An Augmented Deep Koopman Operator-Based MPC for Steering Control of High-Speed Electric Tracked Vehicles

Abstract

With advances in electric drive technology, electric tracked vehicles (ETVs) have emerged as a promising solution for high-mobility ground vehicles. However, under high-speed steering conditions, the equivalent motor load inertia varies significantly, introducing strong nonlinear and time-varying characteristics into the ETV that may induce lateral instability and even rollover. To address this issue, a novel augmented deep Koopman operator-based model predictive control (ADK-MPC) method is proposed. First, a high-order sliding-mode (HOSM) observer is designed to estimate the lumped load disturbances associated with the time-varying equivalent motor load inertia. Then, the estimated disturbances are introduced as an augmented state into the DK operator to construct a data-driven augmented model. The proposed model transforms the nonlinear dynamics into a lifted linear time-invariant representation in the augmented-state space while capturing the dominant nonlinear characteristics. Based on the ADK model, an ADK-MPC controller is developed to convert the nonlinear optimization problem into a quadratic programming problem, thereby improving steering stability and reducing computational complexity. Simulation results under steering conditions indicate that the proposed method achieves better yaw rate tracking and lower computational cost than nonlinear MPC. The yaw rate tracking error is reduced by 45.5%, while the average solving time is shortened by 11.7%.

1. Introduction

With the global energy transition and the rapid development of electric drive technology, electric tracked vehicles (ETVs) have gradually emerged as an important development direction for rescue vehicles and heavy engineering equipment [1]. This is attributed to their high power density, flexible steering capability, and suitability for intelligent control [2]. Each track of ETVs is driven independently or in a coupled manner by electric motors [3]. This configuration provides new possibilities for enhanced maneuverability, rapid torque response, and high-performance control [4,5]. However, during high-speed conditions, the motor load and track–ground interaction of ETVs exhibit complex nonlinear and time-varying characteristics. These factors impose significant challenges on the realization of high-performance drive control [6,7]. Therefore, the development of advanced dynamic control strategies for ETVs has become a critical issue that needs to be addressed in the field of transportation engineering.

Early studies mainly focused on model-free control methods because of limited onboard computational capability and insufficient sensing accuracy. Proportional–Integral–Derivative (PID) control determines motor torque directly according to tracking errors and has been widely applied in low-speed steering control of ETVs [8]. To improve robustness and adaptability, several enhanced PID-based approaches have been developed, such as metaheuristic PID [9] and fuzzy PID [10]. In addition, geometric control methods such as pure pursuit algorithms calculate steering commands based on path geometry and vehicle posture [11]. These methods have simple structures, low computational complexity, and convenient parameter tuning, and they can achieve acceptable control performance under low-speed and flat-road conditions. However, since they do not explicitly consider track slip characteristics and dynamic load transfer, their control performance deteriorates significantly under high-speed steering conditions, especially when large lateral acceleration and strong track–ground coupling occur.

To improve steering performance under complex operating conditions, many studies have introduced model-based control strategies. Kinematic-model-based control methods simplify the track–ground slip problem by establishing geometric relationships among vehicle velocity, heading angle, and track motion. Based on this framework, the model accuracy has been improved by incorporating slip ratio estimation [12] and instantaneous center of rotation estimation [13]. These kinematic models are further combined with active disturbance rejection control [14], sliding mode control [15], and model predictive control (MPC) [16] to enhance steering performance. Such methods avoid complicated force and moment balance equations and maintain relatively low computational burden. Nevertheless, kinematic control methods inherently neglect inertial effects, track–ground force coupling, and yaw dynamic characteristics. As a result, their prediction accuracy and stability deteriorate significantly during high-speed steering, where dynamic effects dominate the system response.

To further improve control accuracy and stability, dynamic-model-based control methods have attracted increasing attention. Dynamic models can more accurately describe the coupling relationships among motor torque, track driving force, ground reaction force, and vehicle motion states [17], thereby providing richer predictive information for controller design and enabling better yaw stability and trajectory tracking performance [18]. In particular, nonlinear model predictive control (NMPC) has been widely studied because it can explicitly handle nonlinear dynamics and actuator constraints [19]. However, NMPC requires solving a non-convex optimization problem online at each sampling instant, which results in high computational cost and severely limits real-time implementation [20]. To reduce computational complexity, linearization-based MPC methods have been proposed by approximating the nonlinear dynamic model using Taylor expansion around operating points [21] or feedback linearization techniques [22]. These methods improve computational efficiency and make online optimization more tractable. However, their performance strongly depends on the selected operating point, and the accumulated linearization error becomes significant when the system operates far from equilibrium, especially under aggressive steering maneuvers.

In recent years, data-driven control methods have provided a new perspective for nonlinear system modeling and predictive control. Among them, Koopman operator theory has attracted considerable attention because it enables a nonlinear dynamical system to be represented as a linear system in a lifted high-dimensional space through observable functions. Combined with dynamic mode decomposition [23] and extended dynamic mode decomposition [24], Koopman-based MPC can transform nonlinear predictive control problems into computationally efficient quadratic programming problems. This framework has been successfully applied in robotics, unmanned aerial vehicles, and autonomous vehicle control, significantly improving the real-time performance of receding-horizon optimization [25]. However, the standard Koopman framework is primarily developed for time-invariant systems [26], making it difficult to directly handle explicit time-varying factors such as abrupt motor load variations and parameter drift. Under high-speed steering conditions of ETVs, the equivalent motor load changes significantly with steering states, which introduces strong time-varying characteristics into the system. When these effects are neglected, the lifted linear model suffers from large prediction errors, which degrades MPC performance and may even lead to closed-loop instability.

To address this problem, an augmented deep Koopman operator-based model predictive control (ADK-MPC) method is proposed for high-speed steering control of ETVs. First, a high-order sliding-mode (HOSM) observer is designed to accurately estimate the time-varying equivalent motor load inertia. Then, the estimated load inertia is introduced into the system as an augmented state, and a data-driven augmented ETV model is constructed by combining the HOSM observer with the deep Koopman operator. This approach transforms the original nonlinear time-varying system into a high-dimensional linear time-invariant model in the augmented-state space while capturing its dominant nonlinear characteristics. Based on the augmented Koopman model, an ADK-MPC controller is developed to convert the nonlinear optimization problem into a quadratic programming problem, thereby improving steering stability and reducing computational complexity. Compared with conventional Koopman MPC methods that directly neglect time-varying load variations, the proposed method explicitly compensates for the parameter variations caused by motor load changes, enabling more accurate yaw rate prediction and better real-time steering control performance under high-speed operating conditions. The proposed method provides a unified framework for simultaneously handling nonlinear dynamics and time-varying load inertia for ETVs. The improved computational efficiency and steering stability also provide practical potential for real-time onboard control of autonomous tracked rescue vehicles, heavy engineering vehicles, and unmanned ground vehicles under high-speed operating conditions.

The remainder of this paper is organized as follows. In Section 2, the dynamic model of the electric transmission tracked vehicle is established. In Section 3, the design procedure of the ADK-MPC method is presented. In Section 4, comparative simulation results and corresponding discussions are provided. Finally, conclusions are drawn in Section 5.

2. Dynamic Model of the Electric Tracked Vehicle

The configuration of the ETV considered in this study is illustrated in Figure 1. Its main components include the energy supply system, DC bus, motor controllers, dual driving motors, coupling transmission mechanism, and tracks. The energy supply system provides DC electrical power, while the motor controllers convert the DC power into AC power to drive the motors. The motor torques are transmitted to the two tracks through the coupling transmission mechanism to propel the ETV. This architecture enables coordinated torque distribution between the two driving motors and provides a foundation for high-performance steering control under high-speed operating conditions.

Figure 1. The configuration of the ETV.

The ETV configuration considered in this study adopts a dual-motor coupled drive architecture. The driving motors transmit power to the sprockets through a reduction planetary gear set, a coupling planetary gear set, and a final drive. The corresponding transmission relationships are described as follows.

$$\begin{array}{l}{\omega }_{zL}={\displaystyle \frac{i}{1+2k}}\left(\left(1+k\right){\omega }_{mL}+k{\omega }_{mR}\right)\\ {\omega }_{zR}={\displaystyle \frac{i}{1+2k}}\left(\left(1+k\right){\omega }_{mR}+k{\omega }_{mL}\right)\\ T_{mL}={\displaystyle \frac{i}{1+2k}}\left(k{T}_{zR}+\left(1+k\right)T_{zL}\right)\\ T_{mR}={\displaystyle \frac{i}{1+2k}}\left(k{T}_{zL}+\left(1+k\right)T_{zR}\right)\end{array}$$

(1)

where ${\omega }_{zL},{\omega }_{zR}$ denote the angular velocities of the left and right sprockets, respectively. ${\omega }_{mL},{\omega }_{mR}$ denote the angular velocities of the left and right driving motors, respectively. $T_{mL},T_{mR}$ represent the torques applied to the left and right driving motors, respectively. $T_{zL},T_{zR}$ represent the torques acting on the left and right sprockets, respectively. $k$ is the characteristic parameter of the coupling planetary gear set, and $i$ is the transmission ratio of the reduction planetary gear set and the final drive.

