Active suspension predictive control for high-speed trains based on disturbance feedforward and neural network residual compensation

Abstract

In high-speed train active suspensions, unmodeled dynamics, wheel-rail wear, and track disturbances can cause dynamic mismatches that degrade the performance of model-dependent predictive control. This study proposes a composite model predictive control approach that integrates neural network residual compensation and equivalent track disturbance feedforward (NNRC-ETDF-MPC) to address this issue. The method firstly estimates the unknown equivalent track disturbances online using an augmented Kalman filter (AKF) and incorporated into the control framework as a physical feedforward compensation term. The nonlinear dynamic residuals are predicted and compensated online using a history-aware physical residual neural network (HA-PRNN), which uses a sliding window to extract multivariate coupling features. Based on these components, a dual feedforward composite control system is developed, along with a constraint-tightening mechanism to prevent actuator saturation. Finally, the performance of the proposed control strategy is evaluated using a co-simulation platform that includes two track spectra, a wide speed range, and cross-service wear conditions. Comparative tests show that the composite control strategy enhances the dynamic vibration suppression capabilities of high-speed trains working in complex, time-varying environments and enhancing robustness against model mismatch. Specifically, under worn wheel conditions, the proposed NNRC-ETDF-MPC achieves average Sperling index improvements of 22.81% and 23.48% across different track excitations.

Keywords

high-speed train active suspension predictive control model decoupling network compensation model mismatch

1. Introduction

The increasing speed of trains inevitably aggravates lateral vibrations caused by track irregularities. As a result, developing active suspension technology has become an effective way to overcome the limitations of traditional passive suspensions while improving vehicle ride comfort.

Early research on active suspension control for automobiles and railway vehicles primarily focused on methods such as skyhook control (Karnopp et al., 1974; Liu and Zuo, 2016; Shi et al., 2025), $H_{\infty}$ robust control (Hua et al., 2022; Zhu et al., 2018), and linear quadratic regulator (LQR) (Orvnäs et al., 2011). However, practical suspension systems frequently face constraints such as actuator output saturation. The traditional methods mentioned above lack the ability to explicitly handle constraints, limiting their performance in complex operating conditions. In contrast, model predictive control (MPC) can explicitly handle multivariable constraints within a receding horizon optimization framework, resulting in superior control effects. In the field of automotive active suspension, researchers have successively introduced linear MPC (LMPC) (Gohrle et al., 2013), nonlinear MPC (NMPC) (Rodriguez-Guevara et al., 2023), and tube-based robust MPC (Tube-MPC) (Hu and Cheng, 2022) to meet the requirements of nonlinear dynamics and real-time calculation. When combined with road preview information, these methods can significantly improve vehicle ride comfort and handling stability (Li et al., 2024; Yu et al., 2025). Similar studies have also been conducted in the railway sector (Gu et al., 2026; Posseckert and Lüdicke, 2023).

However, traditional MPC highly relies on an accurate mathematical model of the controlled plant. When system parameters vary over time or unmodeled dynamics exist, prediction model mismatch often limits control performance (Moradi et al., 2019). For example, in intelligent vehicle trajectory tracking, complex external disturbances, model simplifications, and parameter variations can degrade the tracking accuracy and robustness of MPC (He et al., 2025).

To address this issue, Yang et al. (2021) proposed a dual-loop tube-based robust MPC (DTRMPC) to maintain robustness under parameter uncertainty. However, preserving the uncertainty set may lead to excessive conservatism and reduced suspension performance (Kang et al., 2020). To further mitigate external disturbances, several anti-disturbance control methods have been developed. For example, Shen et al. (2022) proposed an extended disturbance-observer-based data-driven control method to reject unknown inputs. Gu et al. (2024) investigated a dynamic guaranteed-cost event-triggered anti-disturbance control strategy for systems with external uncertainties, while Shen et al. (2020) studied an output anti-disturbance control strategy for stochastic Markov jump systems with multiple disturbances. Alternatively, Meng et al. (2024) proposed a fast iterative MPC (FI-MPC) that iteratively compensates for the error between linear and nonlinear models, achieving performance comparable to NMPC.

With the development of machine learning, learning-based MPC has provided a promising solution to the modeling accuracy limitations of traditional physical models. Rather than constructing highly accurate mathematical models, recent studies increasingly employ frameworks that combine nominal models with residual compensation. In these frameworks, a nominal physical model ensures basic control performance, while residual prediction compensates for complex unmodeled nonlinear dynamics (Amine et al., 2026; Fu et al., 2025; Meng et al., 2025; Xiong et al., 2026). Hancioğlu and Efe (2026) used a neural network to capture nonlinear dynamic residuals from multiple factors and added them to the MPC prediction as a feedforward term. Zhang et al. (2025) developed a temporal residual neural network for vehicles’ complex time-varying nonlinear dynamics in order to learn offline and predict the dynamic residual between the physical mechanism and the actual system, thus overcoming the deviation caused by a single nominal model. Current research shows that, when compared to fully end-to-end control strategies, using a neural network as a residual compensator not only takes advantage of the neural network’s generalization ability, but also preserves the stability of the basic physical model controller (Aswani et al., 2013; Hancioğlu and Efe, 2026; Hewing et al., 2020; Meng et al., 2025; Zhang et al., 2025).

To address the model mismatch in the secondary lateral active suspension of high-speed trains, this paper proposes a mechanism-and-data dual-driven control strategy (NNRC-ETDF-MPC). The architecture first employs a nominal model predictive controller with disturbance feedforward (ETDF-MPC) to manage actuator constraints and ensure baseline stability. A history-aware physical residual neural network (HA-PRNN) is then introduced to estimate unmodeled dynamic residuals online. By incorporating the residual predictions into the MPC framework together with a constraint-tightening mechanism, the proposed strategy enhances control robustness under model mismatch.

