Impacts of the Observation of the Steering Torque Disturbance on the Stability of a Time-Delayed Control System for a Corner Module with Steering

Abstract

Corner modules decouple chassis functions and enable independent wheel steering, but their control is highly sensitive to external disturbances and feedback delays. Disturbance observers (DObs) are often introduced to mitigate such disturbances, yet their additional dynamics can also compromise closed-loop stability when delays are present. This paper establishes the closed-loop control system of a corner module steering system based on its dynamics, designs the corresponding control law, and incorporates a DOb. Classical stability analysis is carried out using D-curve mapping and eigenvalue validation. The results reveal that feedback delay progressively shrinks the stable domain. When a DOb is introduced, disturbance rejection is improved; however, the admissible control gain region becomes narrower, and larger observer gains further constrain the derivative action, generating additional unstable regions. This paper mechanistically elucidates the impact of disturbance observation on the stability of a time-delayed control system for a corner module with steering.

1. Introduction

The corner module (CM) is a highly integrated assembly that combines a drive motor, steering actuator, braking unit, and suspension in each wheel unit. By eliminating mechanical linkages such as the steering column and drive shaft, CMs enable each wheel to operate independently under full X-by-wire control, including steer-by-wire (SbW) and brake-by-wire [1]. This decoupled architecture enhances agility, energy efficiency, and safety and allows novel vehicle maneuvers such as four-wheel steering and lateral “crab” movement [2,3,4,5]. However, the fully electronic and mechanically isolated steering system also introduces new control challenges. Without a physical linkage, steering relies entirely on sensors, actuators, and communication networks, all of which introduce time delays and make the system more sensitive to external disturbances [6,7,8,9].

In a SbW or CM steering system, feedback delay significantly affects stability and responsiveness [10,11,12]. Even modest delays from signal processing or actuator dynamics can add phase lag and cause oscillation. To cope with delay, various compensation strategies have been proposed, including Smith predictor methods that estimate and cancel delay online [13,14], robust or fuzzy controllers ensuring bounded stability [15,16], and delay-tolerant designs based on Lyapunov–Krasovskii or gain-scheduling theory [12].

In addition to delay, the CM steering system or SbW system is exposed to external disturbances such as road unevenness, tire force variations, and other unpredictable forces. To improve disturbance resilience, researchers have proposed various robust control schemes and disturbance observation techniques for SbW systems. One approach is to design disturbance observers (DObs) that estimate the external torque or force acting on the steering mechanism so that the controller can counteract it [17,18,19]. DObs are simple to implement and model-efficient, providing good low-frequency disturbance rejection; however, their fast estimation dynamics may interact with unmodeled high-frequency modes, potentially reducing stability margins under delay. Another approach is to employ robust or adaptive controllers that can reject disturbances actively. These controllers, such as active disturbance rejection control [20,21,22], sliding-mode [23,24], or adaptive schemes [25], offer strong robustness against parameter uncertainty and unmodeled dynamics but often require complex gain tuning and may cause high-frequency chattering or noise amplification. These studies generally confirm that augmenting the steering controller with disturbance observation or rejection capabilities can significantly improve performance.

Recent studies show that while DObs enhance disturbance rejection, they also alter closed-loop dynamics and may compromise stability. DOB fast dynamics can interact with unmodeled dynamics, which makes singular perturbation analysis necessary [26]. Increasing the DOB bandwidth improves estimation but at the same time reduces stability margins [27]. These findings confirm that DOb design introduces nontrivial stability issues, mostly studied with modern control methods.

Unlike Smith predictor or active disturbance rejection frameworks that primarily focus on delay compensation or disturbance suppression, the present study adopts a classical control perspective to reveal the intrinsic mechanism by which a disturbance observer reshapes the delay-dependent stability characteristics of the system. Through the combined use of D-curve stability mapping and eigenvalue-based validation, the proposed approach explicitly quantifies how the observer gain influences the admissible control gain domain in a time-delayed corner module steering system. This classical analytical framework provides deeper mechanistic insight into the trade-off between disturbance rejection and stability robustness, rather than merely offering an alternative controller design. However, such analysis has not yet been extended to SbW systems, and the effect of DObs on closed-loop stability under classical control frameworks remains unexplored.

The CM steering system, as an advanced SbW architecture, inherits delay-related challenges and is more vulnerable to wheel-level disturbances. However, while DObs are widely adopted as a cost-effective means to estimate and compensate for external disturbances, their introduction in time-delayed systems may also induce instability, and the underlying mechanism remains unclear. This study fills this gap with a delay-affected steering control framework. Section 2 builds a physics-based dynamic model including actuator dynamics, Coulomb friction, and tire/disturbance torques. Section 3 introduces the closed-loop architecture with a PD plus feedforward controller and a DOb. Section 4 applies D-curve mapping and eigenvalue analysis to assess stability with and without the observer. Section 5 discusses the results, showing how delay compresses the stability region and how the observer improves disturbance rejection but reduces robustness.

2. Steering System Model of the Single Corner Module

This section develops the dynamic model of the steering system of a single corner module, considering different physical effects step by step, including inertia, damping, Coulomb friction, and tire-to-ground interaction.