In this study, the simplified track–ground interaction modeling approach proposed in [17] is adopted. Each track is discretized into four equivalent contact points uniformly distributed along the contact length. The longitudinal position of them relative to the vehicle’s center of mass are shown as follows.

$$l_j\in \{L/2,L/6,-L/6,-L/2\}$$

(2)

where $L$ denotes the track–ground contact length.

At the j-th contact point, the local velocities of the left and right tracks are determined based on rigid-body kinematics. For the equivalent contact points on the left track, the longitudinal velocity $v_{xl,j}$ and the lateral velocity $v_{yl,j}$ are given as follows.

$$\begin{array}{l}{v}_{xl,j}=v_x-B\omega /2\\ v_{yl,j}=v_y+l_j\omega \end{array}$$

(3)

For the equivalent contact points on the right track, the longitudinal velocity $v_{xr,j}$ and the lateral velocity $v_{yr,j}$ are given as follows.

$$\begin{array}{l}{v}_{xr,j}=v_x+B\omega /2\\ v_{yr,j}=v_y+l_j\omega \end{array}$$

(4)

where $v_x,v_y$ denote the longitudinal and lateral velocities of the vehicle centroid in the body-fixed coordinate system, respectively, $\omega$ denotes the yaw rate, and $B$ denotes the track center distance.

Based on the sprocket angular velocities and the velocities of the equivalent contact points on both tracks, the longitudinal and lateral slip ratios of the left and right tracks at the j-th contact point can be defined as follows.

$$\begin{array}{l}{s}_{xl,j}={\displaystyle \frac{r_w{\omega }_{zL}-v_{xl,j}}{r_w{\omega }_{zL}}}\\ s_{yl,j}=-{\displaystyle \frac{v_{yl,j}}{r_w{\omega }_{zL}}}\\ s_{xr,j}={\displaystyle \frac{r_w{\omega }_{zR}-v_{xr,j}}{r_w{\omega }_{zR}}}\\ s_{yr,j}=-{\displaystyle \frac{v_{yr,j}}{r_w{\omega }_{zR}}}\end{array}$$

(5)

where $s_{xl,j},s_{yl,j}$ denote the longitudinal and lateral slip ratios of the equivalent contact point on the left track, respectively. $s_{xr,j},s_{yr,j}$ denote the corresponding longitudinal and lateral slip ratios on the right track, respectively, and $r_w$ denotes the sprocket radius.

Based on the longitudinal and lateral slip ratios at the equivalent contact points, the magnitude of the resultant slip ratio is further defined as follows.

$$\begin{array}{c}\begin{array}{l}{s}_{l,j}=\sqrt{s_{xl,j}^2+s_{yl,j}^2}\\ s_{r,j}=\sqrt{s_{xr,j}^2+s_{yr,j}^2}\end{array}\end{array}$$

(6)

Under the assumption that longitudinal and lateral load transfer are neglected, the total vehicle weight is assumed to be uniformly distributed among the eight contact points. Accordingly, the normal load at each contact point is given as follows.

$$F_z=mg/8$$

(7)

The tangential force between the track and the ground is described using an exponential saturation model. This formulation reflects that the force exhibits approximately linear behavior in the low-slip region and gradually saturates in the high-slip region. The specific expression is given as follows.

$$\begin{array}{l}{F}_{xl,j}={\displaystyle \frac{s_{xl,j}}{s_{l,j}}}{F}_z\mu \left(1-{\mathrm{e}}^{-C_C{s}_{l,j}}\right)\\ F_{yl,j}={\displaystyle \frac{s_{yl,j}}{s_{l,j}}}{F}_z\mu \left(1-{\mathrm{e}}^{-C_C{s}_{l,j}}\right)\\ F_{xr,j}={\displaystyle \frac{s_{xr,j}}{s_{r,j}}}{F}_z\mu \left(1-{\mathrm{e}}^{-C_C{s}_{r,j}}\right)\\ F_{yr,j}={\displaystyle \frac{s_{yr,j}}{s_{r,j}}}{F}_z\mu \left(1-{\mathrm{e}}^{-C_C{s}_{r,j}}\right)\end{array}$$

(8)

where $\mu$ denotes the road shear coefficient, and $C_c$ denotes the empirical slip gain parameter. By summing the forces at all contact points along the track direction, the resultant forces acting on the left and right tracks can be obtained.

$$\begin{array}{l}{F}_{xl}={\displaystyle \sum _{j=1}^4}{F}_{xl,j}\\ F_{xr}={\displaystyle \sum _{j=1}^4}{F}_{xr,j}\\ F_{yl}={\displaystyle \sum _{j=1}^4}{F}_{yl,j}\\ F_{yr}={\displaystyle \sum _{j=1}^4}{F}_{yr,j}\end{array}$$

(9)

Meanwhile, the equivalent yaw resisting moment can be obtained by summing the moments generated by the lateral forces with respect to the vehicle centroid.

$$M_f={\displaystyle \sum _{j=1}^4}\left(F_{yl,j}{l}_j+F_{yr,j}{l}_j\right)$$

(10)

The ETV planar dynamics equation can be expressed as

$$\begin{array}{l}{\dot{v}}_x=\omega v_y+(F_{xl}+F_{xr})/m-{\mu }_{f}g\\ {\dot{v}}_y=-\omega v_x+(F_{yl}+F_{yr})/m\\ \dot{\omega }=\left(B(F_{xr}-F_{xl})/2-M_f\right)/I_z\end{array}$$

(11)

where the vehicle is modeled as a rigid body. $m$ denotes the weight of the ETV. $I_z$ denotes the yaw moment of inertia. ${\mu }_f$ denotes the longitudinal resistance coefficient, and $g$ denotes the gravitational acceleration.

Based on the driving motor torque, the transmission relationships of the driveline system, and the track–ground interaction, the rotational dynamics of the driving motor rotor can be established as follows.

$$\begin{array}{l}{\dot{\omega }}_{mL}={\displaystyle \frac{1}{J_{L0}}}\left(T_{mL}-{\displaystyle \frac{1+k}{i(1+2k)}}{r}_w{F}_{xl}-{\displaystyle \frac{k}{i(1+2k)}}{r}_w{F}_{xr}\right)\\ {\dot{\omega }}_{mR}={\displaystyle \frac{1}{J_{R0}}}\left(T_{mR}-{\displaystyle \frac{k}{i(1+2k)}}{r}_w{F}_{xl}-{\displaystyle \frac{1+k}{i(1+2k)}}{r}_w{F}_{xr}\right)\end{array}$$

(12)

where $J_{L0},J_{R0}$ denote the inertias of the left and right driving motor rotors, respectively.

Based on the above dynamic equations, a planar dynamic model of the five-degree-of-freedom ETV is established. The schematic diagram is shown in Figure 2.

Figure 2. Dynamic Model of the ETV.

The longitudinal velocity, lateral velocity, yaw rate, and angular velocities of the left and right driving motors are selected as the state vector $x$. The torques of the left and right driving motors are selected as the control input vector $u$. The corresponding formulation is given as follows.

$$\begin{array}{l}\dot{x}=f\left(x,u\right)\\ x={[v_x,v_y,\omega ,{\omega }_{mL},{\omega }_{mR}]}^T\\ u={\left[T_{mL},T_{mR}\right]}^T\end{array}$$

(13)

3. Augmented DK-Based Predictive Steering Control for ETV

To enhance the steering control performance of the ETV, a model predictive steering control strategy based on the deep Koopman operator is proposed in this section, as shown in Figure 3.

Figure 3. Framework of the proposed model predictive steering control strategy.

The blue arrows represent the data flow during the vehicle model training process, corresponding to the offline data flow, whereas the other arrows indicate the data flow during the control process, i.e., the online data flow. During the training stage, in order to construct the training dataset, historical data generated from the ETV and the HOSM observer are first collected. Specifically, the dataset consists of the vehicle states, control inputs, and the observer-estimated states, which are combined to form the augmented state vector. Based on the obtained augmented-state dataset, the encoder, the high-dimensional linear state dynamics, and the decoder are trained. Subsequently, the trained ADK model is employed for the design of the steering controller. Considering the dynamic characteristics of the ETV, an ADK-MPC-based steering control strategy is developed in the lifted high-dimensional space. During the control process, the vehicle states and observer states are first transformed using the trained ADK model, and the tracking error system is updated according to the reference states. Then, the control inputs for the ETV are computed through receding-horizon optimization. The specific implementation details are presented in the following subsections.

