the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Decoupling active disturbance rejection trajectorytracking control strategy for Xbywire chassis system
Haixiao Wu
Yong Zhang
Fengkui Zhao
Pengchang Jiang
Due to the inherent dynamic coupling between mechanical components such as the steering system and suspension system, the vertical external input will affect the lateral movement of the chassis, which makes it difficult to track the ideal trajectory when complex excitation conditions exist. To solve the abovementioned problems, the Xbywire chassis is taken as the research object in this work, and the coupling dynamic model is established. Then, based on proving the reversibility of the coupling dynamic model, a pseudolinear composite system is proposed to decouple the lateral and vertical signals of the chassis system. Next, the decoupling active disturbance rejection (DADR) trajectorytracking control strategy is proposed. And a multiobjective optimization method of the bandwidth parameters of the DADR trajectorytracking controller is proposed according to its convergence conditions. Experiments show that the proposed control strategy can effectively suppress the vehicle roll and yaw motion caused by the lateral–vertical dynamic coupling in the process of trajectory tracking to realize the accurate tracking of the ideal trajectory.
Automatic lanechange maneuvers and obstacleavoidance systems can effectively reduce the incidence of vehicle collision accidents caused by human behavior to further improve traffic safety (Liu et al., 2020; Ji et al., 2019). The achievements of automatic lanechange maneuvers and obstacleavoidance depend on the highprecision tracking of ideal signals in chassis steering, braking, and other subsystems. However, when tracking obstacle avoidance trajectories in complex traffic environments, the trajectorytracking accuracy and response speed will be affected by both sides of external factors, including random road input, lateral wind, etc., and internal factors, such as vertical and lateral dynamic coupling characteristics. Because of these factors, it is difficult to provide accurate and fast actuator responses in the process of vehicle trajectory tracking (Wang et al., 2017). Therefore, how to design the lanechange trajectorytracking control strategy, considering the external disturbance of the vehicle in complex traffic environments and the vehicle chassis dynamic coupling, is the key point for the intelligent vehicle to ensure safety and stability for lanechange maneuvers.
Existing research on trajectorytracking control mainly focuses on local trajectory replanning and hierarchical control under random disturbances (Yue et al., 2019; Wang et al., 2019). In Yue et al. (2019), aiming at a highway tire burst, a hierarchical control framework for vehicle automatic riskavoidance trajectory planning and tracking is proposed. Wang et al. (2019) solve the problem of the fullstate regulation control of asymmetric underactuated surface vehicles under disturbance. The above studies effectively suppress fluctuations in the trajectorytracking process under disturbance. However, due to lateral and vertical dynamic coupling, signal disturbance on lateral or vertical motion cannot be suppressed independently, and active inputs of steering and suspension systems will simultaneously affect vehicle lateral and vertical dynamics (Wang et al., 2018). For instance, the active roll control moments from the chassis controller will not only affect the vertical signals, such as the roll angular velocity, but also change the lateral dynamic parameters, such as yaw angular velocity. Besides, disturbance compensation in trajectorytracking control has been widely studied (Fiengo et al., 2019; Wang et al., 2019). However, if the compensation control signal changes rapidly, then it is difficult to achieve desired control performance due to lateral and vertical dynamic coupling.
Compared with conventional chassis technology without active control, the advanced Xbywire chassis enables the active control of roll torque, yaw torque, and front wheel angle. Therefore, the trajectorytracking control of the Xbywire chassis system is a typical multivariable control system (Ni et al., 2019). Specifically, when the vehicle equipped with an Xbywire chassis tracks the curve trajectory, then the dynamic coupling on the signal layer will cause interference in the mechanical layer. For example, the road excitation signal will affect the yaw motion and roll motion of the vehicle with the Xbywire chassis system. To eliminate the coupling between inputs and outputs, decoupling methods, such as artificial neural network decoupling, robust decoupling, fuzzy decoupling, and inverse system decoupling, are commonly used in multivariable control systems (Zhao et al., 2019; Wu et al., 2020). In Zhao et al. (2019), to solve the vibration problems caused by the nonlinear coupling of longitudinal, lateral, and vertical dynamics, a hierarchical integrated controller for distributed electric vehicles is proposed to improve driving safety. Wu et al. (2020) propose a slidingmode decoupling control strategy for an electric wheel vehicle based on the inverse system method. However, the abovementioned decoupling control strategies are not applicable for curve trajectorytracking problems in real complex traffic environments. Because these strategies do not take the trajectorytracking accuracy and response speed into account in the lanechange obstacleavoidance process, the disturbance suppression of excitation signals under uncertain environmental conditions is also excluded. In recent years, active disturbance rejection control (ADRC) has been proposed and applied to the tracking problem of multivariable control systems. In Luo et al. (2018), a feedforward coupling compensation decoupling control method based on ADRC is proposed to solve the crosscoupling problem caused by variable wind disturbance in the system. In Zhang and Chen (2016), based on ADRC feedback linearization, a feedforward controller based on an equivalent error model is designed to further improve the tracking performance of the system. In Xia et al. (2016), a tracking control method of an automatic land vehicle (ALV) lateral motion is proposed, and the ADRC is used to ensure the control accuracy and robustness. To some extent, the above methods have solved the performance robustness of the controller in the presence of disturbances and the coupling problem caused by the disturbance input but cannot solve the problem of the internal dynamic coupling of the system, so it needs to be further studied.