2.1. Steering System with No Tire-to-Ground Contact

In traditional steering system models such as rack-and-pinion steering, it has been demonstrated by research that the dynamic response of the entire system can be described by a second-order differential equation [28]. The dynamic characteristics of individual components can be ignored.

Considering the structural features of the studied system, and because lateral force transmission parts in traditional steering were omitted, torque is applied directly to the kingpin by the steering motor. Therefore, the use of a second-order differential equation for describing the dynamic response is considered more appropriate.

Thus, if tire forces are ignored (in other words, the wheel is not in contact with ground), the dynamic model of the steering system can be described by the following equation:

$$J_w\ddot{\theta }\left(t\right)+C_w\stackrel{.}{\theta }\left(t\right)=T_{\mathrm{SM}}\left(t\right),$$

(1)

where, $\theta \left(t\right)$ is the steering angle of the CM, $J_w$ is the total moment of inertia of the steering system, $C_w$ is the effective steering damping coefficient, and $T_{\mathrm{SM}}$ is the actuator torque of the steering motor.

2.2. Coulomb Friction in the System

In mechanical contacts between components of the steering system of the CM, Coulomb friction is commonly present. Its main sources include Coulomb friction generated by relative motion between components at the kingpin and by relative motion between gears in the steering motor reducer.

Below, the effect of Coulomb friction is made explicit:

$$T_f=T_{fc}\mathrm{sgn}\left(\stackrel{.}{\theta }\right),$$

(2)

where $T_f$ is the Coulomb friction torque and $T_{fc}$ is the Coulomb friction torque constant. This torque will be compensated via feedforward control.

2.3. Steering Torque Disturbances from Tire Forces

When the tire contacts the ground, tire force is applied to the wheel, generating a torque about the kingpin. In traditional steering system analysis, the torque with the greatest influence is the self-centering torque, and stability of steering is achieved by compensating for this torque in the control. In engineering practice and in most studies, the self-centering torque is expressed explicitly and compensated via feedforward control.

In the steering system of the CM, because the steering tie rod between coaxial wheels is eliminated, the torque produced by longitudinal force about the kingpin on each wheel during driving is not balanced, as shown in Figure 1. Therefore, for a single wheel, this torque must be taken into account. However, owing to the strong nonlinear characteristics of the tire and the requirement for the real-time slip ratio and slip angle to compute longitudinal and lateral forces, tire forces are not modeled here.

An observer will be introduced later in the closed-loop control of the steering system of the CM to estimate the torque from unmodeled tire forces and external disturbances. This torque is denoted as $T_t$ and is given by $T_t=M_{\mathrm{La}}+M_{\mathrm{Lo}}$.

2.4. Steering System with Tire-to-Ground Contact

After the effects of Coulomb friction inside the system and wheel contact with the ground are taken into account, the complete dynamic equation of the steering system of the CM can be established. The dynamic model includes only the degree of freedom of wheel rotation about the kingpin. The effects of the state variables, control input, and disturbances on the steering system are illustrated in Figure 2.

The dynamic model of the steering system with tire-to-ground contact is written by Equation (3).

$$J_w\ddot{\theta }\left(t\right)+C_w\stackrel{.}{\theta }\left(t\right)+K_w\theta \left(t\right)=T_{\mathrm{SM}}\left(t\right)+T_f\left(t\right)+T_t\left(t\right).$$

(3)

where $K_w$ represents the steering stiffness induced by the tire cornering stiffness.

3. Closed-Loop Steering Control Methods

This section introduces the closed-loop steering control architectures. It first presents the baseline PD plus feedforward control framework without a disturbance observer, and then extends it to include a disturbance observer for disturbance estimation and compensation.

3.1. Steering Control System

In the steering system control, the steering motor is governed by a control law to achieve accurate tracking of the wheel steering angle. This control strategy primarily relies on signals measured by sensors, including the wheel angle and its derivative (angular velocity), and constructs a feedback control system based on proportional–derivative (PD) control. However, PD controllers typically struggle to eliminate steady-state errors. To address this issue, a model-based feedforward control is introduced as a predictive compensation mechanism to reduce such errors [28]. The closed-loop control architecture of the steering system is illustrated in the figure below.

For the torque $T_{\mathrm{SM}}\left(t\right)$ in Equation (3), based on the feedforward and PD feedback control law, it can be expressed as follows:

$$T_{\mathrm{SM}}\left(t\right)=\underset{\mathrm{feedforward}\mathrm{term}}{\underbrace{J_w{\ddot{\theta }}_d\left(t\right)+C_w{\stackrel{.}{\theta }}_d\left(t\right)+K_w{\theta }_d\left(t\right)}}+\underset{\mathrm{PD}\mathrm{term}}{\underbrace{K_P\left({\theta }_d\left(t\right)-\theta (t-\tau )\right)+K_D\left({\stackrel{.}{\theta }}_d\left(t\right)-\stackrel{.}{\theta }(t-\tau )\right)}},$$

(4)

where ${\theta }_d\left(t\right)$ denotes the desired steering angle and $K_P$ and $K_D$ represent the proportional gain and derivative gain, respectively. In addition, considering the presence of delays in the feedback process due to sensors and actuators, a time delay $\tau$ is introduced in both the feedback wheel angle and angular velocity signals.

