## Abstract

In automotive drive systems, differential gear backlash degrades the control performance. Specifically, a shock torque, which is generated when the gear runs freely and collides with the backlash, increases the vibration amplitude. Consequently, it is important to develop a vibration control method to suppress the adverse effect of nonlinearity due to backlash. Furthermore, considering implementations on actual vehicles, design at the development site, and mass production, a simple and practical control method is necessary. This paper describes the configuration of a basic experimental device, which abstracts an actual vehicle to focus on the influence due to backlash while reflecting the basic structure of an automotive drive system. Next, a basic controller is designed using a mixed $H2/H∞$ control theory, and a servo system is constructed to track the target value. A simple control mode switching algorithm is proposed for backlash compensation. This algorithm is suited to practical applications because it uses only an output without a state estimation and it compensates for performance deteriorations due to the nonlinearity by operating a single linear controller. Finally, simulations and experiments verify the effectiveness of the proposed control system.

## 1 Introduction

Drivers demand not only higher performances but also increased diversity in automobiles. From the viewpoints of comfort and driving performance, one challenge is to reduce vehicle body vibrations. Acceleration induces a sudden change in the engine torque and causes an automobile body to vibrate. Such vibrations are caused by torsional vibrations of the automobile drive system, which is composed of an engine flywheel, differential gear, drive shaft, tires, etc. Vehicle body vibrations deteriorate the riding comfort and driving performance over time. Additionally, differential gear backlash degrades the control performance. Specifically, the vibration amplitude increases due to the shock torque generated when the gear runs freely and collides with backlash. This is the nonlinearity produced by the dead-zone effect due to backlash, deteriorating the vibration level. In a closed-loop control, the discontinuous property due to backlash produces an uncontrolled time zone with an accumulation of control errors. This situation results in an excessive control input and a deteriorated vibration control performance. Consequently, it is important to develop a vibration control method to suppress the adverse effect of nonlinearity. Furthermore, a simple and intuitive control method that considers actual mountings, design at the development site, and mass production, is necessary.

Previous studies have investigated machine control, including backlash. Dhaouadi et al. [1] studied a motor drive system using the descriptive function method, while Seidl et al. [2] applied a neural network to a positioning system. Azenha and Tenreiro Machado [3] constructed a variable structure controller for a robot with a dead zone. Yang and Fu [4] proposed a nonlinear adaptive control law for a manipulator with gears, while Khan et al. [5] investigated a P-control gain switching law using a two-degree-of-freedom system.

Moreover, Barbosa and coworkers [6,7] examined the robustness of a fractional order proportional-integral-differential controller and a descriptive function method for backlash systems. Wu et al. [8] applied their proposed adaptive fuzzy type $H∞$ control to a tower crane system. Dong and Tan [9] proposed an internal model control strategy based on a pseudo-Hammerstein model with backlash. Kolnik and Agranovich [10] modeled the influence of backlash as a disturbance, leading to compensation by the disturbance observer. Márton and Lantos [11] modeled a nonlinear rotating gear system with a hybrid model approach and applied an linear quadratic control. Wang et al. [12] employed an adaptive sliding mode control to an electric throttle valve of an automobile. Additionally, nonlinear $H∞$ control was applied to a backlash system [13,14]. Aghababa and Aghababa [15] studied adaptive stabilization control to a gyrostat system.

In recent years, vibration control of vehicle powertrains, including those in hybrid electric vehicles, has become increasingly important. Tang et al. [16] established a novel dynamic model of a power-split hybrid powertrain and demonstrated the effectiveness of a dual-mass flywheel. Furthermore, a technique composed of vibration source control and vibration transfer path control has been proposed for start–stop operations [17].

Previous studies have controlled automobile drive systems, including backlash via modeling. In particular, many outcomes are related to model predictive control and state estimations [1825]. These are broadly considered effective techniques. Furthermore, Berriri et al. [26] focused on tuning the design parameter of a control system using a simplified engine model. Wu et al. [27] proposed an expression for the dead zone as disturbance and compensation by a state observer. Fietzek and Rinderknecht [28] considered a nonlinear observer and applied a model-based pole assignment. Abass and Shenton [29] demonstrated a control method by the nonlinear quantitative feedback theory.