To solve the above problems, a decoupling active disturbance rejection (DADR) trajectorytracking control strategy is proposed. First, based on the inverse system theory, the decoupled pseudolinear composite system is constructed to decouple the input and output signals of the original system. Then, for the decoupled pseudolinear system, independent lateral motion and vertical motion DADR trajectorytracking controllers are used. And a multiobjective optimization method of bandwidth parameters of the DADR trajectorytracking controller is proposed according to its convergence conditions. Finally, through the experimental verification on the hardwareintheloop test platform, it is proved that the strategy proposed in this work not only can effectively improve the trajectorytracking performance under road excitation but also significantly inhibits the roll motion of the vehicle when tracking the curve trajectory in order to improve the ride comfort. The decoupling active disturbance rejection (DADR) trajectorytracking control strategy proposed in this work and its hardware in the loop (HIL) test platform are shown in Fig. 1.
The main contributions of this paper are as follows:

The chassis system's yaw signal and roll signal are decoupled on the signal layer so that the yaw motion or roll motion of the chassis system can be compensated and controlled independently. Furthermore, in the mechanical layer, the cooperative control of the chassis mechanical components in the steeringbywire system, brakingbywire system, and active suspension system can be realized.

The system reversibility is analyzed and verified based on the coupled lateral and vertical dynamic model, and the αorder chassis pseudolinear composite system is obtained to decouple the lateral and vertical signals of the chassis system.

A decoupling active disturbance rejection (DADR) trajectorytracking control strategy is proposed, and a multiobjective optimization method of bandwidth parameters of the DADR trajectorytracking controller is proposed according to its convergence conditions.
The rest of this paper is arranged as follows: Sect. 2 is the modeling of the Xbywire chassis and its analysis in the trajectorytracking task. Section 3 proposes the decoupling active disturbance rejection (DADR) trajectorytracking control strategy. Section 4 presents the results and discussion. Section 5 is the conclusion.
To analyze the mechanism of lateral and vertical dynamic coupling, this section first establishes the Xbywire 3 DOF (degrees of freedom) coupled dynamic model, including the input signals of the active roll torque, yaw torque, and front wheel angle, as shown in Eq. (1).
where δ is the differential steering compensation angle signal of the front wheel, T_{z} is the yaw torque signal, and T_{ϕ} is the roll torque signal of the suspension system. m is the vehicle mass, m_{s} is the sprung mass, m_{f} is the front unsprung mass, and m_{r} is the rear unsprung mass. a is the distance from the center of mass to the front axle, b is the distance from the center of mass to the rear axle, I_{z} is the yaw moment of inertia, I_{x} is the roll moment of inertia, I_{xz} is the inertia product of the yaw motion, h is the roll center height, k_{f} is the front wheel yaw stiffness, k_{r} is the rear wheel yaw stiffness, E_{f} is the front roll steering coefficient, E_{r} is the rear roll steering coefficient, v_{x} is the longitudinal speed of the vehicle, β is the sideslip angle of the centroid, ω_{r} is the yaw rate, ϕ_{r} is the roll angle, K_{ϕ} is the suspension roll stiffness coefficient, D_{ϕ} is the suspension roll damping coefficient, and δ_{d} is the front wheel angle applied by the driver.
