High-Order Model-Based Robust Control of a Dual-Motor Steer-by-Wire System with Disturbance Rejection

Abstract

This paper presents a high-order model-based robust control strategy for a dual-motor road wheel actuating system in a steer-by-wire (SbW) architecture. The system consists of a belt-driven and a pinion-driven motor collaboratively actuating the road wheels through mechanically coupled dynamics. To accurately capture the interaction between actuators, structural compliance, and road disturbances, a four-degree-of-freedom (4DOF) lumped-parameter model is developed. Leveraging this high-order dynamic model, a composite control framework is proposed, integrating feedforward model inversion, pole-zero feedback compensation, and a disturbance observer (DOB) to ensure precise trajectory tracking and disturbance rejection. High-fidelity co-simulations in MATLAB/Simulink and Siemens Amesim validate the effectiveness of the proposed control under various steering scenarios, including step and sine-sweep inputs. Compared to conventional low-order control methods, the proposed approach significantly reduces tracking error and demonstrates enhanced robustness and disturbance attenuation. These results highlight the critical role of high-order modeling in the precision control of dual-motor SbW systems and suggest its applicability in real-time, safety-critical vehicle steering applications.

1. Introduction

Electric power steering (EPS) systems have replaced traditional hydraulic steering in modern vehicles due to their improved energy efficiency, flexibility in control design, and ease of integration with advanced driver assistance systems (ADASs) [1,2]. As autonomous and intelligent vehicle technologies advance, steering systems are increasingly required to meet demands for precise control, disturbance rejection, and fault tolerance under diverse road and operational conditions [3,4,5].

To meet these requirements, various model-based and intelligent control strategies have been proposed for EPS systems. Chen et al. [1] proposed a dual-loop control structure combining ${\mathcal{H}}_2$ and ${\mathcal{H}}_{\infty }$ synthesis to address motor torque and steering motion control. Their work demonstrated that reduced-order controllers can provide sufficient performance for industrial applications with a lower computational burden. Similarly, Lee et al. [2] emphasized the importance of stabilizing compensators to mitigate vibration caused by nonlinear torque maps and assist gain dynamics.

Addressing nonlinearities, disturbances, and parameter uncertainties, researchers have developed robust and adaptive controllers. Nguyen [3] introduced a fuzzy-tuned PID-backstepping (FPIDBS) control method capable of dynamically adjusting controller parameters in response to varying road torque. Lin et al. [6] proposed intelligent sliding-mode control using a wavelet fuzzy neural network to estimate uncertainties and suppress chattering. Additionally, Kim et al. [7] developed a torque control strategy based on a steering wheel torque (SWT) model, integrating an extended state observer and nonlinear sliding mode control for enhanced robustness.

Recent developments have shifted toward dual-motor configurations within steer-by-wire (SbW) systems, particularly in commercial vehicles requiring high steering torque and redundancy. Hwang et al. [8] proposed a master–slave control framework using sliding mode control (SMC) with a disturbance observer (DOB) for synchronization and robustness. Zou et al. [9] further enhanced tracking and synchronization by applying active disturbance rejection control (ADRC) with cross-coupled coordination. Xu et al. [4] introduced a layered control architecture combining $\mu /{\mathcal{H}}_2$ robust control and super-twisting sliding mode synchronization to address multiple vehicle uncertainties.

Experimental validation using hardware-in-the-loop (HIL) and physical test platforms has also been actively explored. Jung and Kim [10] validated a finite-preview path-tracking control algorithm using HIL simulation on a dual-motor SbW platform, showing high synchronization accuracy. Chung et al. [11] proposed a novel synchronous control structure based on mean–difference decoupling, demonstrating improved dynamic tracking and vibration suppression. Similarly, Wu et al. [12] demonstrated effective rack force estimation and compensation using DOB-based feedforward/sliding mode control in a SbW system. These contributions collectively underscore the importance of integrated, high-fidelity control strategies in modern steering systems.

1.1. Research Gap and Motivation

Despite extensive prior work, several key limitations remain:

• Most existing EPS and SbW controllers are designed using reduced- or low-order models, which may neglect key dynamic interactions such as structural compliance, coupled actuator dynamics, and rack-load elasticity, factors critical in high-precision applications.
• Dual-motor systems are often treated using master–slave or decoupled frameworks, limiting their ability to optimally distribute load and respond to asynchronous dynamic behaviors.
• Few studies integrate high-order mechanical modeling with composite control strategies, including feedforward model inversion, pole-zero feedback shaping, and disturbance observer design, for fully coordinated dual-actuator control under uncertainty.