Although the effectiveness of model predictive control has been demonstrated in previous studies, two problems remain. First, a state estimator is required. Second, the prediction calculation induces a computational burden in every step. With regard to the backlash discontinuity, switching control systems such as adaptive control are being studied. However, many of these approaches require multiple plant models and controller design. Other methods have their own issues such as implementation difficulty due to complex control systems and intricate design. Hence, many studies have shown the effectiveness via simulations without experiments. Few studies have pursued backlash compensation, which can achieve high control performance by convenient logic for practical use.

Consequently, a simple control logic suitable for implementation that is experimentally tested in practical applications must be developed. This research aims to realize a control system suitable for implementation and verify its effectiveness experimentally. The experiments involve a device, which abstracts actual vehicles to focus on the influence due to backlash while reflecting the basic structure of an automotive drive system. Hence, the influence of backlash can be focused on, allowing the compensation efficiency of control methods to be evaluated easily. Regarding the control system, we focus on a method to operate a linear model-based controller in order to avoid excessive control inputs and to reduce the shock force at a gear collision via a simple technique. This technique represents practical advantages such as using only an output without a state estimation. Operating only a single linear controller compensates for control performance degradation due to nonlinear characteristics. Moreover, functional modeling is used for the control system design. Because this model expresses a nonlinear system as a linear time-varying state equation, model-based linear control theory can be easily applied. For the controller design, a mixed $H2/H∞$ control theory enabling both control performance and robust stability is used. Finally, simulations and experiments confirm that the control system provides a high control performance and can compensate for the backlash effect.

## 2 Basic Experimental Device

### 2.1 Automobile Drive System.

Here, in the automobile drive system modeled is a power transmission system that transmits engine torque to the tires (Fig. 1). Figure 2 shows the structure of the drive system. It is a rotary motion system where an engine flywheel is coupled to a differential gear via a clutch and transmission as well as to the tires and vehicle body via a drive shaft. Mechanically, the engine flywheel is modeled as a moment of inertia. The clutch is modeled as an element with stiffness and friction damping, while the transmission is modeled as a gear ratio. The differential gear combined with torsional stiffness is modeled as an element with a moment of inertia and backlash. The drive shaft is modeled with torsional stiffness and damping. A tire is modeled by the moment of inertia.

Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal

### 2.2 Equivalent Mechanical Model.

Figure 3 shows the structure of an equivalent mechanical model, which emphasizes the influence of backlash of the actual automobile drive system. The structure of the model is a three-degree-of-freedom system. Considering the transmitted force in only one-dimensional direction, the variables defining the system dynamics are the three displacements $XB$, $xG$, and $XE$ at the three mass points $MB$, $mG$, and $ME$ in that direction. The vehicle body $MB$, gear part $mG$, and actuator $ME$ are connected via stiffness and damping. The original automobile drive system is a rotating system in which torque is transmitted from the engine to the tires. Herein, a model where the system is replaced with an equivalent translational system that transfers the force in one-dimensional is considered to simplify the structure of the experimental device. The equivalent moment of inertia from the engine to the input side differential gear is addressed by integrating the actuator mass $ME$. The friction damping $Ccl$ of the clutch is modeled as a damping force, which is proportional to the actuator speed $ẊE$. The torsional stiffness and damping of the gear correspond to the spring $KG$ and damper $CG$ in Fig. 3, respectively. The mass $mG$ corresponds to the output side differential gear. The torsional stiffness and damping of the drive shaft correspond to the spring $KD$ and the damper $CD$, respectively. The mass $MB$ corresponds to the tire of the drive system and the vehicle body. As the control input in the original drive system, the engine torque is reproduced by the actuator thrust $uLM$ in the equivalent mechanical model. Backlash, which is an important nonlinear element, exists in the differential gear. In the equivalent mechanical model, backlash is produced by creating a dead zone between the actuator and $mG$.

Fig. 3
Fig. 3
Close modal

If the actuator thrust changes rapidly from negative to positive, the actuator idles in the gap to the output gear $mG$. Consequently, it collides with the momentum. The shock force is transferred downstream. Finally, the mass of the vehicle body $MB$ vibrates. $MB$ is connected to the wall via the spring $KC$ and damper $CC$ at the most downstream point. In this study, positioning control while suppressing vibrations due to backlash is performed using the vehicle body displacement $XB$ as the observed output.

