# Non-Singular Fast Terminal Sliding Mode Control of High-Speed Train Network System Based on Improved Particle Swarm Optimization Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. The Design of the NFTSM Control Strategy for the High-Speed Train Network Control System

## 3. High-Speed Train Motion Model

## 4. IMPSO-RBF Neural Network

#### 4.1. RBF Neural Network

#### 4.2. Particle Swarm Optimization Algorithm

#### 4.3. Improved Multi-Strategy Particle Swarm Optimization Algorithm

#### 4.3.1. Improved Multi-Strategy Evolutionary Behavior

#### 4.3.2. Multi-Strategy Value Comparison

#### 4.3.3. Strategy Behavioral Mutation Algorithm

## 5. Non-Singular Fast Terminal Sliding Mode Control

#### 5.1. Control Law Design

**Assumption**

**1.**

#### 5.2. Stability Analysis

**Lemma**

**1**

**.**Consider a nonlinear system $\dot{x}=f(x)$, $f(0)=0$, $x\in {R}^{n}$, $x(0)=0$, whereby the equilibrium point $x=0$ is built on the assumption that, as a continuous function, $V(x)$: $D\to R$ is defined on an open neighborhood $U\subseteq D$ of the origin, such that the following conditions hold: (1) $V(x)$ is positive definite; (2) there are real numbers $c>0$ and $\alpha \in $ (0,1), such that $\dot{V}(x)+c{V}^{\alpha}(x)\le 0,x\in U\backslash \left\{0\right\}$

**Lemma**

**2**

**.**Consider a vector $b=\left[{b}_{1},{b}_{2},\dots ,{b}_{n}\right]$, the following inequality holds: $\Vert b\Vert \le {\displaystyle {\sum}_{i=1}^{n}\left|{b}_{i}\right|}$

**Theorem**

**1.**

**Proof.**

#### 5.3. Controller Preprocessing

## 6. Simulation and Analysis

#### 6.1. Real-Time Performance Analysis of the IMPSO-RBFNN

#### 6.2. Delay Compensation Effect of Different Characteristic Periods

#### 6.3. Compared with Other Control Methods

_{∞}control method [13] with $c=5.5$, ${k}_{s}=0.8$, and $\eta =0.8$ are selected to be compared with the proposed method. The speed tracking effect of various control methods when the reference signal is a sine wave is shown in Figure 4. To simulate the real operating condition of the train, the control performances of various methods when the reference signal is a variety of operating modes were analyzed, as shown in Figure 5. In addition, the tracking error statistics are shown in Table 4.

#### 6.4. Discussion

## 7. Conclusions and Prospects

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

RBFNN | RBF Neural Network |

IMPSO | Improved Multi-strategy Particle Swarm Optimization |

NFTSM | Non-Singular Fast Terminal Sliding Mode |

ATO MVB | Automatic Train Operation Multifunction Vehicle Bus |

TCN | Train Communication Network |

SM | Sliding Mode |

PSO | Particle Swarm Optimization |

TSM | Terminal Sliding Mode |

NTSM | Non-Singular Terminal Sliding Mode |

FTSM | Fast Terminal Sliding Mode |

NFTSM AR LMS | Non-Singular Fast Terminal Sliding Mode Auto Regressive Least Mean Square |

CCU | Central Control Unit |

SMAR | Sliding Mode Adaptive Robust |

## Appendix A

Notation | Meaning |
---|---|

$v(k)$ | speed of the train |

$x(k)$ | position of the train |

${v}_{d}(k)$ | the desired speed of the train |

${x}_{d}(k)$ | the desired position of the train |

$F(k)$ | the net force for the train |

$u(k)$ | the control force for the train |

$\eta $ $d$ | acceleration coefficient the train rotary mass coefficient |

${f}_{l}(k)$ | the mutual influence of other vehicles on the reference vehicle |

$f(k)$ | the resistance for the train |

${f}_{0}(v(k))$ | general resistance |

${f}_{r}(k)$ | ramp resistance |

${f}_{c}(k)$ | curve resistance |

${f}_{t}(k)$ | tunnel resistance |

${a}_{0i}(k)$, ${a}_{1i}(k)$ and ${a}_{2i}(k)$ | the resistive coefficients for the ${i}^{th}$ vehicle |