3.1. HOSM Observer for Equivalent Motor Load Inertia

Due to the presence of the coupling planetary gear set in the transmission system, the computation of the transmission ratio differs between straight driving and steering conditions. During straight driving, the transmission ratio is identical to that of a dual-motor independently driven vehicle. Each driving motor actuates only one track, and the transmission ratio is constant. Under this condition, the equivalent motor load inertia can be obtained by dividing the equivalent load inertia by the square of the transmission ratio. In contrast, during steering, the transmission characteristics differ significantly from those of an independently driven configuration. Each track is driven by both motors, and the transmission ratio becomes a dual-input single-output function. As a result, the equivalent motor load inertia cannot be determined using conventional analytical methods.

To address this issue, a nonlinear observer based on the simplified rotational dynamics of the motor rotor is designed in this section. Since the moment of inertia appears in the denominator of the control-related terms, it is not convenient for direct observer design. Therefore, the time-varying component of the inertia is extracted and combined with other terms to form a lumped load disturbance, which is treated as the observable variable. The dynamics model of the motor rotor with load disturbance is thus expressed as follows.

$$\begin{array}{l}{\dot{\omega }}_{mL}=T_{mL}/J_{L0}-{\delta }_{mL}\\ {\dot{\omega }}_{mR}=T_{mR}/J_{R0}-{\delta }_{mR}\end{array}$$

(14)

where ${\delta }_{mL},{\delta }_{mR}$ represent the lumped load disturbances, which include the time-varying load inertia induced by the coupling planetary gear dynamics as well as the load resistance torques.

$$\begin{array}{l}{\delta }_{mL}={\displaystyle \frac{1}{J_{L0}}}\left(\Delta J_L{\dot{\omega }}_{mL}+T_{load,L}\right)\\ {\delta }_{mR}={\displaystyle \frac{1}{J_{R0}}}\left(\Delta J_R{\dot{\omega }}_{mR}+T_{load,R}\right)\\ T_{load,L}={\displaystyle \frac{1+k}{i(1+2k)}}{r}_w{F}_{xl}+{\displaystyle \frac{k}{i(1+2k)}}{r}_w{F}_{xr}\\ T_{load,R}={\displaystyle \frac{k}{i(1+2k)}}{r}_w{F}_{xl}+{\displaystyle \frac{1+k}{i(1+2k)}}{r}_w{F}_{xr}\end{array}$$

(15)

where $\Delta J_L$, $\Delta J_R$ represent the time-varying components of the left and right load inertia, respectively. That is, the complete load inertia on each side is obtained by summing $\Delta J_L$ with $J_{L0}$ for the left side and $\Delta J_R$ with $J_{R0}$ for the right side. $T_{load,L}$, $T_{load,R}$ denote the load torques of the left and right motors, respectively.

According to (15), the estimated value of the disturbance can be expressed as follows.

$$\begin{array}{l}{\widehat{\delta }}_{mL}\left(t\right)={\displaystyle \frac{1}{J_{L0}}}\left[\Delta {\widehat{J}}_L{\dot{\omega }}_{mL}\left(t\right)+T_{load,L}\left(t\right)\right]\\ {\widehat{\delta }}_{mR}\left(t\right)={\displaystyle \frac{1}{J_{R0}}}\left[\Delta {\widehat{J}}_R{\dot{\omega }}_{mR}\left(t\right)+T_{load,R}\left(t\right)\right]\end{array}$$

(16)

By introducing a time delay of $\tau$ to (16), the following expression can be obtained.

$$\begin{array}{l}{\widehat{\delta }}_{mL}\left(t+\tau \right)={\displaystyle \frac{1}{J_{L0}}}\left[\Delta {\widehat{J}}_L{\dot{\omega }}_{mL}\left(t+\tau \right)+T_{load,L}\left(t+\tau \right)\right]\\ {\widehat{\delta }}_{mR}\left(t+\tau \right)={\displaystyle \frac{1}{J_{R0}}}\left[\Delta {\widehat{J}}_R{\dot{\omega }}_{mR}\left(t+\tau \right)+T_{load,R}\left(t+\tau \right)\right]\end{array}$$

(17)

By subtracting (16) from (17), the following result is obtained.

$$\begin{array}{l}{\widehat{\delta }}_{mL}\left(t+\tau \right)-{\widehat{\delta }}_{mL}\left(t\right)={\displaystyle \frac{1}{J_{L0}}}\left[\begin{array}{l}\Delta {\widehat{J}}_L\left({\dot{\omega }}_{mL}\left(t+\tau \right)-{\dot{\omega }}_{mL}\left(t\right)\right)\\ +T_{load,L}\left(t+\tau \right)-T_{load,L}\left(t\right)\end{array}\right]\\ {\widehat{\delta }}_{mR}\left(t+\tau \right)-{\widehat{\delta }}_{mR}\left(t\right)={\displaystyle \frac{1}{J_{R0}}}\left[\begin{array}{l}\Delta {\widehat{J}}_R\left({\dot{\omega }}_{mR}\left(t+\tau \right)-{\dot{\omega }}_{mR}\left(t\right)\right)\\ +T_{load,R}\left(t+\tau \right)-T_{load,R}\left(t\right)\end{array}\right]\end{array}$$

(18)

Accordingly, the time-varying load inertia can be expressed as follows.

$$\begin{array}{l}\Delta {\widehat{J}}_L={\displaystyle \frac{J_{L0}\left[{\widehat{\delta }}_{mL}\left(t+\tau \right)-{\widehat{\delta }}_{mL}\left(t\right)\right]-\left[T_{load,L}\left(t+\tau \right)-T_{load,L}\left(t\right)\right]}{{\dot{\omega }}_{mL}\left(t+\tau \right)-{\dot{\omega }}_{mL}\left(t\right)}}\\ \Delta {\widehat{J}}_R={\displaystyle \frac{J_{R0}\left[{\widehat{\delta }}_{mR}\left(t+\tau \right)-{\widehat{\delta }}_{mR}\left(t\right)\right]-\left[T_{load,R}\left(t+\tau \right)-T_{load,R}\left(t\right)\right]}{{\dot{\omega }}_{mR}\left(t+\tau \right)-{\dot{\omega }}_{mR}\left(t\right)}}\end{array}$$

(19)

Assumption 1: 

For all time instants  $t\ge t_0$, where $t_0$ denotes the initial time, the total load disturbance ${\widehat{\delta }}_{mi}$ of the motor is nonzero. That is, for all $t\ge t_0$, $\left|{\widehat{\delta }}_{mi}\right|>{\widehat{\delta }}_{mi,\min }$, where ${\widehat{\delta }}_{mi,\min }$ is a small positive constant.

Assumption 1 is introduced to guarantee the convergence of the observer. The requirement that the load disturbance remains nonzero at all times is not an inherent limitation of the observer itself, but rather a sufficient condition to ensure accurate disturbance estimation. Under this assumption, the proposed nonlinear observer enables the estimation error of the load disturbance to asymptotically converge to zero, thereby achieving accurate reconstruction of the load disturbance. In addition, the observer simultaneously provides an estimate of the motor speed. This variable contributes to driving the load disturbance estimation error to zero and further enhances the robustness of the overall estimation framework.

Under Assumption 1, and assuming that the angular velocities and angular accelerations of both motors are measurable, an observer is designed to estimate the load disturbances. The observer structure is formulated as follows.

$$\begin{array}{l}{\dot{\widehat{\omega }}}_{mL}=T_{mL}/J_{L0}-{\widehat{\delta }}_{mL}+{\tilde{\omega }}_{mL}+k_1{\left|{\tilde{\omega }}_{mL}\right|}^{1/2}·sign\left({\tilde{\omega }}_{mL}\right)\\ {\dot{\widehat{\omega }}}_{mR}=T_{mR}/J_{R0}-{\widehat{\delta }}_{mR}+{\tilde{\omega }}_{mR}+k_2{\left|{\tilde{\omega }}_{mR}\right|}^{1/2}·sign\left({\tilde{\omega }}_{mR}\right)\\ {\dot{\widehat{\delta }}}_{mL}=-k_3·sign\left({\tilde{\omega }}_{mL}\right)\\ {\dot{\widehat{\delta }}}_{mR}=-k_4·sign\left({\tilde{\omega }}_{mR}\right)\\ {\tilde{\omega }}_{mL}={\omega }_{mL}-{\widehat{\omega }}_{mL},{\tilde{\omega }}_{mR}={\omega }_{mR}-{\widehat{\omega }}_{mR}\end{array}$$

(20)

The convergence of the proposed observer is established as follows.