The proposed NNRC-ETDF-MPC strategy addresses the limitations of conventional predictive control under model mismatch by enabling online adaptive compensation of unmodeled nonlinear dynamics. Unlike conventional disturbance feedforward MPC architectures, it introduces a dedicated data-driven compensation path to account for structural perturbations. Furthermore, while existing residual-based MPC schemes often rely on single-step feedback, the proposed framework incorporates a History-Aware Physical Residual Neural Network (HA-PRNN) to extract deep spatiotemporal features from historical states and disturbances, thereby improving the representation of long-term dynamic dependencies.

The remainder of this paper is organized as follows. Section 2 presents the lateral dynamic model of the high-speed train, the model decoupling strategy, and an AKF-based method for online estimation of equivalent track irregularities. Section 3 describes the design of the ETDF-MPC controller based on the nominal model. Section 4 introduces the residual compensation network and establishes the NNRC-ETDF-MPC framework. Section 5 presents the co-simulation platform, simulation parameters, and validation of the network prediction performance. Finally, the effectiveness and robustness of the proposed strategy are evaluated under two typical track spectra and different wheel-wear conditions.

2. Lateral dynamics model

To balance estimation accuracy and the real-time requirements of MPC in multibody high-speed train systems (Bruni et al., 2011), this paper proposes a framework integrating full-order state observation with reduced-order decoupled control. A high-order observer estimates multivariable states, while modal decoupling provides a reduced-order representation for MPC, thereby lowering computational complexity.

2.1. Lateral dynamics equations

Neglecting the higher-order degrees of freedom of the bogies, a dynamics model is established that includes the carbody lateral displacement, roll, yaw, and the lateral displacements of the front and rear bogies. Figure 1 shows a schematic diagram of the model. The corresponding dynamic equations are given as follows:

{\begin{cases} m_{c} {\ddot{y}}_{c} + 2 c_{p y} {\dot{y}}_{c} + 2 k_{p y} y_{c} - 2 c_{p y} h_{c} {\dot{ϕ}}_{c} - 2 k_{p y} h_{c} ϕ_{c} - c_{p y} (y_{t 1} + y_{t 2}) \\ - k_{p y} (y_{t 1} + y_{t 2}) = u_{1} + u_{2} \\ I_{x} {\ddot{ϕ}}_{c} + (4 c_{s z} d_{s}^{2} + 2 c_{s y} h_{c}^{2}) {\dot{ϕ}}_{c} + (4 k_{s z} d_{s}^{2} + 2 k_{s y} h_{c}^{2}) ϕ_{c} - 2 c_{s y} h_{c} {\dot{y}}_{c} \\ - 2 k_{s y} h_{c} y_{c} + c_{s y} h_{c} ({\dot{y}}_{t 1} + {\dot{y}}_{t 2}) + k_{s y} h_{c} (y_{t 1} + y_{t 2}) = - (u_{1} + u_{2}) h_{c} \\ I_{z} {\ddot{ψ}}_{c} + (2 c_{s y} l^{2} + 4 c_{s x} d_{s}^{2}) {\dot{ψ}}_{c} + (2 k_{s y} l^{2} + 4 k_{s x} d_{s}^{2}) ψ_{c} - c_{s y} l ({\dot{y}}_{t 1} - {\dot{y}}_{t 2}) \\ - k_{s y} l (y_{t 1} - y_{t 2}) = (u_{1} - u_{2}) l \\ m_{t} {\ddot{y}}_{t 1} + (2 c_{s y} + c_{p y}) {\dot{y}}_{t 1} + (2 k_{sy} + k_{p y}) y_{t 1} - c_{p y} {\dot{y}}_{c} - k_{p y} y_{c} + c_{p y} h_{c} {\dot{ϕ}}_{c} \\ + k_{p y} h_{c} ϕ_{c} - c_{p y} l ψ_{c} - k_{p y} l ψ_{c} = - u_{1} + 2 c_{s y} {\dot{y}}_{w 1} + 2 k_{s y} y_{w 1} \\ m_{t} {\ddot{y}}_{t 2} + (2 c_{s y} + c_{p y}) {\dot{y}}_{t 2} + (2 k_{s y} + k_{p y}) y_{c} - c_{p y} {\dot{y}}_{c} - k_{p y} y_{c} + c_{p y} h_{c} {\dot{ϕ}}_{c} \\ + k_{p y} h_{c} ϕ_{c} + c_{p y} l ψ_{c} + k_{p y} l ψ_{c} = - u_{2} + 2 c_{s y} {\dot{y}}_{w 2} + 2 k_{s y} y_{w 2} \end{cases}

(1)

where

m_{c}

I_{x}

, and

I_{z}

are the carbody mass, roll moment of inertia, and yaw moment of inertia, respectively;

m_{t}

is the bogie mass;

k_{p y}

and

c_{p y}

denote the equivalent lateral stiffness and damping of the primary suspension, respectively;

k_{s y}

and

c_{s y}

denote the equivalent lateral stiffness and damping of the secondary suspension, respectively;

k_{s z}

and

c_{s z}

denote the vertical stiffness and damping of the secondary suspension, respectively;

h_{c}

is the vertical distance between the carbody center of gravity and the horizontal plane of the secondary lateral suspension,

d_{s}

is the lateral half-span of the secondary vertical suspension, and

l_{c}

is the half-length of the vehicle;

y_{w 1}

and

y_{w 2}

are the average lateral track irregularities of the front and rear bogie wheels; and

u_{1}

and

u_{2}

are the control forces of the active dampers installed on the front and rear bogies. The main vehicle parameters are listed in Table 1.

Figure 1.

Lateral dynamics model of the high-speed train.

Table 1.

Main parameters of the vehicle.