${\theta}_{i}(k)$ | the gradient angle of the rail for the ${i}^{th}$ vehicle |

$A$ | the parameter obtained through test |

${R}_{i}(k)$ | the radius of the curve for the ${i}^{th}$ vehicle passes |

${L}_{ti}(k)$ | the length of the tunnel for the ${i}^{th}$ vehicle |

$\mathsf{\Delta}{x}_{di}(k)$ | the shaped variable in the elastic coupler of the ${i}^{th}$ vehicle |

${m}_{j}$ | the mass of the ${j}^{th}$ vehicle |

${\omega}_{k}$ | the output weight between the ${k}^{th}$ hidden and output neuron |

${\varphi}_{k}$ | the output of the ${k}^{th}$ hidden neuron |

${c}_{k}$ | the center vector of the ${k}^{th}$ hidden neuron |

${b}_{k}$ | the width of the ${k}^{th}$ hidden neuron |

$E$ | the error function |

$\beta $ | the learning rate |

$\delta $ | the momentum factor |

$Num$ | the swarm size |

${x}_{i}^{(k)}$ | the position of the ${i}^{th}$ particle in the ${k}^{th}$ iteration |

${v}_{i}^{(k)}$ | the velocity of the ${i}^{th}$ particle in the ${k}^{th}$ iteration |

${p}_{i}^{(k)}$ | the best position of the ${i}^{th}$ particle in the ${k}^{th}$ iteration |

${p}_{g}^{(k)}$ | the best position obtained by the swarm in the ${k}^{th}$ iteration |

$w$ | the inertia weight |

${c}_{1}$ and ${c}_{2}$ | the acceleration constants |

$rand$ | the random value uniformly distributed in (0,1) |

$f({v}_{ij}^{(k+1)})$ | the sigmoid function |

${w}_{\mathrm{max}}$ | the maximum inertia weight |

${w}_{\mathrm{min}}$ | the minimum inertia weight |

${k}_{\mathrm{max}}$ | the total number of iterations |

${r}_{1}$,${r}_{2}$ and ${r}_{3}$ | the different integers in $[1,Num]$ |

$value{-\mathrm{I}}_{i}(k)$ | the immediate value |

$value-{F}_{i}(k)$ | the future value |

$valu{e}_{i}(k)$ | the comprehensive value |

$f({x}_{i},k)$ | the fitness in the ${k}^{th}$ iteration |

${N}_{i}^{p}(k)$ | the number of success of the ${i}^{th}$ strategy used by individual before the ${k}^{th}$ iteration |

${M}_{i}^{p}(k)$ | the total number of executions of the ${i}^{th}$ strategy used by individual before the ${k}^{th}$ iteration |

${N}_{i}^{g}(k)$ | the number of success of the ${i}^{th}$ strategy used by all the individual before the ${k}^{th}$ iteration |

${M}_{i}^{g}(k)$ | the total number of executions of the ${i}^{th}$ strategy used by all the individual before the ${k}^{th}$ iteration |

$\mu $ | the constant |

${C}_{0}$ | the constant |

${P}_{i}(k)$ | the probability of each strategy adopted in the ${k}^{th}$ iteration |

${P}_{\mathrm{min}}$ | the minimum selection probability of each strategy |

$M$ | the total number of strategy |

$\overline{f}$ | the mean value of $n$ fitness |

$\lambda $ | threshold |

${T}_{ca}$ | forward channel timestamp |

${\tau}_{ca}$ | forward channel delay |

${T}_{sc}$ | timestamp |

$T$ | sampling period |

${e}_{x}\left(k\right)$ | position tracking error |

${e}_{v}(k)$ | speed tracking error |

${d}_{f}(k)$ | additional disturbance |

$s(k)$ | FTSM surface |

${\alpha}_{1}$ and ${\alpha}_{2}$ | positive diagonal matrixes |

${\gamma}_{1}$ and ${\gamma}_{2}$ | the constant |

${v}_{r}(k)$ | auxiliary variable |

$\epsilon $ | the reconstruction error of the RBFNN |

$\rho $ | known positive constant |

${K}_{1}$ and ${K}_{2}$ | positive diagonal matrices |

${\vartheta}_{z}(k)$ | the singular item |

$\varpi $ | positive constant |

$V(k)$ | Lyapunov function |

$\mathrm{g}(x)$ | fitness function |

$L$ | the train stop time |

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**Figure 1.**Improved network control system of high-speed train based on Improved multi-strategy particle swarm optimization RBF neural network (IMPSO-RBFNN).

