Data-Driven Model-Free Adaptive Sliding Mode Control Based on FFDL for Electric Multiple Units

Abstract

The electric multiple units (EMUs) have become a very convenient and powerful means of transportation in our daily life. Safe and punctual trajectory tracking control is the key to improve the performance of the EMUs system, but it is difficult to realize due to the influence of environmental uncertainty, coupling and nonlinearity. In this paper, a model-free adaptive sliding mode control (MFASMC) method is proposed for the EMUs. This method can solve the dependence of the model-based control method on the train model and eliminate the influence of external disturbances on the robust performance of the system. In this method, the running process of the EMUs is equivalent to a full format dynamic linearization (FFDL) data model, and a model-free adaptive controller (MFAC) is designed based on the data model. Then, to reduce the influence of measurement disturbance and improve the robustness of the system, a discrete sliding mode control (SMC) algorithm is introduced. Furthermore, to prevent the control input from being too large, the parameter estimation error is introduced as an additional correction term of the algorithm. In the end, the simulation experiment is carried out with CRH380A EMUs as the object. Compared with the traditional MFAC and the traditional SMC, the speed tracking effect of each power unit of the MFASMC algorithm is more effective, the change of control force is stable, the acceleration meets the requirements of driving, and has a strong inhibitory effect on external disturbances.

1. Introduction

In recent years, the trajectory tracking control problem of the high-speed EMUs has received wide attention from scholars all over the world. However, it is still difficult to design a traction/brake controller with good performance and high reliability under the circumstances that the operating speed of the EMUs is getting faster and faster, the operating environment is more and more complex and variable, and the dynamics model is more and more difficult to establish [1].

Currently, the research on automatic EMUs operation can be divided into model-based and data-driven control methods. Research on model-based control method includes the following: Reference [2] designs a generalized predictive controller (GPC), adaptively models the model parameters obtained from least squares identification, modifies the tuning parameters of the controller, and then calculates the traction/braking forces to be applied by high-speed trains, further improving the rolling optimization function of GPC; based on the Reference [2], Reference [3] considers the influence of unknown disturbances on the control accuracy by introducing an output feedback controller and designing a GPC method with controller matching. The research objects of the above references are the single-input–single-output (SISO) train system. Compared with the SISO system, the multi-input–multi-output (MIMO) system is more in line with the actual situation of the train. In Reference [4], the dynamic process of train operation is analyzed, and a nonlinear multi-particle model of the train is established considering the carriage type and operation state, then an adaptive robust nonlinear predictive controller is designed for it; in Reference [5], a model parameter estimation method based on adaptive update law is introduced to avoid wheel idling or skidding during train operation, and a dynamic surface-based terminal SMC strategy is proposed; Reference [6] established a coupling model of high-speed EMUs and designed a distributed neural network SMC strategy; similarly, References [7,8] established the prediction controllers based on the ANFIS model and distributed model for trains, respectively. Although the subspace identification methods used in References [6,7,8] require less model structure information, they are still essentially model-based control methods. The above studies often need to obtain the system model structure and parameters when designing control algorithms and stability analysis, or need to estimate the nonlinear part of the system. If the system model information is unknown or the model structure is complex, it is difficult to apply the above methods.

Firstly, for a multi-power unit EMUs system, its accurate dynamic model is difficult to construct, but its input and output data can be obtained easily. Secondly, for the existing multi-particle train model, decoupling must be considered when designing the controller, which undoubtedly increases the difficulty of design. Therefore, this paper adopts the data-driven algorithm to realize the operation control of the EMUs.

MFAC is a typical data-driven control method proposed by Professor Hou in his doctoral dissertation [9]. The general idea of this method is that, based on satisfying certain assumptions, an equivalent dynamic linearized model is established at each sampling point of the discrete system by introducing the concept of pseudo partial derivatives, and then the model is used for controller design, structural adaptive law design, stability analysis, etc. After years of development, MFAC has been successfully applied in many practical systems, such as network services [10,11], robotic systems [12,13], and aircraft systems [14,15]. The studies applied to the field of rail transportation include the following: Reference [16] used the traditional MFAC in a single-particle model of train operation control and achieved better control performance than the traditional PID method. Based on this, Reference [17] combined predictive control to achieve energy-efficient train operation by solving the optimal control problem online for each sampling point of the train system, making full use of the advantages of MFAC and predictive control. In addition, it is difficult for the controller designed based on the dynamics model to guarantee the safety of the train, when the actuator has some failure problems. Hence, MFAC has been widely applied in train fault-tolerant control in recent years: Reference [18] converted the train input and output data into a compact form dynamic linearization (CFDL) model, and proposed a model-free adaptive fault-tolerant control method based on actuator failure under the speed and traction/braking force constraints; References [19,20] further improved based on Reference [18], and proposed the model-free adaptive fault-tolerant control based on a partial form dynamic linearization (PFDL) model for high-speed trains with traction/braking force constraints and actuator faults, using methods such as neural networks to approximate the nonlinear terms caused by faults. In the above studies, the dynamic linearization of the train system is performed by using the CFDL or PFDL method, which only considers the relationship between the amount of variation in the system output and the amount of variation in the current or multiple inputs. All possible complex behaviors in the system, such as nonlinearity, parametric time variation, and structural time variation, are compressed into time-varying parameters. As a result, when the above methods are used in practical systems, more complex time-varying parameter estimation algorithms need to be designed to obtain better control effects. Moreover, the existing studies on the application of MFAC to trains do not consider the multi-particle train as the controlled object, nor do they consider the impact of disturbances on the system.

Based on the above analysis and to further improve the robustness of the system, a data-driven MFASMC scheme for high-speed EMUs is proposed in this paper.

(1)

Compared with the MFAC methods used in References [16,18], this paper further incorporates a discrete sliding mode control algorithm to reduce the effect of measurement perturbations. To prevent the output from being too large, the parameter estimation error is introduced as a limiting term, which improves the robustness of the system;

(2)

Compared with the CFDL model used in References [16,17,18,19,20], the FFDL method used in this paper integrates the relationship between the output change at the next moment and the input and output within a fixed-length sliding time window. Although the dimensionality of the FFDL data model increases, the dynamic behavior of each component becomes simpler, and the design and selection of the parameter estimation algorithm become easier. It is not sensitive to the time-varying structure and parameters of the system.

