Fyflf−Fyrlr,(3)Ixx+msh2ϕ··−Ixzψ··=msghϕ+msvhβ·+ψ·−kfϕ−ϕuf−bfϕ·−ϕuf·+MARf+Uf−krϕ−ϕur−brϕ·−ϕur·+MARr+Ur,(4)−rFyf=mufvr−hufβ·+ψ·+mufghufϕuf−ktfϕuf+kfϕ−ϕuf+bfϕ·−ϕuf·+MARf+Uf,(5)−rFyr=murvr−hurβ·+ψ·−murghurϕur−ktrϕur+krϕ−ϕur+brϕ·−ϕur·+MARr+Ur,where Uf, Ur are the torques at the two axles; Fyf, Fyr the lateral tyre forces; MARf, MARr the moments of the passive antiroll bar, which impact the unsprung and sprung masses at the front and rear axles [14].By using the previous equation, the truck can be represented by the linear system in the state space representation (6):(6)x·=Ax+B1w+B2u,y=Cx,with the state vector: x=βψ·φφ·φufφurT, the disturbance input: w=δf, the control inputs: u=UfUrT, and the output vector: y=βψ·φφ·φufφurT. The matrices A, B1, and B2 are defined as in Appendix.


### 2.2. Actuator of the Active Antiroll Bar System

Figure3 illustrates the diagram of a hydraulic cylinder in combination with an electronic servo-valve. The symbols of the actuator are shown in Table 2. The spool valve of the electronic servo-valve is controlled by a current, which generates a displacement Xv. The high-pressure oil supply Ps is always stored outside the electronic servo valve, and the moving spool valve distributes the high-pressure oil into two chambers of the hydraulic cylinder. The difference of pressure ∆P = P1 −P2 between the two chambers produces the output force Fact given by(7)Fact=APΔP,where AP is the area of the piston, the equations for each chamber of the hydraulic cylinder can be written as(8)dV1dt+V1βedP1dt=Q1−CipP1−P2−CepP1,dV2dt+V2βedP2dt=CipP1−P2−CepP2−Q2,where βe is the effective bulk modulus of the oil, Cep and Cip are the hydraulic cylinder’s external and internal leakage coefficients.Figure 3
Diagram of the actuator for the active antiroll bar system [14, 17].Table 2
Symbols of the actuator [14, 17].
SymbolsDescriptionAPArea of the pistonKxValve flow gain coefficientKPTotal flow pressure coefficientCtpTotal leakage coefficient of the actuatorVtTotal volume of trapped oilβeEffective bulk modulus of the oilτTime constant of the servo-valveKvServo-valve gainThe volume in each chamber varies with the piston displacementya, as(9)V1=V01+Apya,V2=V02−Apya,where V01 and V02 are the initial volumes in each chamber. In assuming that V01 = V02 = V0, the total volume of trapped oil is given by (10)(10)Vt=V1+V2=V01+V02=2V0.Therefore, the equations in each chamber become(11)APdyadt+V0+APyaβedP1dt=Q1−CipP1−P2−CepP1,−APdyadt+V0−APyaβedP2dt=CipP1−P2−CepP2−Q2.Subtracting the second equation from the first one leads to(12)2QL=Q1+Q2=2CtpΔP+2Apdyadt+V0βedΔPdt,where Ctp = 2Cip + Cep is the total leakage coefficient of the hydraulic cylinder.From equations (8) to (12), the dynamic equation of the servo-valve hydraulic cylinder is obtained as follows:(13)Vt4βedΔPdt+KP+CtpΔP−KxXv+APdyadt=0,where ya is the displacement of the piston inside the hydraulic cylinder.The three-land-four-way spool valve is used in the actuator. The displacement of the spool valveXv is controlled by the electrical current u. The effects of hysteresis and flow forces on the servo-valve are neglected here, and then the dynamical behavior of the electronic servo-valve can be approximated by a first-order model [14] as(14)dXvdt+1τXv−Kvτu=0,where τ is the time constant and Kv the gain of the servo-valve model.From equations (7), (13), to (14), the dynamical equations of the actuator are summarized in equation (15). Here, the input signal is the current u and the output is the force Fact.(15)Fact=APΔP,Vt4βedΔPdt+KP+CtpΔP−KxXv+APdyadt=0,dXvdt+1τXv−Kvτu=0.The torque generated by the active antiroll bar system at each axle is given by(16)Ui=lactFactri−lactFactli.Here,lact is half the distance between the two actuators, Facti the actuator forces on the left and on the right.The combination of equations (1)–(5) is the general equation to control the active antiroll bar system. The parameters of this system are found in [2, 4, 14].