### 2.3 Experimental Device.

As a theoretical basis to transform the rotating system into the translation system, the equations of motion for rotation of an actual vehicle drivetrain after simplification can be written in the same form as the equations of motion of the translation system described later.

In this study, the experimental device is aimed at focusing on backlash and its influence on the vehicle body in an actual vehicle. Therefore, the experimental device is constructed to purposely simplify the structure of an actual vehicle. In other words, the device does not completely reproduce an actual vehicle and its dynamics. Specifically, in order to make it easy to evaluate the adverse effect due to backlash and the improvement by the control system, we constructed the experimental device for basic study.

Figure 4 shows the basic experimental device and relates it to the equivalent mechanical model shown in Sec. 2.2. The actuator uses a linear shaft motor, and the slider mass on the rail is $ME$. The vehicle body mass $MB$ is connected by coil springs from both sides and moves on the linear guide installed in parallel with the motor. At an intermediate point between the guide and the motor, a gear mass $mG$ is coupled to the vehicle body. $KG$ and $KD$ are reproduced by a leaf spring in the experimental device. The damping coefficients $CG$, $CD$, and $CC$ in Fig. 3 are identified experimentally. For backlash, creating an interval of leaf springs on both sides of $mG$ causes a gap (Fig. 4). Adjusting the interval between the two leaf springs changes the backlash length. Table 1 shows the device specifications. Here, each of these parameters was empirically designed as a unique value of the experimental device so as to more prominently reproduce the influence of backlash of an actual vehicle even in the device.

Fig. 4
Fig. 4
Close modal
Table 1

Parameters of experimental device

SymbolValueUnit
$MB$$0.232$$kg$
$mG$$0.039$$kg$
$ME$$1.04$$kg$
$KC$$660.0$$N/m$
$KD$$1.5×104$$N/m$
$KG$$2.7×104$$N/m$
$CC$$0.31$$N·s/m$
$CD$$5.55$$N·s/m$
$CG$$20.12$$N·s/m$
$Ccl$$3.58$$N·s/m$
SymbolValueUnit
$MB$$0.232$$kg$
$mG$$0.039$$kg$
$ME$$1.04$$kg$
$KC$$660.0$$N/m$
$KD$$1.5×104$$N/m$
$KG$$2.7×104$$N/m$
$CC$$0.31$$N·s/m$
$CD$$5.55$$N·s/m$
$CG$$20.12$$N·s/m$
$Ccl$$3.58$$N·s/m$

Figure 5 shows the displacement response of the vehicle body and the motor thrust. These are experimental results simulating a series of phenomena from a sudden change in engine torque to idling, a collision of gears in backlash, and longitudinal vibration of the vehicle body. The lower graph of thrust gives a step that changes from –4 N to 15 N at 2 s. This corresponds to the driving condition with a sudden change in the engine torque. This condition shows the most prominent adverse effect of shock due to backlash. The upper diagram in Fig. 5 shows the vehicle body displacement for various backlash lengths. The black, blue, and red lines are the responses for a length of 0 mm, 10 mm, and 20 mm, respectively. The difference clearly depends on the backlash width. The longer the length, the larger the shock force due to a collision. Both the vibration amplitude and convergence worsen. Specifically, the effect due to backlash can be evaluated in the form of the amplitude increase after coupling in backlash or the overshoot during the closed-loop control under the driving conditions of tip-in and tip-out. These are the target dynamics that the experimental device must reproduce. The simulation results, which are obtained from actual vehicle data provided by the automobile manufacturer and the model shown in Fig. 2, were compared with the experimental results (Fig. 5). The increase rate in the amplitude of the vehicle body vibration due to backlash in the experimental device was equal to or higher than that of the actual vehicle. Consequently, Fig. 5 means that the experimental device was developed with an emphasis on the reproduction of the nonlinearity due to backlash in an actual vehicle, and we can directly evaluate it.

Fig. 5
Fig. 5
Close modal

## 3 System Modeling

### 3.1 Functional Model.

The experimental device is modeled for the control system design and simulations using the Functional model [3033]. Figure 6 shows the modeled structure of the experimental device.