(3)

Compared with the existing literature on the application of MFAC to trains, the controlled object of this paper is a multi-power unit train, which is more in line with the actual operational requirements. The control method in this paper can be extended to other types of train systems (Such as CRH380AL with 14 motors and 2 trailers, CRH380B with 4 motors and 4 trailers, TR08 maglev train with 3–5 sections, and so on).

(4)

Compared with References [4,5], the algorithm in this paper does not depend on the dynamic model of the EMUs and is a data-driven control algorithm. The main structure of this paper is as follows: Section 1 presents the model structure of the three-power unit EMUs (which only provides data support for train operation simulation); Section 2 proposes the dynamic linearization method and the model-free adaptive sliding mode controller for the EMUs, and its stability is rigorously proved; Section 3 and Section 4 are simulation analysis and conclusion, respectively.

2. Dynamic Analysis of EMUs Operation Process

The traction/braking system of EMUs is composed of several relatively independent power units [21]. Each power unit is associated with an adjacent power unit and couples each other, its longitudinal dynamic is described in Figure 1.

In the Figure 1, the middle carriage is the powered locomotive equipped with a traction unit, and the first and last carriages are trailers. According to Newton’s law of kinematics, the forces on the train power units are analyzed, and the traction force or braking force, basic resistance, and carriage force are applied to each power unit. According to the analysis of the dynamic process of each power unit, the mathematical model of the three power units of the train can be expressed as [22]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{m}_1{\dot{v}}_1\left(t\right)=u_1-F_{N1}\left(t\right)-F_{Z1}\left(t\right),\\ ⋮\\ m_i{\dot{v}}_i\left(t\right)=u_i-F_{Ni}\left(t\right)+F_{Z(i-1)}\left(t\right)-F_{Zi}\left(t\right),\\ ⋮\\ m_n{\dot{v}}_n\left(t\right)=u_n-F_{Nn}\left(t\right)+F_{Z(n-1)}\left(t\right),\\ {\dot{x}}_i\left(t\right)=v_i\left(t\right).\end{array}\right.\end{array}$$

(1)

where

$$F_{Ni}\left(t\right)=a_i+b_i{v}_i\left(t\right)+c_i{v}_i^2\left(t\right),$$

(2)

$$F_{Zi}\left(t\right)=k\left(x_i\left(t\right)-x_{i+1}\left(t\right)\right)+d\left(v_i\left(t\right)-v_{i+1}\left(t\right)\right),$$

(3)

$u_i(i=1,\cdots ,n)$ is the traction/braking force generated by the power unit i under different working conditions of the train; $m_i\left(t\right),v_i\left(t\right),x_i\left(t\right)$ is the mass, speed, and displacement of power unit i, respectively; $F_{Ni}\left(t\right)$ is the basic resistance of power unit i; $a_i,b_i,c_i$ are the operating resistance coefficients; ${F_Z}_i\left(t\right)$ is the force between power unit i and power unit $i+1$, k is the elastic coefficient of the adjacent power unit, and d is the damping coefficient.

Taking the CRH380A EMUs (with six motors and two trailers) as the object for research, which has three independent traction power units (T1 + M1 + M2, M3 + M4, M5 + M6 + T2), and the grouping is shown in Figure 2.

Substituting ${F_Z}_i\left(t\right),F_{Ni}\left(t\right)$ into (1), the longitudinal dynamics model of the high-speed EMUs are obtained as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{m}_1{\dot{v}}_1\left(t\right)=u_1-\left(a_1+b_1{v}_1\left(t\right)+c_1{v}_1^2\left(t\right)\right)\\ -\left(k\left(x_1\left(t\right)-x_2\left(t\right)\right)+d\left(v_1\left(t\right)-v_2\left(t\right)\right)\right),\\ m_2{\dot{v}}_2\left(t\right)=u_2-\left(a_2+b_2{v}_2\left(t\right)+c_2{v}_2^2\left(t\right)\right)\\ +\left(k\left(x_1\left(t\right)-x_2\left(t\right)\right)+d\left(v_1\left(t\right)-v_2\left(t\right)\right)\right)\\ -k\left(x_2\left(t\right)-x_3\left(t\right)\right)-d\left(v_2\left(t\right)-v_3\left(t\right)\right),\\ m_3{\dot{v}}_3\left(t\right)=u_3-\left(a_3+b_3{v}_3\left(t\right)+c_3{v}_3^2\left(t\right)\right)\\ +\left(k\left(x_2\left(t\right)-x_3\left(t\right)\right)+d\left(v_2\left(t\right)-v_3\left(t\right)\right)\right),\\ {\dot{x}}_i\left(t\right)=v_i\left(t\right)(i=1,2,3).\end{array}\end{array}$$

(4)

The control force of each power unit is used as the input signal, and the speed is used as the output signal, which together constitute the MIMO system of the EMUs. In the actual operation of EMUs, it is easily affected by environmental and road changes. There are uncertainties in the mass parameters, drag coefficient, spring coefficient, and damping coefficient in the train model, and there are also nonlinear terms in the model [23]. If the data-driven control method is not used to design the controller, the coupling relationship of each power unit needs to be considered, and the amount of calculation is huge. Therefore, a model-free adaptive sliding mode controller is designed in this paper to perform speed tracking control for EMUs.

Remark 1.

The common traction/braking models of trains are divided into empirical models [1,2,3], fluid dynamics [4,24,25], and fluid-empirical dynamics models [24]. Empirical models are widely used in train dynamics simulation. A fluid dynamics model is a better model to study the behavior of brake systems, but it is more complex and slow in calculation. The fluid–empirical dynamics model combines the fluid dynamics brake pipeline model and the empirical brake valve model. The model of this paper is the empirical model.

3. Model-Free Adaptive Sliding Mode Controller for EMUs

MFASMC does not require the accurate train dynamics model, as it is a data-driven control algorithm suitable for nonlinear and coupled systems.