The state variable of the coupled chassis system is $x=[\mathit{\beta},{\mathit{\omega}}_{r},{\mathit{\varphi}}_{r},{\dot{\mathit{\varphi}}}_{r}{]}^{\mathrm{T}}$, the control input signal is $u=[\mathit{\delta},{T}_{z},{T}_{\mathit{\varphi}}{]}^{\mathrm{T}}$, and the system output signal is $y=[\mathit{\beta},{\mathit{\omega}}_{r},{\mathit{\varphi}}_{r}{]}^{\mathrm{T}}$. The state space of the system is shown in Eq. (2).
Then, the trajectorytracking task is analyzed based on the above dynamic model, which is shown in Eq. (2). For the trajectorytracking task, the goal is to eliminate the error between the actual trajectory and the ideal trajectory e. For the chassis motion control task, the goal is to eliminate the error between the ideal front wheel angle and the actual front wheel angle. Therefore, it is necessary to establish the relationship between the trajectorytracking error and the ideal front wheel angle.
The required front wheel angle to track a given path is calculated based on a pure pursuit algorithm (Yamasaki and Balakrishnan, 2010), as shown in Eq. (3).
where δ_{i} is the ideal front wheel angle, L is the wheelbase, e is the lateral error between the current vehicle attitude and the target waypoint, v_{x} is the longitudinal speed of the vehicle, and k_{v} is the adjustment factor for the tracking process.
Next, ideal signals for the trajectorytracking task are analyzed and listed as follows: tracking the desired trajectory is done by controlling the front wheel rotation angle and maintaining the roll angle as 0. For the Xbywire chassis system, the target trajectory should be converted into the target yaw angle, as shown in Eq. (4) (Wang et al., 2018).
Based on the dynamic model without any decoupling transformation, the above trajectorytracking control tasks are analyzed, and the results are shown in Fig. 3. It can be seen that the lateral motion and vertical motion of the chassis system cannot be controlled separately and are coupled with each other. In other words, the lateral motion cannot be corrected separately without affecting the vertical motion, and vice versa.
In this section, to suppress the random roll motion caused by the coupling of lateral and vertical dynamic signals during the trajectory tracking, a decoupling active disturbance rejection (DADR) trajectorytracking control strategy is proposed. The control strategy is shown in Fig. 4. First, the αorder pseudolinear composite chassis system is obtained, which decouples the input signal and output signal of the coupling system. Then, a trajectorytracking control strategy with DADR for the pseudolinear composite system is proposed, which can realize the independent antiinterference control of the lateral or vertical motion of the chassis system. Next, the convergence of extendedstate observer (ESO) is verified, and a multiobjective optimization method based on bandwidth parameters of the DADR trajectorytracking controller is proposed according to its convergence conditions, which can demonstrate that the system has better antiinterference performance in the process of trajectory tracking.
3.1 DADR trajectorytracking control strategy for the pseudolinear composite system
To complete the abovementioned control tasks, a pseudolinear composite system is established by constructing an inverse system, which realizes the decoupling of lateral and vertical input and output. To build the inverse system, the system reversibility is first proved by introducing the following definitions and lemmas.
Definition 1 (Wu et al., 2020) is that nonlinear systems will be reversible if the system has a relative order vector, as in Eq. (5).
where α_{i} is the relative order vector of the original system, and i is the ith output signal of the system.
If $\sum _{i=\mathrm{1}}^{i=q}{\mathit{\alpha}}_{i}\le n$, then the system is reversible. n is the matrix order of the original system.
Lemma 1 (Wang et al., 2018) is created, for which the existence condition of the relative order vector is presented. We continuously derive the system output variable $y=h(x,u)$ until each component in ${\mathit{Y}}_{\mathit{q}}=[{y}_{\mathrm{1}}^{{(}^{{\mathit{\alpha}}^{\mathrm{1}}})},{y}_{\mathrm{2}}^{{(}^{{\mathit{\alpha}}^{\mathrm{2}}})},\mathrm{\dots},{y}_{q}^{{(}^{{\mathit{\alpha}}^{q}})}{]}^{\mathrm{T}}$ can contain the input u, and ${\mathit{Y}}_{\mathit{q}}=[{y}_{\mathrm{1}}^{{(}^{{\mathit{\alpha}}^{\mathrm{1}}})},{y}_{\mathrm{2}}^{{(}^{{\mathit{\alpha}}^{\mathrm{2}}})},\mathrm{\dots},{y}_{q}^{{(}^{{\mathit{\alpha}}^{q}})}{]}^{\mathrm{T}}$ is the full rank of the Jacobian matrix of the input u; then, the system has a relative order vector, as in Eq. (6).
Based on Definition 1 and Lemma 1, the system reversibility of the coupled chassis system is proved.