Parameter	Value
$m_{c}$ (kg)	33,700
$I_{x}$ (kg·m²)	1.25 × 10⁵
$I_{z}$ (kg·m²)	1,570,000
$m_{t}$ (kg)	2300
$k_{s y}$ (N/m)	3.2 × 10⁵
$c_{s y}$ (N·s/m)	4 × 10⁴
$k_{s x}$ (N/m)	5 × 10⁶
$c_{s x}$ (N·s/m)	4.5 × 10⁴
$k_{p y}$ (N/m)	2.2 × 10⁷
$c_{p y}$ (N·s/m)	3 × 10⁴
$h_{c}$ (m)	1.2
$l_{c}$ (m)	8.75

2.2. Augmented Kalman filter

Based on the dynamic equations established in Section 2.1, a continuous-time state-space model is constructed to facilitate the subsequent state estimation and controller design. By defining the state vector $x_{p} = {[y_{c}, ϕ_{c}, ψ_{c}, y_{t 1}, y_{t 2}, {\dot{y}}_{c}, {\dot{ϕ}}_{c}, {\dot{ψ}}_{c}, {\dot{y}}_{t 1}, {\dot{y}}_{t 2}]}^{T}$ , the second-order differential equations in Section 2.1 are transformed into the standard state-space form:

{\dot{x}}_{p} (t) = A_{p} x_{p} (t) + B_{p} u_{p} (t) + G_{p} w (t)

(2)

where

B_{p} \in R^{10 \times 2}

and

G_{p} \in R^{10 \times 4}

are the coefficient matrices;

u_{p} (t) = {[u_{1}, u_{2}]}^{T}

is the active control force vector; and

w (t) = {[y_{w 1}, y_{w 2}, {\dot{y}}_{w 1}, {\dot{y}}_{w 2}]}^{T}

is the equivalent track irregularity vector containing the wheelsets’ lateral displacements and velocities.

To estimate the equivalent track irregularities without prior knowledge of their exact time-varying spectra, their temporal dynamics are modeled as a stochastic process, where the irregularities’ accelerations are driven by zero-mean Gaussian white noise $α (t)$ (Bejarano et al., 2009; Kim et al., 2021). An augmented state vector is created by augmenting both the physical state $x_{p}$ and the irregularity state: $X_{aug} = {[x_{p}^{T}, w^{T}]}^{T} \in R^{14}$ . The continuous augmented system is expressed as follows:

{\dot{X}}_{aug} (t) = [\begin{array}{c} A_{p} & G_{p} \\ 0_{2 \times 10} & [\begin{array}{l} 0_{2 \times 2} & I_{2 \times 2} \\ 0_{2 \times 2} & 0_{2 \times 2} \end{array}] \end{array}] X_{aug} (t) + [\begin{array}{l} B_{p} \\ 0_{4 \times 2} \end{array}] u_{p} (t) + [\begin{array}{c} 0_{12 \times 1} \\ α (t) \end{array}]

(3)

To ensure the system’s observability, the measurement vector is selected as follows:

Y_{k} = {[y_{c} - h_{c} ϕ_{c} + l_{c} ψ_{c} - y_{t 1}, y_{c} - h_{c} ϕ_{c} - l_{c} ψ_{c} - y_{t 2}, {\dot{y}}_{c}, {\dot{ϕ}}_{c}, {\dot{ψ}}_{c}, {\dot{y}}_{t 1}, {\dot{y}}_{t 2}]}^{T}

(4)

which includes the lateral relative displacements of the secondary suspensions of the front and rear bogies, the lateral velocities of the carbody and bogies, as well as the carbody’s yaw and roll velocities. The observability of the continuous augmented system is verified via the observability matrix.

The discrete Kalman filter (KF) form is obtained by discretizing the continuous model described above:

{\begin{cases} X_{aug} (k + 1) = A_{kf} X_{aug} (k) + B_{kf} u_{p} (k) + v (k) \\ Y (k) = C_{kf} X_{aug} (k) + η (k) \end{cases}

(5)

where

A_{kf}

B_{kf}

, and

C_{kf}

are the discretized state transition matrix, input matrix, and observation matrix, respectively;

v (k)

and

η (k)

are the process and measurement noises corresponding to the covariance matrices

Q

and

R

, respectively.

Q

and

R

are configured as diagonal matrices, with their elements calibrated based on sensor precision and system disturbance variances. Specifically, to enable effective identification of unknown equivalent track excitations, the diagonal elements in the process noise matrix

Q

corresponding to the track disturbance states are assigned relatively large weights, ensuring the observer’s rapid dynamic tracking capability for time-varying excitations. Through the standard KF update equations, this observer can effectively estimate the physical states

{\hat{x}}_{p} (k)

and the equivalent track irregularities

\hat{w} (k)

, providing the necessary information for the subsequent controller.

2.3. Lateral model decoupling

Although the high-order KF can provide accurate state information, solving it directly in a high-dimensional state space is computationally costly. As a result, by ignoring the roll mode, this paper applies the concept of modal decoupling and introduces a homogeneous coordinate transformation to decompose the system into two independent motion modes (Foo and Goodall, 2000; Fu et al., 2020; Orvnäs et al., 2011), as illustrated in Figure 2.

Figure 2.

Decomposition of lateral motion: (a) pure lateral motion mode (b) pure yaw motion mode.

The motions of the bogies are decomposed into common-mode lateral motion and differential-mode yaw motion:

y_{tc} = \frac{y_{t 1} + y_{t 2}}{2}, y_{psit} = \frac{y_{t 1} - y_{t 2}}{2}

(6)

where the yaw angle of the carbody is equivalently converted into linear displacement,

y_{ψ} = l_{c} ψ_{c}

. Through this transformation, the coupled system can be split into two mutually independent and structurally symmetric low-dimensional subsystems.