3.1. Dynamic Linearization

The CFDL method only considers the relationship between the output change of the system at the next moment and the input change at the current moment [9]. However, the output signal of the EMUs operation system depends not only on the control input at a certain moment. Based on the above considerations, the influence of the input signal and output signal in a fixed length sliding time on the output signal at the next moment can be taken into account when the data is linearized, which is the FFDL data processing method. Theoretically, the complex dynamics existing in the system can be well-obtained by using this method, and the dynamic linearization method can effectively reduce the complexity of the system [26].

The input and output data set of the EMUs can be equivalent to the following MIMO discrete-time nonlinear system:

$$\mathit{v}(t+1)=g(\mathit{v}\left(t\right),\cdots ,\mathit{v}(t-n_v),\mathit{u}\left(t\right),\cdots ,\mathit{u}(t-n_u)).$$

(5)

where $\mathit{v},\mathit{u}$ are the input and output of the system at time t, respectively; $n_u$, $n_v$ are the order of input and output, respectively; and $g(·)$ is an unknown nonlinear time-varying function.

Define $\overline{\mathit{H}}\left(t\right)\in {\mathit{R}}^{m{L}_v+m{L}_u}$ as a matrix consisting of control inputs within the sliding time window $[t-L_u+1,t]$ and outputs within the sliding time window $[t-L_v+1,t]$, as follows:

$$\overline{\mathit{H}}\left(t\right)=[{\mathit{v}}^T\left(t\right),\cdots ,{\mathit{v}}^T(t-L_y+1),{\mathit{u}}^T\left(t\right),\cdots ,{\mathit{u}}^T(t-L_u+1){]}^T,$$

(6)

where $0\le L_v\le n_v,1\le L_u\le n_u$ are the length constants of the linearization of the system output and control input, respectively, which are also called pseudo-orders.

For the system of the form (5), the following two assumptions are given.

Assumption 1.

([26]) The partial derivatives of the nonlinear function $g(·)$ with respect to all variables in the system are continuous.

Assumption 2.

([27]) System (5) satisfies the following generalized Lipschitz condition, for any $0\le t_1\ne t_2$ and ${\overline{\mathit{H}}}_{L_v.L_u}\left(t_1\right)\ne {\overline{\mathit{H}}}_{L_v.L_u}\left(t_2\right)$:

$$∥\mathit{v}(t_1+1)-\mathit{v}(t_2+1)∥\le b∥\overline{\mathit{H}}\left(t_1\right)-\overline{\mathit{H}}\left(t_2\right)∥,$$

where b is a positive constant. Denoting $\Delta \overline{\mathit{H}}\left(t\right)=\overline{\mathit{H}}\left(t\right)-\overline{\mathit{H}}(t-1)$.

Theorem 1.

([27]) Consider the discrete-time nonlinear system (5) satisfies Assumptions 1 and 2, for any $0\le L_v\le n_v,1\le L_u\le n_u$, and $∥\Delta \overline{\mathit{H}}\left(t\right)∥\ne 0$. Then, there exists a time-varying matrix $\Phi \left(t\right)\in {\mathit{R}}^{m{L}_v+m{L}_u}$, which is called a pseudo-partitioned jacobian matrix (PPJM), such that system (5) can be transformed into the following FFDL data model:

$$\begin{array}{c}\hfill \mathit{v}(t+1)=\mathit{v}\left(t\right)+\Phi \left(t\right)\Delta \overline{\mathit{H}}\left(t\right).\end{array}$$

(7)

The parameter matrix $\Phi \left(t\right)=[{\Phi }_1\left(t\right),\cdots ,{\Phi }_{L_v+L_u}\left(t\right)]$ is bounded at any time t, where ${\Phi }_n\left(t\right)\in {\mathit{R}}^{m\times m}$, $n=1,\cdots ,L_v+L_u$.

Remark 2.

Theorem 1 has been proved rigorously in [26,27], and different FFDL data models can be obtained by choosing different linearization length constants $L_v,L_u$. Reasonable choice of ${\Phi }_n\left(t\right)$ and $L_v,L_u$ can improve the flexibility of the data model to describe the system.

3.2. Model-Free Adaptive Controller

Consider the following criterion function:

$$J\left(\mathit{u}\left(t\right)\right)={∥{\mathit{v}}_d(t+1)-\mathit{v}(t+1)∥}^2+\lambda {∥\mathit{u}\left(t\right)-\mathit{u}(t-1)∥}^2,$$

(8)

where ${\mathit{v}}_d$ is the desired output, and $\lambda >0$ is a weighting factor that limits the change in control input. Substituting (7) into (8) yields:

$$J\left(\mathit{u}\left(t\right)\right)={∥{\mathit{v}}_d(t+1)-\mathit{v}\left(t\right)-\Phi \left(t\right)\Delta \overline{\mathit{H}}\left(t\right)∥}^2+\lambda {∥\mathit{u}\left(t\right)-\mathit{u}(t-1)∥}^2.$$

(9)

Minimizing (9) with respect to the input signal, yields the following control law:

$$\begin{array}{cc}\hfill {\mathit{u}}_{MFA}\left(t\right)=& {\mathit{u}}_{MFA}(t-1)+\frac{{\Phi }_{L_v+1}^T\left(t\right){\rho }_{L_v+1}({\mathit{v}}_d(t+1)-\mathit{v}\left(t\right))}{\lambda +{∥{\Phi }_{L_v+1}\left(t\right)∥}^2}\hfill \\ \hfill -& \frac{{\Phi }_{L_v+1}^T\left(t\right)\left({\displaystyle \sum _{i=1}^{L_v}}{\rho }_i{\Phi }_i\left(t\right)\Delta \mathit{v}(t-i+1)+{\displaystyle \sum _{i=L_v+2}^{L_v+L_u}}{\rho }_i{\Phi }_i\left(t\right)\Delta {\mathit{u}}_{MFA}(t-i+1)\right)}{\lambda +{∥{\Phi }_{L_v+1}\left(t\right)∥}^2}.\hfill \end{array}$$

(10)

where ${\rho }_i\in (0,1]$ is a step-size constant, which is used to make the FFDL-MFAC algorithm general.

Then, we need to estimate the unknown PPJM $\Phi \left(t\right)$. Introduce the following PPJM criterion function:

$$J\left(\Phi \left(t\right)\right)={∥\Delta \mathit{v}\left(t\right)-\Phi \left(t\right)\Delta \overline{\mathit{H}}(t-1)∥}^2+\mu {∥\Phi \left(t\right)-\widehat{\Phi }\left(t\right)∥}^2.$$

(11)

where $\mu >0$ is a weighting factor that limits the rate of change of adjacent parameters.