## 2.1. Yaw-Roll Model of a Truck

Figure2 illustrates the combined yaw-roll dynamics of a truck modeled by a three-body system, where ms is the sprung mass, muf the unsprung mass at the front, including the front wheels and axle, and mur the unsprung mass at the rear, with the rear wheels and axle. The model variables are given in Table 1 and the parameters are in [2].Figure 2
Yaw-roll model of a truck [2, 4].Table 1
Variables of the yaw-roll model [2, 4].
SymbolsDescriptionValuesmsSprung mass12487 kgmu;fUnsprung mass on the front axle706 kgmu;rUnsprung mass on the rear axle1000 kgmThe total vehicle mass14193 kghHeight of CG of sprung mass from the roll axis1.15 mhu;iHeight of CG of unsprung mass from the ground0.53 mrHeight of roll axis from the ground0.83 mCfTire cornering stiffness on the front axle582 × 103 kN/radCrTire cornering stiffness on the rear axle783 × 103 kN/radkfSuspension roll stiffness on the front axle380 × 103 kNm/radkrSuspension roll stiffness on the rear axle684 × 103 kNm/radbfSuspension roll damping on the front axle100 × 103 kN/radbrSuspension roll damping on the rear axle100 × 103 kN/radktfTire roll stiffness on the front axle2060 × 103 kNm/radktrTire roll stiffness on the rear axle3337 × 103 kNm/radIxxRoll moment of inertia of sprung mass24201 kgm2IxzYaw-roll product of inertia of sprung mass4200 kgm2IzzYaw moment of inertia of sprung mass34917 kgm2lfLength of the front axle from the CG1.95 mlrLength of the rear axle from the CG1.54 mlwHalf of the vehicle width0.93 mµRoad adhesion coefficient1vForward velocity—vwiComponents of the forward velocity—ayLateral acceleration—βSide-slip angle at the center of mass—ψHeading angle—ψ·Yaw rate—αSide slip angle—ϕSprung mass roll angle—ϕu,iUnsprung mass roll angle—δfSteering angle—In the vehicle modeling, the differential equations of motion of the yaw-roll dynamics of the truck, i.e., the lateral dynamics (1), the yaw moment (2), the roll moment of the sprung mass (3), the roll moment of the front (4), and the rear and unsprung masses (5). They are formalized in the equations (6).(1)mvβ·+ψ·−mshϕ··=Fyf+Fyr,(2)−Ixzϕ··+Izzψ··=Fyflf−Fyrlr,(3)Ixx+msh2ϕ··−Ixzψ··=msghϕ+msvhβ·+ψ·−kfϕ−ϕuf−bfϕ·−ϕuf·+MARf+Uf−krϕ−ϕur−brϕ·−ϕur·+MARr+Ur,(4)−rFyf=mufvr−hufβ·+ψ·+mufghufϕuf−ktfϕuf+kfϕ−ϕuf+bfϕ·−ϕuf·+MARf+Uf,(5)−rFyr=murvr−hurβ·+ψ·−murghurϕur−ktrϕur+krϕ−ϕur+brϕ·−ϕur·+MARr+Ur,where Uf, Ur are the torques at the two axles; Fyf, Fyr the lateral tyre forces; MARf, MARr the moments of the passive antiroll bar, which impact the unsprung and sprung masses at the front and rear axles [14].By using the previous equation, the truck can be represented by the linear system in the state space representation (6):(6)x·=Ax+B1w+B2u,y=Cx,with the state vector: x=βψ·φφ·φufφurT, the disturbance input: w=δf, the control inputs: u=UfUrT, and the output vector: y=βψ·φφ·φufφurT. The matrices A, B1, and B2 are defined as in Appendix.


## 2.2. Actuator of the Active Antiroll Bar System

Figure3 illustrates the diagram of a hydraulic cylinder in combination with an electronic servo-valve. The symbols of the actuator are shown in Table 2. The spool valve of the electronic servo-valve is controlled by a current, which generates a displacement Xv. The high-pressure oil supply Ps is always stored outside the electronic servo valve, and the moving spool valve distributes the high-pressure oil into two chambers of the hydraulic cylinder. The difference of pressure ∆P = P1 −P2 between the two chambers produces the output force Fact given by(7)Fact=APΔP,where AP is the area of the piston, the equations for each chamber of the hydraulic cylinder can be written as(8)dV1dt+V1βedP1dt=Q1−CipP1−P2−CepP1,dV2dt+V2βedP2dt=CipP1−P2−CepP2−Q2,where βe is the effective bulk modulus of the oil, Cep and Cip are the hydraulic cylinder’s external and internal leakage coefficients.Figure 3
Diagram of the actuator for the active antiroll bar system [14, 17].Table 2
Symbols of the actuator [14, 17].
SymbolsDescriptionAPArea of the pistonKxValve flow gain coefficientKPTotal flow pressure coefficientCtpTotal leakage coefficient of the actuatorVtTotal volume of trapped oilβeEffective bulk modulus of the oilτTime constant of the servo-valveKvServo-valve gainThe volume in each chamber varies with the piston displacementya, as(9)V1=V01+Apya,V2=V02−Apya,where V01 and V02 are the initial volumes in each chamber. In assuming that V01 = V02 = V0, the total volume of trapped oil is given by (10)(10)Vt=V1+V2=V01+V02=2V0.Therefore, the equations in each chamber become(11)APdyadt+V0+APyaβedP1dt=Q1−CipP1−P2−CepP1,−APdyadt+V0−APyaβedP2dt=CipP1−P2−CepP2−Q2.Subtracting the second equation from the first one leads to(12)2QL=Q1+Q2=2CtpΔP+2Apdyadt+V0βedΔPdt,where Ctp = 2Cip + Cep is the total leakage coefficient of the hydraulic cylinder.From equations (8) to (12), the dynamic equation of the servo-valve hydraulic cylinder is obtained as follows:(13)Vt4βedΔPdt+KP+CtpΔP−KxXv+APdyadt=0,where ya is the displacement of the piston inside the hydraulic cylinder.The three-land-four-way spool valve is used in the actuator. The displacement of the spool valveXv is controlled by the electrical current u. The effects of hysteresis and flow forces on the servo-valve are neglected here, and then the dynamical behavior of the electronic servo-valve can be approximated by a first-order model [14] as(14)dXvdt+1τXv−Kvτu=0,where τ is the time constant and Kv the gain of the servo-valve model.From equations (7), (13), to (14), the dynamical equations of the actuator are summarized in equation (15). Here, the input signal is the current u and the output is the force Fact.(15)Fact=APΔP,Vt4βedΔPdt+KP+CtpΔP−KxXv+APdyadt=0,dXvdt+1τXv−Kvτu=0.The torque generated by the active antiroll bar system at each axle is given by(16)Ui=lactFactri−lactFactli.Here,lact is half the distance between the two actuators, Facti the actuator forces on the left and on the right.The combination of equations (1)–(5) is the general equation to control the active antiroll bar system. The parameters of this system are found in [2, 4, 14].