The lateral subsystem dominates the lateral motion of the train. Defining the state vector $X_{lat} = {[y_{c, lat}, {\dot{y}}_{c, lat}, y_{t, lat}, {\dot{y}}_{t, lat}]}^{T}$ , the equivalent carbody lateral mass is $M_{c} = m_{c} / 2$ , and its continuous state-space equation has the following form:

{\dot{X}}_{lat} = A_{lat} X_{lat} + B_{lat} U_{lat} + B_{w, lat} W_{lat}

(7)

The corresponding equivalent lateral disturbance is extracted from the augmented states of the KF as follows:

W_{lat} = {[\frac{y_{w 1} + y_{w 2}}{2}, \frac{{\dot{y}}_{w 1} + {\dot{y}}_{w 2}}{2}]}^{T}

(8)

The yaw subsystem dominates the yaw motion of the train. Defining the state vector $X_{yaw} = {[y_{c, yaw}, {\dot{y}}_{c, yaw}, y_{t, yaw}, {\dot{y}}_{t, yaw}]}^{T}$ , the equivalent carbody yaw mass is $M_{yaw} = I_{z} / (2 l_{c}^{2})$ , and its continuous state-space equation has the following form:

{\dot{X}}_{yaw} = A_{yaw} X_{yaw} + B_{yaw} U_{yaw} + B_{w, yaw} W_{yaw}

(9)

The corresponding equivalent differential-mode disturbance is extracted as follows:

W_{yaw} = {[\frac{y_{w 1} - y_{w 2}}{2}, \frac{{\dot{y}}_{w 1} - {\dot{y}}_{w 2}}{2}]}^{T}

(10)

In terms of the arrangement of active dampers, only one secondary lateral passive damper at the front and rear ends of the carbody is replaced by an active damper. The front and rear active dampers provide the active control forces $u_{1}$ and $u_{2}$ .

Defining the output control forces of the lateral and yaw controllers as $U_{lat}$ and $U_{yaw}$ , respectively, the relationship between the actuator forces and the modal equivalent forces follows:

[\begin{array}{l} U_{lat} \\ U_{yaw} \end{array}] = [\begin{array}{l} 1 / 2 & 1 / 2 \\ 1 / 2 & - 1 / 2 \end{array}] [\begin{array}{l} u_{1} \\ u_{2} \end{array}]

(11)

Considering the saturation limit of the maximum thrust $u_{\max}$ of the actuators in practical execution, the inverse mapping to the actuator thrusts yields:

[\begin{array}{l} u_{1} \\ u_{2} \end{array}] = sat ([\begin{array}{l} 1 & 1 \\ 1 & - 1 \end{array}] [\begin{array}{l} U_{lat} \\ U_{yaw} \end{array}], [\begin{array}{l} \pm u_{\max} \\ \pm u_{\max} \end{array}])

(12)

The above distribution mechanism successfully decouples the lateral and yaw controllers and calculates the actuator thrusts.

3. Nominal equivalent track disturbance feedforward MPC

To achieve effective control under complex track conditions, this section develops a feedforward incremental MPC (FI-MPC) based on the nominal linear dynamics model. By incorporating the online estimated equivalent track irregularities as feedforward compensation, the system’s ability to suppress external disturbances is enhanced (Gu et al., 2026).

3.1. Incremental predictive model and disturbance augmentation

Since the pure lateral and pure yaw subsystems decoupled in Section 2 share the same structure, a unified state variable is adopted to represent the discretized physical states of either subsystem ( $X_{lat} (k)$ or $X_{yaw} (k)$ ). This simplifies the subsequent controller design. Meanwhile, $u (k)$ denotes the active control input of the corresponding subsystem.

Due to the bandwidth limitations of practical actuators, an incremental state-space formulation is employed to penalize actuator force increments (Meng et al., 2024, 2025). The physical state at the current time step $x (k)$ and the control force at the previous time step $u (k - 1)$ are combined to form the augmented state vector $ξ (k) = {[x^{T} (k), u^{T} (k - 1)]}^{T} \in R^{5}$ . Based on the decoupled discretized nominal system matrices, the augmented state-space equation that incorporating physical track irregularities is established as follows:

ξ (k + 1) = \tilde{A} ξ (k) + \tilde{B} Δ u (k) + {\tilde{B}}_{w} W (k)

(13)

where

Δ u (k) = u (k) - u (k - 1)

is the control force increment at the current time step and

W (k) \in R^{2}

is the equivalent track irregularity.

The corresponding augmented matrices are defined as:

\tilde{A} = [\begin{array}{l} A_{d} & B_{d} \\ 0_{1 \times 4} & 1 \end{array}], \tilde{B} = [\begin{array}{l} B_{d} \\ 1 \end{array}], {\tilde{B}}_{w} = [\begin{array}{l} B_{w d} \\ 0_{1 \times 2} \end{array}]

(14)

where

A_{d}

B_{d}

, and

B_{w d}

are the nominal matrices derived from the discretization of the continuous matrices of the respective decoupled subsystems.

The system output is defined by the physical variables to be optimized. In this study, the carbody displacement, velocity, and acceleration are selected as optimization variables, $y (k) = {[y_{m} (k), {\dot{y}}_{m} (k), {\ddot{y}}_{m} (k)]}^{T} = C_{aug} ξ (k) + D_{aug} Δ u (k)$ , where $y_{m}$ denotes the carbody’s lateral displacement $y_{c}$ or the corresponding yaw displacement $y_{ψ}$ .

3.2. Prediction equations and constraints

Designate the prediction horizon of the system as $N_{p}$ and the control horizon as $N_{c}$ . It is assumed that the track irregularities remain constant at the present state within the prediction horizon, $W (k + i | k) \approx W (k)$ . Utilizing the state-space equation from Section 3.1, recursive prediction yields the output sequence over the prediction horizon $Y = {[y {(k + 1 | k)}^{T}, . . ., y {(k + N_{p} | k)}^{T}]}^{T}$ :

Y = Ψ ξ (k) + Θ Δ U + Γ_{w} W (k)

(15)

where

Δ U = {[Δ u (k), . . ., Δ u (k + N_{c} - 1)]}^{T}

is the control increment sequence;

Ψ

and

Θ

are the standard MPC state prediction matrix and control prediction matrix, respectively. The feedforward matrix

Γ_{w}

propagates the current track irregularity disturbance to future prediction steps, thereby enhancing disturbance rejection capability.