First, the complete observation error vector is defined as follows.

$$\begin{array}{l}{\tilde{\omega }}_{mL}={\omega }_{mL}-{\widehat{\omega }}_{mL}\\ {\tilde{\omega }}_{mR}={\omega }_{mR}-{\widehat{\omega }}_{mR}\\ {\tilde{\delta }}_{mL}={\delta }_{mL}-{\widehat{\delta }}_{mL}\\ {\tilde{\delta }}_{mR}={\delta }_{mR}-{\widehat{\delta }}_{mR}\end{array}$$

(21)

To ensure that the derivation can be closed, the following assumption is introduced.

Assumption 2: 

In the stability proof, the yaw rate  $\omega$ is assumed to be piecewise constant or slowly varying and bounded over the observer convergence timescale. There exist positive constants ${\omega }_{\min }$, ${\omega }_{\max }$ such that the following conditions are satisfied.

$$0<{\omega }_{\min }\le \left|\omega \left(t\right)\right|\le {\omega }_{\max }$$

(22)

Moreover, the left and right load disturbances are assumed to remain constant during the short convergence interval, namely,

$$\begin{array}{l}{\dot{\delta }}_{mL}=0\\ {\dot{\delta }}_{mR}=0\end{array}$$

(23)

It should be noted that Assumption 2 is introduced only for the finite-time convergence proof of the HOSM observer. In the actual steering process, the yaw rate varies continuously according to the nonlinear ETV dynamics and is not strictly constant. Although this assumption is not completely equivalent to the practical operating condition, it provides a reasonable local approximation within the short observer convergence timescale. Since the proposed controller employs a relatively small control update step, the yaw-rate variation within each short control interval is limited compared with the fast observer dynamics. Therefore, treating the yaw rate as piecewise constant or slowly varying is reasonable for the theoretical stability analysis and provides theoretical guidance for practical observer implementation.

By substituting the observation errors of the left motor speed and the load disturbance into the observer equations, the following expressions can be obtained.

$$\begin{array}{l}{\dot{\tilde{\omega }}}_{mL}=\left(T_{mL}/J_{L0}-{\delta }_{mL}\right)-\left(T_{mL}/J_{L0}-{\widehat{\delta }}_{mL}+{\tilde{\omega }}_{mL}+k_1{\left|{\tilde{\omega }}_{mL}\right|}^{1/2}sign({\tilde{\omega }}_{mL})\right)\\ {\dot{\tilde{\delta }}}_{mL}={\dot{\delta }}_{mL}-{\dot{\widehat{\delta }}}_{mL}\end{array}$$

(24)

By expanding the above expressions and eliminating the common terms, the error subsystem of the left motor can be obtained as follows.

$$\begin{array}{l}{\dot{\tilde{\omega }}}_{mL}=-{\tilde{\delta }}_{mL}-{\tilde{\omega }}_{mL}-k_1{\left|{\tilde{\omega }}_{mL}\right|}^{1/2}sign\left({\tilde{\omega }}_{mL}\right)\\ {\dot{\tilde{\delta }}}_{mL}=k_{3}sign\left({\tilde{\omega }}_{mL}\right)\end{array}$$

(25)

Similarly, the error subsystem of the right motor can be derived as follows.

$$\begin{array}{l}{\dot{\tilde{\omega }}}_{mR}=-{\tilde{\delta }}_{mR}-{\tilde{\omega }}_{mR}-k_2{\left|{\tilde{\omega }}_{mR}\right|}^{1/2}sign\left({\tilde{\omega }}_{mR}\right)\\ {\dot{\tilde{\delta }}}_{mR}=k_{4}sign\left({\tilde{\omega }}_{mR}\right)\end{array}$$

(26)

It can be observed from the above error equations that both error subsystems possess the same HOSM structure. In order to achieve a unified proof, the following standard form is considered.

$$\begin{array}{l}{\dot{z}}_1=z_2-z_1-{\lambda }_1{\left|z_1\right|}^{1/2}sign\left(z_1\right)\\ {\dot{z}}_2=-{\lambda }_{2}sign\left(z_1\right)\end{array}$$

(27)

where ${\lambda }_1>0$ and ${\lambda }_2>0$ denote the state variables of the unified error system. The correspondence between the parameters in this standard form and those of the original observer is established as follows. For the left motor subsystem, the parameters are selected as follows.

$$\begin{array}{l}{z}_1={\tilde{\omega }}_{mL}\\ z_2=-{\tilde{\delta }}_{mL}\\ {\lambda }_1=k_1\\ {\lambda }_2=k_3\end{array}$$

(28)

For the right motor subsystem, the parameters are selected as follows.

$$\begin{array}{l}{z}_1={\tilde{\omega }}_{mR}\\ z_2=-{\tilde{\delta }}_{mR}\\ {\lambda }_1=k_2\\ {\lambda }_2=k_4\end{array}$$

(29)

A Lyapunov function is constructed below for the unified error system. Since the system simultaneously contains the terms $z_1$, ${\left|z_1\right|}^{1/2}sign\left(z_1\right)$ and $z_2$, the direct selection of a quadratic form such as $V=z_1^2/2+z_2^2/2$ is not sufficient to effectively handle the non-smooth term induced by ${\left|z_1\right|}^{1/2}sign\left(z_1\right)$. Therefore, it is necessary to introduce transformed variables that are commonly employed in the analysis of HOMS observers.

$$\begin{array}{l}{\xi }_1={\left|z_1\right|}^{1/2}sign\left(z_1\right)\\ {\xi }_2=z_2\end{array}$$

(30)

Therefore

$$\begin{array}{l}{z}_1={\xi }_1\left|{\xi }_1\right|\\ \left|z_1\right|={\xi }_1^2\end{array}$$

(31)

Based on this transformation, the following quadratic Lyapunov function candidate is constructed.

$$\begin{array}{l}V={\zeta }^{\mathrm{T}}P\zeta \\ \zeta ={\left[{\xi }_1,{\xi }_2\right]}^T\end{array}$$

(32)

where

$$P={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}4{\lambda }_2+{\lambda }_1^2& -{\lambda }_1\\ -{\lambda }_1& 2\end{array}\right]$$

(33)

Since ${\lambda }_1>0$ and ${\lambda }_2>0$, the leading principal minors of the matrix $P$ satisfy the following conditions.

$$P_{11}={\displaystyle \frac{1}{2}}\left(4{\lambda }_2+{\lambda }_1^2\right)>0$$

(34)

Moreover, the second-order leading principal minor of matrix $P$, namely the determinant of the matrix itself, satisfies the following condition.

$$\begin{array}{ll}\det \left(P\right)& ={\displaystyle \frac{1}{4}}\left(2\left(4{\lambda }_2+{\lambda }_1^2\right)-{\lambda }_1^2\right)\\ & ={\displaystyle \frac{1}{4}}\left(8{\lambda }_2+{\lambda }_1^2\right)>0\end{array}$$

(35)

Therefore, $P\succ 0$, and $V$ is positive definite. Consequently, there exist positive constants $c_1$ and $c_2$ such that the following inequality holds.

$$c_1{‖\zeta ‖}^2\le V\le c_2{‖\zeta ‖}^2$$

(36)

$\dot{V}$ is then evaluated. In the region where $z_1\ne 0$, it follows from (30) that

$${\dot{\xi }}_1={\displaystyle \frac{1}{2{\left|z_1\right|}^{1/2}}}{\dot{z}}_1$$

(37)

By substituting (37) into (27), the following expression is obtained.

$${\xi }_1={\displaystyle \frac{1}{2{\left|z_1\right|}^{1/2}}}\left(z_2-z_1-{\lambda }_1{\left|z_1\right|}^{1/2}sign\left(z_1\right)\right)$$

(38)

By substituting (31) into (38), the following expression is obtained.

$${\dot{\xi }}_1={\displaystyle \frac{z_2}{2{\left|z_1\right|}^{1/2}}}-{\displaystyle \frac{1}{2}}{\xi }_1-{\displaystyle \frac{{\lambda }_1}{2}}sign\left(z_1\right)$$

(39)

Since

$${\dot{\xi }}_2={\dot{z}}_2=-{\lambda }_{2}sign\left(z_1\right)$$

(40)

then

$$\dot{\zeta }=\left[\begin{array}{c}{\displaystyle \frac{z_2}{2{\left|z_1\right|}^{1/2}}}-{\displaystyle \frac{1}{2}}{\xi }_1-{\displaystyle \frac{{\lambda }_1}{2}}sign\left(z_1\right)\\ -{\lambda }_{2}sign\left(z_1\right)\end{array}\right]$$

(41)

and

$$\dot{V}={\dot{\zeta }}^{\mathrm{T}}P\zeta +{\zeta }^{\mathrm{T}}P\dot{\zeta }=2{\zeta }^{\mathrm{T}}P\dot{\zeta }$$