1.2. Contribution of This Study

To bridge the identified gaps, this study proposes a robust, high-order control framework for a dual-motor SbW system. The key contributions are as follows:

• Development of a four-degree-of-freedom (4DOF) dynamic model capturing actuator coupling, compliance, and external disturbances.
• Design of a composite controller integrating feedforward inversion, pole-zero feedback, and a DOB.
• Validation through high-fidelity co-simulations in MATLAB/Simulink R2020b [13] and Simcenter Amesim [14], showing improved tracking and robustness over low-order methods.

This work provides a scalable, high-performance control solution for multi-actuator SbW systems and lays the groundwork for future experimental implementation in safety-critical autonomous driving applications.

The remainder of this paper is organized as follows. Section 2 describes the dual-motor SbW system architecture and its mechanical configuration. Section 3 presents the high-order lumped-parameter dynamic model and high-fidelity simulation model in Siemens Amesim. Section 4 details the design of the composite controller, including feedforward inversion, feedback compensation, and disturbance observer design. Section 5 provides simulation-based performance evaluations under various steering scenarios. Finally, Section 6 summarizes the findings and discusses potential directions for future research.

2. System Description

This study investigates a dual-motor road wheel actuation system within a SbW architecture, implemented on a commercial vehicle platform. The proposed system aims to enhance steering precision, improve robustness under fault conditions, and enable cooperative control. Unlike traditional single-motor EPS configurations, this architecture employs two independent electric actuators: a belt-driven motor coupled to a ball screw mechanism and a pinion-driven motor interfaced via a worm gear. This mechanically decoupled dual-path configuration enables distributed torque control and introduces redundancy, which is vital for safety-critical applications such as highly automated driving.

2.1. Motivation and Architectural Comparison

Conventional EPS systems rely on a single brushless AC (BLAC) motor to provide steering assistance via a belt-driven ball screw. While effective in standard driving conditions, this architecture suffers from limited fault tolerance and represents a single point of failure. In scenarios requiring fail-operational behavior—such as SAE Level 3+ automated vehicles—the absence of actuator-level redundancy poses a critical risk to system safety and reliability. The proposed dual-motor actuation system mitigates these limitations by introducing a secondary, mechanically isolated actuation path. The primary actuator, a belt motor, delivers torque through a ball screw, while the secondary pinion motor transmits torque via a worm gear directly to the steering pinion. This architecture allows torque to be delivered independently through two mechanical paths, thereby enabling redundancy, torque sharing, and cooperative control. For reference, the standard BLAC motor outputs up to 6 Nm of torque with a torque constant of 0.07 Nm/A_rms and a peak current of 90 A_rms. However, under high-load or fault conditions, its lack of redundancy becomes a liability. In contrast, the proposed dual-motor configuration supports fail-operational capability by ensuring continued steering actuation via the secondary path even if one actuator is degraded or offline.

2.2. Functional Modeling Framework

To support model-based control design, the dual-motor SbW system is modeled as a 4DOF lumped-parameter mechanical system. The free-body diagram and its equivalent 4DOF representation are shown in Figure 1. The model captures critical dynamic interactions between the belt motor, pinion motor, and steering rack, incorporating structural compliance, gear dynamics, and external disturbances such as road forces.

The system dynamics include the following:

• Belt Motor Path: Torque $T_{bm}$ is applied through inertia $J_{bm}$ and damping $B_b$ to a ball screw with inertia $J_{sc}$ and damping $B_{sc}$, connected to the rack via stiffness $K_{sc}$ and gear ratios $g_b$ and $g_{sc}$.
• Pinion Motor Path: Torque $T_{pm}$ is delivered via worm gear mechanics, including motor-side inertia $J_{pm}$, damping $B_g$, stiffness $K_g$, and gear ratio $g_g$.
• Rack Dynamics: The rack has mass $M_r$, damping $B_r$, and is subjected to external disturbance $F_r$, modeling tire–road interactions. Nonlinear friction effects $T_{f,bm},T_{f,pm},T_{f,sc},F_{f,r}$ are lumped at each stage.