## 3.H∞ Control Synthesis of the Active Antiroll Bar System of Trucks

### 3.1. Background onH∞ Control

TheH∞ control problem is formulated according to the generalized control structure shown in Figure 4 [4, 5, 23].Figure 4
Generalized control structure.WithP partitioned as(17)zy=P11sP12sP21sP22sdu,and u=Ks.y, which yields(18)zd=FlP,K≔=P11+P12KI−P22K−1P21.The aim is to design a controllerK that stabilizes the closed-loop system and also reduces the signal transmission path from disturbances d to performance outputs z. This problem is then to find a controller K that minimizes γ such that(19)FlP,K∞<γ.By minimizing a suitably weighted version ofFlP,K the control aim is achieved. The controller yK=KsuK in any LTI system is represented as a state-space form as flows:(20)xK·=AKxK+BKuK,yK=CKxK+DKuK,where, AK,BK,CK,DK are the matrices of the controller.The interconnection of the controller and the open-loop system asuK=ys, yK=us.


### 3.2. Control Objective and Problem Statement

The objective of the active antiroll bar control system is to maximize the roll stability of the vehicle. Usually, an imminent rollover is detected when the calculated normalized load transfer (Rf, r) reaches 1 (or −1) [2]. The normalized load transfer R = ±1 corresponds to the most significant possible load transfer. In that case, the inner wheel in the bend lifts off.While attempting to minimize the load transfer, it is also necessary to constrain the roll angles between the sprung and unsprung massesϕ−ϕtf,r so that they stay within the limits of the suspension travel (7-8 deg) [2, 4, 5].The performance characteristic that is most interesting when designing the active antiroll bar system is the normalized load transfer. The chosen control objective is to minimize the effect of the steering angle on the normalized load transferRf, r, in the H∞ framework. As explained later, the limitation of the torques Uf, r generated by the actuators is also crucial for practical implementation.


### 3.3. The FirstH∞ Control Synthesis for the Active Antiroll Bar System

The closed-loop system is considered in Figure5, which includes the feedback structure of the nominal model G, the controller K and the weighting functions Wij. In this diagram, Uf and Ur are the control inputs, y1 and y2 are the measured outputs, n1 and n2 are the measurement noises. δf is the steering angle considered as a disturbance signal, which the driver sets. The variables e1, e2, e3, e4, and e5 represent the performance outputs.Figure 5
G-K control structure of the first H∞ active antiroll bar control.According to Figure5, the concatenation of the linear model (6 and 15) with performance weighting functions leads to the state space representation of Ps [24]:(21)x·tztyt=AB1B2C1D11D12C2D21D22xtwtut,with the disturbance input: wt=d1d2d3T, the control input: ut=UfUrT, the performance output: zt=e1e2e3e4e5T, the measured output: yt=ayϕ·T. A, B1, and B2 are the matrices of the concatenation of the linear model. Meanwhile, C1, D11, and D12 are the matrices defined for the performance outputs of the control goals; C2, D21, and D22 are the determination matrices for the measured signals.The weighting functions used in Figure5 are the most important for designing the robust controller.The input scaling weightWd1, chosen as Wd1=π/180, normalizes the steering angle δf to the maximum expected value, corresponding to a 10 steering angle command.The weighting functionsWd2 and Wd3 are selected as: Wd2=Wd3=0.01, which accounts for small sensor noise models in the control design. The noise weights are chosen as 0.01 (m/s2) for the lateral acceleration and 0.01(°/sec) for the derivative of the roll angle ϕ· [2, 4]. Note that other low pass filters could be selected if needed.The weighting functionsWpi represent the performance outputs (Wp1, Wp2, Wp3, Wp4, and Wp5). The purpose of the weighting functions is to keep small the control inputs, normalized load transfers, and the lateral acceleration over the desired frequency range. The weighting functions chosen for performance outputs can be considered as penalty functions. That is, weights should be prominent in the frequency range where small signals are desired and small where more significant performance outputs can be tolerated.The weighting functionsWp1 and Wp2 corresponding to the front and rear control torques generated by active antiroll bars are chosen as(22)Wp1=1P1;Wp2=1P2.The weighting functionsWp3 and Wp4 corresponding to the normalized load transfers at the front and rear axles are selected as(23)Wp3=1P3;Wp4=1P4.The weighting functionWp5 is selected as(24)Wp5=P51P52s+P53P54s+P55.Here, the weighting functionWp5 corresponds to a design that avoids the rollover with the driver’s bandwidth in the frequency range of up to more than 4rad/s [2, 23]. This weighting function will directly minimize the lateral acceleration when it reaches the critical value to avoid rollover.The parametersPij are constant. From equations (22) to (24), the following variables P1, P2, P3, P4, P51, P52, P53, P54, and P55 are chosen as Table 3 [22, 23]. The selection of values of Pij in the weight functions Wpi can be selected by the experience of the designer and must be paid through the process of testing, evaluation, and comparison to choose a reasonable value. In addition, it can also be combined with other optimization methods, such as genetic algorithms to determine their values [25–27].Table 3
The weighting functions of theH∞ active antiroll bar controller.
Wp1Wp2Wp3Wp4Wp5P1P2P3P4P51P52P53P54P551500002000000.7601.05010.0005501000.01Therefore, in the closed-loop system, theK1 controller has the specified state-space matrices as follows:(i)
AK1 = [2.672 −1.402 0.2084 −0.08157 −3.922 −6.447 5.901e−06; 93.1 −6.57 1.157 −0.5325 −21.78 −35.79 3.276e−05; 0.0002613 −3.783e−06 1.73e−06 1 −3.256e−05 −5.352e−05 4.899e−11; 64.56 −9.706 5.034 −1.922 −94.74 −155.7 0.0001425; 9.579 −0.8747 3.823 0.8872 −26.02 −2.79 8.114e−06; −3.006 −0.2695 6.753 0.8712 −0.1192 −40.58 1.003e−05; −0.000241 3.489e−06 −1.596e−06 −2.535e−07 3.003e−05 4.936e−05 −0.0001](ii)
BK1 = [0.1724 0.01091; 2.032 0.205; 5.4e − 06 2.585e − 07; 2.587 0.1765; 0.266 0.01443; 0.01437 0.002626; 0.907 1.509e − 06];(iii)
CK1 = [1.032e + 05 –1.834e + 04 −3901 −3899 6473 6908 0.3002 1.802e + 05 –3.287e + 04 −5764 −6675 1.145e + 04 1.211e + 04 0.5256];(iv)
DK1 = [6.18e − 06 0; 1.099e − 05 0].