(42)

By substituting (33), (32) and (41) into (42), the following expression is obtained.

$$\begin{array}{ll}\dot{V}& =\left(4{\lambda }_2+{\lambda }_1^2\right){\xi }_1\left({\displaystyle \frac{z_2}{2{\left|z_1\right|}^{1/2}}}-{\displaystyle \frac{1}{2}}{\xi }_1-{\displaystyle \frac{{\lambda }_1}{2}}sign\left(z_1\right)\right)\\ & -{\lambda }_1{\xi }_2\left({\displaystyle \frac{z_2}{2{\left|z_1\right|}^{1/2}}}-{\displaystyle \frac{1}{2}}{\xi }_1-{\displaystyle \frac{{\lambda }_1}{2}}sign\left(z_1\right)\right)\\ & -{\lambda }_1{\xi }_1\left(-{\lambda }_{2}sign\left(z_1\right)\right)+2{\xi }_2\left(-{\lambda }_{2}sign\left(z_1\right)\right)\end{array}$$

(43)

By substituting

$$\begin{array}{l}{\xi }_1={\left|z_1\right|}^{1/2}sign\left(z_1\right)\\ {\xi }_{1}sign\left(z_1\right)={\left|z_1\right|}^{1/2}\\ {\xi }_1^2=\left|z_1\right|\\ {\xi }_2=z_2\end{array}$$

(44)

Into (43), then

$$\dot{V}=-{\displaystyle \frac{1}{2\left|{\xi }_1\right|}}{\zeta }^{\mathrm{T}}Q\zeta -\left(2{\lambda }_2+{\displaystyle \frac{{\lambda }_1^2}{2}}\right)\left|z_1\right|$$

(45)

with

$$Q={\lambda }_1\left[\begin{array}{cc}2{\lambda }_2+{\lambda }_1^2& -{\lambda }_1\\ -{\lambda }_1& 1\end{array}\right]$$

(46)

Since ${\lambda }_1>0$ and ${\lambda }_2>0$, the leading principal minors of the matrix $Q$ satisfy the following conditions.

$$Q_{11}={\lambda }_1\left(2{\lambda }_2+{\lambda }_1^2\right)>0$$

(47)

Moreover, the second-order leading principal minor of matrix $Q$, namely the determinant of the matrix itself, satisfies the following condition.

$$\det \left(Q\right)={\lambda }_1^2\left(\left(2{\lambda }_2+{\lambda }_1^2\right)·1-{\lambda }_1^2\right)=2{\lambda }_1^2{\lambda }_2>0$$

(48)

Therefore, $Q\succ 0$, and there exists a positive constant ${\lambda }_{\min }\left(Q\right)>0$ such that the following inequality holds.

$${\zeta }^{\mathrm{T}}Q\zeta \ge {\lambda }_{\min }\left(Q\right){‖\zeta ‖}^2$$

(49)

Since $\left|{\xi }_1\right|\le ‖\zeta ‖$, then

$$\dot{V}\le -{\displaystyle \frac{{\lambda }_{\min }\left(Q\right)}{2\left|{\xi }_1\right|}}{‖\zeta ‖}^2\le -{\displaystyle \frac{{\lambda }_{\min }\left(Q\right)}{2‖\zeta ‖}}{‖\zeta ‖}^2=-{\displaystyle \frac{{\lambda }_{\min }\left(Q\right)}{2}}‖\zeta ‖$$

(50)

By substituting (36) into (50), the following expression is obtained.

$$\begin{array}{l}\dot{V}\le -{\displaystyle \frac{{\lambda }_{\min }\left(Q\right)}{2}}\sqrt{{\displaystyle \frac{V}{c_2}}}=-c{V}^{1/2}\\ \hspace{0.33em}c={\displaystyle \frac{{\lambda }_{\min }\left(Q\right)}{2\sqrt{c_2}}}>0\end{array}$$

(51)

By integrating both sides, the following result can be obtained.

$${\displaystyle \frac{dV}{dt}}\le -c{V}^{1/2}\Rightarrow {\displaystyle \frac{d\left(V^{1/2}\right)}{dt}}\le -{\displaystyle \frac{c}{2}}$$

(52)

By integrating from the initial time 0 to $t$, the following expression is obtained.

$$V^{1/2}\left(t\right)\le V^{1/2}\left(0\right)-{\displaystyle \frac{c}{2}}t$$

(53)

with $t\ge T^{\star }=2V^{1/2}\left(0\right)/c$, then $V\left(t\right)=0$. Since $V$ is positive definite, it follows that

$$\zeta (t)=0,\hspace{1em}\forall t\ge T^{\star }$$

(54)

Therefore, the unified error system converges to the origin in finite time. Consequently, the observation errors of the motor speeds and the load disturbances on both sides asymptotically converge to zero within a finite time interval.

3.2. Augmented Deep Koopman Model for ETV

The deep Koopman operator is capable of extracting features from data and mapping nonlinear dynamics into a high-dimensional space, where the system can be represented as a linear system. The standard Koopman operator is originally developed for time-invariant systems. However, under high-speed steering conditions, the equivalent motor load inertia of the ETV varies significantly, causing the system to exhibit pronounced time-varying characteristics. If the standard Koopman operator is directly applied, the time-varying load inertia appears as unmodeled dynamics in the lifted space, resulting in accumulated prediction errors and degraded MPC performance. To address this issue, the lumped load disturbances estimated by the HOSM observer are introduced as augmented state variables, leading to the formulation of the augmented system equations.

$$\begin{array}{l}{\dot{v}}_x=\omega v_y+{\displaystyle \frac{1}{m}}(F_{xl}+F_{xr})-{\mu }_{f}g\\ {\dot{v}}_y=-\omega v_x+{\displaystyle \frac{1}{m}}(F_{yl}+F_{yr})\\ \dot{\omega }={\displaystyle \frac{1}{I_z}}\left({\displaystyle \frac{B}{2}}(F_{xr}-F_{xl})-M_f\right)\\ {\dot{\omega }}_{mL}={\displaystyle \frac{1}{J_{L0}}}{T}_{mL}-{\widehat{\delta }}_{mL}\\ {\dot{\omega }}_{mR}={\displaystyle \frac{1}{J_{R0}}}{T}_{mR}-{\widehat{\delta }}_{mR}\\ {\dot{\widehat{\omega }}}_{mL}={\displaystyle \frac{1}{J_{L0}}}{T}_{mL}-{\widehat{\delta }}_{mL}+{\omega }_{mL}-{\widehat{\omega }}_{mL}+k_1{\left|{\omega }_{mL}-{\widehat{\omega }}_{mL}\right|}^{1/2}·sign\left({\omega }_{mL}-{\widehat{\omega }}_{mL}\right)\\ {\dot{\widehat{\omega }}}_{mR}={\displaystyle \frac{1}{J_{R0}}}{T}_{mR}-{\widehat{\delta }}_{mR}+{\omega }_{mR}-{\widehat{\omega }}_{mR}+k_2{\left|{\omega }_{mR}-{\widehat{\omega }}_{mR}\right|}^{1/2}·sign\left({\omega }_{mR}-{\widehat{\omega }}_{mR}\right)\\ {\dot{\widehat{\delta }}}_{mL}=-k_3·sign\left({\omega }_{mL}-{\widehat{\omega }}_{mL}\right)\\ {\dot{\widehat{\delta }}}_{mR}=-k_4·sign\left({\omega }_{mR}-{\widehat{\omega }}_{mR}\right)\end{array}$$

(55)

In this augmented formulation, the time-varying load characteristics associated with the equivalent motor load inertia are incorporated into the augmented state through the HOSM-estimated lumped load disturbances. As a result, the dominant time-varying effects are represented by the evolution of the augmented states rather than by explicit time-varying coefficients. This facilitates the learning process of the deep Koopman operator. To avoid the adverse effects of numerical discontinuities introduced by the sign function on network training, a smooth approximation is employed to replace the sign function. The states, inputs, and outputs of the augmented model are defined as follows.

$$\begin{array}{l}\dot{z}=\Psi \left(z,u\right)\\ y=G\left(z\right)\\ z={\left[v_x,v_y,\omega ,{\omega }_{mL},{\omega }_{mR},{\widehat{\omega }}_{mL},{\widehat{\omega }}_{mR},{\widehat{\delta }}_{mL},{\widehat{\delta }}_{mR}\right]}^T\\ u={\left[T_{mL},T_{mR}\right]}^T\\ y={\left[v_x,v_y,\omega ,{\omega }_{mL},{\omega }_{mR}\right]}^T\end{array}$$

(56)

The training dataset for the ADK model is constructed using the same state variables as those defined in the augmented model. Each sample consists of the vehicle states, control inputs, and the observer-estimated variables generated by the HOSM observer. These estimated variables are obtained from measurable motor signals and are incorporated into the augmented state to characterize the time-varying load dynamics. During online control, the same observer provides the required estimates in real time, ensuring consistency between the offline training dataset and the online control implementation.

