# Automatic Train Control with Actuator Saturation Using Contraction Theory

## Abstract

Application of contraction theory provides a platform to analyze the exponential stability of nonlinear system. To effective compensate the time-varying parametric uncertainties exist in the longitudinal dynamics of the train, this paper proposes a contraction based saturated adaptive robust control for automatic train operation, which is subject to input limit of actuator. With consideration of actuator saturation, a recently developed robust modification is used to saturate the control input in each step, while the global stability is preserved. The resulting saturated adaptive robust control renders the transformed error enter the predefined region, furthermore, inside the predefined region, the contracting behaviour of closed-loop dynamics is regain and the tracking error exponentially converge to a residual set subsequently. The results of comparative experiments under different control strategies also verify the effectiveness of the proposed saturated adaptive robust control.

## Keywords

Automatic train operation Actuator saturation Contraction theory Adaptive robust control## 1 Introduction

Railway transportation has been widely constructed and utilized all over the world, especially the high-speed railway in recent decades. To meet the high safety and efficiency requirements of high-speed train, the automaticity in train control is continuously raising. As an essential part of automatic train control (ATC), automatic train operation (ATO) is an on-board equipment whose main function is speed regulation to render the train tracking with the desired trajectory [1].

The desired trajectory is generated by certain optimal algorithm in accordance with prescribed operating criteria, which mainly includes safety, ride-comfort, timetable, energy-efficiency, etc. The discussion of optimal algorithm can be found in [2, 3, 4], and this paper focus on improving the tracking performance of ATO. As travelling speed increases, the effect caused by the inherent time-varying uncertainties in the longitudinal dynamics of the train keeps growing, so excellent tracking performance is not easy to achieve. There are much research had been done to reduce the dependence on inaccurate modeling process, including [5, 6, 7, 8, 9].

In this paper, based on the treatment of actuator saturation is motivated by [11], we further improved in [10], where the actual state is replaced by desired velocity in both model compensation and parameter adaptation. The overall saturated adaptive robust control is designed to preserve the contracting behaviour of closed-loop dynamics during normal working range while guarantee global stability for large magnitude of modeling uncertainties.

## 2 Saturated Adaptive Robust Control

- Step 1.
By considering the saturation problem, the control law \( \alpha_{1} \) is re-designed as

- (i)
\( \left| {\sigma_{1} } \right.\left. {(z_{1} )} \right| = M_{1} , \) if \( \left| {z_{1} } \right| \ge L_{12} ; \)

- (ii)
\( z_{1} \sigma_{1} (z_{1} ) > 0, \) for all \( z_{1} \ne 0; \)

- (iii)
\( \left\{ {\begin{array}{*{20}l} {\frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\left( {z_{1} } \right) = \iota_{1} ,} \hfill & {\text{if}\,\left| {z_{1} } \right| < L_{11} ,} \hfill \\ {\frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\left( {z_{1} } \right) \le \iota_{1} ,} \hfill & {\text{if}\,L_{11} \le \left| {z_{1} } \right| < L_{12} ,} \hfill \\ {\frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\left( {z_{1} } \right) = 0,} \hfill & {\text{if}\left| {z_{1} } \right| \ge L_{12} .} \hfill \\ \end{array} } \right. \)

*Step 1*, \( \upiota_{1} ,L_{11} ,L_{12} ,M_{1} \), are controller parameters.

- Step 2.
For the control law given by [11], both \( u_{a} \) and \( u_{s} \) should to be bounded to satisfy the actual input constraints. Due to the difference introduced by \( \sigma_{1} \), we re-expressed Eq. (3) as

- Step 3.$$ \dot{z}_{2} { = }\frac{1}{m}\left[ {u_{a} - \theta^{T} \phi - m\left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right)} \right] + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}z_{2} + \frac{1}{m}u_{s} . $$(4)

In *Step 2*, \( \upiota_{2} ,L_{2} ,M_{2} \), are controller parameters.

### Theorem 1

*If the saturated adaptive robust control*Eqs. (13)

*and*(6)

*is applied to train model, while the selection of design parameters satisfying (a)*\( \upiota_{2} > \bar{m}\upiota_{1} \),

*(b)*\( \upiota_{1} L_{11} > L_{22} \)

*, (c)*\( M_{1} \le \left( {\underline{c}_{\upsilon } /\bar{c}_{a} } \right) \)

*, (d)*\( M_{2} > \bar{m}(h + k_{1} M_{1} ) \)

*, (e)*\( - \bar{B} - \underline{u}_{a} \le M_{2} \le \bar{F} - \bar{u}_{a} \)

*, then the following results can be guaranteed*:

- i)
The closed-loop system is global stable.