### 3.4. The SecondH∞ Control Synthesis for the Active Antiroll Bar System

The design of the secondH∞ active antiroll bar controller is based on the first one, i.e., the control structure diagram is extended from Figure 5. The addition of two main contents includes two suspension roll angle sensors of the front and rear suspension systems, and these two signals are also considered as the performance outputs of the closed-loop control system.Figure6 shows the control structure diagram of the active antiroll bar system using the second H∞ controller, where e6, e7 is the performance outputs for the two suspension roll angles. In order to reduce these two roll angles, two weighting functions Wp6, Wp7 are used with values defined as Wp6 = Wp7 = 1/0.5.Figure 6
G-K control structure of the second H∞ active antiroll bar control.Based on the firstH∞ controller and the control structure diagram in Figure 6, the state space representation of Ps has the disturbance input: wt=d1d2d3d4d5T, the control input: ut=UfUrT, the performance output: zt=e1e2e3e4e5e6e7T, the measured output: yt=ayϕ−ϕtfϕ−ϕtrϕ·T. Considering the performance outputs, including the suspension roll angle, make perfect sense. Because when studying the roll stability of automobiles in general and trucks in particular, the limit value of the suspension roll angle is from 7 to 8 degrees [2, 4, 5]. Therefore, in addition to reducing the normalized load transfer, it is also necessary to reduce the roll angle of the suspension system.Therefore, in the closed-loop system, theK2 controller has the specified state-space matrices as follows:(i)
AK2 = [3.272 −1.383 −0.6932 −0.2283 −3.584 −6.127 4.228e − 06; 96.43 −6.465 −3.854 −1.347 −19.9 −34.01 2.347e − 05; 9.586e − 11 −1.305e − 12 −0.002227 1 0.001259 0.0009676 1.264e − 17; 79.04 −9.249 −16.79 −5.466 −86.55–148 0.0001021; 10.39 −0.8458 2.648 0.6878 −25.68 −2.282 5.873e − 06;−1.977 −0.2401 5.149 0.6195 0.5803–40.1 7.123e − 06;−8.844e − 11 1.204e − 12 −0.01636 −1.93e − 05 0.008129 0.008233 −0.0001];(ii)
BK2 = [7.505 −0.1252 −0.05453 0.4748; 88.47 −0.5415 −0.05149 8.923; 8.461e − 11 0.09942 0.0764 9.562e − 05; 112.6 −0.6281 −0.1142 7.683; 11.58 −0.04497 0.002496 0.6282; 0.6254 0.02552 0.02093 0.1143; 39.48 0.6418 0.65 0.001524];(iii)
CK2 = [3057 −395.7–1061 −258 390.4 617.8 0.005017; 5382 −720 −2067 −456.7 1113 852.2 0.008576];(iv)
DK2 = [6.18e − 06 0 0 0; 1.099e − 05 0 0 0].


## 3.1. Background onH∞ Control

TheH∞ control problem is formulated according to the generalized control structure shown in Figure 4 [4, 5, 23].Figure 4
Generalized control structure.WithP partitioned as(17)zy=P11sP12sP21sP22sdu,and u=Ks.y, which yields(18)zd=FlP,K≔=P11+P12KI−P22K−1P21.The aim is to design a controllerK that stabilizes the closed-loop system and also reduces the signal transmission path from disturbances d to performance outputs z. This problem is then to find a controller K that minimizes γ such that(19)FlP,K∞<γ.By minimizing a suitably weighted version ofFlP,K the control aim is achieved. The controller yK=KsuK in any LTI system is represented as a state-space form as flows:(20)xK·=AKxK+BKuK,yK=CKxK+DKuK,where, AK,BK,CK,DK are the matrices of the controller.The interconnection of the controller and the open-loop system asuK=ys, yK=us.


## 3.2. Control Objective and Problem Statement

The objective of the active antiroll bar control system is to maximize the roll stability of the vehicle. Usually, an imminent rollover is detected when the calculated normalized load transfer (Rf, r) reaches 1 (or −1) [2]. The normalized load transfer R = ±1 corresponds to the most significant possible load transfer. In that case, the inner wheel in the bend lifts off.While attempting to minimize the load transfer, it is also necessary to constrain the roll angles between the sprung and unsprung massesϕ−ϕtf,r so that they stay within the limits of the suspension travel (7-8 deg) [2, 4, 5].The performance characteristic that is most interesting when designing the active antiroll bar system is the normalized load transfer. The chosen control objective is to minimize the effect of the steering angle on the normalized load transferRf, r, in the H∞ framework. As explained later, the limitation of the torques Uf, r generated by the actuators is also crucial for practical implementation.