Fig. 6
Fig. 6
Close modal

The functional model diagram gives a nonlinear model collectively as one linear state equation. As the parameters in the linear state equation change from moment to moment, the nonlinear characteristics of the model can be represented equivalently. The yellow blocks in the model diagram (Fig. 6) are called a “mechanism model.” The nonlinear characteristics of the system are integrated internally into an equivalent lumped constant and updated as a nonlinear parameter at every time-step. The nonlinear parameter in the mechanism model is calculated based on physical laws, theoretical expressions, experimental formulas, and data from each field. The nonlinear parameters in Fig. 6 are $Sw$, $OKG$, $err$, $Oer$, and $Fr$.

The nonlinear plant obtained from the block diagram is a time-varying system, which can be represented by a linear state equation at each time-step. Therefore, linear control theory is easily applied to a nonlinear system. In addition, since a state equation can be obtained in real-time for discontinuous nonlinear characteristics such as backlash, multiple models for each mode do not need to be prepared in advance.

### 3.2 State Equation.

For the three vibration systems in the functional model diagram, the following equations of motion are obtained from Newton's second law. $ẌB$, $ẍG,$ and $ẌE$ are described as
$ẌB=1MB{KDxG−XB+CDẋG−ẊB−errKCXB−Oer−CCẊB+Fr}$
(1)
$ẍG=1mG{SwKGXE−xG+OKG+SwCGẊE−ẋG+KDXB−xG+CD(ẊB−ẋG)}$
(2)
$ẌE=1MEuLM−CclẊE−SwCGẊE−ẋG−SwKGXE−xG−OKG$
(3)
Accordingly, the state equation and the output equation of the plant for an experimental device are derived as
$ẋp=Apxp+Bp1wp+Bp2up$
$y=Cpxp+Dp1wp+Dp2up$
(4)
Here, each coefficient matrix is expressed as
$Ap=[000100000010000001−(KD+errKC)MBKDMB0−(CD+CC)MBCDMB0KDmG−(SwKG+KD)mGSwKGmGCDmG−(SwCG+CD)mGSwCGmG0SwKGME−SwKGME0SwCGME−(SwCG+Ccl)ME]Bp1=[ 0000000000−1MB1MB1mG00−1ME00 ], Bp2=[ 000001ME ], Cp=[ 100000 ]Dp1=0, Dp2=0$
(5)
Each external input in Eq. (4) is as follows:
$wp= OKGOerFr, up=uLM$
(6)
The state equation contains nonlinear parameters. In particular, the nonlinear characteristics of backlash, which are important in this research, are expressed as $Sw$ in the A-matrix and $OKG$ in the external input. Backlash acts as a dead band. Assuming the relative displacement of the motor slider and the gear mass is $ΔX$, then the relation of the nonlinear parameter and transferred spring force $F$ is expressed as
(7)

When the relative displacement is within the range of the backlash width $B$, the motor and the gear mass do not make contact, and the force is not transferred. On the other hand, if the relative displacement exceeds the backlash width, the motor and gear make contact via the stiffness $KG$ and the spring force is transferred. $Sw$ is a switching variable to determine contact, while $OKG$ is the offset of the spring force due to the backlash width.

The structure induces other nonlinear characteristics in the experimental device. One is the nonlinear characteristic of the coil spring constant $KC$. This is given by the parameters $err$ and $Oer$ of the functional model diagram. The dynamic friction, which exists in the linear guide, is modeled as Coulomb's frictional force $Fr$. The nonlinearities other than backlash are intentionally not reproduced as elements of an actual vehicle. Namely, these are just constraints, which are originally included in the experimental device.

## 4 Control System Design

### 4.1 Controller Design.

In this study, a controller is designed based on the mixed $H2/H∞$ control theory, which has excellent vibration control performance and a robust stability [3334]. Figure 7 shows a block diagram of the control system. Here, $P(s)$ is the plant and $K(s)$ is the controller. $u$ is the control input, and in this research, it is the motor thrust $uLM$. $e$ is the control error and $r$ is the target displacement. $w′$ is the disturbance acting on the plant due to the shock force generated by backlash and the dynamic friction of the linear guide. $z2$ is the controlled variable related to the transient vibration response. $z∞$ is the controlled variable related to the control input restriction, which is the performance index for the robust stability. For each controlled variable, the $H2$ norm of the transfer function matrix $Tz2w$ from an external input $w=rw′TT$ to $z2$ and the $H∞$ norm of the transfer function matrix $Tz∞w$ from $w$ to $z∞$ correspond to the performance indices.