Minimizing (11) with respect to $\Phi \left(t\right)$ yields the following parameter estimation algorithms:

$$\widehat{\Phi }\left(t\right)=\widehat{\Phi }(t-1)+\left(\Delta \mathit{v}\left(t\right)-\widehat{\Phi }(t-1)\Delta \overline{\mathit{H}}(t-1)\right)\Delta \overline{\mathit{H}}{(t-1)}^T{\left(\mu +\Delta \overline{\mathit{H}}(t-1)\Delta \overline{\mathit{H}}{(t-1)}^T\right)}^{-1},$$

(12)

The PPJM estimation algorithm (12) without inversion can be further given:

$$\widehat{\Phi }\left(t\right)=\widehat{\Phi }(t-1)+\frac{\beta \left(\Delta \mathit{v}\left(t\right)-\widehat{\Phi }(t-1)\Delta \overline{\mathit{H}}(t-1)\right)\Delta \overline{\mathit{H}}{(t-1)}^T}{\mu +{∥\Delta \overline{\mathit{H}}(t-1)∥}^2}.$$

(13)

where $\widehat{\Phi }\left(t\right)$ is the estimation value of $\Phi \left(t\right)$; $\beta \in (0,2]$ is a step-size constant.

In summary, control law (10) can be rewritten as:

$$\begin{array}{cc}\hfill {\mathit{u}}_{MFA}\left(t\right)=& {\mathit{u}}_{MFA}(t-1)+\frac{{\left({\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)\right)}^T{\rho }_{L_v+1}\left({\mathit{v}}_d(t+1)-\mathit{v}\left(t\right)\right)}{\lambda +{∥{\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)∥}^2}\hfill \\ \hfill -& \frac{{\left({\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)\right)}^T\left({\displaystyle \sum _{i=1}^{L_v}}{\rho }_i{\left({\Phi }_i\left(t\right)-{\delta }_i\left(t\right)\right)}^T\Delta \mathit{v}(t-i+1)\right)}{\lambda +{∥{\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)∥}^2}\hfill \\ \hfill -& \frac{{\left({\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)\right)}^T\left({\displaystyle \sum _{i=L_v+2}^{L_v+L_u}}{\rho }_i{\left({\Phi }_i\left(t\right)-{\delta }_i\left(t\right)\right)}^T\Delta {\mathit{u}}_{MFA}(t-i+1)\right)}{\lambda +{∥{\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)∥}^2}.\hfill \end{array}$$

(14)

where ${\delta }_i$ is parameter estimation errors.

Remark 3.

The PPJM $\Phi \left(t\right)$ in (7) is considered as differential signals in a certain sense and is bounded at any time t. The interaction between $\Phi \left(t\right)$ and $\Delta \mathit{u}\left(t\right)$ can be neglected when the system sampling period and $\Delta \mathit{u}\left(t\right)$ both are small [28].

Remark 4.

Equations (12) and (13) are independent of the mathematical model of the controlled system, and the PPJM estimate is related only to the online I/O data of the system.

3.3. Discrete Sliding Mode Controller

When there is no external disturbance and parameter estimation error in the system, the conventional MFAC scheme converges the system output error to zero. On the contrary, when the system is subject to disturbances and parameter estimation errors, the error will converge to a non-zero constant [29]. Therefore, a discrete SMC is incorporated into the controller design to improve the robustness of the system in this paper.

The output error is defined as:

$$\mathit{e}\left(t\right)={\mathit{v}}_d-\mathit{v}\left(t\right).$$

(15)

Let $\mathit{s}\left(t\right)=\mathit{e}\left(t\right)$, so $\mathit{s}(t+1)$ can be set as:

$$\mathit{s}(t+1)=\mathit{e}(t+1)={\mathit{v}}_d(t+1)-\mathit{v}\left(t\right)-\left(\Phi \left(t\right)-\delta \left(t\right)\right)\Delta \overline{\mathit{H}}\left(t\right).$$

(16)

The SMC law is designed as:

$$\mathit{s}(t+1)=(1-qT)\mathit{s}\left(t\right)-\epsilon T\mathrm{sgn}\left(\mathit{s}\right(t\left)\right),$$

(17)

where $\mathrm{sgn}(·)$ is a symbolic function. $\epsilon >0$, $q>0$, and satisfies $1-qT>0$.

Substituting (16) into (17) yields the following sliding mode control law:

$$\begin{array}{cc}\hfill \Delta \mathit{u}\left(t\right)=& \frac{{\mathit{v}}_d(t+1)-\mathit{v}\left(t\right)}{{\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)}-\frac{(1-qT)\mathit{s}\left(t\right)-\epsilon T\mathrm{sgn}\left(\mathit{s}\right(t\left)\right)}{{\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)}\hfill \\ \hfill -& \frac{{\displaystyle \sum _{i=1}^{L_v}}{\left({\Phi }_i\left(t\right)-{\delta }_i\left(t\right)\right)}^T\Delta \mathit{v}(t-i+1)+{\displaystyle \sum _{i=L_v+2}^{L_v+L_u}}{\left({\Phi }_i\left(t\right)-{\delta }_i\left(t\right)\right)}^T\Delta \mathit{u}(t-i+1)}{{\Phi }_{L_v+1}\left(t\right)-{\delta }_{L_v+1}\left(t\right)}.\hfill \end{array}$$

(18)

Let $\Delta \mathit{u}\left(t\right)={\mathit{u}}_{SM}\left(t\right)$. Then, the MFASMC algorithm is as follows:

$$\mathit{u}\left(t\right)={\mathit{u}}_{MFA}\left(t\right)+\mathit{P}{\mathit{u}}_{SM}\left(t\right).$$

(19)

where $\mathit{P}=diag\{p_1,p_2,p_3\}\in {\mathit{R}}^{3\times 3}$, $p_1,p_2,p_3>0$ are weighting factors.