3.2. DOb in the Closed-Loop

To eliminate the effects of friction torque and external torque acting on the kingpin—denoted collectively as $T_d\left(t\right)=T_f\left(t\right)+T_t\left(t\right)$—a DOb is incorporated into the closed-loop control system shown in Figure 3. This observer estimates the disturbance torque $T_d\left(t\right)$ applied at node 3 and compensates for it at node 2, thereby enhancing the disturbance rejection capability of the system. The modeling of the DOb is based on the work of Chen [29]. The block diagram of the steering control system with the integrated DOb is illustrated below.

With the introduction of the DOb, Equation (3) can be reformulated according to the control logic shown in Figure 4.

$$\begin{array}{cc}\hfill J_w\ddot{\theta }\left(t\right)+C_w\stackrel{.}{\theta }\left(t\right)+K_w\theta \left(t\right)& =T_{\mathrm{SM}}\left(t\right)-{\widehat{T}}_d\left(t\right)+T_d\left(t\right)\hfill \\ \hfill & =u\left(t\right)+T_d\left(t\right),\hfill \end{array}$$

(5)

where ${\widehat{T}}_d\left(t\right)$ denotes the estimated disturbance torque.

Let $\mathbf{x}\left(t\right)=\left[\begin{array}{c}\stackrel{.}{\theta }\left(t\right)\\ \theta \left(t\right)\end{array}\right]$; then Equation (5) can be expressed in the form of a state-space equation:

$$\left\{\begin{array}{cc}\underset{\stackrel{.}{\mathbf{x}}\left(t\right)}{\underbrace{\left[\begin{array}{c}\ddot{\theta }\left(t\right)\\ \stackrel{.}{\theta }\left(t\right)\end{array}\right]}}\hfill & =\underset{\mathbf{A}}{\underbrace{\left[\begin{array}{cc}-{\displaystyle \frac{C_w}{J_w}}& -{\displaystyle \frac{K_w}{J_w}}\\ 1& 0\end{array}\right]}}\underset{\mathbf{x}\left(t\right)}{\underbrace{\left[\begin{array}{c}\stackrel{.}{\theta }\left(t\right)\\ \theta \left(t\right)\end{array}\right]}}+\underset{\mathbf{B}}{\underbrace{\left[\begin{array}{c}{\displaystyle \frac{1}{J_w}}\\ 0\end{array}\right]}}\left(u\left(t\right)+T_d\left(t\right)\right)\hfill \\ \mathbf{y}\left(t\right)\hfill & =\mathbf{x}(t-\tau )\hfill \end{array}\right.$$

(6)

Define the observation error of the DOb as follows:

$$e\left(t\right)=T_d\left(t\right)-{\widehat{T}}_d\left(t\right).$$

(7)

According to the structure of the DOb, an intermediate variable $z\left(t\right)$ and the observer gain $\mathbf{L}={\left[\begin{array}{cc}L& 0\end{array}\right]}^{\top }$ are defined, and the estimated disturbance ${\widehat{T}}_d\left(t\right)$ can therefore be expressed as follows:

$$\left\{\begin{array}{cc}\stackrel{.}{z}\left(t\right)\hfill & =-\mathbf{LB}\left(z\left(t\right)+\mathbf{Ly}\left(t\right)\right)-\mathbf{LAy}\left(t\right)-\mathbf{LB}u\left(t\right)\hfill \\ {\widehat{T}}_d\left(t\right)\hfill & =z\left(t\right)+\mathbf{Ly}\left(t\right).\hfill \end{array}\right.$$

(8)

Taking its derivative and combining it with Equations (6) and (8) yields

$$\begin{array}{cc}\hfill \stackrel{.}{e}\left(t\right)& ={\stackrel{.}{T}}_d\left(t\right)-{\stackrel{.}{\widehat{T}}}_d\left(t\right)\hfill \\ \hfill & ={\stackrel{.}{T}}_d\left(t\right)-\stackrel{.}{z}\left(t\right)-\mathbf{L}\stackrel{.}{\mathbf{x}}(t-\tau )\hfill \\ \hfill & ={\stackrel{.}{T}}_d\left(t\right)+\mathbf{L}\mathbf{B}{\widehat{T}}_d\left(t\right)+\mathbf{L}\mathbf{A}\mathbf{y}\left(t\right)+\mathbf{L}\mathbf{B}u\left(t\right)\hfill \\ \hfill & -\mathbf{L}\left(\mathbf{A}\mathbf{x}(t-\tau )+\mathbf{B}\left(u(t-\tau )+T_d(t-\tau )\right)\right).\hfill \end{array}$$

(9)