Fig. 7
Fig. 7
Close modal
The vehicle body displacement is the controlled variable that must follow the target value while suppressing the transient response vibrations. For this reason, the displacement is evaluated by the $H2$ norm. The corresponding frequency weighting function $W2(s)$ is described as the following function to contain an integrator in a third-order low-pass filter:
$W2s=2.481×107s3+125.7s2+7896s+2.481×105·10s$
(8)
The weighting function corresponding to the control input evaluated in the $H∞$ norm uses the following third-order high-pass filter:
$W∞s=10s3s3+1257s2+7.896×105s+2.481×108$
(9)
This provides a high-gain filter in the high-frequency band, realizing a robust stability. Because $W2(s)$ in Eq. (8) has an unstable origin pole and the generalized plant satisfies neither the stability condition nor the detectability of a closed-loop system when designing the controller, $W2(s)$ should be divided into a proportional-integral (PI) compensator $M(s)$ with an origin pole and a transfer function $W̃2(s)$ of the stable and minimum phase
$W2s=2.481×107s3+125.7s2+7896s+2.481×105·10s=2.481×107s3+125.7s2+7896s+2.481×105·10s+0.1·s+0.1s=W̃2(s)·M(s)$
(10)
$W̃2s=2.481×107s3+125.7s2+7896s+2.481×105·10s+0.1Ms=s+0.1s$
(11)
The performance weighting function $W̃2(s)$ and PI compensator $M(s)$ are used in the generalized plant in Fig. 8, which equivalently converts into the generalized plant in Fig. 7. Thus, the suboptimal controller $K̃(s)$ is designed. Each frequency weighting function and the PI compensator in Fig. 8 are given by the following state space representation:
$W̃2s=(A2,B2,C2,D2),W∞s=(A∞,B∞,C∞,D∞), and Ms=(AM,BM,CM,DM)$
(12)
Fig. 8
Fig. 8
Close modal
The state equation of the generalized plant $G(s)$ considering these evaluation specifications is described as:
$ẋG=AxG+B1w+B2uz∞=C∞xG+D∞1w+D∞2uz2=C2xG+D21w+D22uẽ=CyxG+Dy1w+Dy2u$
(13)
Here, each coefficient matrix is given as
$A=[ Ap000−BMCpAM00−B2DMCpB2CMA20000A∞ ], B1=[ 0Bp1BM−BMDp1B2DM−B2DMDp100 ] B2=[ Bp2−BMDp2−B2DMDp2B∞ ]C∞=[ 000C∞ ], C2=[ −D2DMCpD2CMC20 ]Cy=[ −DMCpCM00 ]D∞1=0, D∞2=D∞, D21=[D2DM−D2DMDp1], D22=−D2DMDp2Dy1=[DM−DMDp1], Dy2=−DMDp2$
(14)
For a closed-loop system of $G(s)$ and $K̃s$ and for the given $γ$, we designed a controller $K̃s$ to minimize the $H2$ norm $Tz2w(s)22$ while satisfying the $H∞$ norm restriction $Tz∞w(s)∞<γ$ [34]. As a design method, the linear matrix inequality approach is used. From the equivalent conversion of the block diagram using the suboptimal $H2/H∞$ controller $K̃s$ and the PI compensator, the optimal $H2/H∞$ controller $K(s)$ with an integrator is obtained as shown below. $K(s)$ is designed using MATLAB's “Control System Toolbox” and “Robust Control Toolbox”
$Ks=K∼sMs=K∼(s)s+0.1s$
(15)

### 4.2 Feedforward Input.

The servo system is constructed using the controller $K(s)$ from Sec. 4.1. Since the controller includes an integrator, the vehicle body displacement may be delayed with respect to the increased target displacement. Therefore, a feedforward input is co-used to improve the transient response. Specifically, the motor thrust necessary for bending the spring $Kc$ to the target displacement is added to the input $u$.