The dataset used for training the ADK model is generated from 560 simulated ETV trajectories. Each trajectory contains 220 sampling points with a sampling interval of 0.01 s. Among them, 50% correspond to straight-driving conditions and 50% correspond to steering conditions. The straight-driving dataset consists of acceleration and steady-speed driving samples, while the steering dataset consists of steering establishment, steering holding, steering recovery, and steering-direction switching samples. After trajectory generation, a sliding-window method with a sequence length of 15 is employed to construct the training samples. The resulting samples are randomly divided into training, validation, and test sets with proportions of 80%, 10%, and 10%, respectively. To obtain the training data, the continuous system is discretized using the fourth-order Runge–Kutta method, resulting in the following discrete augmented model.

$$z_{k+1}=\Xi \left(z_k,u_k\right)$$

(57)

To enhance numerical stability, the original states and inputs are normalized through a standardization transformation.

$$\begin{array}{l}{z}_n={\displaystyle \frac{z-{\mu }_z}{{\sigma }_z}}\\ u_n={\displaystyle \frac{u-{\mu }_u}{{\sigma }_u}}\end{array}$$

(58)

where ${\mu }_z$, ${\sigma }_z$, ${\mu }_u$ and ${\sigma }_u$ are derived from the statistics of the training data.

To construct the Koopman linear representation, an encoder $\varphi \left(·\right)$ is introduced to implement a nonlinear lifting mapping, which maps the original state into a high-dimensional space.

$${\delta }_k=\varphi \left(z_{n,k}\right)$$

(59)

In this space, the system is assumed to evolve according to a linear dynamic law.

$${\delta }_{k+1}=A{\delta }_k+B{u}_k+c$$

(60)

where ${\delta }_k\in {ℝ}^{n_{\sigma }}$, matrix $A\in {ℝ}^{n_{\sigma }\times n_{\sigma }}$, $B\in {ℝ}^{n_{\sigma }\times n_{\sigma }}$ and $c\in {ℝ}^{n_{\sigma }}$ represent the parameters to be learned. Then, a decoder $\psi \left(·\right)$ is employed to map the high-dimensional states back to the original state space.

$${\widehat{z}}_k=\psi \left({\delta }_k\right)$$

(61)

Both the encoder and the decoder are implemented using multilayer feedforward neural networks. The encoder has an input dimension of 9 and an output dimension of 24. Its hidden structure consists of three fully connected layers with 128, 128, and 64 neurons, respectively. The hyperbolic tangent function is adopted as the activation function in each layer. The encoder can thus be expressed as follows.

$$\begin{array}{l}{h}^{\left(1\right)}=\tanh \left(W_1{z}_n+b_1\right)\\ h^{\left(2\right)}=\tanh \left(W_2{h}^{\left(1\right)}+b_2\right)\\ h^{\left(3\right)}=\tanh \left(W_3{h}^{\left(2\right)}+b_3\right)\\ \delta =W_4{h}^{\left(3\right)}+b_4\end{array}$$

(62)

The decoder is designed symmetrically, with hidden layers of 64, 128, and 128 neurons, and its output layer restores the original state dimension. This architecture ensures sufficient capability for nonlinear embedding and inverse mapping, while avoiding the training instability that may be induced by excessively deep networks.

By jointly considering reconstruction accuracy, high-dimensional space consistency loss, one-step prediction, multi-step prediction, output accuracy, and stability constraints, the following multi-objective loss function is constructed.

$$\begin{array}{ll}L& =w_{rec}{L}_{rec}+w_{latent}{L}_{latent}+w_{state}{L}_{state}\\ & +w_{roll}{L}_{roll}+w_{out}{L}_{out}+w_{stable}{L}_{stable}\end{array}$$

(63)

The definitions of the individual terms are given as follows. The reconstruction loss is introduced to ensure the consistency of the encoding–decoding process, and it is formulated as follows.

$$L_{rec}={‖\psi \left(\varphi \left(z\right)\right)‖}^2$$

(64)

The high-dimensional space consistency loss is used to constrain the validity of linear dynamics, and it is formulated as follows.

$$L_{latent}={‖A{\delta }_k+B{u}_k+c-{\delta }_{k+1}‖}^2$$

(65)

The one-step prediction loss is used to improve the accuracy of short-term predictions, and it is formulated as follows.

$$L_{state}={‖{\widehat{z}}_{k+1}-z_{k+1}‖}^2$$

(66)

The multi-step prediction loss is used to improve the long-term prediction accuracy and is crucial for the application of rolling optimization, and it is formulated as follows.

$$L_{roll}={\displaystyle \frac{1}{H}}{\displaystyle \sum _{i=1}^H{‖{\widehat{z}}_{k+i}-z_{k+i}‖}^2}$$

(67)

For the output variable $y=C_{y}z$, define the output accuracy loss, and it is formulated as follows.

$$L_{out}={‖C_y\widehat{z}-C_{y}z‖}^2$$

(68)

To suppress the divergence of high-dimensional systems, the spectral radius constraint is used as the stability constraint loss, and it is formulated as follows.

$$L_{stable}=\max {\left(\rho \left(A\right)-{\rho }_{\mathrm{target}},0\right)}^2$$

(69)

where $\rho \left(A\right)$ represents the spectral radius of matrix $A$. The approximate calculation is performed through power iteration, and the target value is set to ${\rho }_{\mathrm{target}}=0.98$.

All parameters are optimized using the Adam algorithm based on first-order gradients. Let the parameter set be denoted by $\theta$, and its update rule is expressed as follows.

$$\begin{array}{l}{m}_t={\beta }_1{m}_{t-1}+(1-{\beta }_1){\nabla }_{\theta }\mathcal{L},\hspace{1em}\\ v_t={\beta }_2{v}_{t-1}+(1-{\beta }_2){({\nabla }_{\theta }\mathcal{L})}^2\\ {\theta }_{t+1}={\theta }_t-\alpha {\displaystyle \frac{m_t}{\sqrt{v_t}+ϵ}}\end{array}$$

(70)

where $\alpha$ denotes the learning rate and ${\beta }_1$, ${\beta }_2$ denotes the momentum coefficient. To prevent gradient explosion, L2-norm clipping is applied to the parameters of the linear system. During training, a mini-batch stochastic gradient descent strategy is adopted, and an early stopping mechanism based on the multi-step prediction error on a validation set is introduced. The training process is terminated when the performance no longer improves over several consecutive epochs, thereby avoiding overfitting.

3.3. Augmented Deep Koopman Operator-Based Model Predictive Control

Based on the training results of the ADK model, an MPC controller is constructed. First, the states and inputs are standardized, and then lifted to a higher-dimensional space through the encoder $\varphi \left(·\right)$. The resulting predictive model is expressed as follows.

$${\delta }_{k+1}=A{\delta }_k+B{u}_k+c$$

(71)

To facilitate the formulation of the optimization problem as a quadratic program, the nonlinear decoder is locally linearized, yielding its first-order approximation form.

$$z_{n,k+i}=C_{\delta ,i}{\delta }_{k+i}+d_i$$

(72)

where $C_{\sigma ,i}$ denotes the local Jacobian approximation of the decoder with respect to the lifted state, and $d_i$ represents the affine term. The Jacobian is obtained using a central difference numerical approximation.

$$C_{\delta ,i}\approx {\displaystyle \frac{\psi \left({\delta }_{k+i}+\epsilon \right)-\psi \left({\delta }_{k+i}-\epsilon \right)}{2\epsilon }}$$

(73)

Restore the standardized state to its original state and calculate the output quantity prediction

$$\begin{array}{l}{y}_{k+i}\approx C_{y\delta ,i}{\delta }_{k+i}+d_{y,i}\\ C_{y\delta ,i}=C_y{S}_z{C}_{\delta ,i},d_{y,i}=C_y\left({\mu }_z+S_z{d}_i\right)\\ S_z=diag\left({\sigma }_z\right)\end{array}$$

(74)

The prediction horizon $N_p$ and the control horizon $N_c$ are selected, and the input sequence is taken as the optimization variable.

$$U={\left[u_k^T,u_{k+1}^T,\cdots ,u_{k+N_c-1}^T\right]}^T$$

(75)

The optimization problem is formulated as follows. The first term represents the output tracking cost, the second term denotes the input penalty cost, the third term corresponds to the terminal cost, and the fourth term represents the torque difference cost between the left and right motors, which is introduced as a safety constraint. The formulation is given as follows.

$$\begin{array}{l}\underset{u}{\min }J={\displaystyle {\displaystyle \sum _{i=1}^{N_p}}{‖y_{k+i}-y_{k+i}^{ref}‖}_{Q_y}^2}+{\displaystyle {\displaystyle \sum _{i=1}^{N_c-1}}{‖u_{k+i}‖}_{R_u}^2}+{‖y_{k+N_p}-y_{k+N_p}^{ref}‖}_{Q_f}^2+{‖M_{diff}U‖}_{{\lambda }_{diff}}^2\\ s.t.\hspace{1em}{A}_{ineq}U\le b_{ineq}\end{array}$$

(76)

where $M_{diff}=I_{N_c}\otimes E_{\Delta },E_{\Delta }={\left[1,-1\right]}^T$ and the inequality constraints include saturation constraints and slope constraints.