For the delayed control input $u(t-\tau )$, the feedforward term is not time-shifted by $\tau$. As shown in Figure 4, the desired steering angle is treated as a known quantity in this study—such as in autonomous driving applications where the wheel follows a pre-planned reference steering path. In this case, the feedforward control can compensate in advance for the impending actuator delay. Therefore, Equation (9) can be further expressed as follows:

$$\begin{array}{cc}\hfill \stackrel{.}{e}\left(t\right)& ={\stackrel{.}{T}}_d\left(t\right)+\mathbf{L}\mathbf{B}{T}_{\mathrm{SM}}\left(t\right)-\mathbf{L}\mathbf{B}{T}_{\mathrm{SM}}(t-\tau )-\mathbf{L}\mathbf{B}e(t-\tau )\hfill \\ \hfill & ={\stackrel{.}{T}}_d\left(t\right)-\mathbf{L}\mathbf{B}\left(K_P\theta (t-\tau )+K_D\stackrel{.}{\theta }(t-\tau )\right)+\mathbf{L}\mathbf{B}\left(K_P\theta (t-2\tau )+K_D\stackrel{.}{\theta }(t-2\tau )\right)\hfill \\ \hfill & -\mathbf{L}\mathbf{B}e(t-\tau )\hfill \end{array}$$

(10)

4. Stability Analysis

This section analyzes the stability of the steering control system under feedback delay. Both the baseline system and the DOb-augmented system are studied, and their stability boundaries are derived using D-curve mapping and characteristic root analysis.

4.1. Steering Control System Without DOb

The stability of the steering control system shown in Figure 3 is first analyzed, with the objective of identifying the feasible gain region for the PD controller under feedback delay. The analysis begins with linearization of the system.

Based on Equations (3) and (4), the feedback control system in Figure 3 can be reformulated into a state-space representation.

$$\underset{\stackrel{.}{\stackrel{~}{\mathbf{x}}}\left(t\right)}{\underbrace{\left[\begin{array}{c}\ddot{\theta }\left(t\right)-{\ddot{\theta }}_d\left(t\right)\\ \stackrel{.}{\theta }\left(t\right)-{\stackrel{.}{\theta }}_d\left(t\right)\end{array}\right]}}=\underset{\mathbf{A}}{\underbrace{\left[\begin{array}{cc}-{\displaystyle \frac{C_w}{J_w}}& -{\displaystyle \frac{K_w}{J_w}}\\ 1& 0\end{array}\right]}}\underset{\tilde{\mathbf{x}}\left(t\right)}{\underbrace{\left[\begin{array}{c}\stackrel{.}{\theta }\left(t\right)-{\stackrel{.}{\theta }}_d\left(t\right)\\ \theta \left(t\right)-{\theta }_d\left(t\right)\end{array}\right]}}+\underset{\tilde{\mathbf{B}}}{\underbrace{\left[\begin{array}{cc}-{\displaystyle \frac{K_P}{J_w}}& -{\displaystyle \frac{K_D}{J_w}}\\ 0& 0\end{array}\right]}}\underset{\tilde{\mathbf{x}}(t-\tau )}{\underbrace{\left[\begin{array}{c}\stackrel{.}{\theta }(t-\tau )-{\stackrel{.}{\theta }}_d\left(t\right)\\ \theta (t-\tau )-{\theta }_d\left(t\right)\end{array}\right]}}$$

(11)

Substituting the trial solution $\tilde{\mathbf{x}}\left(t\right)=\mathbf{C}{e}^{\lambda t}$, where $\mathbf{C}\in {\mathbb{C}}^2$ and $\lambda \in \mathbb{C}$, into Equation (11) yields the corresponding characteristic equation. Here, $\mathbf{A}$ and $\tilde{\mathbf{B}}$ denote the system and delayed feedback matrices defined in Equation (11). The characteristic equation can thus be written as

$$\Delta :=\det (\lambda \mathbf{I}-\mathbf{A}-\tilde{\mathbf{B}}{e}^{-\lambda \tau })=0,$$

(12)

Expanding the determinant of Equation (12) and substituting the coefficients from Equation (11) yield the following expression (the detailed derivation is provided in Appendix A):

$$D\left(\lambda \right):={\lambda }^2+{\displaystyle \frac{C_w}{J_w}}\lambda +\left({\displaystyle \frac{K_D}{J_w}}\lambda +{\displaystyle \frac{K_P}{J_w}}\right)e^{-\lambda \tau }+{\displaystyle \frac{K_w}{J_w}}=0.$$

(13)

To examine stability boundaries, let $\lambda =\gamma +i\omega$, where $\gamma$ and $\omega$ represent the real and imaginary parts of the eigenvalue, respectively, and use the Euler identity $e^{-i\omega \tau }=cos\left(\omega \tau \right)-isin\left(\omega \tau \right)$. Separating Equation (13) into real and imaginary parts gives

$$\begin{array}{cc}\hfill & R\left(\omega \right)={\gamma }^2-{\omega }^2+{\displaystyle \frac{C_w}{J_w}}\gamma +e^{-\gamma \tau }\left({\displaystyle \frac{K_D}{J_w}}\gamma +{\displaystyle \frac{K_P}{J_w}}\right)\cos \left(\omega \tau \right)+e^{-\gamma \tau }{\displaystyle \frac{K_D}{J_w}}\omega \sin \left(\omega \tau \right)+{\displaystyle \frac{K_w}{J_w}}\hfill \\ \hfill & S\left(\omega \right)=2\gamma \omega +{\displaystyle \frac{C_w}{J_w}}\omega -e^{-\gamma \tau }\left({\displaystyle \frac{K_D}{J_w}}\gamma +{\displaystyle \frac{K_P}{J_w}}\right)\sin \left(\omega \tau \right)+e^{-\gamma \tau }{\displaystyle \frac{K_D}{J_w}}\omega \cos \left(\omega \tau \right)\hfill \end{array}$$

(14)

where $\omega \in [0,+\infty )$.