## 3.3. The FirstH∞ Control Synthesis for the Active Antiroll Bar System

The closed-loop system is considered in Figure5, which includes the feedback structure of the nominal model G, the controller K and the weighting functions Wij. In this diagram, Uf and Ur are the control inputs, y1 and y2 are the measured outputs, n1 and n2 are the measurement noises. δf is the steering angle considered as a disturbance signal, which the driver sets. The variables e1, e2, e3, e4, and e5 represent the performance outputs.Figure 5
G-K control structure of the first H∞ active antiroll bar control.According to Figure5, the concatenation of the linear model (6 and 15) with performance weighting functions leads to the state space representation of Ps [24]:(21)x·tztyt=AB1B2C1D11D12C2D21D22xtwtut,with the disturbance input: wt=d1d2d3T, the control input: ut=UfUrT, the performance output: zt=e1e2e3e4e5T, the measured output: yt=ayϕ·T. A, B1, and B2 are the matrices of the concatenation of the linear model. Meanwhile, C1, D11, and D12 are the matrices defined for the performance outputs of the control goals; C2, D21, and D22 are the determination matrices for the measured signals.The weighting functions used in Figure5 are the most important for designing the robust controller.The input scaling weightWd1, chosen as Wd1=π/180, normalizes the steering angle δf to the maximum expected value, corresponding to a 10 steering angle command.The weighting functionsWd2 and Wd3 are selected as: Wd2=Wd3=0.01, which accounts for small sensor noise models in the control design. The noise weights are chosen as 0.01 (m/s2) for the lateral acceleration and 0.01(°/sec) for the derivative of the roll angle ϕ· [2, 4]. Note that other low pass filters could be selected if needed.The weighting functionsWpi represent the performance outputs (Wp1, Wp2, Wp3, Wp4, and Wp5). The purpose of the weighting functions is to keep small the control inputs, normalized load transfers, and the lateral acceleration over the desired frequency range. The weighting functions chosen for performance outputs can be considered as penalty functions. That is, weights should be prominent in the frequency range where small signals are desired and small where more significant performance outputs can be tolerated.The weighting functionsWp1 and Wp2 corresponding to the front and rear control torques generated by active antiroll bars are chosen as(22)Wp1=1P1;Wp2=1P2.The weighting functionsWp3 and Wp4 corresponding to the normalized load transfers at the front and rear axles are selected as(23)Wp3=1P3;Wp4=1P4.The weighting functionWp5 is selected as(24)Wp5=P51P52s+P53P54s+P55.Here, the weighting functionWp5 corresponds to a design that avoids the rollover with the driver’s bandwidth in the frequency range of up to more than 4rad/s [2, 23]. This weighting function will directly minimize the lateral acceleration when it reaches the critical value to avoid rollover.The parametersPij are constant. From equations (22) to (24), the following variables P1, P2, P3, P4, P51, P52, P53, P54, and P55 are chosen as Table 3 [22, 23]. The selection of values of Pij in the weight functions Wpi can be selected by the experience of the designer and must be paid through the process of testing, evaluation, and comparison to choose a reasonable value. In addition, it can also be combined with other optimization methods, such as genetic algorithms to determine their values [25–27].Table 3
The weighting functions of theH∞ active antiroll bar controller.
Wp1Wp2Wp3Wp4Wp5P1P2P3P4P51P52P53P54P551500002000000.7601.05010.0005501000.01Therefore, in the closed-loop system, theK1 controller has the specified state-space matrices as follows:(i)
AK1 = [2.672 −1.402 0.2084 −0.08157 −3.922 −6.447 5.901e−06; 93.1 −6.57 1.157 −0.5325 −21.78 −35.79 3.276e−05; 0.0002613 −3.783e−06 1.73e−06 1 −3.256e−05 −5.352e−05 4.899e−11; 64.56 −9.706 5.034 −1.922 −94.74 −155.7 0.0001425; 9.579 −0.8747 3.823 0.8872 −26.02 −2.79 8.114e−06; −3.006 −0.2695 6.753 0.8712 −0.1192 −40.58 1.003e−05; −0.000241 3.489e−06 −1.596e−06 −2.535e−07 3.003e−05 4.936e−05 −0.0001](ii)
BK1 = [0.1724 0.01091; 2.032 0.205; 5.4e − 06 2.585e − 07; 2.587 0.1765; 0.266 0.01443; 0.01437 0.002626; 0.907 1.509e − 06];(iii)
CK1 = [1.032e + 05 –1.834e + 04 −3901 −3899 6473 6908 0.3002 1.802e + 05 –3.287e + 04 −5764 −6675 1.145e + 04 1.211e + 04 0.5256];(iv)
DK1 = [6.18e − 06 0; 1.099e − 05 0].


## 3.4. The SecondH∞ Control Synthesis for the Active Antiroll Bar System

The design of the secondH∞ active antiroll bar controller is based on the first one, i.e., the control structure diagram is extended from Figure 5. The addition of two main contents includes two suspension roll angle sensors of the front and rear suspension systems, and these two signals are also considered as the performance outputs of the closed-loop control system.Figure6 shows the control structure diagram of the active antiroll bar system using the second H∞ controller, where e6, e7 is the performance outputs for the two suspension roll angles. In order to reduce these two roll angles, two weighting functions Wp6, Wp7 are used with values defined as Wp6 = Wp7 = 1/0.5.Figure 6
G-K control structure of the second H∞ active antiroll bar control.Based on the firstH∞ controller and the control structure diagram in Figure 6, the state space representation of Ps has the disturbance input: wt=d1d2d3d4d5T, the control input: ut=UfUrT, the performance output: zt=e1e2e3e4e5e6e7T, the measured output: yt=ayϕ−ϕtfϕ−ϕtrϕ·T. Considering the performance outputs, including the suspension roll angle, make perfect sense. Because when studying the roll stability of automobiles in general and trucks in particular, the limit value of the suspension roll angle is from 7 to 8 degrees [2, 4, 5]. Therefore, in addition to reducing the normalized load transfer, it is also necessary to reduce the roll angle of the suspension system.Therefore, in the closed-loop system, theK2 controller has the specified state-space matrices as follows:(i)
AK2 = [3.272 −1.383 −0.6932 −0.2283 −3.584 −6.127 4.228e − 06; 96.43 −6.465 −3.854 −1.347 −19.9 −34.01 2.347e − 05; 9.586e − 11 −1.305e − 12 −0.002227 1 0.001259 0.0009676 1.264e − 17; 79.04 −9.249 −16.79 −5.466 −86.55–148 0.0001021; 10.39 −0.8458 2.648 0.6878 −25.68 −2.282 5.873e − 06;−1.977 −0.2401 5.149 0.6195 0.5803–40.1 7.123e − 06;−8.844e − 11 1.204e − 12 −0.01636 −1.93e − 05 0.008129 0.008233 −0.0001];(ii)
BK2 = [7.505 −0.1252 −0.05453 0.4748; 88.47 −0.5415 −0.05149 8.923; 8.461e − 11 0.09942 0.0764 9.562e − 05; 112.6 −0.6281 −0.1142 7.683; 11.58 −0.04497 0.002496 0.6282; 0.6254 0.02552 0.02093 0.1143; 39.48 0.6418 0.65 0.001524];(iii)
CK2 = [3057 −395.7–1061 −258 390.4 617.8 0.005017; 5382 −720 −2067 −456.7 1113 852.2 0.008576];(iv)
DK2 = [6.18e − 06 0 0 0; 1.099e − 05 0 0 0].