The four dynamic states correspond to the rotational angles of the belt motor, screw, pinion motor, and the linear displacement of the steering rack. Since both motors are mechanically coupled through the rack, any torque applied by one actuator influences the entire system. Hence, coordinated control is essential to prevent internal conflicts and to ensure consistent steering performance. Unlike simplified decoupled models, the proposed 4DOF formulation captures structural flexibilities, resonance effects, and inter-actuator dynamics critical for high-bandwidth and high-precision control applications. These dynamic equations are used to derive transfer function representations for subsequent controller synthesis.

3. System Model

To design a robust and high-performance controller for the dual-motor SbW system, an accurate understanding of the underlying dynamics is essential. This section presents the development of a lumped-parameter analytical model and a high-fidelity simulation model using Siemens LMS Amesim. These models are used for controller synthesis, simulation, and validation.

3.1. Lumped-Parameter Dynamic Modeling with 4DOFs

The dual-motor SbW system comprises two independently actuated motors: a belt motor and a pinion motor. Each applies torque through separate mechanical transmission paths to a common steering rack. Figure 1 shows the physical system and the corresponding free-body diagram used to derive a 4DOF mechanical model.

The following second-order differential equations govern the 4DOF system dynamics:

$$\begin{array}{cc}\hfill J_{bm}{\ddot{\theta }}_{bm}+B_b{\dot{\theta }}_{bm}+K_b({\theta }_{bm}-{\theta }_{sc})& =T_{bm}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill J_{sc}{\ddot{\theta }}_{sc}+B_{sc}{\dot{\theta }}_{sc}+K_{sc}({\theta }_{sc}-x_r/g_{sc})+K_b({\theta }_{sc}-{\theta }_{bm})& =0\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill J_{pm}{\ddot{\theta }}_{pm}+B_g{\dot{\theta }}_{pm}+K_g({\theta }_{pm}-x_r/g_g)& =T_{pm}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill M_r{\ddot{x}}_r+B_r{\dot{x}}_r+K_{sc}(x_r/g_{sc}-{\theta }_{sc})+K_g(x_r/g_g-{\theta }_{pm})& =F_r\hfill \end{array}$$

(4)

These equations represent coupled rotational and translational dynamics among the belt motor, screw, pinion motor, and rack.

3.2. Transfer Function Representation

Applying Laplace transforms under zero initial conditions yields transfer functions from motor torque inputs to motor angle outputs. For each actuator, the transfer functions can be written as follows:

$$P_{nb}\left(s\right)={\displaystyle \frac{{\theta }_{bm}\left(s\right)}{T_{bm}\left(s\right)}}={\displaystyle \frac{a_{b2}{s}^2+a_{b1}s+a_{b0}}{b_{b4}{s}^4+b_{b3}{s}^3+b_{b2}{s}^2+b_{b1}s+b_{b0}}}$$

(5)

$$P_{np}\left(s\right)={\displaystyle \frac{{\theta }_{pm}\left(s\right)}{T_{pm}\left(s\right)}}={\displaystyle \frac{a_{p2}{s}^2+a_{p1}s+a_{p0}}{b_{p4}{s}^4+b_{p3}{s}^3+b_{p2}{s}^2+b_{p1}s+b_{p0}}}$$

(6)

Each transfer function exhibits a relative order of 2 and is derived from the full 4DOF system, which results in an overall 8th-order plant. These models incorporate compliance, damping, and cross-coupling between the subsystems and form the basis for the control design.

3.3. High-Fidelity Modeling with Amesim

To complement the analytical model, a high-fidelity model of the system was developed using Siemens LMS Amesim. This platform supports the simulation of mechanical and electrical subsystems based on physical parameters and experimental calibration. The Amesim model is organized into modular components representing the following:

• Belt motor and ball screw, including elastic and frictional effects;
• Pinion motor and worm gear, including backlash and nonlinearity;
• Rack load with mass and external disturbance;
• Electrical power supply (12.6 V battery) with current limitations.