Under the assumption of meeting physical constraints, the controller’s optimization goal is to minimize the carbody’s dynamic response while limiting control energy consumption. The cost function $J (k)$ for finite-horizon receding optimization is defined as:

J (k) = \sum_{i = 1}^{N_{p}} y^{T} (k + i | k) Q_{y} y (k + i | k) + \sum_{j = 0}^{N_{c} - 1} Δ u^{T} (k + j) R_{y} Δ u (k + j)

(16)

where

Q_{y} \in R^{3 \times 3}

is the single-step state penalty weight matrix and

R_{y} \in R

is the single-step control increment penalty scalar.

Defining the full-horizon weighting matrices as ${\bar{Q}}_{y} = I_{N_{p}} \otimes Q_{y}$ and $\bar{R} = I_{N_{c}} \otimes R_{y}$ , and substituting the prediction equation into the cost function $J (k)$ , the original optimization problem can be reformulated into a standard quadratic programming (QP) form:

\min_{Δ U} \frac{1}{2} Δ U^{T} H_{qp} Δ U + F_{qp}^{T} Δ U

(17)

where the matrix

H_{qp} = 2 (Θ^{T} {\bar{Q}}_{y} Θ + \bar{R})

is only related to the nominal parameters of the system; the

F_{qp} = 2 Θ^{T} {\bar{Q}}_{y} (Ψ ξ (k) + Γ_{w} W (k))

incorporates the state feedback and disturbance feedforward terms at each control step.

In practical engineering, the active damper thrust is constrained by absolute thrust limits and slew rate restrictions (Sun et al., 2012). To prevent actuator saturation and reserve control authority for nonlinear compensation, constraint tightening margins ( $M_{U}$ and $M_{Δ U}$ ) are integrated into the standard MPC formulation (Mayne et al., 2005; Sun et al., 2012). Consequently, the initial actuator constraints are transformed into QP linear inequalities:

[\begin{array}{l} A_{Δ u} \\ A_{u} \end{array}] Δ U ⩽ [\begin{array}{l} b_{Δ u} \\ b_{u} \end{array}]

(18)

The increment constraint matrices $(A_{Δ u}, b_{Δ u})$ and absolute force constraint matrices $(A_{u}, b_{u})$ are constructed as follows:

{\begin{cases} A_{Δ u} = [\begin{array}{l} I_{N_{c}} \\ - I_{N_{c}} \end{array}], b_{Δ u} = [\begin{array}{l} (Δ U_{\max} - M_{Δ U}) I_{c} \\ (- Δ U_{\min} - M_{Δ U}) I_{c} \end{array}] \\ A_{u} = [\begin{array}{l} L \\ - L \end{array}], b_{u} = [\begin{array}{l} (U_{\max} - M_{U}) I_{c} - I_{c} u (k - 1) \\ - (U_{\min} + M_{U}) I_{c} + I_{c} u (k - 1) \end{array}] \end{cases}

(19)

where

I_{N_{c}}

is an

N_{c}

-dimensional identity matrix,

I_{c} = {[1, 1, . . ., 1]}^{T} \in R^{N_{c}}

is a column vector, and

Δ U_{\max / \min}

and

U_{\max / \min}

are the single-step slew rate constraints and absolute thrust constraints of the actuator, respectively.

While the nominal MPC effectively suppresses track-induced vibrations under ideal conditions, inevitable parameter perturbations cause model mismatches and dynamic residuals (Meng et al., 2024, 2025). These unmodeled dynamics cause deviations between the predicted trajectory and the actual vehicle response, degrading control performance (He et al., 2025; Sun et al., 2014). To overcome these uncertainties, Section 4 introduces a residual compensation neural network to enhance the controller’s adaptive capacity (Aswani et al., 2013; Hewing et al., 2020).

4. Neural network residual compensation and composite control

4.1. History-aware physical residual neural network design

To overcome the nominal model’s limitations, this section proposes a history-aware physical residual neural network (HA-PRNN). Rather than predicting absolute states, HA-PRNN estimates unmodeled nonlinear residuals using independent parallel networks tailored to the distinct decoupled lateral and yaw subsystems.

To enable the network to fully extract the true mapping relationships under varying conditions, a training dataset with different speeds, track spectrum excitations, and wheel profile wear states is constructed. For a single decoupled mode, the single-step base feature vector $\tilde{E} (k)$ is defined as:

\tilde{E} (k) = {[v (k), x^{T} (k), W^{T} (k), u (k - 1)]}^{T} \in R^{8}

(20)

where

v (k)

is the vehicle speed,

x (k)

is the decoupled state,

W (k)

is the track disturbance, and

u (k - 1)

is the control force at the previous time step.

The network label is defined as the single-step dynamic residual $F_{res} (k)$ between the physical system and the nominal model. Using the posterior state $x^{+} (k)$ estimated by the KF and the state at the next time step $x_{true} (k + 1)$ , this physical residual can be calculated using the following equation:

F_{res} (k) = x_{true} (k + 1) - (A_{d} x^{+} (k) + B_{d} u (k) + B_{w d} W (k))

(21)

Due to the strong temporal memory effects inherent in the vehicle’s dynamic responses, single-step features are insufficient for accurate residual estimation. Therefore, a sliding window of length $N$ is employed to extract history-aware sequences. To ensure the generalization capability of the network, a comprehensive dynamic dataset was constructed based on co-simulations covering a multi-dimensional grid of operating conditions. Specifically, the dataset encompasses a speed range from 80 to 440 km/h (with a step increment of 20 km/h). Regarding track excitations, four typical track spectra were introduced: the Wuhan-Guangzhou (WG) spectrum, the Beijing-Tianjin (JJ) spectrum, the German high-speed spectrum, and the US track spectrum. Furthermore, in terms of wheel-rail contact states, five measured wheel profiles from different service wear stages were selected, with equivalent conicities widely distributed between 0.04 and 0.47. This mechanism yields approximately 130,000 samples for each decoupled network (lateral and yaw). To enhance the network’s generalization ability under different operating conditions, the data is split into training (80%), validation (10%), and testing (10%) sets based on individual sliding-window samples rather than distinct operating conditions. The resulting 2D spatiotemporal feature tensor $E_{k}$ is constructed as follows:

E_{k} = [\tilde{E} (k - N + 1), . . ., \tilde{E} (k)] \in R^{8 \times N}

(22)

To process the 2D spatiotemporal tensor, a multi-scale convolutional neural network (CNN) is developed, leveraging its proven capability in handling multivariate time series (Bai et al., 2018; Zheng et al., 2014). The network comprises modules for local temporal extraction, cross-physical channel fusion, deep dynamic feature extraction, and fully connected mapping. This architecture outputs a single-step prediction vector representing the unmodeled nonlinear physical residual ${\hat{F}}_{res} (k)$ . Figure 3 and Table 2 detail the architecture and hyperparameter configuration of the network, providing the structural basis for the subsequent residual compensation strategy. Figure 4 illustrates the training loss curve of the HA-PRNN for both lateral and yaw modal. The results demonstrate that the loss values for both models converge rapidly and remain stable after approximately 1000 iterations, indicating effective learning of the unmodeled dynamics.

Figure 3.

Multi-scale convolutional neural network architecture.

Table 2.

Network structure parameters.

Network layer	Kernel/Node size	Activation function/Extra operation	Output tensor dimension
Input layer	/	/	8 × N
Conv1D (intra-channel)	1 × 3	BN and ReLU	8 × N × 16
Conv2D (cross-channel)	8 × 5	BN and ReLU	8 × N × 32
Conv1D (deep temporal)	1 × 3	BN and ReLU	8 × N × 64
MaxPooling1D	1 × 2	/	8 × N/2 × 64
FC 1 (hidden layer)	128	Dropout (0.2)	128
FC 2 (hidden layer)	64	ReLU	64
FC 3 (output layer)	4	Regression	4

Figure 4.

Training loss curve: (a) lateral modal (b) yaw modal.

The network’s core computational layers operate as follows:

(1) Intra-channel local temporal convolution: Uses [1 × 3] 1D kernels to extract independent temporal features from each state and disturbance, yielding an $(8 \times N \times 16)$ tensor.

(2) Cross-channel physical fusion: Utilizes [8 × 5] kernels across all eight spatial channels to capture multivariate coupling effects, raising the dimension to $(8 \times N \times 32)$ .

(3) Deep global dynamic extraction: Employs [1 × 3] kernels to capture long-range dependencies of the nonlinear residual evolution, outputting an $(8 \times N \times 64)$ tensor.

(4) Max pooling and flattening: A [1 × 2] pooling kernel halves the time axis to mitigate overfitting $(8 \times N / 2 \times 64)$ , followed by a flattening operation to unroll the feature maps into a 1D vector.

(5) Fully-connected mapping: Sequentially maps the 1D vector into a (4 × 1) output, generating the final prediction for the unmodeled physical residual ${\hat{F}}_{res} (k)$ .

4.2. Residual feedforward integration and control framework

While the nominal state space (Section 3) relies solely on the current state, control increments and track irregularities, nonlinear compensation is achieved by incorporating the predicted residual vector into the augmented state equation as an independent external input:

ξ (k + 1) = \tilde{A} ξ (k) + \tilde{B} Δ u (k) + {\tilde{B}}_{w} W (k) + {\tilde{B}}_{res} {\hat{F}}_{res} (k)

(23)

where

{\tilde{B}}_{res}

is the coefficient matrix, defined as:

{\tilde{B}}_{res} = [\begin{array}{l} I_{4 \times 4} \\ 0_{1 \times 4} \end{array}]

(24)

To achieve real-time feedforward compensation within the prediction horizon $N_{p}$ , it is assumed that the nonlinear residuals at future moments within a single prediction window remain constant at the estimated value of the current moment, ${\hat{F}}_{res} (k + i | k) \approx {\hat{F}}_{res} (k)$ . By performing receding optimization iterative expansion on the above state equation in the time domain, the output prediction sequence $\tilde{Y}$ can be derived:

\tilde{Y} = Ψ ξ (k) + Θ Δ U + Γ_{w} W (k) + Γ_{res} {\hat{F}}_{res} (k)

(25)

where

Γ_{res}

is the residual feedforward matrix. Functioning similarly to the

Γ_{w}

matrix, the

Γ_{res}

matrix enables the residuals perceived by the network to be transmitted to future moments. The linear coefficient vector is updated to

{\tilde{F}}_{q p}

{\tilde{F}}_{qp} = - 2 Θ^{T} {\bar{Q}}_{y} (Y_{ref} - Ψ ξ (k) - Γ_{w} W (k) - Γ_{res} {\hat{F}}_{res} (k))

(26)

Because the network introduces additional nonlinear compensation control efforts, the actual output of the actuator may approach its physical limits. However, the constraint tightening margins $M_{U}$ and $M_{Δ U}$ predefined in Section 3.2 precisely reserve a certain execution space for the MPC controller to perform residual compensation.

As shown in Figure 5, the closed-loop NNRC-ETDF-MPC architecture first employs the AKF to estimate vehicle states and track excitations using SIMPACK feedback. The augmented states are then decoupled into independent lateral and yaw subsystems, where parallel neural networks perform forward propagation to predict unmodeled dynamic residuals. The estimated disturbances and residuals are incorporated as dual feedforward inputs into the MPC, which performs receding-horizon optimization to generate constrained optimal control sequences. Finally, the optimized control forces are transformed through the corresponding transfer functions (TF) and applied to the vehicle dynamic model.

Figure 5.

Operation framework of NNRC-ETDF-MPC.

5. Co-simulation verification and result analysis

5.1. Co-simulation setup and test conditions

To validate the proposed control algorithm, a SIMPACK-MATLAB/Simulink co-simulation platform was established. The multibody train model operates in SIMPACK, while the controllers and neural network are executed in Simulink, enabling real-time closed-loop interaction between system states and control forces.

To account for long-term wheel-rail degradation (Gao et al., 2025a; Polach, 2006), new and worn wheel profiles are defined for comparison, yielding equivalent conicities of 0.04 and 0.34, respectively (Figure 6(a)). Additionally, a TF model implemented in Simulink characterizes the active damper’s step response (Figure 6(b)).