The stability properties of delay differential equations can be represented in the form of stability charts constructed by the so-called D-subdivision method. The curves on which the number of unstable characteristic roots changes are known as the D-curves (also called exponent-crossing or transition curves) and are defined by the conditions $R\left(\omega \right)=0$ and $S\left(\omega \right)=0$, where $R\left(\omega \right)=\Re \left[D\right(i\omega \left)\right]$ and $S\left(\omega \right)=\Im \left[D\right(i\omega \left)\right]$ [30].

In the present study, the D-curve represents the set of proportional–derivative gain pairs $(K_P,K_D)$ that satisfy $R\left(\omega \right)=S\left(\omega \right)=0$ for a given delay $\tau$, forming the boundary between stable and unstable regions in the $K_P$-$K_D$ plane.

To begin with, the static instability boundary can be analyzed. In this case, the system is at marginal static stability. By expanding Equation (13) around $\omega =0$ (i.e., considering the low-frequency limit) and setting $\gamma =0$, the first-order approximation yields the static boundary conditions. The D-curve originates from the point $(K_P\left(0\right),K_D\left(0\right))$, whose detailed derivation is provided in Appendix B.

$$K_P\left(0\right)=-K_w,K_D\left(0\right)=-(C_w+K_w\tau ).$$

(15)

At this point, the instability boundary becomes a vertical line with respect to the $K_P$ axis. The region on the left-hand side of this line corresponds to static instability.

When $\omega \ne 0$ and increases, the dynamic instability boundary can be determined by simultaneously solving $R\left(\omega \right)$ and $S\left(\omega \right)$ in Equation (14), as shown below:

$$\begin{array}{cc}\hfill & K_P\left(\omega \right)={\displaystyle \frac{C_w{\omega }^2+\left(J_w{\omega }^2-K_w\right)sin\left(\omega \tau \right)cos\left(\omega \tau \right)-C_w\omega {cos}^2\left(\omega \tau \right)}{\omega sin\left(\omega \tau \right)}}\hfill \\ \hfill & K_D\left(\omega \right)={\displaystyle \frac{\left(J_w{\omega }^2-K_w\right)sin\left(\omega \tau \right)-C_w\omega cos\left(\omega \tau \right)}{\omega }}\hfill \end{array}$$

(16)

4.2. Steering Control System with DOb

Following the inclusion of the estimation error of the DOb, the closed-loop model in Figure 4 is reformulated into a third-order time-delay system. To capture both the tracking error and the observer dynamics, an augmented state vector is defined as ${\tilde{\mathbf{x}}}_a\left(t\right)={\left[\begin{array}{ccc}\stackrel{.}{\theta }\left(t\right)-{\stackrel{.}{\theta }}_d\left(t\right)& \theta \left(t\right)-{\theta }_d\left(t\right)& e\left(t\right)\end{array}\right]}^{\top }$, where the first two components denote the velocity and position tracking errors, while the third component represents the observer error.

Based on this definition, the linearized dynamics of the augmented system can be written in the compact state-space form

$${\stackrel{.}{\stackrel{~}{\mathbf{x}}}}_a\left(t\right)={\mathbf{A}}_a{\tilde{\mathbf{x}}}_a\left(t\right)+{\tilde{\mathbf{B}}}_{a1}{\tilde{\mathbf{x}}}_a(t-\tau )+{\tilde{\mathbf{B}}}_{a2}{\tilde{\mathbf{x}}}_a(t-2\tau ),$$

(17)

where the coefficient matrices are

$${\mathbf{A}}_a=\left[\begin{array}{ccc}-{\displaystyle \frac{C_w}{J_w}}& -{\displaystyle \frac{K_w}{J_w}}& -{\displaystyle \frac{1}{J_w}}\\ 1& 0& 0\\ 0& 0& 0\end{array}\right], {\tilde{\mathbf{B}}}_{a1}=\left[\begin{array}{ccc}-{\displaystyle \frac{K_D}{J_w}}& -{\displaystyle \frac{K_P}{J_w}}& 0\\ 0& 0& 0\\ -{\displaystyle \frac{L{K}_D}{J_w}}& -{\displaystyle \frac{L{K}_P}{J_w}}& -{\displaystyle \frac{L}{J_w}}\end{array}\right], {\tilde{\mathbf{B}}}_{a2}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ {\displaystyle \frac{L{K}_D}{J_w}}& {\displaystyle \frac{L{K}_P}{J_w}}& 0\end{array}\right].$$

(18)

Introducing the trial solution ${\tilde{\mathbf{x}}}_a\left(t\right)=\mathbf{C}{e}^{{\lambda }^{★}t}$$(\mathbf{C}\in {\mathbb{C}}^3)$ into Equation (17) yields

$${\Delta }_a:=det\left({\lambda }^{★}\mathbf{I}-{\mathbf{A}}_a-{\tilde{\mathbf{B}}}_{a1}{e}^{-{\lambda }^{★}\tau }-{\tilde{\mathbf{B}}}_{a2}{e}^{-2{\lambda }^{★}\tau }\right)=0.$$

(19)

To distinguish the augmented system (with the DOb) from the case without the DOb, all characteristic roots and frequencies associated with the DOb framework are denoted with a superscript ★.