The control block diagram is shown in Figure 3. From (18), it is possible ${\mathit{u}}_{SM}$ to become large or even unbounded when $\widehat{\Phi }\left(t\right)$ is small. To avoid this situation, $\delta \left(t\right)$ is introduced as an additional correction term for $\widehat{\Phi }\left(t\right)$.

From (14) and (18), it can be seen that the design of the controller process requires only the output data generated by each power unit model without any information from the model. Therefore, the MFASMC algorithm is data-driven.

3.4. Stability Analysis

Assumption 3.

The EMUs iare not affected by serious external factors during operation (such as derailment caused by debris flow and rail damage). Since the train system is subject to strict speed regulation, the speed variation of the train tends to be infinitely small when the system is stable and the sampling time is small enough.

Remark 5.

For the convenience of description, let $L_v=L_u=1$, other cases are similar.

Theorem 2.

If the system (6) satisfying Assumptions 1–3 is controlled by FFDL-MFASMC schemes (12), (13), and (19) with the desired output signal ${\mathit{v}}_d(t+1)={\mathit{v}}_d=const$, then there exists ${\lambda }_{min}$, when $\lambda >{\lambda }_{min}$, the control system guarantees:

(1) 

The estimated value $\widehat{\Phi }\left(t\right)=[{\widehat{\Phi }}_1\left(t\right),{\widehat{\Phi }}_2\left(t\right)]$ of PPJM is bounded;

(2) 

$\underset{t\to \infty }{lim}∥\mathit{e}\left(t\right)∥=0$;

(3) 

The output sequence and the input sequence are bounded.

Proof 1.

$\widehat{\Phi }\left(t\right)=[{\widehat{\Phi }}_1\left(t\right),{\widehat{\Phi }}_2\left(t\right)]$ is bounded.

Firstly, rewrite (13) into the following form:

$$\widehat{\Phi }\left(t\right)=\widehat{\Phi }(t-1)+\frac{\beta \left(\mathit{v}\left(t\right)-\mathit{v}(t-1)\right)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}-\frac{\beta \widehat{\Phi }(t-1)\Delta \overline{\mathit{H}}(t-1)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}.$$

(20)

Subtracting $\Phi \left(t\right)$ from both sides of (20) yields:

$$\begin{array}{cc}\hfill \delta \left(t\right)=& \delta (t-1)+\Phi (t-1)-\Phi \left(t\right)+\frac{\beta \left(\mathit{v}\left(t\right)-\mathit{v}(t-1)\right)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}\hfill \\ \hfill -& \frac{\beta \widehat{\Phi }(t-1)\Delta \overline{\mathit{H}}(t-1)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}.\hfill \end{array}$$

(21)

Substituting $\widehat{\Phi }(t-1)=\delta \left(t\right)+\Phi (t-1)$ into (21) yields:

$$\begin{array}{cc}\hfill \delta \left(t\right)=& \delta (t-1)+\Phi (t-1)-\Phi \left(t\right)+\frac{\beta \left(\mathit{v}\left(t\right)-\mathit{v}(t-1)\right)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}\hfill \\ \hfill -& \frac{\beta \left(\delta (t-1)+\Phi (t-1)\right)\Delta \overline{\mathit{H}}(t-1)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}.\hfill \end{array}$$

(22)

Substituting data model (7) into (22) yields:

$$\delta \left(t\right)=\delta (t-1)\left(I-\frac{\beta \Delta \overline{\mathit{H}}(t-1)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}\right)+\Phi (t-1)-\Phi \left(t\right).$$

(23)

From Theorem 1, b is a positive constant and $∥\Phi \left(t\right)∥\le b$. Taking the norm on both sides of (23), we have:

$$\begin{array}{c}\hfill ∥\delta \left(t\right)∥=∥\delta (t-1)\left(I-\frac{\beta \Delta \overline{\mathit{H}}(t-1)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}\right)∥+2b.\end{array}$$

(24)

Squaring the first term on the right side of (24), one obtains:

$$\begin{array}{cc}\hfill & {∥\delta (t-1)\left(I-\frac{\beta \Delta \overline{\mathit{H}}(t-1)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}\right)∥}^2={∥\delta (t-1)∥}^2\hfill \\ & +\left(\frac{\beta \Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}-2\right)\frac{\beta {∥\delta (t-1)∥}^2\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}.\hfill \end{array}$$

(25)

Since the step size factor $\beta \in (0,2]$, it is not difficult to obtain:

$$\frac{\beta \Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}<\frac{\beta \Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}{\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}=2.$$

(26)

Then, it can be obtained from (25) and (26):

$$\begin{array}{c}\hfill {∥\delta (t-1)\left(I-\frac{\beta \Delta \overline{\mathit{H}}(t-1)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(k-1){\Vert }^2}\right)∥}^2<{∥\delta (t-1)∥}^2.\end{array}$$

(27)

From (27), there must be $0<M<1$ such that the following inequality holds:

$$\begin{array}{c}\hfill ∥\delta (t-1)\left(I-\frac{\beta \Delta \overline{\mathit{H}}(t-1)\Delta {\overline{\mathit{H}}}^T(t-1)}{\mu +\Vert \Delta \overline{\mathit{H}}(t-1){\Vert }^2}\right)∥\le M∥\delta (t-1)∥.\end{array}$$

(28)

Substituting (28) into (24), one obtains:

$$\begin{array}{cc}\hfill ∥\delta \left(t\right)∥\le & M∥\delta (t-1)∥+2b\le M\left(M∥\delta (t-2)∥+2b\right)+2b\hfill \\ \hfill \le & \cdots \le M^{t-1}∥\delta \left(1\right)∥+\frac{1-M^{t-1}}{1-M}2b.\hfill \end{array}$$

(29)

It can be obtained from (29) that the parameter estimation error $\delta \left(t\right)=[{\delta }_1\left(t\right),{\delta }_2\left(t\right)]$ is bounded, and since $\Phi \left(t\right)$ is bounded, the estimation value $\widehat{\Phi }\left(t\right)$ is bounded too. The proof of conclusion (1) is completed. □

Since ${\Phi }_1\left(t\right),{\Phi }_2\left(t\right)$ is unknown, ${\delta }_1\left(t\right),{\delta }_2\left(t\right)$ cannot be accurately obtained. The introduction of ${\Phi }_1\left(t\right),{\Phi }_2\left(t\right)$ prevents the control input from being too large, and for the convenience of the controller design, let the element in ${\delta }_1\left(t\right),{\delta }_2\left(t\right)$ be adjustable constants.