Figure 6.

Simulation settings: (a) comparison of new and worn wheel profiles (b) step force response of the TF.

The simulations are conducted using the WG and JJ track spectra. The proposed controller is benchmarked against a passive suspension, a classical skyhook control (damping coefficient set to 40,000 N·s/m), and the nominal ETDF-MPC (constructed in Section 3). For a fair comparison, both MPC strategies use identical prediction, control, and penalty matrices. All controllers operate with a 0.01 s sampling period, and the neural network utilizes a history-aware window length of 50. The baseline ETDF-MPC employs the same tightened constraints as the NNRC-ETDF-MPC. Although this setup inevitably restricts the maximum output capacity of the baseline controller, it ensures that both control strategies operate under identical physical execution boundaries. This design isolates and evaluates the contribution of the neural network to compensating for unmodeled dynamics.

5.2. Verification of residual network prediction

To evaluate the residual prediction capability of the network, tests were conducted using the WG spectrum at 100 and 350 km/h for both new and worn wheel profiles. Figure 7(a) and (b) compare the true and predicted residuals (Dim 2 and Dim 4) for new wheels, demonstrating fundamental consistency. As speed increases, the nonlinear residuals become more pronounced, with the peak lateral velocity residual (Dim 4) increasing by approximately 0.03 m/s. Nevertheless, the network accurately captures these dynamic variations.

Figure 7.

Comparison between predicted and true values: (a) WG new wheel, 100 km/h (b) WG new wheel, 350 km/h (c) WG worn wheel, 100 km/h (d) WG worn wheel, 350 km/h.

Figure 7(c) and (d) present the prediction results under worn-wheel conditions, where the contact geometry between the wheel and rail exacerbate nonlinear dynamics (Gao et al., 2025b; Li et al., 2024; Zeng et al., 2025). At 350 km/h, wheel wear amplifies the residual amplitudes, with the peak yaw velocity residual (Dim 2) increasing from 1.2 mm/s to nearly 4.0 mm/s. Despite slight deviations near extreme peaks, the network accurately reconstructs the overall trajectories. These results demonstrate that the network effectively captures time-varying dynamics, providing reliable nonlinear feedforward compensation for the composite MPC.

5.3. Dynamic responses under typical conditions

To evaluate the vibration suppression performance, four suspension configurations are compared on the WG track spectrum at 350 km/h under both new and worn wheel conditions. Figures 8 and 9, alongside Table 3, illustrate the dynamic responses and acceleration RMS values of the vehicle under various control strategies for both new and worn wheel profiles.

Figure 8.

Dynamic responses under the Wu-Guang spectrum at 350 km/h: (a) New wheel – Carbody lateral acceleration (b) New wheel – Yaw angular acceleration (c) New wheel – Carbody lateral acceleration PSD (d) New wheel – Yaw angular acceleration PSD.

Figure 9.

Dynamic responses under the Wu-Guang spectrum at 350 km/h: (a) Worn wheel – Carbody lateral acceleration (b) Worn wheel – Yaw angular acceleration (c) Worn wheel – Carbody lateral acceleration PSD (d) Worn wheel – Yaw angular acceleration PSD.

Table 3.

Acceleration RMS under WG spectrum at 350 km/h.

Condition	Strategy	Lateral RMS(m/s²)	Yaw RMS (rad/s²)	Improvement
New profile	Passive	0.102	0.0107	/
	Skyhook	0.042	0.0043	58.82%/59.81%
	ETDF-MPC	0.045	0.0043	55.88%/59.81%
	NNRC-ETDF-MPC	0.026	0.0024	74.51%/77.57%
Worn profile	Passive	0.160	0.0157	/
	Skyhook	0.108	0.0108	32.50%/31.21%
	ETDF-MPC	0.091	0.0106	43.13%/32.48%
	NNRC-ETDF-MPC	0.072	0.0072	55.00%/54.14%

Figure 8(a)–(d) show the lateral acceleration of the front carbody center and the carbody yaw acceleration under the new wheel condition. Owing to the absence of active control, the passive suspension exhibits the largest vibration amplitudes, with lateral and yaw acceleration RMS values of 0.102 m/s² and 0.0107 rad/s², respectively. Compared with the passive suspension, Skyhook control and ETDF-MPC reduce the RMS values by 55%-59%. In contrast, NNRC-ETDF-MPC provides the best control performance, reducing the RMS values to 0.026 m/s² and 0.0024 rad/s², corresponding to improvements of 74.51% and 77.57%, respectively.

Figure 8(c) and (d) show power spectral density (PSD) curves that demonstrate the frequency-domain effects of various controllers. The lateral and yaw resonance peaks are mainly concentrated in the low-frequency range of 1-2 Hz. Within this range, NNRC-ETDF-MPC provides the greatest attenuation, particularly at the 1.22 Hz resonance peak. In contrast, the performance of ETDF-MPC deteriorates above 1.2 Hz, where its PSD curve approaches that of Skyhook control. Nevertheless, NNRC-ETDF-MPC maintains effective vibration suppression in the 4-12 Hz frequency band.

Figure 9(a)–(d) show the time-frequency domain comparison results under the worn wheel condition. Wheel wear increases the vibration levels of the passive suspension and degrades the performance of both Skyhook control and ETDF-MPC. As shown in Table 3, the lateral and yaw RMS improvement rates of Skyhook control decrease to 32.50% and 31.21%, respectively, while the yaw improvement rate of ETDF-MPC drops to 32.48%. In the frequency domain, the vibration suppression capability of Skyhook control below 2 Hz is weakened. Meanwhile, ETDF-MPC exhibits limited attenuation of lateral vibrations in the 8-12 Hz range and yaw vibrations in the 4-12 Hz range. In contrast, NNRC-ETDF-MPC compensates for model mismatch through neural network-based residual correction while maintaining strong low-frequency vibration suppression. Consequently, its lateral and yaw improvement rates remain above 54% under worn-wheel conditions. Figure 9(c) and (d) demonstrate that NNRC-ETDF-MPC maintains a lower vibration energy density across the dominant frequency bands.