## 5 Compensation for Backlash

If a controller, which is nonvariant for normal operations, is used at all times, the control error accumulates due to backlash and the motor collides. This deteriorates the vibration control performance. The behavior of the vehicle body is determined by the accelerator operations of the driver in an actual vehicle. There is dead time before these operations are reflected on the actual vehicle body behavior. This dead time, which is not related to backlash, is an inherent property that is originally included in a vehicle system. The proposed control system positively utilizes this feature to reduce backlash beforehand and to avoid accumulation of the control error during dead time. Hence, the control performance can be improved.

Therefore, we introduce this dead time into the experimental device and make the dead time before the target vehicle-body displacement increases. The value of this dead time is defined as 65 ms in this study. Switching the control modes avoids control error accumulation and reduces backlash beforehand. Figure 9 shows the switching law.

Fig. 9
Fig. 9
Close modal

Four control modes (I–IV) are switched at each step. However, with regard to the controller itself, only a same mixed $H2/H∞$ controller, which is designed in Sec. 4, is used in each mode.

(Control mode I)
$ek=rk−XBk$
(16)
$xKdk+1=AKdxKdk+BKdek$
(17)
$u1k=CKdxKdk+DKdek+KCr(k)$
(18)
(Control mode II)
$eprek=rprek−XBk=rpre−XB(k)$
(19)
$xAWk+1=AKdxKdk+BKdepre(k)$
(20)
$xAWk=z−1xAWk+1 ( xAWk=xKdk )$
(21)
$xKdk+1=xAWk ( substitute xAWk for xKdk+1 )$
(22)
$u2k=CKdxKdk+DKdeprek+KCrpre$
(23)
(Control mode III)
$xKdk+1=AKdxKdk+BKdeprek$
(24)
$u3k=CKdxKdk+DKdeprek+KCrpre$
(25)
(Control mode IV)
$xAWk+1=AKdxKdk+BKde(k)$
(26)
$xAWk=z−1xAWk+1$
(27)
$xKdk+1=xAWk$
(28)
$u4k=CKdxKdk+DKdek+KCr(k)$
(29)

The mixed $H2/H∞$ controller of each control mode implements $(AKd, BKd, CKd, DKd)$, which are obtained by discretizing the continuous time state space representation $(AK, BK, CK, DK)$ of the controller in Eq. (15). $xKd(k)$ is the internal state variable of the controller.

Control mode I is the normal control mode. It does not compensate for backlash. Using the target value $r(k)$ and vehicle body displacement $XBk$, the error $ek$ is calculated and the feedforward input $uF=KCr(k)$ is used.

Control mode II is a control mode where compensation for backlash occurs 65 ms before the target value increases. This process reduces backlash beforehand, avoiding accumulation in the control error. At 65 ms before the target displacement increases, the target is switched to a small positive value $rpre$ to calculate the error $eprek$. $rpre$ is also used for the feedforward input. Since the controller outputs a small thrust force to follow this target value, backlash is reduced. It is impossible to transfer the thrust to the vehicle body because the motor and the gear part do not make contact while backlash is reduced. That is, the system is in an uncontrolled state. This uncontrolled state causes errors to accumulate in the controller, and unnecessarily large control inputs are calculated. Consequently, the motor accelerates during the process to reduce backlash. A collision occurs at the time of coupling, generating a shock force. To prevent such a process, anti-windup should be applied to the controller during the backlash-reducing process. The state variable of the controller is calculated at each step with a discrete-time state equation and is updated by the difference equation. Normally, the control input is calculated with the updated state variable. However, in anti-windup, once backlash starts to be reduced, the state variable is no longer updated. By ending the update and calculating the input with an internal state of the same value, error accumulation can be avoided. $z−1=e−hs$ in Eq. (21) is a one control cycle $h$ delay operator. Accordingly, control mode II simultaneously performs backlash-reducing process and anti-windup.

Target value switching ends when the original target value $r(k)$ exceeds $rpre$, and the original target value is restored. Anti-windup to the controller stops at the moment when backlash is eliminated. This moment is identified by a jerk, which is the time differentiation of vehicle body acceleration. At the moment of backlash elimination, the vehicle body switches discontinuously from a free vibration to a forced vibration. That is, the acceleration of the vehicle body changes suddenly, and the jerk is larger than the values at other points in time, generating a steep increase. Herein a threshold value is set, and the time that backlash elimination occurs is defined as the time where the absolute value of the jerk exceeds the threshold.