On the stability boundary ${\lambda }^{★}=i{\omega }^{★}$$({\omega }^{★}\ge 0)$, write

$${\Delta }_a\left(i{\omega }^{★}\right)=R\left({\omega }^{★}\right)+iS\left({\omega }^{★}\right)=0,$$

(20)

with $R\left({\omega }^{★}\right)=\Re \Delta \left(i{\omega }^{★}\right)$ and $S\left({\omega }^{★}\right)=\Im \Delta \left(i{\omega }^{★}\right)$.

Since the controller gains $K_P$ and $K_D$ appear linearly in the matrices ${\mathbf{B}}_{a1}$ and ${\mathbf{B}}_{a2}$, the boundary conditions $R\left(\omega \right)=0$ and $S\left(\omega \right)=0$ can be expressed as affine equations in terms of $(K_P,K_D)$:

$$\mathbf{J}\left({\omega }^{★}\right)\left[\begin{array}{c}{K}_P\\ K_D\end{array}\right]+\mathbf{h}\left({\omega }^{★}\right)=\mathbf{0},{\omega }^{★}\in [0,\infty ),$$

(21)

where the Jacobian matrix and offset vector are defined by

$$\mathbf{J}\left({\omega }^{★}\right)=\left[\begin{array}{cc}{\displaystyle \frac{\partial R\left({\omega }^{★}\right)}{\partial K_P}}& {\displaystyle \frac{\partial R\left({\omega }^{★}\right)}{\partial K_D}}\\ {\displaystyle \frac{\partial S\left({\omega }^{★}\right)}{\partial K_P}}& {\displaystyle \frac{\partial S\left({\omega }^{★}\right)}{\partial K_D}}\end{array}\right],\mathbf{h}\left({\omega }^{★}\right)=\left[\begin{array}{c}-R\left({\omega }^{★}\right)|K_P=K_D=0\\ -S\left({\omega }^{★}\right)|K_P=K_D=0\end{array}\right].$$

(22)

This affine formulation is adopted because the analytical expressions of $K_P\left(\omega \right)$ and $K_D\left(\omega \right)$ are too complex to derive explicitly, whereas their numerical values can be readily obtained by solving $J\left({\omega }^{★}\right){\left[K_P{K}_D\right]}^{\top }=-h\left({\omega }^{★}\right)$ at each sampled frequency, which greatly simplifies the D-curve construction.

To delineate the dynamic boundary of the D-curve corresponding to a prescribed delay $\tau$ and observer gain L, the frequency domain is swept on the interval ${\omega }^{★}\in (0,{\omega }_{max}^{★}]$ using a sufficiently dense mesh ${\left\{{\omega }_k^{★}\right\}}_{k=1}^N$, where k denotes the index of the sampled frequency point and N is the total number of samples. For each sampled frequency ${\omega }_k^{★}$, the Jacobian matrix $\mathbf{J}\left({\omega }_k\right)$ and the offset vector $\mathbf{h}\left({\omega }_k^{★}\right)$ are evaluated with the plant parameters $\left(J_w,C_w,K_w\right)$. Whenever $det\mathbf{J}\left({\omega }_k^{★}\right)\ne 0$, the affine relation (21) is solvable, and the admissible controller gains follow from

$$\left[\begin{array}{c}{K}_P\left({\omega }_k^{★}\right)\\ K_D\left({\omega }_k^{★}\right)\end{array}\right]=-{\mathbf{J}}^{-1}\left({\omega }_k^{★}\right)\mathbf{h}\left({\omega }_k^{★}\right).$$

(23)

The collection of points ${\{K_P\left({\omega }_k^{★}\right),K_D\left({\omega }_k^{★}\right)\}}_{k=1}^N$, ordered by increasing frequency, furnishes a parametric representation of the dynamic stability boundary.

In the static limit ${\omega }^{★}=0$, the static instability boundary of the steering control system with the DOb is expressed below:

$$K_P\left(0\right)=-K_w,K_D\left(0\right)=-(C_w+K_w\tau ).$$

(24)

5. Result and Discussion

This chapter presents the stability analysis results of the steering control system with feedback delay, with a particular focus on the impact of introducing a DOb on system stability. The system parameters used in the following result analysis are listed in

Table 1. Model parameters of the steering system.

Parameter	Value	Unit
$J_w$	6.5	kg·m2
$C_w$	35	N·m/s
$K_w$	8000	Nm

.

5.1. Stability Charts of the Steering Control System Without a DOb

Based on the analysis of the delay differential equation without a DOb in Section 4.1, the static and dynamic stability boundaries have been derived in Equations (15) and (16), respectively. As a result, the stability chart of the system can be constructed in the $K_P-K_D$ plane, as shown in Figure 5.

When no feedback delay $\tau$ is present, the system remains stable in the region where $K_P>-K_w$ and $K_D>0$ (see Equation (15)). In this case, $e^{-\omega \tau }$ is identically 1, and with $K_D=0$, the value of $\omega$ does not affect the stability of the system.