Proof 2.

$\underset{t\to \infty }{lim}∥\mathit{e}\left(t\right)∥=0$.

Substituting (7) and (14) into (15) yields:

$$\begin{array}{c}\hfill \mathit{e}\left(t+1\right)=[\mathit{I}-\frac{\rho ({\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)){({\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right))}^T}{\lambda +{∥{\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)∥}^2}\left]\right[\mathit{e}\left(t\right)-({\widehat{\Phi }}_1\left(t\right)-{\delta }_1\left(t\right))\Delta v\left(t\right)].\end{array}$$

(30)

Since $\rho \in (0,1]$ and $\lambda >0$, we have:

$$\begin{array}{c}\hfill ∥\mathit{I}-\frac{\rho ({\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)){({\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right))}^T}{\lambda +{∥{\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)∥}^2}∥=\epsilon \in (0,1).\end{array}$$

(31)

Since ${\widehat{\Phi }}_1\left(t\right)-{\delta }_1\left(t\right)$ is bounded, then combining (30), (31), and Assumption 3, we have:

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}∥\mathit{e}\left(t+1\right)∥=\epsilon \underset{t\to \infty }{lim}∥\mathit{e}\left(t\right)∥,\epsilon \in (0,1).\end{array}$$

(32)

This means that $\underset{t\to \infty }{lim}∥\mathit{e}\left(t\right)∥=0$. The proof of conclusion (2) is completed. □

Proof 3.

The output sequence and the input sequence are bounded.

Since $\mathit{e}\left(t\right)$ and ${\mathit{v}}_d$ are bounded, it is easy to find that the output sequence is bounded. According to (15) and (32), using $\sqrt{ab}\le a+b$ and ${\lambda }_{min}<\lambda$, we have:

$$\begin{array}{cc}\hfill ∥\Delta {\mathit{u}}_{MFA}\left(t\right)∥\le & ∥\rho \frac{{({\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right))}^T}{\lambda +{∥{\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)∥}^2}\mathit{e}\left(t\right)∥\le \left|\rho \frac{∥{\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)∥}{\lambda +{∥{\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)∥}^2}\right|∥\mathit{e}\left(t\right)∥\hfill \\ \hfill \le & \left|\frac{\rho ∥{\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)∥}{2\sqrt{\lambda {∥{\widehat{\Phi }}_2\left(t\right)-{\delta }_2\left(t\right)∥}^2}}\right|∥\mathit{e}\left(t\right)∥\le \left|\frac{\rho }{2\sqrt{{\lambda }_{min}}}\right|∥\mathit{e}\left(t\right)∥\hfill \end{array}$$

(33)

Further, from (32) and (33), we have:

$$\begin{array}{cc}\hfill ∥{\mathit{u}}_{MFA}\left(t\right)∥& \le ∥{\mathit{u}}_{MFA}\left(t\right)-{\mathit{u}}_{MFA}(t-1)∥+∥{\mathit{u}}_{MFA}(t-1)∥\hfill \\ & \le ∥\Delta {\mathit{u}}_{MFA}\left(t\right)∥+∥{\mathit{u}}_{MFA}(t-1)-{\mathit{u}}_{MFA}(t-2)∥+∥{\mathit{u}}_{MFA}(t-2)∥\hfill \\ & ⋮\hfill \\ & \le ∥\Delta {\mathit{u}}_{MFA}\left(t\right)∥+∥\Delta {\mathit{u}}_{MFA}(t-1)∥+\cdots +∥\Delta {\mathit{u}}_{MFA}\left(2\right)∥+∥{\mathit{u}}_{MFA}\left(1\right)∥\hfill \\ & \le \left|\frac{\rho }{2\sqrt{{\lambda }_{min}}}\right|(∥\mathit{e}\left(t\right)∥+∥\mathit{e}(t-1)∥+\cdots +∥\mathit{e}\left(2\right)∥)+∥{\mathit{u}}_{MFA}\left(1\right)∥\hfill \\ & <\left|\frac{\rho }{2\sqrt{{\lambda }_{min}}}\right|({\epsilon }^{k-1}∥\mathit{e}\left(1\right)∥+{\epsilon }^{t-1}∥\mathit{e}\left(1\right)∥+\cdots +\epsilon ∥\mathit{e}\left(1\right)∥)+∥{\mathit{u}}_{MFA}\left(1\right)∥\hfill \end{array}$$

(34)

Since $\left|\frac{\rho }{2\sqrt{{\lambda }_{min}}}\right|$ and the $\mathit{e}\left(t\right)$ are bounded, (34) can show that $∥{\mathit{u}}_{MFA}\left(t\right)∥$ is bounded. The boundedness of $∥{\mathit{u}}_{SM}\left(t\right)∥$ is similar to the boundedness of $∥{\mathit{u}}_{MFA}\left(t\right)∥$. It is not difficult to conclude that the input sequence is bounded. The proof of conclusion (3) is completed.

In summary, the proof of Theorem 2 is completed. □

4. Simulation Experiment and Analysis

4.1. Simulation Environment

The semi-physical simulation platform of the high-speed EMU running process equipped in the existing laboratory, including the CRH380 EMU driving simulator, and its related virtual visual equipment, is shown in Figure 4. By inputting the corresponding control strategy through the reserved programming interface of the platform, the train operation effect can be displayed on the virtual visual equipment in real time, and the running speed, position and distance from the target point of the train can be recorded to simulate the real running environment. The basic parameters of CRH380A high-speed EMUs are shown in

Table 1. The CRH380A EMUs model parameters.

Parameters	Parameter Values	Unit
Power unit mass $M_1$	$1.836\times {10}^5$	T
Power unit mass $M_2$	$1.123\times {10}^5$	T
Power unit mass $M_3$	$1.836\times {10}^5$	T
Drag coefficient $a_0$	$5.2$	N/kg
Drag coefficient $a_1$	$3.6\times {10}^{-2}$	N· s2/kg· m
Drag coefficient $a_2$	$1.2\times {10}^{-3}$	N· s2/kg· m2
Elastic coefficient k	$2\times {10}^7$	N/m
Damping coefficient d	$5\times {10}^6$	N· s/m

[7,22].