- (ii)
At steady-state, the location and velocity tracking errors of the train converge to the following residual set

## 3 Comparative Experiments

### 3.1 System Setup and Implementation

*CRH3*series high-speed trains, which have been operating in

*Beijing-Tianjin Intercity Railway*for more than five years, and the nominal values of their primary parameters are listed in Table 1, where, \( \omega \) denotes the running resistance of mass per unit quality, \( \bar{F} \) denotes the traction capacity of actual motors, B1 denotes the capacity limit of regenerating braking, \( \bar{B}_{2} \) denotes the capacity limit of electropneumatic braking. The force of traction and regenerating braking that train traction motors can supply are correlated with travelling velocity, as shown in paper [11]. The electropneumatic braking is a reserve equipment, which is enabled in the situation of regenerating braking is deficiency (usually in low travelling velocity range), so the total braking force can be calculated from \( \bar{B} = \bar{B}_{1} + \bar{B}_{2} \).

Parameters of *CRH-300*

Category | Value | Condition | Unit |
---|---|---|---|

\( m \) | 475 | \( {\text{t}} \) | |

\( \omega \) | \( 7.75 + 0.06327\,\upsilon + 0.00128\;\upsilon^{2} \) | \( {\text{N}}/{\text{t}} \) | |

\( \bar{F} \) | \( - 0.285\,\upsilon + 300 \) | \( \upsilon < 119\;{\text{km/h}} \) | \( {\text{kN}} \) |

\( \frac{31500}{\upsilon } \) | \( \upsilon \ge 119\;{\text{km}}/{\text{h}} \) | ||

\( \bar{B}_{1} \) | \( 59.8\;\upsilon \) | \( \upsilon < 5\;{\text{km}}/{\text{h}} \) | \( {\text{kN}} \) |

\( - 0.285\;\upsilon + 300 \) | \( 5\;{\text{km}}/{\text{h}} \le \upsilon \,106.7\;{\text{km}}/{\text{h}} \) | ||

\( \frac{28800}{\upsilon } \) | \( \upsilon \ge 106.7\;{\text{km}}/{\text{h}} \) | ||

\( \bar{B}_{2} \) | 450 | \( {\text{kN}} \) |

The controller parameters are selected as follows, \( \upiota_{1} = 0.5,{\text{L}}_{11} = 2m \) (\( m \): denotes distance unit meter), \( {\text{L}}_{12} = 3\;{\text{m}},M_{1} = 1.25,\upiota_{2} = 0.6\;{\bar{\text{m}}},{\text{L}}_{2} = 0.3\;{\text{m}}/{\text{s}}, \) \( M_{2} = \left\{ {\begin{array}{*{20}c} {\bar{F} - u_{a} ,} & {\text{if}\,u\text{ > }0} \\ { - B - u_{a} ,} & {\text{if}\,u < 0} \\ \end{array} } \right. \)

### 3.2 Comparable Results

*Gaussian white noises*are added to the nominal values of resistance coefficients to obtain time-varying parameters, and abrupt changes are imposed at s = 17 km, s = 22 km to simulate the train travelling through a tunnel, as shown in Fig. 3. To give a comparison, we construct the following two cases:

## 4 Conclusion

A Bead on contraction theory, a saturated adaptive robust control strategy is proposed to improve the tracking performance of ATO which is subject to the capacity limit of practical actuator. By using contraction analysis, the selection of Lyapunov-like energy function is not needed, and this dramatically facilitate the controller design of nonlinear systems, furthermore, it provides a platform to analyze the exponential stability of the closed-loop dynamics. With consideration of actuator saturation, a robust modification of saturation is developed, which preserves the global stability of the closed-loop system and renders the transformed output error with any initial state converging to the prescribed normal region. When the transformed error enters the normal region, the contraction behaviour of the closed-loop dynamics is regain, and train tracking errors exponential converge to a residual set subsequently. Comparative experiments under different control strategies are constructed, and the results con.rm the superiority of the proposed saturated adaptive robust control.

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