As the feedback delay $\tau$ increases, the characteristics of the stability chart for the steering control system without a DOb change significantly. When $\tau$ is less than 0.06 s (this value is illustrative and not a strict threshold), the stable region is enclosed by the black curve representing the dynamic stability boundary (as $\omega$ varies) and the red dashed line representing the static stability boundary (at $\omega =0$). It can be observed that as $\tau$ increases, the terminal value of the imaginary-axis frequency ${\omega }_{\mathrm{E}}$ defining the edge of stability becomes smaller, leading to a continuous reduction in the area of the stable region.

When the feedback delay $\tau$ increases further, such as at $\tau =0.08\mathrm{s}$ and $0.1\mathrm{s}$, the stable region is formed by a self-intersection of the dynamic stability boundary, enclosing a closed area. The intersection point is referred to here as a node, marking the onset and terminal frequencies of stability. In this type of stability chart, the onset frequency ${\omega }_{\mathrm{S}}$ is no longer zero. As $\tau$ increases, the onset frequency ${\omega }_{\mathrm{S}}$ rises while the terminal frequency ${\omega }_{\mathrm{E}}$ decreases, leading to an obvious reduction in the overall area of the stable region.

It is worth noting that a feedback delay of 0.1 s, which lies within the upper limit of typical actuator–sensor communication latency in steer-by-wire systems, already causes a visible contraction of the stable region in Figure 5. For a lightweight corner module steering system with relatively low damping and stiffness, such a moderate delay can shift a pair of complex roots across the imaginary axis, leading to oscillatory behavior. This tendency is further confirmed by a sensitivity analysis with respect to the steering stiffness $K_w$ and damping $C_w$, where further reductions in damping and stiffness lead to a more pronounced shrinkage of the stable region. This observation highlights that even short delays can significantly affect closed-loop stability unless specific delay compensation strategies are applied.

5.2. Stability Charts of the Steering Control System with a DOb

Based on the discretized relationship among ${\omega }^{★}$, $K_P$, and $K_D$ formulated in Equations (23) and (24) in Section 4.2, the stability chart of the steering control system with a DOb under feedback delay can be obtained. The computation involves frequency sweeping over ${\omega }^{★}\in (0,120]$ rad/s with 3000 samples, calculation of the Jacobian matrix $\mathbf{J}\left({\omega }^{★}\right)$ and offset vector $\mathbf{h}\left({\omega }^{★}\right)$ at each frequency point, and numerical solution of Equation (21).

In practical tuning, the proportional and derivative gains can first be chosen within the stable domain without the DOB (Figure 5) and then adjusted using Figure 6 to account for the selected observer gain L. Under a feedback delay of $\tau =0.04\mathrm{s}$, the stability charts of the system for different DOb gains L are illustrated in Figure 6. The introduction of the DOb affects the $K_D$ value at the onset frequency point. According to Equation (24), regardless of the DOb gain L, $K_D\left({\omega }_S^{★}\right)$ always starts at $-C_w$ and is independent of the delay $\tau$. This is because the observer gain L effectively cancels out the first-order delayed term in Equation (15), leaving only the damping term $C_w$.

As the DOb gain L increases, the stable region gradually shrinks inward while keeping $K_D\left({\omega }_S^{★}\right)$ fixed. The static stability boundary remains unchanged, but the overall shape of the stability region evolves from a “D-shape” into a closed “O-shape.” This trend is evident in two aspects: (1) the terminal frequency ${\omega }_E^{★}$, where the dynamic stability boundary intersects the static boundary, decreases with increasing L, and (2) the corresponding $K_D$ value at ${\omega }_E^{★}$ also becomes smaller. The specific values for the above two aspects can be found in

Table 2. Terminal frequency ${\omega }_E^{★}$ and corresponding gain $K_D\left({\omega }_E^{★}\right)$ under different observer gains L. Units: ${\omega }_E^{★}$ in $\mathrm{rad}/\mathrm{s}$; $K_D\left({\omega }_E^{★}\right)$ in $\mathrm{N}·\mathrm{m}$.

Symbol	$\mathit{L}=0.1$	$\mathit{L}=10$	$\mathit{L}=20$	$\mathit{L}=30$	$\mathit{L}=50$
${\omega }_E^{★}$	58.0	57.7	57.4	57.0	56.5
$K_D\left({\omega }_E^{★}\right)$	198.8	188.5	178.0	167.5	146.5

.

The parameters in the system matrix A also affect the overall stability of the system. Taking the nominal stiffness $K_w=8000$ Nm/rad as the baseline, a sensitivity analysis was performed by varying $K_w$ by $\pm 20\%$. The results are shown in Figure 7. When the steering stiffness changes, the static instability boundary remains vertical but shifts along the $K_P$-axis according to the variation in $K_w$, since its intersection with the $K_P$-axis is given by $-K_w$. Moreover, as $K_w$ decreases, the terminal frequency ${\omega }_E^{★}$ gradually decreases, leading to a progressive reduction in the stable region. The terminal frequencies and their corresponding $K_D$ values are summarized in

Table 3. Terminal frequency ${\omega }_E^{★}$ and corresponding gain $K_D\left({\omega }_E^{★}\right)$ under different $K_w$. Units: $K_w$ in $\mathrm{Nm}/\mathrm{rad}$.