The experiment is divided into two groups: the first group of experiments uses the proposed algorithm for simulation experiments without considering external disturbances to verify the effectiveness of the algorithm in this paper; the second group of experiments adds disturbance signals to verify the robustness of the algorithm in this paper and compares with MFAC and adaptive sliding mode control, analyzes the speed tracking effect of each power unit, the change of control force, the change of acceleration and other indicators to verify the advantages of the algorithm in this paper.

4.2. Simulation 1: No External Disturbance

The first set of experiments is simulated using the MFASMC algorithm without considering external disturbances to verify the effectiveness of the proposed algorithm. The initial conditions of the system are set as ${\mathit{v}}_1\left(1\right)={\mathit{v}}_2\left(1\right)={\mathit{v}}_3\left(1\right)={\mathit{v}}_d$, ${\Phi }_1\left(1\right)={\Phi }_2\left(1\right)=diag\{0.5,0.5,0.5\}$. The controller parameters are set to $\lambda =1$, $\rho =0.85$, $\mu =1$, $\beta =0.2$, $\epsilon =10$, $q=0.5$, ${\delta }_1\left(1\right)={\delta }_2\left(1\right)=-0.6$, $\mathit{P}=diag\{2,2,2\}$.

Figure 5 is the speed-displacement tracking curve of each power unit by MFASMC method, Figure 6 is the speed tracking error curve, and Figure 7 shows the corresponding velocity level error curve. It can be seen that the whole train speed tracking effect using MFASMC method is better, and the tracking error range of each power unit is between [$-0.245$ km/h, $0.175$ km/h], which meets the speed tracking accuracy requirement. According to CTCS-3 [30] (China Train Control System Level 3) speed error requirements (that is, speed error range should be ≤30 km/h, no more than $\pm 2$ km/h; when the speed is greater than 30 km/h, it should be within $2\%$ of the speed). In addition, the speed change of the EMUs from Tai’an to Xu’zhou East is stable and the error fluctuation is small.

Remark 6.

The errors of individual sampling points are too large to represent the overall control performance of the system, so the data of points with large errors are excluded in the experiment.

Figure 8 is the control force change curve of each power unit using the MFASMC algorithm. The control force range of each power unit is between [$-47$ N/kN, 41 N/kN], and the working condition switching process is relatively stable. Figure 9 shows the acceleration curve of each power unit. The acceleration range of each power unit is between $[-0.923$ m/s², 0.812 m/s²]. The acceleration change is stable, which can meet the comfort standard of the train operation.

This section verifies the effectiveness of the MFASMC algorithm by speed tracking error, the control force variation case, and the acceleration variation case without considering external disturbances.

4.3. Simulation 2: External Disturbance

The second group of experiments incorporates disturbance signals to verify the robustness of the proposed algorithm and compares it with MFAC and DITSMC methods. During the experiment, white noise is introduced to simulate the external disturbance encountered in the actual operation of the train. The control strategy and vehicle information are input into the simulation experimental bench, then to record the train operation speed, position, and control force of the three methods.

(1)

MFASMC: The initial conditions of the system are set as ${\mathit{v}}_1\left(1\right)={\mathit{v}}_2\left(1\right)={\mathit{v}}_3\left(1\right)={\mathit{v}}_d$, ${\Phi }_1\left(1\right)={\Phi }_2\left(1\right)=diag\{0.5,0.5,0.5\}$. The controller parameters are set to $\lambda =1$, $\rho =0.85$, $\mu =1$, $\beta =0.2$, $\epsilon =10$, $q=0.5$, ${\delta }_1\left(1\right)={\delta }_2\left(1\right)=-0.6$, $\mathit{P}=diag\{2,2,2\}$.

(2)

MFAC: The FFDL-MFAC method control law scheme is as follows:

$$\mathit{u}\left(t\right)=\mathit{u}(t-1)+\frac{\rho {\widehat{\Phi }}_2^T\left(t\right)({\mathit{v}}_d(t+1)-\mathit{v}\left(t\right)-{\widehat{\Phi }}_1\left(t\right)\Delta \mathit{v}\left(t\right))}{\lambda +{∥{\widehat{\Phi }}_2\left(t\right)∥}^2}.$$

(35)

The initial conditions of the system are set as ${\mathit{v}}_1\left(1\right)={\mathit{v}}_2\left(1\right)={\mathit{v}}_3\left(1\right)={\mathit{v}}_d$, ${\Phi }_1\left(1\right)={\Phi }_2\left(1\right)=diag\{0.5,0.5,0.5\}$. The controller parameters are set to $\lambda =1$, $\rho =0.85$, $\mu =1$, $\beta =0.2$, $\epsilon =10$.

(3)

DITSMC [28]: The CFDL-DITSMC method control law scheme is as follows:

$$\begin{array}{c}\hfill \Delta \mathit{u}\left(t\right)=\frac{1}{\widehat{\Psi }\left(t\right)}[{\mathit{K}}_1^{-1}\mathit{s}\left(t\right)-{\mathit{K}}_1^{-1}{\mathit{K}}_2\mathit{E}\left(t\right)+{\mathit{v}}_d(t+1)-\mathit{v}\left(t\right)-\mathrm{sgn}\left(\mathit{s}\left(t\right)\right)]\end{array}$$

(36)

where ${\mathit{K}}_1=diag\{K_{11},K_{12},K_{13}\}$, ${\mathit{K}}_2=diag\{K_{21},K_{22},K_{23}\}$. The terminal sliding mode function $\mathit{s}\left(t\right)$ and the integral error term are $\mathit{E}\left(t\right)$, respectively,

$$\mathit{s}\left(t\right)={\mathit{K}}_1\mathit{e}\left(t\right)+{\mathit{K}}_2\mathit{E}(t-1)$$

(37)

$$\mathit{E}\left(t\right)=\mathit{E}(t-1)+\mathit{e}\left(t\right)$$

(38)

The initial conditions of the system are set as ${\mathit{v}}_1\left(1\right)={\mathit{v}}_2\left(1\right)={\mathit{v}}_3\left(1\right)={\mathit{v}}_d$, $\widehat{\Psi }\left(1\right)=diag\{0.5,0.5,0.5\}$. The controller parameters are set to $\lambda =1$, $\rho =0.85$, $\mu =1$, $\beta =0.2$, $\epsilon =10$, $q=0.5$, $\mathit{P}=diag\{2,2,2\}$.