First, the output signal of the lateral and vertical coupled chassis system is obtained from Eq. (2). According to Lemma 1, the derivative of the output vector $\mathit{y}=[\mathit{\beta},{\mathit{\omega}}_{r},{\mathit{\varphi}}_{r}{]}^{\mathrm{T}}$, with the input vector u, is calculated, as shown in Eq. (7).
The relative order vector is discussed according to Lemma 1. Assuming ${Y}_{\mathrm{1}}={\dot{y}}_{\mathrm{1}}$, the rank to the Jacobian matrix of the control input u is shown in Eq. (8). The first derivative equation of y_{1} contains the control input u for the first time, and the Jacobian matrix of Y_{1} to u is full rank, so that α_{1}=1.
Next, assuming ${Y}_{\mathrm{2}}=[{\dot{y}}_{\mathrm{1}},{\dot{y}}_{\mathrm{2}}{]}^{\mathrm{T}}$, the rank of the Jacobian matrix of u is shown in Eq. (9). The first derivative equation of y_{2} contains the input u for the first time, and the Jacobian matrix of Y_{2} to u is full rank, so that α_{2}=1.
Assuming ${Y}_{\mathrm{3}}=[{\dot{y}}_{\mathrm{1}},{\dot{y}}_{\mathrm{2}},{\ddot{y}}_{\mathrm{3}}{]}^{\mathrm{T}}$, the rank of the Jacobian matrix to u is shown in Eq. (10). The second derivative equation of y_{3} contains the input u for the first time, and the Jacobian matrix of Y_{3} has full rank to u, so that α_{3}=2.
From Lemma 1, since the Jacobian matrix of ${Y}_{\mathrm{3}}=[{\dot{y}}_{\mathrm{1}},{\dot{y}}_{\mathrm{2}},{\ddot{y}}_{\mathrm{3}}{]}^{\mathrm{T}}$ to u is full rank, the system has a relative order vector, as shown in Eq. (11).
According to Definition 1, ${\mathit{\alpha}}_{\mathrm{1}}+{\mathit{\alpha}}_{\mathrm{2}}+{\mathit{\alpha}}_{\mathrm{3}}=\mathrm{4}=n$. Therefore, the original integrated chassis system is reversible. The decoupling of the coupled signals in the original chassis system can be achieved by establishing an inverse system.
According to the above analysis, the coupled chassis system is reversible, which means the input and output signals of the original system can be decoupled by establishing a pseudolinear composite system to achieve the compensation control for trajectory tracking. Then, the inverse system model of the coupled chassis system and the αorder pseudolinear composite system will be established and analyzed. The following definitions and lemmas are introduced to present the derivation process.
For Lemma 2, in the αorder inverse system, the operator θ is used to represent the input–output relationship of a nonlinear system Σ with pdimensional input and qdimensional output, as shown in Eq. (12).
where u is the control input of the original system, and y is the control output of the original system.
Assuming Π_{α} is a system with q input and p output, the mapping relationship between input and output is shown in Eq. (13).
where $[{\mathit{\phi}}_{\mathrm{1}},{\mathit{\phi}}_{\mathrm{2}},\mathrm{\dots},{\mathit{\phi}}_{q}{]}^{\mathrm{T}}$ is a differentiable function vector whose initial value satisfies the initial value condition of the original nonlinear system Σ, and φ_{i} is defined as the α_{i}order derivative of y_{di}, as Eq. (14).
And if the operator ${\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathit{\alpha}}$ in Eq. (11) satisfies the condition in Eq. (15), then the following applies:
Then, the system Π_{α} is regarded as the αorder inverse system of the original nonlinear system Σ.
Definition 2 is the αorder pseudolinear composite system. If the system Ω is a composite system $\mathit{\theta}{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathit{\alpha}}$, with a similar linear transfer relationship composed of the αorder inverse system and the nonlinear original system Σ in series, then the system Ω is called the αorder pseudolinear composite system.
The original multiinput–multioutput nonlinear system has been linearized and decoupled into q independent linear integral systems. The composite system $\mathit{\theta}{\stackrel{\mathrm{\u203e}}{\mathit{\theta}}}_{\mathit{\alpha}}$ is equivalent to several connected integrators in series, and its input and output are shown in Eq. (16).
Based on Definition 2 and Lemma 2, the inverse system model of the coupled chassis system is deduced, and then the input and output characteristics of the αorder pseudolinear composite system are analyzed.