## 4. Roll Stability Analysis in the Frequency Domain

In this section, the authors present the simulation results in the frequency domain with the forward velocity are considered at 70 km/h. The simulation result of the secondH∞ controller (continuous line) is compared with the first H∞ controller (continuous-dotted line) and the passive antiroll bar system (dash line).Figure7 shows the transfer functions magnitude of the sprung mass roll angle ϕ due to the steering angle in the frequency domain. It can be seen that with two H∞ controllers for the active antiroll bar system, the roll angle of the sprung mass decreases in the frequency range up to 9 rad/s. But with the second H∞ controller for the active antiroll bar system, the roll angle of the sprung mass decreases more than the first one. Figures 8 and 9 show the transfer functions magnitude of the normalized load transfer at the front axle Rf and the rear axle Rr due to the steering angle in the frequency domain. It can be seen that with the two H∞ controllers for the active antiroll bar system, the normalized load transfers at the front axle decrease in the frequency range up to 7 rad/s.Figure 7
Transfer function magnitude of the sprung mass roll angle (ϕ) due to the steering angle.Figure 8
Transfer function magnitude of the normalized load transfer at the front axle (Rf) due to the steering angle.Figure 9
Transfer function magnitude of the normalized load transfer at the rear axle (Rr) due to the steering angle.Since the bandwidth of the steering angle when studying the roll stability of the vehicle reaches a maximum of 4 rad/s [2, 4], the corresponding comparison results from Figures 7–9 in the frequency domains at 10−5, 10−4, 1, and 4 rad/s are summarized in Table 4. In the last row, the authors make a comparison of the transfer functions magnitude of the considered signals at each frequency when comparing the case of using an active antiroll bar system with a second H∞ controller and the subject of using a passive antiroll bar system. The comparison results show that the second H∞ controller has significantly reduced the value of the transfer function magnitude from the steering angle. Thus, the consideration of using the suspension roll angle sensors has contributed substantially to reducing the value of the transfer function magnitude, thereby increasing the roll stability of the truck.Table 4
Comparison results in the frequency domain.
10−5 (rad/s)10−4 (rad/s)1 (rad/s)4 (rad/s)ϕ (dB)Rf (dB)Rr (dB)ϕ (dB)Rf (dB)Rr (dB)ϕ (dB)Rf (dB)Rr (dB)ϕ (dB)Rf (dB)Rr (dB)Passive7.52727.57.52727.57.52727.502425FirstH∞7.52727.5524260.52020−111415SecondH∞3222312021−41717−12138Reduction (compared with passive)4.554.56.576.511.51010.5121117


## 5. Roll Stability Analysis in the Time Domain

In this section, the authors evaluate the roll stability of the truck using theH∞ active antiroll bar system in the time domain with three scenarios: (1) truck in a double lane change to avoid obstacle manoeuver; (2) truck in a double lane change to overtake manoeuver; and (3) truck in a cornering manoeuver.


### 5.1. Truck in a Double Lane Change to Avoid Obstacle Manoeuver

The truck manoeuver is a double lane change to avoid obstacles, which is often used to avoid an obstacle in an emergency. The manoeuver has a 2.5 m path deviation over 100 m. The size of the path deviation is chosen to test real obstacle avoidance in an emergency. The forward velocity is considered at 70 km/h. Figure10 shows the steering angle, the roll angle of the sprung mass, the roll angle of the unsprung mass at the front axle and the roll angle of the unsprung mass at the rear axle. We can see that the first H∞ active antiroll bar controller has significantly reduced (approximately 40%) the roll angles of the sprung mass, the unsprung mass at the front axle, and the roll angle of the unsprung mass at the rear axle compared to the passive antiroll bar system. And the second H∞ active antiroll bar controller has decreased dramatically (approximately 55%), respectively.Figure 10
Time responses of the steering angle, roll angle of the sprung mass, and unsprung masses of the truck: in a double lane change to avoid obstacle manoeuver. (a) Steering angle. (b) Roll angle of sprung mass. (c) Roll angle of unsprung mass of the front axle. (d) Roll angle of unsprung mass of the rear axle.
(a)(b)(c)(d)Figure11 shows the normalized load transfer and the roll angle of the suspension at the front and at the rear axle, respectively. We can see that the value of the normalized load transfer at the rear axle exceeds −1 at 2.8 seconds in the case of the passive antiroll bar system, but this value is within the limitation at the front axle. In two cases of H∞ active antiroll bar controller, the roll stability is achieved because the limitation of the normalized load transfer is in the range from −1 to 1. The maximum of absolute values of the roll angle of the suspensions is always less than 7-8 (deg), so they are within the limitations of the suspension travel. When compared with the passive system, we can see that the first H∞ active antiroll bar controller has significantly reduced (approximately 40%) and the second H∞ active antiroll bar controller has decreased substantially (about 55%) the normalized load transfer and the roll angle of suspension at the front and the rear axles, respectively.Figure 11
Time responses of normalized load transfer and roll angle of suspension on two axles of the truck: in a double lane change to avoid obstacle manoeuver. (a) Normalized load transfer at the front axle. (b) Normalized load transfer at the rear axle. (c) Roll angle of the suspension at the front axle. (d) Roll angle of the suspension at the rear axle.
(a)(b)(c)(d)The simulation results when the truck in a double lane change to avoid obstacle manoeuver clearly shows the effectiveness of the secondH∞ active antiroll bar controller in using the sensors of the suspension roll angle.