Figure 2 illustrates a comparison between the high-fidelity Amesim model and the physical dual-motor SbW hardware system. The model, developed in Siemens LMS Amesim, is organized into modular subsystems, each of which directly corresponds to components in the real system. These include the belt motor and pulley (highlighted in red), the pinion motor and worm gear (blue), the rack and pinion mechanism (yellow), and the battery power supply (black). The color-coded representation facilitates intuitive mapping between the simulation and physical implementation, ensuring that key dynamic and nonlinear effects—such as elasticity, friction, backlash, and power limitations—are accurately captured in the simulation environment.

As illustrated in Figure 3, the Amesim model was calibrated through open-loop torque excitation applied to each motor. A series of reference inputs—including step, sinusoidal, and chirp signals—were introduced to evaluate the system’s dynamic response. The corresponding outputs, such as motor angles, were measured and compared with experimental data from the real hardware. The close agreement between simulation and physical results confirms the validity of the model for capturing key electromechanical behaviors.

Figure 4 shows the frequency response comparison between the Amesim model, the low-order model, and the high-order model. The input to the system is the motor torque, and the output is the rotational speed. As observed in the Bode plots, the high-order model exhibits excellent agreement with the Amesim simulation results across the entire frequency range. Both magnitude and phase responses of the high-order model closely follow those of the Amesim model, accurately capturing the system dynamics including the resonance and phase lag behavior. In contrast, the low-order model shows noticeable discrepancies, particularly in the mid- to high-frequency range, failing to capture certain dynamic characteristics present in the Amesim model. Therefore, the high-order model is validated to be a more accurate representation of the system dynamics when compared to the low-order approximation.

4. Design of a Robust High-Order Controller

To enable accurate and robust control of the dual-motor SbW system, a high-order controller is designed based on a fourth-order dynamic model that captures actuator coupling, gear compliance, and frictional effects. This model provides a detailed representation of the physical system, going beyond simplified second-order approximations. The controller is synthesized using a generalized plant formulation that supports the analytical design of both feedforward and feedback components. The feedforward path uses filtered nominal model inversion for improved responses, while the feedback path employs pole-zero shaping to enhance tracking and robustness. A DOB complements the structure by estimating and rejecting external disturbances and model uncertainties.

Figure 5 shows the overall control architecture. Each actuator—belt and pinion—is controlled independently through a parallel configuration of feedforward, feedback, and DOB blocks. The DOB integrates an inverse nominal model with a second-order low-pass Q-filter, enabling disturbance compensation. Control inputs are computed from the reference steering angle ${\theta }_m^{*}$ and applied separately to each motor for coordinated actuation.

4.1. Generalized Plant Model and System Identification

The actuator system is modeled as a fourth-order linear time-invariant system with a second-order numerator, resulting in the following general form of the transfer function:

$$P_n\left(s\right)={\displaystyle \frac{\theta \left(s\right)}{T\left(s\right)}}={\displaystyle \frac{a_2{s}^2+a_{1}s+a_0}{b_3{s}^4+b_2{s}^3+b_1{s}^2+b_{0}s}}$$

(7)

This model structure captures second-order dynamics, which is consistent with mechanical systems exhibiting inertia and compliance.

Two specific models are identified for the belt and pinion motors through system identification in the Amesim model. These are denoted as $P_{nb}\left(s\right)$ and $P_{np}\left(s\right)$, respectively, as follows:

$$P_{nb}\left(s\right)={\displaystyle \frac{{\theta }_{bm}\left(s\right)}{T_{bm}\left(s\right)}}={\displaystyle \frac{3030s^2+141100s+16290000}{s^4+44.14s^3+5322s^2+28780s}}$$

(8)

$$P_{np}\left(s\right)={\displaystyle \frac{{\theta }_{pm}\left(s\right)}{T_{pm}\left(s\right)}}={\displaystyle \frac{9276s^2+397600s+20320000}{s^4+152.6s^3+6874s^2+36450s}}$$

(9)

These transfer functions are used as the basis for controller synthesis and frequency-domain analysis. The numerator captures the motor torque-to-angle dynamics, while the denominator models the high-order dynamics of mechanical coupling and load behavior.