As shown in Eq. (2), the output of the nonlinear original system Σ is $y\left(u\right)=[{v}_{\mathrm{1}},{v}_{\mathrm{2}},{v}_{\mathrm{3}}{]}^{\mathrm{T}}={\left[\dot{\mathit{\beta}},{\dot{\mathit{\omega}}}_{r},{\ddot{\mathit{\varphi}}}_{r}\right]}^{\mathrm{T}}$. From Lemma 2, the output of the αorder inverse system Π_{α} is the control input u of the original system, as shown in Eq. (17).
where
Then the state space of the inverse system is shown as Eq. (18).
where
Next, based on the inverse system derived above and the original coupling system, the decoupled pseudolinear composite system is obtained. The input and output characteristics of the αorder pseudolinear composite system Π_{α} are presented and analyzed, as shown in Fig. 8. As shown in Fig. 5, the αorder pseudolinear composite system Π_{α} is composed of the αorder inverse system Ω and the nonlinear original system Σ in series. So the original system can be equivalent to the series integral system, which is easier to design the tracking controller.
For instance, with a single yaw signal input in the αorder pseudolinear composite system, only the corresponding output value changes accordingly, and the other outputs are not affected, as shown in Fig. 6. However, although the coupled system was decoupled, some fluctuations in the other outputs still exist. And such fluctuations will be worse under road disturbance, which should be should be taken into consideration in the controller design.
Next, the trajectorytracking control strategy is designed. The purpose of the trajectorytracking control task is to track the desired trajectory by controlling the front wheel angle and suppressing the random roll motion caused by road disturbance. To realize the above control task, a trajectorytracking control strategy with decoupling active disturbance rejection (DADR) for the pseudolinear composite system is proposed, as shown in Fig. 7. The lateral and vertical tracking controllers are designed separately through an active disturbance rejection control algorithm to achieve antiinterference tracking of the ideal trajectory and suppress the vertical motion at the same time.
The standard form of the active disturbance rejection control algorithm for αorder systems is first discussed. As shown in Eq. (19),
where α is the order of the system, b is the control variable gain, y is the system output, w is system interference, and t is the timevarying state of the system. The interference sources are analyzed, including the external road input, crosswind, and other interference, as is the interference caused by the measurement error within the sensor inside the system.
Since the control variable gain b in Eq. (19) is difficult to obtain in the real control system, the estimated value b_{0} is commonly used. And the disturbance caused by the inaccurate estimation is added to the total disturbance to realize the suppression of interference signals, such as road noise in the trajectorytracking process, as shown in Eq. (20).
The state space of the αorder controlled object is shown in Eq. (21).
where
The observer of the controlled object is shown in Eq. (22).
where L is an adjustable parameter in the state observer. A, B, L, and C are integrated into Eq. (20) to obtain the extendedstate observer, as shown in Eq. (23).
As in Eq. (23), when ESO reaches the design goal, the last output of the extendedstate observer z_{α} can track the real error f of the system. Therefore, the control law can be designed, as shown in Eq. (24), to achieve the control task.
By substituting Eq. (24) into Eq. (20), Eq. (25) can be obtained.
From the input–output relationship of the αorder pseudolinear composite system Π_{α}, the yaw rate control can be regarded as the firstorder system control, and the roll angle control is the secondorder system control. Therefore, for the yaw rate control, the state error feedback (SEF) control law SEF_ω_{r} is shown in Eq. (26).
The yaw rate control extendedstate observer ESO_ω_{r} is shown in Eq. (27).
For the roll angle control, the state error feedback control law SEF_φ_{r} is shown in Eq. (28).
The roll angle control extendedstate observer ESO_φ_{r} is shown in Eq. (29).
When the roll angle control extendedstate observer ESO_φ_{r} converges, then ${z}_{{\mathit{\phi}}_{\mathrm{r}}\mathrm{1}}$ will converge to ${y}_{{\mathit{\phi}}_{\mathrm{r}}}$, and ${z}_{{\mathit{\phi}}_{\mathrm{r}}\mathrm{2}}$ will converge to ${\dot{y}}_{{\mathit{\phi}}_{\mathrm{r}}}$. The roll angle tracking control state error feedback control law SEF_φ_{r} and the stabilized controlled object are shown in Eq. (30).
From Eq. (30), the closedloop transfer function is shown in Eq. (31). The roll angle control extendedstate observer ESO_φ_{r} works in an ideal state, so the transfer function between r and y is only related to the state error feedback control law. k_{p} and k_{d} directly determine the dynamic system response.