Figure 10 is the speed-displacement tracking curve of each power unit of MFASMC, MFAC and DITSMC, and Figure 11 is the corresponding speed tracking error curve. Due to the addition of external disturbances, the control performance of MFAC and DITSMC methods is greatly affected, and the convergence is difficult to guarantee. In most of the road sections, the tracking error range of each power unit of MFAC is between [$-0.474$ km/h, 0.468 km/h], and the tracking error range of each power unit of DITSMC is between [$-0.528$ km/h, 0.513 km/h]. However, when passing through special road sections, due to the rapid change of expected speed, the errors of MFAC and DITSMC can only be stabilized between [$-1.247$ km/h, 1.348 km/h] and [$-0.851$ km/h, 0.762 km/h], respectively. In contrast, MFASMC uses the discrete sliding mode exponential reaching law to further ensure convergence. The tracking error range of each power unit is stable between [$-0.251$ km/h, 0.206 km/h], which is less affected by disturbance and meets the speed tracking accuracy requirements.

In contrast, MFASMC uses the discrete sliding mode exponential reaching law to further ensure convergence. The tracking error range of each power unit is stable between [−0.251km/h, 0.206km/h], which is less affected by disturbances and meets the speed tracking accuracy requirements.

Figure 12 shows the unit control force curves of MFASMC, MFAC, and DITSMC. The unit control force range of each power unit of MFAC and DITSMC is between [$-53$ N/kN, 48 N/kN] and [$-54$ N/kN, 50 N/kN], respectively, and the control force of MFAC changes frequently and is greatly affected by external disturbances. The control force of MFASMC is relatively stable, and the amplitude is smaller than that of MFAC. The unit control force range of each power unit is between [$-49$ N/kN, 44 N/kN].

Figure 13 shows the acceleration curves of MFASMC, MFAC, and DITSMC. It is not difficult to see that the acceleration of MFAC and DITSMC methods changes too fast, and the range is between [$-1.045$ m/s², 0.947 m/s²] and [$-1.056$ m/s², 1.024 m/s²]. The acceleration of MFASMC’s high-speed EMUs changes gently, ranging from [$-0.942$ m/s², 0.858 m/s²], which meets the comfort requirements of passengers [31].

Further, in order to analyze the control performance of each controller more intuitively, the following several performance indicators are considered to evaluate the controller.

(1)

Mean square error performance (MSE)

$$MSE=\frac{1}{nT}\sum _{i=1}^n\sum _{t=1}^T{\left|v_d-v_i\left(t\right)\right|}^2.$$

(39)

(2)

Integral of absolute value of error criterion (IAE)

$$IAE=\frac{{\displaystyle \sum _{i=1}^n}{\int }_0^{\infty }\left|e_i\left(t\right)\right|dt}{n}.$$

(40)

(3)

Maximum acceleration/deceleration (MAXA)

$$MAXA=\underset{t\in \left[2,T\right]i\in \left[1,n\right]}{sup}\left|v_i\left(t\right)-v_i(t-1)\right|.$$

(41)

The performance index values for the three control methods are recorded in

Table 2. Comparison of several performance indexes of each control method.

Methods	MSE	IAE	MAXA
MFASMC	$0.052$	437	0.942
MFAC	$0.248$	1180	1.045
DITSMC	$0.476$	2810	1.056

. The MSE indicator measures the deviation between the observed value and the true value and is more sensitive to outliers in the data. The smaller the value, the better the system tracking effect, and the IAE index is similar to the MSE measurement effect. The MSE and IAE values of MFAC and DITSMC were greater than those of MFASMC. Input stability of the maximum acceleration/deceleration reaction system. It can be seen that the maximum acceleration/deceleration of the train using the MFASMC control scheme is 0.942 m/s², and the change is small; the maximum acceleration/deceleration of MFASMC and DITSMC is 1.045 m/s² and 1.056 m/s², respectively, which means that MFAC and DITSMC are not conducive to the comfort requirements of passengers, while MFASMC can meet the requirements.

In summary, this section introduces external disturbances, and verifies the robustness and superiority of the proposed algorithm through velocity tracking error, control force changes, and acceleration changes.

5. Conclusions

In this paper, an improved MIMO-FFDL-MFASMC method is proposed for the automatic driving system of high-speed EMUs. In this method, the operation process of the EMUs is equivalent to a full format dynamic data model, and a model-free adaptive controller is designed based on the data model. Then, to reduce the influence of measurement disturbance and improve the robustness of the system, a discrete sliding mode control algorithm is introduced. Furthermore, to prevent the sliding mode control input from being too large, the parameter estimation error is introduced as an additional correction term of the algorithm. The simulation results demonstrate that compared with MFAC and DITSMC, the MFASMC control scheme:

(1)

Achieves higher precision tracking control than the traditional MFAC method for given speed-displacement curve, and the error range is between [$-0.251$ km/h, 0.206 km/h], which meets the requirements of train speed error and achieves safe and punctual operation of the train;

(2)

The control force changes more smoothly, and the acceleration range of each power unit is [$-0.942$ m/s², 0.858 m/s²], which meets the comfort requirements of passengers;

(3)

It has a strong suppression effect on external disturbances and good robustness. In conclusion, the speed tracking effect of each power unit of the control algorithm in this paper is favorable, the control force changes smoothly, the acceleration meets the driving safety requirements, and has a strong inhibitory effect on external disturbances.

Further, based on this paper, the author will do the following research:

(1)

The heavy-haul combined trains composed of different types of locomotives and the delay are considered as [31];

(2)

Using the intelligent control method to optimize the parameters designed by the control method in this paper as [32];

(3)

The heat of multi-agent formation control has been high in recent years, and it aims to accomplish the task efficiently through the collaboration of each agent in the system [33,34], so the authors will study the model-free adaptive control of high-speed trains with multi-agent in the future;

(4)

Design an algorithm to improve train operation safety [35].