### 5.2. Truck in a Double Lane Change to Overtake Manoeuver

In this subsection, the authors introduce the time responses of a truck in a double lane change to overtake a manoeuver. Figure12 shows the steering angle, the roll angle of the sprung mass, and the roll angles of the unsprung mass at the front and rear axles. When compared with the passive antiroll bar system, we can see that the first H∞ active antiroll bar controller has significantly reduced (approximately 41%) and the second H∞ active antiroll bar controller has diminished considerably (about 53%) for the roll angle of the sprung mass, the roll angle of the unsprung mass at the front and rear axles.Figure 12
Time responses of the steering angle and roll angle of the sprung mass and unsprung masses of the truck: in a double lane change to overtake manoeuver. (a) Steering angle. (b) Roll angle of sprung mass. (c) Roll angle of unsprung mass at the front axle. (d) Roll angle of unsprung mass at the rear axle.
(a)(b)(c)(d)Figure13 shows the normalized load transfer and the suspension roll angle at the front and rear axles. We can see that the values of normalized load transfer are always in the range from −1 to 1 and the maximum absolute values of the roll angle of the suspensions are always less than 7-8 (deg). When we compare it with the passive system, we can see that the first H∞ active antiroll bar controller has significantly reduced (approximately 38%) and the second H∞ active antiroll bar controller has significantly reduced (approximately 54%) the normalized load transfer and the roll angle of suspension at the front and rear axles, respectively.Figure 13
Time responses of normalized load transfer and roll angle of suspension on the axles of the truck: in a double lane change to overtake manoeuver. (a) Normalized load transfer at the front axle. (b) Normalized load transfer at the rear axle. (c) Roll angle of the suspension at the front axle. (d) Roll angle of the suspension at the rear axle.
(a)(b)(c)(d)


### 5.3. Truck in a Cornering Maneuver

The final scenario used to evaluate the performance of the proposedH∞ controller is when the truck is in a cornering maneuver, which is a very common case of the rollover phenomenon. In this case, the lateral acceleration is inversely proportional to the radius of the trajectory, and the truck has a high center of gravity and a heavy load, the possibility of a rollover accident also increases. Figure 14 shows the steering angle, the roll angle of the sprung mass, and the roll angles of the unsprung masses at the front and rear axles. When the steering angle increases, i.e., the truck turns around, the values of the angles also increase and reach a stable value when the driver keeps the steering wheel at 2.5 (deg). From 2.5 s, the roll angle of the sprung mass reaches 3.8 (deg) with the passive antiroll bar system, while with the first H∞ active antiroll bar controller, this value reaches 2.6 (deg), and with the second H∞ active antiroll bar controller, this value reaches 1.8 (deg). Thus, compared to the first H∞ active antiroll bar controller, the roll angle of the sprung mass using the second H∞ active antiroll bar controller has been reduced by about 20%.Figure 14
Time responses of the steering angle and roll angle of the sprung mass and unsprung masses of the truck: in a cornering manoeuver. (a) Steering angle. (b) Roll angle of sprung mass. (c) Roll angle of unsprung mass at the front axle (d) Roll angle of unsprung mass at the rear axle.
(a)(b)(c)(d)Figure15 shows the results of the evaluation of the normalized load transfer at the two axles and the roll angles of the suspension system. The response according to the steering angle of the above signals is very suitable. When the truck used the second H∞ active antiroll bar controller, the normalized load transfer is reduced by 50% compared with the passive antiroll bar system and about 20% compared with the first H∞ active antiroll bar controller. Similar results are evident in the roll angle of the suspension system at the two axles.Figure 15
Time responses of normalized load transfer and roll angle of suspension on the axles of the truck: in a cornering manoeuver. (a) Normalized load transfer at the front axle. (b) Normalized load transfer at the rear axle. (c) Roll angle of the suspension at the front axle. (d) Roll angle of the suspension at the rear axle.
(a)(b)(c)(d)Simulation results in the time domain with different scenarios reinforce the results in the frequency domain. Therefore, it can be asserted that the use of the suspension roll angle sensors has improved the efficiency of the active antiroll bar system using the robust control method.


## 5.1. Truck in a Double Lane Change to Avoid Obstacle Manoeuver

The truck manoeuver is a double lane change to avoid obstacles, which is often used to avoid an obstacle in an emergency. The manoeuver has a 2.5 m path deviation over 100 m. The size of the path deviation is chosen to test real obstacle avoidance in an emergency. The forward velocity is considered at 70 km/h. Figure10 shows the steering angle, the roll angle of the sprung mass, the roll angle of the unsprung mass at the front axle and the roll angle of the unsprung mass at the rear axle. We can see that the first H∞ active antiroll bar controller has significantly reduced (approximately 40%) the roll angles of the sprung mass, the unsprung mass at the front axle, and the roll angle of the unsprung mass at the rear axle compared to the passive antiroll bar system. And the second H∞ active antiroll bar controller has decreased dramatically (approximately 55%), respectively.Figure 10
Time responses of the steering angle, roll angle of the sprung mass, and unsprung masses of the truck: in a double lane change to avoid obstacle manoeuver. (a) Steering angle. (b) Roll angle of sprung mass. (c) Roll angle of unsprung mass of the front axle. (d) Roll angle of unsprung mass of the rear axle.
(a)(b)(c)(d)Figure11 shows the normalized load transfer and the roll angle of the suspension at the front and at the rear axle, respectively. We can see that the value of the normalized load transfer at the rear axle exceeds −1 at 2.8 seconds in the case of the passive antiroll bar system, but this value is within the limitation at the front axle. In two cases of H∞ active antiroll bar controller, the roll stability is achieved because the limitation of the normalized load transfer is in the range from −1 to 1. The maximum of absolute values of the roll angle of the suspensions is always less than 7-8 (deg), so they are within the limitations of the suspension travel. When compared with the passive system, we can see that the first H∞ active antiroll bar controller has significantly reduced (approximately 40%) and the second H∞ active antiroll bar controller has decreased substantially (about 55%) the normalized load transfer and the roll angle of suspension at the front and the rear axles, respectively.Figure 11
Time responses of normalized load transfer and roll angle of suspension on two axles of the truck: in a double lane change to avoid obstacle manoeuver. (a) Normalized load transfer at the front axle. (b) Normalized load transfer at the rear axle. (c) Roll angle of the suspension at the front axle. (d) Roll angle of the suspension at the rear axle.
(a)(b)(c)(d)The simulation results when the truck in a double lane change to avoid obstacle manoeuver clearly shows the effectiveness of the secondH∞ active antiroll bar controller in using the sensors of the suspension roll angle.