The controller is configured using the bandwidth method. For the controller parameters, the relationship between the bandwidth ω_{c}, k_{p}, and k_{d} is shown in Eq. (32).
It can be seen that ω_{c} corresponds to the cutoff frequency of the secondorder system, since the closedloop cutoff frequency can be applied to evaluate the closedloop system transient response speed. A positive correlation is shown between the closedloop cutoff frequency and dynamic system response speed. In addition, by adjusting the closedloop cutoff frequency, the system is equivalent to a bandpass filter to remove road noise signals in specific frequency bands.
3.2 Extendedstate observer (ESO) convergence proof
The DADRbased trajectorytracking algorithm proposed in this paper has the ability of an antiinterference system because the error in the trajectorytracking process can be estimated by the extendedstate observers, ESO_ω_{r} and ESO_φ_{r}, and will be converted to an integrator series system, which is shown in Fig. 5, to achieve the control objectives of removing the interference. In this section, the convergence of ESO is proved, and a multiobjective optimization method of bandwidth parameters of the DADR trajectorytracking controller is proposed according to its convergence conditions.
The following definitions are introduced to present the proposed controller derivation in detail.
For Definition 3 (Li and Lam, 2013), the linear timeinvariant system is shown in Eq. (33).
The necessary and sufficient condition for a system to be asymptotically stable at the origin is that all the characteristic roots of matrix A have negative real parts, and the characteristic polynomial is shown in Eq. (34).
Its nzero real points λ_{i} are completely located on the left half of the plane, as shown in Eq. (35).
Taking the secondorder system as an example, the convergence of the roll angle control extendedstate observer ESO_φ_{r} is proved.
From the equation of the roll angle extendedstate observer ESO_φ_{r}, the differential equation state estimation can be obtained, as in Eq. (36).
The state estimation error differential equation is established as Eq. (37). With increasing time, the estimation error value will tend to be 0.
The state estimation error is defined as Eq. (38).
From Eqs. (37) and (38), we obtain the following:
The characteristic equation is obtained as Eq. (40). For the differential equation shown in Eq. (40), β_{1}, β_{2}, and β_{3} are configured according to Definition 3. The eigenvalue can be less than 0 to ensure its convergence and system stability.
The controller parameters are configured using the bandwidth method (Fu and Tan, 2018), as shown in Eq. (41).
The system will be stable if, and only if, ω_{0}>0.
According to Eq. (41), the relationship between the extendedstate observer ESO parameter and the observer bandwidth ω_{o} is shown in Eq. (42).
The extendedstate observer bandwidth ω_{o} and the controller bandwidth ω_{c} are further discussed. ESO can convert the original problem into a series integrator control problem because the z_{α} of ESO can rapidly suppress the disturbance of the control object, which is equivalent to the inner loop of the controller, so the response speed is faster than the SEF of the outer loop. Therefore, the observer bandwidth ω_{o} should be larger than the controller bandwidth ω_{c}, but the excessive observer bandwidth is difficult to be realized in practical engineering and wastes the control output. In addition, for the controller bandwidth ω_{c}, the excessive bandwidth will also waste the control energy, and it is difficult to suppress the road noise signal of too small a bandwidth (MosqueraSánchez et al., 2017; Sahu et al., 2014). To sum up, the objective function in Eq. (43) is established to optimize the extendedstate observer bandwidth ω_{o}.
where ω_{c}, a_{c}, and u are design variables to achieve the design goals so that the inner loop responds faster than the outer loop by adjusting the weight parameters. The objective function f_{1}(ω_{c}) represents the difference between the integral of the target signal and the integral of the actual system output. The objective function f_{1}(ω_{c}) is defined to evaluate the system response speed, overshoot, and steadystate error. The objective function f_{2}(a_{c}) is the difference between ∫f_{ideal} and ∫z_{α}, which represent the integral of the priorknown noise signal and the integral of the noise observation, respectively. The objective function f_{2}(a_{c}) is minimized to realize fast and accurate noise estimation. The objective function f_{3}(u) is defined as the integral of the controller output. The upper and lower bounds of the constraint are obtained through bench testing and then corrected through multiple tests. The schematic diagrams of these three objective functions are shown in Fig. 8. Due to the relative contradiction between the three objectives, the Pareto solution sets are obtained through iterative optimization, and then the required optimal solution is selected (RodríguezMolina et al., 2020; Wu et al., 2019).