Control mode III is a control mode where anti-windup stops because backlash is eliminated. It is a waiting state until the original target value $r(k)$ exceeds the $rpre$.

By contrast, control mode IV is a control mode where anti-windup is continuously applied since backlash is not fully eliminated, although the original target value $r(k)$ increases.

Backlash length information is unnecessary for the control system design and the compensation. The compensation can cope with unknown backlash, which is a nominal backlash with length fluctuations, by the two switching routes shown in Fig. 9. There are two types of routes: branching to the left and branching to the right after mode II (Fig. 9). The route depends on whether the jerk threshold condition ($Jerk≥Threshold$) or the target switching condition ($rk≥rpre$) is satisfied first. The left route (I, II, III, I) is for a short backlash width. Backlash has been eliminated, and it is in a controllable state before the original target value exceeds $rpre$. For this reason, the vehicle body is stopped at $rpre$. When the original target value exceeds it, the vehicle body displacement can smoothly shift from $rpre$ to $rk$ and immediately follow. The right route (I, II, IV, I) is for a long backlash width. The target value is switched to the original target value before backlash is fully reduced. Since backlash remains, it is necessary to prevent the motor from idling rapidly and colliding at the moment of the switching. Hence, the motor is driven gently by continuously applying anti-windup to the controller. The compensation algorithm roughly classifies unknown backlashes into long and short, and the two switching routes shown in Fig. 9 deal with them. The order in which $rk≥rpre$ or $Jerk≥Threshold$ is initially satisfied depends on the length (long or short) of the unknown backlash. Thus, the controller automatically selects (I, II, III, I) or (I, II, IV, I) for the switching routes in real-time according to the length without knowing the backlash length.

## 6 Simulations and Experiments

### 6.1 Configuration of the Experimental System.

Figures 10 and 11 outline the experimental system. The vehicle body displacement is measured with a laser displacement sensor. A signal amplifier amplifies the displacement signal and fed it back to the controller. The control input is calculated by a digital signal processor. The control input signal output from the digital signal processor is sent to a servo amplifier. According to the control input, the motor slider is driven to perform vibration and positioning control of the vehicle body. In addition, the displacement signals and thrust indication signals are recorded by a data-collecting device.

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

In actual vehicles, the engine serves as the actuator to realize control input, and an engine control unit calculates the proper torque value. Hence, the proposed control system should be implemented on the engine control unit when actual vehicle running tests are performed. Future research involves this task.

### 6.2 Control Simulation Results.

The control simulation was performed using MATLAB/SIMULINK. The plant model runs in real-time based on the time-varying linear state equation in Eq. (4), which is derived from the Functional model shown in Fig. 6. The control system was implemented with a sampling frequency of 50 kHz (discretization method: zeroth-order hold). The closed-loop system for the simulation is composed of the plant and the controller. The verification conditions were the same as those of the experiments.

Figure 12 shows the simulation results for backlash widths of 5 mm (graphs: (a)) and 10 mm (graphs: (b)). The upper graphs in (a) and (b) show the time responses of the vehicle body displacement, while the lower graphs indicate the control inputs.

Fig. 12
Fig. 12
Close modal

Regarding the displacement, the blue line indicates the response without control, while the green line denotes the target displacement. This is a static deflection by pushing the spring $KC$ with the step-thrust in Fig. 5. The black line represents the control results only by the mixed $H2/H∞$ controller without backlash compensation. The red line depicts the control results using backlash compensation by the switching control modes.

Regarding the control input, the blue and black lines indicate the step thrust in Fig. 5 and the thrust when controlled using only the mixed $H2/H∞$ controller without backlash compensation, respectively. The red line shows the thrust with backlash compensation by switching the control modes.

### 6.3 Control Experimental Results.

Figure 13 shows the results of control experiments. Backlash widths of 5 mm (graphs: (a)) and 10 mm (graphs: (b)) were considered. The upper and lower graphs as well as the waveforms of each line are the same as those in the simulations (Fig. 12).