This optimization problem can be reformulated into a standard quadratic programming form.

$$\begin{array}{l}\underset{U}{\min }{\displaystyle \frac{1}{2}}{U}^T{H}_{sym}U+f^{T}U\\ H_{sym}={\displaystyle \frac{1}{2}}\left(H+H^T\right)\\ H=2\left(G_y^T{Q}_{\mathrm{bar}}{G}_y+R_{\mathrm{bar}}+G_{y,N_p}^T{Q}_f{G}_{y,N_p}+{\lambda }_{\mathrm{diff}}{M}_{\mathrm{diff}}^T{M}_{\mathrm{diff}}\right)\\ f_0=2G_y^{\top }{Q}_{\mathrm{bar}}{e}_{\mathrm{aff}}+2G_{y,N_p}^{\top }{Q}_f{e}_{f,Y}\\ Q_{\mathrm{bar}}=I_{N_p}\otimes Q_y,\hspace{1em}{R}_{\mathrm{bar}}=I_{N_c}\otimes R_u,e_{\mathrm{Y}}=Y-Y_{\mathrm{ref}}\\ e_{f,Y}=y_{{\mathrm{N}}_p}-y_{{\mathrm{k}+\mathrm{N}}_p,\mathrm{ref}}\end{array}$$

(77)

Since $Q_y\succ 0,Q_f\succ 0,R_u\succ 0,{\lambda }_{\mathrm{diff}}\ge 0$, then $H_{sym}\succ 0$. Therefore, the resulting optimization problem is strictly convex.

4. Results and Discussion

In this section, the proposed observer is first validated through simulation experiments, which reveal the time-varying characteristics of the equivalent motor load inertia. Subsequently, the proposed steering control method is evaluated through simulation studies. The detailed results are presented as follows.

4.1. Estimation Performance of the HOSM Observer

In this section, the proposed HOSM observer is validated using the state of the ETV under high-speed steering conditions. The motor angular velocities and load disturbances are estimated through nonlinear sliding-mode laws, where gain terms containing square-root components and sign functions are employed to capture system nonlinearities and time-varying parameters. The high-speed steering scenario is defined as follows: the vehicle is first accelerated to 8 m/s, and then sequential steering maneuvers with yaw rate references of −0.1 rad/s, −0.2 rad/s, −0.3 rad/s, and −0.15 rad/s are applied. As illustrated in Figure 4, hollow squares denote the initial states, while red pentagrams indicate the terminal states.

Figure 4. Observer verification condition.

As shown in Figure 5, Figure 6 and Figure 7, the estimated angular velocities of both the left and right motors converge rapidly. After the initial step variations, the estimated speeds quickly track the actual values, indicating that the proposed sliding-mode observer achieves fast response under highly dynamic inputs. This also verifies the effectiveness of the combined square-root gain terms and sign functions in handling nonlinear dynamics. The disturbance estimation results demonstrate that the load disturbances of both motors are accurately captured, and the estimated values closely follow the actual disturbance variations. Moreover, the observer outputs are able to quickly adjust to new steady-state levels during multiple steering phases, which indicates strong robustness in response to load variations induced by steering maneuvers. The convergence rate of the disturbance estimates is consistent with that of the angular velocity estimates, further confirming the fast convergence capability of the proposed sliding-mode observer.

Figure 5. The actual value and the observed value of the left motor speed.

Figure 6. The actual value and the observed value of the right motor speed.

Figure 7. Estimated load disturbances.

To quantitatively evaluate the effectiveness of the HOSM-observed load disturbance, a comparative prediction-based validation method is introduced. Since no widely accepted analytical method currently exists for directly calculating the equivalent motor load inertia in dual-motor coupled transmission systems, two prediction models are considered for comparison. The first is the nominal model, which directly uses the original vehicle dynamics, while the second is the compensated model, which incorporates the HOSM-observed ${\delta }_{mL}$ and ${\delta }_{mR}$ as compensation terms into the vehicle dynamics. Both models employ the same states and control inputs of the reference system to predict the motor rotational speeds. Here, the reference system denotes the actual control object to which the observer and controller are applied. The predicted motor speeds are then compared with the actual motor speeds of the reference system. If the compensated model achieves smaller prediction errors than the nominal model, the effectiveness of the HOSM-observed load disturbance can be verified.

Figure 8 and Figure 9 present the speed validation results of the proposed HOSM-based observed load disturbance compensation method. In Figure 8, the motor speeds predicted by the nominal model and the compensated model are compared with those of the reference system, while Figure 9 shows the corresponding prediction errors. Here, the reference system represents the actual control object in which both the observer and the controller are implemented. Figure 9 further illustrates the prediction errors of the two models. Since the nominal model neglects the time-varying characteristics of the motor load, relatively large prediction errors are observed in both steady-state and transient conditions. In contrast, after introducing the HOSM-based observed load disturbance compensation, the prediction errors are significantly reduced and remain close to zero over most operating intervals. The root-mean-square prediction error of the left motor is reduced from 0.5612 rad/s to 0.0831 rad/s, while that of the right motor decreases from 0.5889 rad/s to 0.0867 rad/s. These results demonstrate that the load disturbances observed by the HOSM observer can effectively capture the time-varying characteristics of motor load dynamics and enhance the prediction accuracy of the vehicle model.

Figure 8. Comparison of motor speed prediction: (a) left motor speed; (b) right motor speed.

Figure 9. Comparison of motor speed prediction errors: (a) left motor speed prediction error; (b) right motor speed prediction error.

Based on (19) and Figure 7, the estimated values of the load inertia can be obtained, as illustrated in Figure 10.

Figure 10. The estimation value of load inertia. The five local enlarged views, labeled (a–e), correspond to the load inertia during straight driving and steering conditions with yaw rates of −0.1 rad/s, −0.2 rad/s, −0.3 rad/s, and −0.15 rad/s, respectively.

It can be observed that during the straight-driving phase, the equivalent motor load inertia remains constant. In contrast, during continuous steering phases, the equivalent motor load inertia exhibits pronounced time-varying characteristics. In particular, short-term variations occur around 100 s, 150 s, 200 s, and 250 s during steering, followed by a return to steady values. Moreover, these steady-state values differ from those observed during straight driving. This indicates that the equivalent motor load inertia is not constant, but varies dynamically with changing load and driving conditions during steering. The designed observer is capable of capturing these inertia variations and accurately estimating them, thereby providing reliable data support for subsequent motor load compensation and controller design.

The above simulation results demonstrate that the designed HOSM observer can achieve fast convergence in the estimation of motor angular velocities and load disturbances under steering conditions. Meanwhile, the time-varying characteristics of the equivalent motor load inertia are revealed. These findings provide experimental support for subsequent modeling and control strategy development.

4.2. Control Performance of the Proposed ADK-MPC

In this section, the control performance of conventional NMPC, DK-MPC, and ADK-MPC is compared. First, the convergence behaviors of the loss functions during the training processes of ADK and DK are presented, as shown in Figure 11 and Figure 12, respectively. From the training curves, it can be observed that the overall loss decreases rapidly during the initial iterations, indicating that the network quickly captures the dominant dynamic characteristics of the data at the early stage. As training proceeds, all sub-loss components exhibit a steady downward trend, and the oscillation amplitude gradually diminishes in the later stages. This suggests that the network progressively learns an effective linearized latent-space representation and the corresponding state–output relationships. Moreover, the multi-step prediction errors are effectively reduced during training, demonstrating that both models achieve satisfactory fitting performance with respect to the original nonlinear system.

Figure 11. The convergence process of DK loss.

Figure 12. The convergence process of ADK loss.

Figure 12 further presents the convergence behavior of the ADK loss function. By comparing Figure 11 and Figure 12, the influence of the augmented-state framework on the training convergence characteristics can be more clearly observed.