The above analysis shows that it will inevitably lead to large controller output if the trajectorytracking performance needs to be improved. Therefore, a multiobjective particle swarm optimization (PSO) algorithm is applied to optimize these contradictory objectives. The main steps are listed as follows (Zhao et al., 2018).
Step 1 – problem definition. This includes the model definition and algorithm parameters definition.

Model definition. This is the optimization objectives, constraints, and design variables.

Algorithm parameters definition. This includes the maximum iterations I_{te}, the number of particles n_{ori}, inertia weight w, weight descent rate w_{damp}, individual learning factor c_{1}, global learning factor c_{2}, and Pareto set threshold n_{TPareto}.
Step 2 – initialization. This is the position and velocity that are assigned for each particle, and the fitness function is calculated.
Step 3 – calculation cycle. This algorithm consists of a PSO module, decomposition module, and Pareto module. The particle velocity, position, and fitness functions are updated by the PSO module. The decomposition module is used to decompose and search the updated particles provided by the PSO module. Finally, the termination conditions are checked, and the Pareto set is obtained (Mac et al., 2018).
Step 4. If the termination conditions are not satisfied, then the iteration process will return to Step 3. Otherwise, the Pareto solution set will be obtained.
The time–frequency domain analysis is conducted in multiple working conditions, and the system dynamic performance of the proposed control strategy is discussed. Then, a hardwareintheloop chassis test platform is designed based on dSPACE AutoBox and MATLAB/CarSim software. The proposed control strategy is verified for the lanechanging trajectorytracking conditions in the established virtual traffic environment. And the simulation results are compared with the neural network proportional integral derivative (PID) control strategy in Wang et al. (2018). It proves that the proposed control strategy can effectively enhance trajectorytracking accuracy and system antidisturbance performance.
According to the analysis in Sect. 4, the system's dynamic response performance can be improved by optimizing controller bandwidth parameters and observer bandwidth parameters using a multiobjective optimization algorithm. The multiobjective optimization algorithm parameters are listed in Table 1.
The system dynamic behaviors are discussed in the time–frequency domain first. Then, based on the HIL test results, the system response of the proposed control strategy and the neural network PID method are compared and analyzed for the lanechanging trajectorytracking conditions.
4.1 System dynamic performance simulation in the time–frequency domain
The trajectorytracking performance and system robustness of the proposed control strategy are discussed based on the time–frequency simulation results. First, from the perspective of the frequency domain analysis, the natural frequencies of lateral vibration and vertical vibration are from 2 to 6 Hz (Morioka and Griffin, 2010; Arnold and Griffin, 2018). The road excitation within this frequency range will result in vehicle resonance, so the controller is required to suppress the signals within the natural frequencies. In this work, the bandwidth method is applied to design the controller's cutoff frequency. The original and optimum Bode diagrams are shown in Fig. 9.
Figure 10 presents the system response in the time domain with the inputs of step signal and the priorknown interference, which verifies the improvement in the system's antiinterference performance. Compared with the benchmark, the dynamic system response speed is dramatically improved by 43.64 %. In addition, the accuracy of the observer state error estimation is improved by 11.93 %. The optimized system can estimate the interference signal faster, which is the foundation of the system simplification and the improvement of trajectorytracking accuracy.
As shown in Fig. 10, the baseline observer has a delay of 0.3 s compared with the target signal. Within 1 to 1.3 s, the system cannot be regarded as a multistage series system, so highprecision control performance is unlikely to be achieved in the baseline observer. However, the optimized system can effectively balance the contradiction of controller bandwidth, observer bandwidth, and control outputs to improve dynamic system performance.
4.2 Trajectorytracking performance of the hardwareintheloop test
In this section, the hardwireintheloop test platform for an Xbywire chassis is established, based on the MicroAutoBox 1401/1501 (dSPACE GmbH). And the lanechanging trajectorytracking conditions are established in the virtual traffic environment of MATLAB/CarSim. The effectiveness of the proposed trajectorytracking control strategy is verified and compared with the neural network PID method in this test platform. The test platform is shown in Fig. 11. The chassis controller output of the platform is collected by the data acquisition card and is transmitted to the vehicle model in the virtual traffic environment. After that, the yaw rate and dynamic model parameters are calculated based on the vehicle model in the virtual traffic environment, which is transmitted to the chassis controller of the real vehicle in real time through the controller area network (CAN) network. This procedure realizes the closedloop interaction between the real vehicle chassis controller and the virtual traffic environment.