4.2. Feedforward Controller Design

The feedforward controller is designed to compensate the nominal plant dynamics and linearize the system response with respect to the reference input. To prevent improper transfer functions and avoid amplifying high-frequency noise, the inverse model is regularized by a second-order low-pass filter. The resulting feedforward controller takes the following form:

$$C_{ff}\left(s\right)={\displaystyle \frac{{\omega }_{ff}^2{P}_n{\left(s\right)}^{-1}}{s^2+2\xi {\omega }_{ff}s+{\omega }_{ff}^2}}$$

(10)

Here, ${\omega }_{ff}$ is the desired feedforward cutoff frequency, and $\xi$ is the damping ratio, typically set between 0.6 and 0.8. The denominator of the filter enforces causality, while the multiplication by ${\omega }_{ff}^2$ ensures unity DC gain at a low frequency. This formulation enables effective pre-compensation of the reference trajectory and reduces the steady-state load on the feedback loop.

4.3. Feedback Controller Design via Pole-Zero Shaping

To enhance closed-loop robustness and meet transient performance specifications, a feedback controller is designed to shape the closed-loop dynamics. The desired closed-loop transfer function is specified as a second-order reference model:

$${\displaystyle \frac{C\left(s\right)P_n\left(s\right)}{1+C\left(s\right)P_n\left(s\right)}}\approx {\displaystyle \frac{{\omega }_{fb}^2}{s^2+2{\xi }_{fb}{\omega }_{fb}s+{\omega }_{fb}^2}}$$

(11)

Solving this expression for $C\left(s\right)$ yields the feedback controller:

$$C\left(s\right)={\displaystyle \frac{{\omega }_{fb}^2}{P_n\left(s\right)\left(s^2+2{\xi }_{fb}{\omega }_{fb}s+{\omega }_{fb}^2\right)}}$$

(12)

By substituting the generalized model of $P_n\left(s\right)$ from Equation (7), the controller becomes

$$C\left(s\right)={\displaystyle \frac{{\omega }_{fb}^2\left(b_3{s}^4+b_2{s}^3+b_1{s}^2+b_{0}s\right)}{a_2{s}^4+(2{\xi }_{fb}{\omega }_{fb}{a}_2+a_1)s^3+(2{\xi }_{fb}{\omega }_{fb}{a}_1+a_0)s^2+2{\xi }_{fb}{\omega }_{fb}{a}_{0}s}}$$

(13)

This structure cancels dominant plant dynamics while introducing desired second-order behavior. The damping ratio ${\xi }_{fb}$ and natural frequency ${\omega }_{fb}$ are selected based on desired settling time, overshoot, and stability margin. Since the feedback controller relies directly on the full plant transfer function $P_n\left(s\right)$, the accuracy of the model is critical to achieving the intended pole-zero placement.

The use of a high-order nominal model enables precise design of both feedforward and feedback controllers. The proposed control law achieves fast and accurate trajectory tracking, while maintaining robustness against system uncertainty and disturbances. In the following sections, disturbance observers will be introduced to further enhance the system’s disturbance rejection capability.

4.4. Disturbance Observer Design

To enhance robustness against external disturbances and model uncertainties, a model-based DOB is designed using the fourth-order nominal plant models identified in Section 4.1.

As shown in Figure 6, the DOB structure estimates disturbances by comparing the actual output response with the predicted response from the nominal plant model. Specifically, the disturbance estimate $\widehat{d}\left(s\right)$ is obtained as follows:

$$\widehat{d}\left(s\right)=Q\left(s\right)\left[P_n^{-1}\left(s\right)y\left(s\right)-u\left(s\right)\right]$$

(14)

where $y\left(s\right)$ is the measured plant output, $u\left(s\right)$ is the control input, $P_n\left(s\right)$ is the nominal plant transfer function, and $Q\left(s\right)$ is the disturbance observer filter.

The filter $Q\left(s\right)$, commonly referred to as the Q-filter, is implemented as a second-order low-pass filter:

$$Q\left(s\right)={\displaystyle \frac{{\omega }_c^2}{s^2+2\xi {\omega }_{c}s+{\omega }_c^2}}$$

(15)

where ${\omega }_c$ is the cutoff frequency and $\xi$ is the damping ratio. This filter allows low-frequency disturbances to pass through for compensation, while attenuating high-frequency components such as sensor noise and model uncertainties.