5.4. Ride quality across speed and wear conditions

To evaluate the controllers’ robustness against dynamic mismatches under various operational severities, co-simulations were conducted over a wide speed range (100-400 km/h with a step of 20 km/h). The test scenarios include two typical track excitations and two wheel-rail contact states. Figure 10 shows the variation trends of the vehicle lateral Sperling index under these conditions. In the low-speed range, where the discrepancy between the nonlinear vehicle dynamics and the nominal reduced-order model is limited, both ETDF-MPC and NNRC-ETDF-MPC outperform the Skyhook controller in vibration suppression. As speed increases and wheel wear becomes more severe, model mismatch becomes more pronounced. This is evident in Figure 10(b) and (d), where the difference between the Sperling index curves of the nominal ETDF-MPC and the NNRC-ETDF-MPC becomes more prominent. Under 400 km/h worn wheel conditions (Figure 10(b) and (d)), the mismatch causes the lateral Sperling index of ETDF-MPC to increase, resulting in hunting instability of the vehicle. In contrast, NNRC-ETDF-MPC, with residual feedforward compensation, maintains a smooth and stable trend.

Figure 10.

Sperling index under different track spectra and wheel profiles: (a) WG spectrum new wheel (b) WG spectrum worn wheel (c) JJ spectrum new wheel (d) JJ spectrum worn wheel.

To quantify performance within the operating speed range, Table 4 summarizes the average improvement rates of the Sperling index between 100 and 380 km/h. NNRC-ETDF-MPC achieves strong vibration reduction under both WG and JJ spectra across different wheel-rail conditions. Notably, as the contact state changes from new to worn, the control performance of both the skyhook control and ETDF-MPC deteriorates. For example, under the WG spectrum, the average improvement of Skyhook control decreases from 23.39% to 11.00%. In contrast, NNRC-ETDF-MPC maintains stable performance, with average improvements of 22.81% and 23.48% under worn-wheel conditions for the two spectra.

Table 4.

Average improvement of Sperling index (100-380 km/h).

Strategy	WG (new) (%)	WG (worn) (%)	JJ (new) (%)	JJ (worn) (%)
Skyhook	23.39	11.00	21.42	12.43
ETDF-MPC	25.22	17.44	31.82	20.53
NNRC-ETDF-MPC	36.50	22.81	35.34	23.48

To investigate the instability mechanism under extreme operating conditions, time-domain and time-frequency responses are analyzed for the WG and JJ spectra at 400 km/h (Figures 11 and 12). As shown in Figure 11(a), when the simulation time exceeds 13.5 s, the peak lateral carbody acceleration under the ETDF-MPC strategy increases to 0.6 m/s². Combined with the time-frequency energy distribution in Figure 12(a), the hunting motion energy is mainly concentrated in the 4-6 Hz band. For the JJ spectrum (Figure 11(b)), both Skyhook control and ETDF-MPC exhibit high-frequency vibrations, with acceleration peaks of approximately 0.4 m/s². The corresponding time-frequency results in Figure 12(b) further confirm the presence of high-frequency hunting instability. In particular, ETDF-MPC shows a pronounced energy concentration in the 4-6 Hz band after 12 s under JJ excitation. In contrast, NNRC-ETDF-MPC achieves effective suppression of hunting oscillations. The proposed composite controller consistently attenuates energy in the 4-6 Hz band for both track spectra, maintaining carbody acceleration within safe limits.

Figure 11.

Time-domain responses with worn wheels at 400 km/h: (a) WG (b) JJ.

Figure 12.

Time-frequency responses with worn wheels at 400 km/h: (a) WG (b) JJ.

The instability of the vehicle dynamic response is also reflected in abnormal actuator force outputs. Figure 13(a) and (b) compare the actual output forces of the front bogie active damper under the ETDF-MPC and the NNRC-ETDF-MPC strategies at 400 km/h with worn wheels. During high-frequency hunting instability, the force commands generated by ETDF-MPC cause actuator saturation. In contrast, NNRC-ETDF-MPC employs physical residual feedforward to provide early corrective action during disturbance evolution, thereby effectively suppressing energy accumulation associated with hunting motion.

Figure 13.

Active forces of the front active damper with worn wheels at 400 km/h. (a) WG (b) JJ.

6. Conclusions

This paper proposes a composite model predictive control strategy (NNRC-ETDF-MPC) to address the nominal model mismatch and control performance degradation in high-speed trains. The proposed method is evaluated through co-simulations over a speed range of 100-400 km/h under two wheel-rail contact conditions. The main conclusions are as follows:

(1) The residual neural network effectively approximates transient dynamic residuals that cannot be captured by the nominal linear model, using local states and temporal disturbance features within a sliding window. Multi-condition tests demonstrate good generalization and prediction capability, enabling reliable residual feedforward compensation.

(2) Both ETDF-MPC and NNRC-ETDF-MPC effectively reduce vibrations under new wheel conditions. NNRC-ETDF-MPC achieves superior vibration attenuation in both the 1-2 Hz and 4-12 Hz frequency bands, thereby improving the Sperling index. Under worn-wheel conditions, it maintains robust broadband vibration suppression and improves ride comfort.

(3) When operating at 400 km/h with wheel-rail wear, the ETDF-MPC’s performance degrades significantly, leading to high-frequency hunting instability. Using neural network residual feedforward compensation, NNRC-ETDF-MPC provides early corrective control through neural-network-based residual feedforward compensation, effectively suppressing the growth of unstable energy.

Footnotes

ORCID iDs

Haofeng Xie

Liangcheng Dai

Pengcheng Zeng

Ethical considerations

This article does not contain any studies with human or animal participants.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Key R&D Program of China [grant numbers 2022YFB4301303, 2025YFB4304002].

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The datasets generated during and/or analyzed during the current study are available from the corresponding author on request.*

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