Fig. 13
Fig. 13
Close modal

## 7 Discussion

Using only a mixed $H2/H∞$ controller, after the target value increases at 2 s, the motor collides and the large overshoot occurs in the vehicle body displacement because backlash is not compensated. Additionally, the influence of the shock force induces residual vibrations. There is an uncontrollable time zone in which the force is not transferred from the motor to the vehicle body and gear mass. Even when the target value increases, the vehicle body cannot immediately follow the target value, resulting in the accumulation of the control error. Consequently, the calculated control input is unnecessarily large. When backlash is eliminated, a thrust that exceeds the force necessary to follow the target value is inputted to the motor. Since the motor collides vigorously and the shock force is transferred, the vehicle body displacement exceeds the target value. This is why the nonlinearity of backlash degrades the control performance and cannot be dealt with using a traditional linear controller. The control simulation results obtained by applying only a mixed $H2/H∞$ controller to the actual vehicle model also confirmed that a large overshoot similar to that shown in the black line of Figs. 13(a) and 13(b) occurred in the actual vehicle body behavior.

Using control mode switching as the backlash compensation, a high control performance is achieved in the simulations and the experiments. The thrust waveforms show that the thrust to reduce backlash is inputted 65 ms before the target value increases with the controller applied anti-windup. By this thrust, backlash can be reduced without a rapid acceleration of the motor. When shifting to a controllable state, the errors from the target value barely accumulate and the excess control input does not need to be calculated. Consequently, the shock due to the collision is greatly reduced, improving overshooting and residual vibrations. Immediately after returning to control mode I, it is possible to quickly follow the target value. Figure 14 shows the vehicle body jerk in the experiments. This waveform corresponds to the experimental result in Fig. 13(b). The blue line indicates the waveform without control, while the red line represents the waveform when backlash compensation is applied. The red line indicates that the jerk can be drastically reduced at the moment of eliminating backlash after at 2 s and the influence of backlash on the vehicle body can be suppressed.

Fig. 14
Fig. 14
Close modal

To achieve the above control performance, the process to reduce backlash beforehand and anti-windup to the controller must be used simultaneously. In the case of reducing backlash without anti-windup, the calculated control input during the reduction process is unnecessarily large, degrading the performance. Figure 15 (red line) shows the experimental result controlled without anti-windup. Due to the uncontrollable state while reducing backlash, the control error accumulates, causing the calculated thrust to be excessively large.

Fig. 15
Fig. 15
Close modal

Both Figs. 13(a) and 13(b) are obtained by the same control system parameters, demonstrating that the proposed technique is robust against unknown backlash. This study evaluated two different backlash widths: short (5 mm) and long (10 mm). Switching of the control mode in Fig. 9 is divided into two routes according to the backlash width. Figure 16 shows the switching time history of the control modes for backlash widths of 5 mm (graph: (a)) and 10 mm (graph: (b)). The blue dashed and red solid lines denote the simulated and experimental values, respectively. The numbers 1–4 on the vertical axis correspond to the control modes in Sec. 5. The 5-mm case assumes the route in Fig. 9, left (I, II, III, I), while the 10-mm case takes the route on the right (I, II, IV, I). Accordingly, even when the backlash length fluctuates, the performance can be maintained with these two switching routes.

Fig. 16
Fig. 16
Close modal

## 8 Conclusion

In this research, we propose a simple and efficient control logic suitable for mounting on actual vehicles to control the vibrations of an automobile drive system considering backlash explicitly. First, the paper describes the experimental device, which abstracts actual vehicles to focus on the backlash influence while reflecting the basic structure of an automotive drive system. The policy is given to model the system for controller design and simulations via the functional model, allowing a nonlinear plant to be handled in the form of a linear state equation. Next, a simple control technique, which uses only a control output and compensates for backlash performance degradation using only a single controller, is proposed. We designed a mixed $H2/H∞$ servo controller and demonstrated control mode switching to prevent control error accumulation in backlash and to reduce the collision shock. Finally, the effectiveness of the control system is verified using both simulations and experiments. The high control performance confirms that the system can compensate for the backlash effect.

In the future, we will further improve the control performance. Moreover, we intend to consider multiple engine constraints, including the upper and lower limits of the control inputs, air system delay, and control cycle constraints, to model systems and to design control systems.

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