**Figure 3.**Comparison of delay compensation effect of different characteristic periods: (

**a**) Characteristic period 64 ms, (

**b**) characteristic period 128 ms.

**Figure 5.**Control performances of various methods under multiple operating modes: (

**a**) Speed tracking under multiple operating modes; (

**b**) position tracking under multiple operating modes; (

**c**) control input under multiple operating modes.

Parameters | Value |
---|---|

Total train weight (ton) | 400 |

Maximum operating speed (km/h) | 350 |

Sustained operating speed (km/h) | 300 |

Rotary mass coefficient | $d=0.06$ |

Unit general resistance (N/KN) | ${f}_{0}(v(k))=0.53+0.0039v(k)+0.000114{v}^{2}(k)$ |

Parameters | Value | Unit |
---|---|---|

${m}_{i}$ | $50+\mathsf{\Delta}{m}_{i},\mathsf{\Delta}{m}_{i}\in [-7,7]$ | ton |

${\theta}_{i}$ | ${\theta}_{i}\in [0.9,3]$ | $\circ $ |

${a}_{0i},{a}_{1i},{a}_{2i}$ | ${a}_{0i}\in [0.052,0.057]$ ${a}_{1i}\in [0.0038,0.00393]$ ${a}_{2i}\in [0.000112,0.0001156]$ | - |

${R}_{i}$ | ${R}_{i}\in [2,7.2]$ | km |

${L}_{ti}$ | ${L}_{ti}\in [0,6]$ | km |

$\mathsf{\Delta}{x}_{di}$ | $\mathsf{\Delta}{x}_{di}\in [1,1.5]$ | mm |

$A$ | 600 | - |

${K}_{1},{K}_{2}$ | ${K}_{1}=30.85,{K}_{2}=31.56$ | - |

${\gamma}_{1},{\gamma}_{2}$ | ${\gamma}_{1}=1.06,{\gamma}_{2}=0.06$ | - |

${\alpha}_{1},{\alpha}_{2}$ | ${\alpha}_{1}=0.81,{\alpha}_{2}=-2.31$ | - |

$\varpi $ | 0.45 | - |

$\beta $ | 0.05 | - |

$\delta $ | 0.35 | - |

neurons | 13 | - |

Neurons | Approximation Error | Training Time/s |
---|---|---|

3 | 0.9962 | 1.7044 × 10^{−5} |

6 | 0.5955 | 2.7368 × 10^{−5} |

9 | 0.5128 | 2.8476 × 10^{−5} |

12 | 0.4171 | 3.0462 × 10^{−5} |

13 | 0.3580 | 3.2155 × 10^{−5} |

14 | 0.3862 | 3.3030 × 10^{−5} |

17 | 0.4019 | 3.7213 × 10^{−5} |

Tracking Error | Our Method | RBFNN [5] | NFTSM [29] | SMAR [13] |
---|---|---|---|---|

Maximum speed tracking error (km/h) | 4.888 | 9.3379 | 4.506 | 5.39 |

Minimum speed tracking error (km/h) | 9.724 × 10^{−8} | 7.639 × 10^{−9} | 2.614 × 10^{−4} | 6.669 × 10^{−5} |

Mean speed tracking error (km/h) | 0.048 | 0.32 | 0.2 | 0.229 |

Maximum position tracking error (km) | 0.005 | 0.038 | 0.039 | 0.037 |

Minimum position tracking error (km) | 2.5 × 10^{−4} | 2.833 × 10^{−5} | 2.5 × 10^{−4} | 2.5 × 10^{−4} |

Mean position tracking error (km) | 0.003 | 0.018 | 0.025 | 0.02 |

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**MDPI and ACS Style**

Kong, X.; Zhang, T.
Non-Singular Fast Terminal Sliding Mode Control of High-Speed Train Network System Based on Improved Particle Swarm Optimization Algorithm. *Symmetry* **2020**, *12*, 205.
https://doi.org/10.3390/sym12020205

**AMA Style**

Kong X, Zhang T.
Non-Singular Fast Terminal Sliding Mode Control of High-Speed Train Network System Based on Improved Particle Swarm Optimization Algorithm. *Symmetry*. 2020; 12(2):205.
https://doi.org/10.3390/sym12020205

**Chicago/Turabian Style**

Kong, Xiangyu, and Tong Zhang.
2020. "Non-Singular Fast Terminal Sliding Mode Control of High-Speed Train Network System Based on Improved Particle Swarm Optimization Algorithm" *Symmetry* 12, no. 2: 205.
https://doi.org/10.3390/sym12020205