Symbol	${\mathit{K}}_{\mathit{w}}=6400(-20\%)$	${\mathit{K}}_{\mathit{w}}=8000$	${\mathit{K}}_{\mathit{w}}=9600(+20\%)$
${\omega }_E^{★}$	54.4	57.4	60.2
$K_D\left({\omega }_E^{★}\right)$	193.8	178.0	161.2

.

The effect of steering damping variation on system stability is illustrated in Figure 8. As $C_w$ decreases, the $K_D$ value corresponding to the onset frequency point becomes smaller, while the $K_D$ value at the terminal frequency point increases, and the associated frequency gradually decreases (as summarized in

Table 4. Terminal frequency ${\omega }_E^{★}$ and corresponding gain $K_D\left({\omega }_E^{★}\right)$ under different $C_w$. Units: $C_w$ in $\mathrm{Nms}/\mathrm{rad}$.

Symbol	${\mathit{C}}_{\mathit{w}}=28.0(-20\%)$	${\mathit{C}}_{\mathit{w}}=35$	${\mathit{C}}_{\mathit{w}}=42(+20\%)$
${\omega }_E^{★}$	57.1	57.4	57.6
$K_D\left({\omega }_E^{★}\right)$	173.0	178.0	183.1

). Consequently, the stable region becomes smaller, although the influence is not very significant. Since the static stability boundary depends only on the stiffness $K_w$, it remains unchanged throughout the variation.

The DOb enhances robustness by estimating and compensating for external torques and model uncertainties, thereby improving low-frequency disturbance rejection. From a stability perspective, however, it eliminates the delay-induced damping term only at the expense ofstronger high-frequency phase lag, which advances the dynamic instability boundary and reduces the stable region. Careful tuning of the observer gain L is therefore required to balance disturbance rejection with delay-robust stability. Therefore, careful tuning of the observer gain L is essential to balance disturbance rejection with delay-robust stability. In practical tuning, moderate observer gains ($L\le 20$) offer a good compromise, providing satisfactory low-frequency disturbance attenuation without significantly shrinking the stable domain. Conversely, excessively large observer gains ($L>30$) tend to narrow the admissible $K_D$ range and may induce high-frequency oscillations.

5.3. Comparison of Eigenvalue Characteristics

To validate the system characteristics revealed by the stability charts, the characteristic equation is numerically solve in MATLAB R2024b. A comparative analysis is performed on the characteristic roots of the steering control system with and without the DOb under various control gains.

The following analysis is based solely on the stability charts of the system under a feedback delay of $\tau =0.04,\mathrm{s}$, focusing on the characteristics of system stability rather than the influence of delay. As shown in Figure 9a, when the DOb gain is $L=30$, the system with the observer exhibits a noticeably smaller stable region compared to the one without, primarily due to the upward shift of the onset frequency, as demonstrated in Section 5.1. Both systems, however, share the same static instability region where $K_P<-K_w$.

To further validate these stability characteristics, two representative paths are selected from the stability charts: one where $K_D=-150$ and $K_P$ increases from the static instability region toward the right and another where $K_P=-5500$ and $K_D$ decreases from the upper dynamic instability region to the lower dynamic instability region.

The final root locus results are shown in Figure 9b. For Path 1, both systems begin with a pair of purely real positive roots that gradually approach the origin as $K_P$ increases. However, after crossing the origin on the real axis, the two systems exhibit distinct behaviors.

In the system without a DOb, a Hopf bifurcation occurs at this point, generating a pair of complex conjugate roots with negative real parts that move rightward (indicating dynamic stability). As the gain continues to increase, this root pair eventually crosses into the right half-plane, resulting in dynamic instability.

In contrast, the system with a DOb generates a pair of unstable complex conjugate roots directly as the real root approaches zero. This pair first crosses into the left half-plane (indicating stability) and then re-enters the right half-plane, following a dynamic evolution from instability to stability and back to instability.

This finding confirms the stability chart’s prediction: when $K_D=-150$ and $K_P>-K_w$, the system with the DOb experiences a region of dynamic instability, while the system without the observer remains dynamically stable under the same conditions.

For Path 2, the root locus characteristics of both systems show no significant differences. In both cases, the system transitions from dynamic instability to dynamic stability and then back to dynamic instability.

6. Conclusions

This paper provides a detailed mechanistic interpretation. In a corner module steering control system with time delay, DObs offer a cost-effective means for disturbance estimation and compensation, but their embedding within the closed-loop controller also alters the system’s stability characteristics. Focusing on this trade-off, the main work and conclusions of this study are summarized as follows:

• In the dynamic modeling of the corner module steering system, the sources of steering disturbance torque are explicitly identified, and a DOb is incorporated into the closed-loop control framework to estimate such disturbances.
• Incorporating a DOb enhances disturbance rejection but reduces the admissible gain domain. Higher observer gains further constrain derivative action and give rise to additional unstable regions. Root locus checks along representative gain paths confirm these characteristics.
• Increasing delay progressively shrinks the stable domain, shifts the terminal stability frequency downward, and can transform the boundary from a D-curve into a closed ring.