In the DOB structure (Figure 6), the inverse nominal model $P_n^{-1}\left(s\right)$ predicts the disturbance-free output. The difference between this prediction and the actual control input $u\left(s\right)$ reflects the influence of disturbances, which is filtered by $Q\left(s\right)$ to produce a stable disturbance estimate $\widehat{d}\left(s\right)$. This estimate is then used to compensate the input, resulting in a closed-loop system that closely tracks the nominal behavior while maintaining robustness.

By appropriately tuning the filter parameters, the DOB effectively distinguishes between disturbance and uncertainty components, thereby improving tracking performance and enhancing overall system resilience.

4.5. Frequency-Domain Analysis of Robust Performance and Stability

To validate the robust control design under model uncertainty and frequency-dependent performance requirements, a frequency-domain analysis is conducted for both the pinion and belt motor subsystems. Figure 7 summarizes this analysis through Bode plots of model variations, sensitivity functions, and uncertainty bounds.

The top row of Figure 7 shows the Bode magnitude plots of the nominal plant model and its perturbed variations under different operating conditions, including friction changes, load variations, and structural compliance. The shaded region formed by the upper and lower bounds represents the model uncertainty envelope. The nominal model (blue line) closely follows the average frequency response, while deviations become more pronounced above 10 Hz. This increasing uncertainty highlights the need for robust stabilization, especially in high-frequency regions susceptible to unmodeled dynamics and resonance. Under typical driving conditions, road-induced disturbances range from 1 to 10 Hz and may reach 20 Hz on rough surfaces. The proposed controller ensures robust rejection within this frequency range.

To assess robust performance and stability, the bottom plots illustrate the closed-loop sensitivity function $S\left(j\omega \right)$ and complementary sensitivity function $T\left(j\omega \right)$, along with the inverse weighting functions $W_S^{-1}\left(j\omega \right)$ and $W_T^{-1}\left(j\omega \right)$. The weighting functions were designed to shape the closed-loop frequency response as follows:

• Stability weighting function $W_T\left(s\right)$:

  $$W_T\left(s\right)={\displaystyle \frac{(s+2\pi ·25){(s+2\pi ·60)}^2}{290000(s+2\pi ·13)}},M_T=max\left|T\left(j\omega \right)\right|<3\mathrm{dB},$$

  (16)
  This condition suppresses high-frequency gain and limits sensitivity to model uncertainty.
• Performance weighting function $W_S\left(s\right)$:

  $$W_S\left(s\right)=0.5012·\left({\displaystyle \frac{s+2\pi ·25}{s}}\right),M_S=max\left|S\left(j\omega \right)\right|<6\mathrm{dB}$$

  (17)
  This criterion enhances low-frequency disturbance rejection and reference tracking.

The controller is verified to satisfy the following robust performance conditions across all frequencies:

$$\left|S\left(j\omega \right)\right|<|W_S^{-1}\left(j\omega \right)|,|T\left(j\omega \right)|<|W_T^{-1}\left(j\omega \right)|$$

(18)

$$|W_T\left(j\omega \right)T\left(j\omega \right)|+|W_S\left(j\omega \right)S\left(j\omega \right)|<1,\forall \omega$$

(19)

These inequalities confirm that the mixed sensitivity criterion is met, ensuring robustness to both structured disturbances and model uncertainties. The nominal loop crossover frequency, marked around 20 Hz in the plots, aligns with the designed bandwidth transition between $S\left(j\omega \right)$ and $T\left(j\omega \right)$. The absence of gain peaking indicates stable loop behavior and a smooth trade-off between tracking accuracy and noise attenuation. Also shown in the bottom plots are the inverse model uncertainty magnitudes (black lines), representing frequency-dependent mismatches between the nominal inverse plant $P_n^{-1}\left(s\right)$ and actual system behavior. Notably, the inverse uncertainties remain below $W_T^{-1}\left(j\omega \right)$, even in the 10–20 Hz range where resonance and perturbations may occur. This validates the safe application of $P_n^{-1}\left(s\right)$ in both the disturbance observer and feedforward control paths, without compromising closed-loop stability.

Overall, the frequency-domain results confirm that the proposed controller achieves robust performance across the defined operating range, while maintaining sufficient stability margins and resilience to modeling error and external disturbances.