The experimental results are analyzed below and compared with benchmarks. Figure 12 presents the performance of the proposed control strategy on decoupling with uncertain road interference. Both the benchmark and the proposed control strategy can realize the decoupling of lateral and vertical signals. However, in the case of interference, the variance between the actual signal and the ideal signal obtained by the proposed control strategy can be reduced by 51.61 % when compared with that of the benchmark. The mechanism of the above phenomenon is analyzed. The neural network PID inverse system requires a large number of priorknown training data. As for the model uncertainty from road interference, the training dataset is unlikely to cover all the working conditions. Therefore, the neural network PID inverse system cannot eliminate the coupling on the signal layer, which will result in the chassis system interference on the mechanical layer.
Then, the effectiveness of the proposed trajectorytracking control strategy is discussed. The active control signals to the controlled object are shown in Fig. 13. The active yaw and roll moment can effectively resist road interferences and enhance the trajectorytracking performance. Compared with the benchmark, the proposed control strategy can provide a faster response of active yaw moment and roll moment when road disturbances exist. The peak values of active yaw and roll moment obtained by the proposed control strategy can be reduced by 61.45 % and 70.49 %, respectively.
The benefits of the proposed control strategy on the antiinterference ability are compared from the perspectives of the lateral distance and yaw rate in trajectory tracking in Fig. 14. Compared with the benchmark, the proposed control strategy can reduce the trajectorytracking error by 10.03 %, and the variance of the error within the yaw rate can be reduced by 34.54 %.
From the perspective of vertical motion control, compared with the benchmark, the proposed control strategy can reduce the peak roll angular velocity by 43.64 %. The trajectorytracking performance on stability is further discussed. The suspension system force and displacement are shown in Fig. 15. The vertical relative displacement of the proposed control strategy is smaller compared with the benchmark. Therefore, the vehicle body behaves more stably with the proposed control strategy in the trajectorytracking process.
The conclusion can be further verified in Fig. 16. It shows the front wheel angle and lateral acceleration in trajectory tracking. With the same front wheel angle, the lateral acceleration obtained by the proposed control is relatively smaller compared to that of the benchmark. The proposed control strategy can effectively reduce the chassis system interference on the mechanical layer induced by the coupling on the signal layer.
In this paper, a coupled dynamic model of the Xbywire chassis system is established, and the αorder chassis pseudolinear composite system is obtained. On this basis, a decoupling active disturbance rejection (DADR) trajectorytracking control strategy is proposed, and a multiobjective optimization method of bandwidth parameters of the DADR trajectorytracking controller is proposed, according to its convergence conditions.
As far as the results of the simulation and hardwareintheloop test is concerned, the trajectorytracking error within the proposed control strategy can be reduced by 10.03 %, compared with the benchmark, and the variance of the error within the yaw rate can be reduced by 34.54 %. In particular, when road interference existed, the chassis system with the proposed control strategy can track the target trajectory more stably. In addition, the peak value of vehicle roll angular velocity with the proposed strategy can be reduced by 43.64 %. And the relative vertical displacement is remarkably reduced, which demonstrates that the proposed strategy makes the vehicle body more stable.
In the future, first, the vehicle trajectorytracking model with higher degrees of freedom should be studied, and the trajectorytracking control method under the condition of actuator failure should be studied. Moreover, a large number of tests should be carried out through a hardwareintheloop test and real vehicle road tests to support the theoretical research.
The data are available upon request from the corresponding author.
WH, ZY, ZF, and JP discussed and decided on the methodology of the study and prepared the paper. WH contributed to the prototype and test. ZY and ZF contributed to the model building.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors would like to thank the National Natural Science Foundation of China (grant nos. 52072175 and 51875279) and Jiangsu Outstanding Youth Fund Project (grant no. BK20220078).
This work has been supported by the National Natural Science Foundation of China (grant nos. 52072175 and 51875279) and Jiangsu Outstanding Youth Fund Project (grant no. BK20220078).
This paper was edited by Peng Yan and reviewed by Yangming Zhang and one anonymous referee.
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 Abstract
 Introduction
 Modeling and analysis of Xbywire chassis trajectorytracking task
 Decoupling active disturbance rejection (DADR) trajectorytracking control strategy
 Results and discussion
 Conclusion
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Modeling and analysis of Xbywire chassis trajectorytracking task
 Decoupling active disturbance rejection (DADR) trajectorytracking control strategy
 Results and discussion
 Conclusion
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References