The key parameters of the three MPC approaches are set identically, with the prediction horizon and control horizon chosen as 16 and 15, respectively. The sampling time is set to 0.02 s, and all optimization problems are solved using an interior-point method. The weighting matrices are selected as $Q_y=\mathrm{diag}\left(0.08,1\right)$, $Q_f=10Q_y$, $R_u=\mathrm{diag}\left({10}^{-5},{10}^{-5}\right)$, and ${\lambda }_{\mathrm{diff}}={10}^{-7}$, respectively.

The test scenarios include two conditions: a single steering condition and a double-lane-change steering condition. In both experiments, the vehicle first accelerates in a straight line to the target speed, followed by the corresponding steering maneuvers and subsequent recovery phases.

The longitudinal velocity and yaw rate profiles of the single steering condition are shown in Figure 13 and Figure 14. All three methods are capable of rapidly reaching the desired speed and yaw rate during the initial acceleration phase. However, ADK-MPC exhibits significantly smaller tracking errors than NMPC and DK-MPC when maintaining the steady-state values of the target yaw rate and vehicle speed. During the steering-hold phase, ADK-MPC demonstrates the smallest fluctuations in both vehicle speed and yaw rate. This indicates that the introduction of the augmented-state Koopman model improves the prediction accuracy of vehicle dynamics, resulting in control inputs that better match the actual system response. Figure 13 and Figure 14 present the comparison of the absolute errors in longitudinal velocity and yaw rate for the three methods. ADK-MPC achieves the best performance in both metrics. In terms of longitudinal velocity, the error is reduced by approximately 20.8% and 31.5% compared with NMPC and DK-MPC, respectively. For yaw rate, the error is reduced by approximately 45.5% and 65% relative to NMPC and DK-MPC, respectively. These results verify the effectiveness of the augmented Koopman state in improving both prediction accuracy and control performance.

Figure 13. Reference and actual velocity of single steering condition: (a) comparison of velocity among the three strategies; (b) comparison of velocity error among the three strategies.

Figure 14. Reference and actual yaw rate of single steering condition: (a) comparison of yaw rate among the three strategies; (b) comparison of yaw rate error among the three strategies.

To further evaluate the steering control performance, the corresponding yaw-rate tracking results and yaw-rate errors of the three methods are presented in Figure 14.

Figure 15 and Figure 16 illustrate the comparisons of the angular velocities and torques of the left and right driving motors, further demonstrating the superiority of ADK-MPC. Under ADK-MPC, the motor speeds exhibit the smallest fluctuations around the reference values, and the torque outputs are smoother, with reduced instantaneous peaks. This indicates improved performance in actuator coordination and the handling of saturation constraints. In contrast, NMPC shows relatively larger fluctuations in motor speed and torque during the steering phase, which can be attributed to the characteristics of the conventional optimization solver and linearization approximations. Although DK-MPC achieves relatively high prediction accuracy, small oscillations still appear when handling multiple states.

Figure 15. Comparison of motor speed of single steering condition: (a) comparison of left motor speed among the three strategies; (b) comparison of right motor speed among the three strategies.

Figure 16. Comparison of motor torque of single steering condition: (a) comparison of left motor torque among the three strategies; (b) comparison of right motor torque among the three strategies.

Besides the motor speed responses, the corresponding motor torque outputs under the three control strategies are further illustrated in Figure 16 to analyze the actuator coordination performance.

To further validate the robustness and generalization capability of the proposed method, an additional double-lane-change steering scenario is considered. In this condition, the reference yaw rate changes continuously with alternating steering directions, resulting in stronger nonlinear coupling and more rapid variations in the equivalent motor load inertia compared with the previous single steering maneuver. Figure 17 and Figure 18 present the comparison of the absolute errors in longitudinal velocity and yaw rate for the three methods. ADK-MPC achieves the best performance in both metrics. In terms of longitudinal velocity, the error is reduced by approximately 3.6% and 25.5% compared with NMPC and DK-MPC, respectively. For yaw rate, the error is reduced by approximately 69.2% and 46.0% relative to NMPC and DK-MPC, respectively. These results verify the effectiveness of the augmented Koopman state in improving both prediction accuracy and control performance.

Figure 17. Reference and actual velocity of double-lane-change steering condition: (a) comparison of velocity among the three strategies; (b) comparison of velocity error among the three strategies.

Figure 18. Reference and actual yaw rate of double-lane-change steering condition: (a) comparison of yaw rate among the three strategies; (b) comparison of yaw rate error among the three strategies.

To further evaluate the steering control performance, the corresponding yaw-rate tracking results and yaw-rate errors of the three methods are presented in Figure 18.

Figure 19 and Figure 20 illustrate the comparisons of the speeds and torques of the left and right driving motors under the double-lane-change steering condition, further demonstrating the superiority of ADK-MPC. Under ADK-MPC, the motor speeds exhibit the smallest fluctuations around the reference values during repeated steering transitions, and the torque outputs are smoother, with reduced instantaneous peaks. This indicates improved performance in actuator coordination and the handling of saturation constraints under dynamically varying conditions. In contrast, NMPC shows relatively larger fluctuations in motor speed and torque during rapid steering reversals, which can be attributed to the characteristics of the conventional optimization solver and linearization approximations. Although DK-MPC achieves relatively high prediction accuracy, small oscillations still appear when handling multiple states, indicating limited robustness under complex dynamic conditions.

Figure 19. Comparison of motor speed of double-lane-change steering condition: (a) comparison of left motor speed among the three strategies; (b) comparison of right motor speed among the three strategies.

Figure 20. Comparison of motor torque of double-lane-change steering condition: (a) comparison of left motor torque among the three strategies; (b) comparison of right motor torque among the three strategies.

Besides the motor speed responses, the corresponding motor torque outputs under the three control strategies are further illustrated in Figure 20 to analyze the actuator coordination performance.

Figure 21 presents the comparison of computational time among the three methods under two different conditions. Figure 21a corresponds to the single steering condition, while Figure 21b corresponds to the double-lane-change steering condition. In the single steering condition, NMPC exhibits the largest solving time, with an average of approximately 15.14 ms, whereas ADK-MPC achieves an average of about 13.36 ms, corresponding to an improvement of 11.7%. DK-MPC requires slightly less time, approximately 12.65 ms. In the double-lane-change condition, the computational times of all methods slightly increase due to the higher dynamic complexity. NMPC shows an average execution time of 13.38 ms, DK-MPC 11.73 ms, and ADK-MPC 11.91 ms. Despite the additional augmented states in ADK-MPC, the computational time remains within an acceptable range for real-time implementation and is lower than NMPC. These results demonstrate that the proposed ADK-MPC maintains a balanced trade-off between computational efficiency and control performance even under more complex, dynamically varying steering conditions.

Figure 21. Comparison of execution time: (a) single steering condition; (b) double-lane-change steering condition.

Compared with NMPC and standard DK-MPC, ADK-MPC provides superior tracking accuracy and effectively reduces the steady-state errors in both yaw rate and vehicle speed. Meanwhile, the computational time is significantly reduced relative to NMPC. Although the introduction of augmented states leads to a slight increase in solving time compared with DK-MPC, this increase remains within an acceptable range for real-time implementation. Therefore, the proposed method achieves a balanced trade-off between control performance and computational efficiency, making it suitable for high-dynamic steering control in practical applications.

The proposed ADK-MPC approach offers lower online computational complexity compared with conventional NMPC-based steering methods [27] as the nonlinear optimization problem is reformulated as a quadratic program. Existing deep Koopman methods typically assume time-invariant models [28]. For systems with time-varying parameters, prior studies often require additional techniques to handle parameter variations [29]. In contrast, the present study explicitly incorporates the time-varying equivalent motor load inertia within the augmented-state framework, thereby improving prediction accuracy under high-speed and dynamically changing steering conditions. Furthermore, compared with kinematics-based steering controllers [30], the proposed method more effectively captures the nonlinear characteristics of ETVs.

5. Conclusions

In this paper, a model predictive steering control strategy is proposed. A HOSM observer is developed to capture the variations in the equivalent motor load inertia, and the nonlinear model of the ETV is compensated accordingly to obtain an augmented model. Based on the deep Koopman framework, an augmented dynamic model of the ETV is established, and a model predictive steering control strategy is constructed using the trained ADK model. The observer and the control strategy are validated through simulation studies. The results demonstrate that the HOSM observer can effectively capture the time-varying characteristics of the equivalent motor load inertia. Furthermore, the ADK-MPC-based steering control achieves superior performance. Compared with NMPC, the yaw rate tracking error is reduced by 45.5%, and the computational time is decreased by 11.7%. These results indicate that the proposed steering control strategy has strong potential for practical application in ETV systems.

One limitation of the proposed method is its reliance on offline training based on historical data. Although, in principle, online updating could be realized using data collected during vehicle operation to enable continuous model improvement, such an online updating mechanism has not yet been implemented. This may limit the adaptability of the ETV to varying operating conditions and road environments.