Through comprehensive Bode analysis, the robust controller is shown to satisfy all design constraints imposed by the mixed-sensitivity framework. Both the belt and pinion motor control loops achieve (1) robust tracking within the desired bandwidth, (2) rejection of high-frequency noise and unmodeled dynamics, and (3) safe application of inverse model-based compensation. These results validate the theoretical synthesis conditions and confirm the effectiveness of the proposed high-order model-based controller architecture.

5. Simulation Results

To evaluate the performance and robustness of the proposed control strategies, simulation tests were conducted using reference steering scenarios that reflect typical operating conditions in steering systems. All simulations were carried out in a co-simulation environment integrating MATLAB/Simulink and Siemens Amesim, allowing for dynamic interactions between control algorithms and high-fidelity physical plant models.

Two main steering scenarios were used to assess different aspects of the controller’s performance: a step input and a frequency-swept sine input. The specific conditions used in each case are summarized in

Table 1. Simulation conditions for controller performance evaluation.

Test Type	Input Signal	Amplitude/Range
Step Steering Test	Step input	120 deg
	Slew rate limit	500 deg/s
Sine Steering Test	Chirp signal	0–3 Hz, 45 deg peak

.

The first scenario, the step steering input, is designed to test the transient response, disturbance rejection, and low-pass filtering characteristics of the controller. A sharp position reference with a maximum steering angle of 120 degrees and a slew rate of 500 deg/s is applied. To suppress high-frequency content while preserving responsiveness, a 10 Hz low-pass filter is applied to the input. This scenario enables us to carry out an evaluation of time-domain tracking accuracy, the DOB’s effectiveness, and controller responsiveness to abrupt commands. The second scenario involves a sine-sweep (chirp) input, which assesses the closed-loop system’s performance across a wide frequency spectrum. The chirp signal has an amplitude of ±45 degrees and a frequency range from 0 to 3 Hz over an 8-second period. This test is useful for identifying potential resonance amplification, validating bandwidth, and comparing the performance of control designs under various excitation frequencies.

5.1. Time-Domain Tracking Comparison

Figure 8 presents the time-domain tracking errors between the reference inputs and motor responses for both belt and pinion motors, under step (left column) and sine-swept (right column) inputs. High-order control (red or blue lines) is compared against conventional low-order model-based control (black lines). Under step inputs, the high-order controller yields significantly lower overshoot and faster settling. For the belt motor, transient spikes from the low-order controller exceed ±2°, whereas the high-order controller keeps errors within ±1°. In the sine-sweep case, where input frequency increases over time, the position error of the low-order model grows noticeably, especially after 4 s, due to its inability to capture high-frequency dynamics. In contrast, the high-order controller consistently maintains lower tracking errors with reduced phase lag and better amplitude tracking, demonstrating superior frequency-domain performance and robustness to dynamic variations.

5.2. Quantitative Performance Evaluation

Figure 9 presents the root mean square (RMS) tracking errors of the belt and pinion motors under both step and sine steering tests. The results compare the performance of the proposed high-order controller against a conventional low-order controller.

In the step steering test, the high-order controller reduced the RMS tracking error by 27% for the belt motor and 65% for the pinion motor. Under sine steering input, which spans a wider frequency range, the improvements were even more pronounced—a 50% reduction for the belt motor and a 91% reduction for the pinion motor. These results confirm the superior accuracy and robustness of the high-order controller, particularly in scenarios involving dynamic coupling and resonance.

The simulation results confirm that the high-order model-based controller delivers substantial performance advantages over conventional low-order methods. It ensures accurate tracking, robust frequency responses, and improved resilience to input variation and model uncertainty. The observed improvements in both the RMS and peak tracking errors validate the use of high-fidelity plant modeling in controller design.

6. Conclusions

This paper proposed a robust, high-order model-based control strategy for a dual-motor SbW system. A 4DOF dynamic model was developed to capture coupled actuator dynamics, including inertia, compliance, damping, and gear effects. Based on this model, a composite controller was designed using feedforward inversion, pole-zero feedback, and a disturbance observer. Simulation results showed that the high-order controller significantly outperformed a conventional low-order design. RMS tracking errors were reduced by up to 65% under step input and 91% under frequency-sweep input for the pinion motor, with similar improvements for the belt motor. The controller also reduced overshoot and improved phase responses. Future work will focus on real-time implementation and experimental validation, with a potential extension to multi-actuator steering systems in ADAS and autonomous vehicles.