## 5.2. Truck in a Double Lane Change to Overtake Manoeuver

In this subsection, the authors introduce the time responses of a truck in a double lane change to overtake a manoeuver. Figure12 shows the steering angle, the roll angle of the sprung mass, and the roll angles of the unsprung mass at the front and rear axles. When compared with the passive antiroll bar system, we can see that the first H∞ active antiroll bar controller has significantly reduced (approximately 41%) and the second H∞ active antiroll bar controller has diminished considerably (about 53%) for the roll angle of the sprung mass, the roll angle of the unsprung mass at the front and rear axles.Figure 12
Time responses of the steering angle and roll angle of the sprung mass and unsprung masses of the truck: in a double lane change to overtake manoeuver. (a) Steering angle. (b) Roll angle of sprung mass. (c) Roll angle of unsprung mass at the front axle. (d) Roll angle of unsprung mass at the rear axle.
(a)(b)(c)(d)Figure13 shows the normalized load transfer and the suspension roll angle at the front and rear axles. We can see that the values of normalized load transfer are always in the range from −1 to 1 and the maximum absolute values of the roll angle of the suspensions are always less than 7-8 (deg). When we compare it with the passive system, we can see that the first H∞ active antiroll bar controller has significantly reduced (approximately 38%) and the second H∞ active antiroll bar controller has significantly reduced (approximately 54%) the normalized load transfer and the roll angle of suspension at the front and rear axles, respectively.Figure 13
Time responses of normalized load transfer and roll angle of suspension on the axles of the truck: in a double lane change to overtake manoeuver. (a) Normalized load transfer at the front axle. (b) Normalized load transfer at the rear axle. (c) Roll angle of the suspension at the front axle. (d) Roll angle of the suspension at the rear axle.
(a)(b)(c)(d)


## 5.3. Truck in a Cornering Maneuver

The final scenario used to evaluate the performance of the proposedH∞ controller is when the truck is in a cornering maneuver, which is a very common case of the rollover phenomenon. In this case, the lateral acceleration is inversely proportional to the radius of the trajectory, and the truck has a high center of gravity and a heavy load, the possibility of a rollover accident also increases. Figure 14 shows the steering angle, the roll angle of the sprung mass, and the roll angles of the unsprung masses at the front and rear axles. When the steering angle increases, i.e., the truck turns around, the values of the angles also increase and reach a stable value when the driver keeps the steering wheel at 2.5 (deg). From 2.5 s, the roll angle of the sprung mass reaches 3.8 (deg) with the passive antiroll bar system, while with the first H∞ active antiroll bar controller, this value reaches 2.6 (deg), and with the second H∞ active antiroll bar controller, this value reaches 1.8 (deg). Thus, compared to the first H∞ active antiroll bar controller, the roll angle of the sprung mass using the second H∞ active antiroll bar controller has been reduced by about 20%.Figure 14
Time responses of the steering angle and roll angle of the sprung mass and unsprung masses of the truck: in a cornering manoeuver. (a) Steering angle. (b) Roll angle of sprung mass. (c) Roll angle of unsprung mass at the front axle (d) Roll angle of unsprung mass at the rear axle.
(a)(b)(c)(d)Figure15 shows the results of the evaluation of the normalized load transfer at the two axles and the roll angles of the suspension system. The response according to the steering angle of the above signals is very suitable. When the truck used the second H∞ active antiroll bar controller, the normalized load transfer is reduced by 50% compared with the passive antiroll bar system and about 20% compared with the first H∞ active antiroll bar controller. Similar results are evident in the roll angle of the suspension system at the two axles.Figure 15
Time responses of normalized load transfer and roll angle of suspension on the axles of the truck: in a cornering manoeuver. (a) Normalized load transfer at the front axle. (b) Normalized load transfer at the rear axle. (c) Roll angle of the suspension at the front axle. (d) Roll angle of the suspension at the rear axle.
(a)(b)(c)(d)Simulation results in the time domain with different scenarios reinforce the results in the frequency domain. Therefore, it can be asserted that the use of the suspension roll angle sensors has improved the efficiency of the active antiroll bar system using the robust control method.


## 6. Conclusions

This article has focused on studying the influence of the suspension roll angle sensors on the efficiency of the active antiroll bar system using robust controllers. A truck model was first introduced to solve that goal, and the dynamics equation was written as a state space representation. Then, a basic structure of an active antiroll bar system with the robust control method (firstH∞ active antiroll bar controller) was described in detail. Then, a full controller for the active antiroll bar system (second H∞ active antiroll bar controller) is proposed considering the information from the suspension roll angle sensors, and this is also the performance output considered. Simulation results in the frequency domain and time domain with different scenarios have clearly shown that the effectiveness of the active antiroll bar system using the suspension roll angle sensors has improved the efficiency compared to the basic controller by about 20% and about 50% when compared with the passive antiroll bar system.Further research can be conducted to examine the effect of measurement noise on the performance of an active antiroll bar system.



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*Source: 2715787-2023-01-17.xml*