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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

With consideration of actuator saturation, the control law should to be synthesized to satisfy input constraint. By using the same structure of contraction adaptive control developed in [10], the saturated adaptive robust control is developed as follows.
  1. Step 1.

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

     
$$ \left\{ {\begin{array}{*{20}l} {\alpha_{1} = \alpha_{1a} + \alpha_{{ 1\text{s}}} } \hfill \\ {\alpha_{1a} = \dot{x}_{d} } \hfill \\ {\alpha_{{ 1 {\text{s}}}} = - \sigma_{1} (z_{1} )} \hfill \\ \end{array} } \right. $$
(1)
where \( \sigma_{1} (z_{1} ) \) is a smooth saturation function with respect to \( z_{1} \), and it is designed to have the following properties:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
Since \( \alpha_{1a} \) is bounded, the overall \( \alpha_{1} \) can be bounded. By applying Eq.  (1), we obtain
$$ \dot{z}_{1} = - \sigma_{ 1} ( {\text{z}}_{ 1} )+ {\text{z}}_{ 2} , $$
(2)
$$ \dot{z}_{2} { = }\frac{1}{m}u - \theta^{T} \phi - \ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}( - \sigma_{1} + z_{2} ), $$
(3)
In Step 1, \( \upiota_{1} ,L_{11} ,L_{12} ,M_{1} \), are controller parameters.
  1. 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

     
  2. 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)
     
Intuitively, \( u_{a} \) should to be designed to cancel the model dynamics \( - \hat{\theta }^{T} \phi - \hat{m}\left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right), \) however, since \( \phi (x) \) depends on \( x_{2} \), it is impossible to put a bound on \( u_{a} \). Motivated by [11], we re-formulate the regressor as a function of the desired trajectory only and re-design \( u_{a} \) as
$$ u_{a} = \hat{\theta }^{T} \phi_{d} + \hat{m}\left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right), $$
(5)
where \( \phi_{d} = \phi (\dot{x}_{d} ) = \left[ {1,\dot{x}_{d} ,\dot{x}_{d}^{2} } \right]^{T} \). Automatically, the adaptive law is modified as
$$ \left\{ {\begin{array}{*{20}l} {\dot{\hat{\theta }} = \Pr \left[ { - \varGamma \phi_{d} z_{2} } \right],} \hfill \\ {\dot{\hat{m}} = \Pr \left[ { - \gamma \left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right)z_{2} } \right],} \hfill \\ \end{array} } \right. $$
(6)
where \( \varGamma = \varGamma^{T} > 0 \), \( \gamma > 0 \) are adaptive gain. Due to the property of \( \Pr ( \bullet ) \), the lower bound \( \underline{u}_{a} \) and upper bound \( \bar{u}_{a} \) of \( u_{a} \) can be easily determined. Furthermore, the desired compensation control Eqs. (5) and (6) also enhance the capacity of noise-rejection in implementation, and large adaptation gain can be used to improve train tracking performance, as discussed in [11].
By applying Eq.  (5), we obtain
$$ \dot{z}_{2} { = }\frac{1}{m}\left[ {\hat{\theta }^{T} \phi_{d} - \theta^{T} \phi + \tilde{m}\left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right)} \right] + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}z_{2} + \frac{1}{m}u_{s} . $$
(7)
Noting that \( \hat{\theta }^{T} \phi_{d} - \theta^{T} \phi \) can be re-written as \( \tilde{\theta }^{T} \phi_{d} + \theta^{T} (\phi_{d} - \phi ) \). So Eq. (7) can be forward expressed as
$$ \dot{z}_{2} { = }\frac{1}{m}\left[ {\tilde{\theta }^{T} \phi_{d} + \tilde{m}\left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right) + \theta^{T} (\phi_{d} - \phi )} \right] + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}z_{2} + \frac{1}{m}u_{s} . $$
(8)
Equations (1) and (2), we have
$$ \begin{aligned} \theta^{T} (\phi_{d} - \phi ) & = \theta^{T} \left[ {0,\sigma_{1} - z_{2} ,(\sigma_{1} - z_{2} )(\dot{x}_{d} + x_{2} )} \right]^{T} \\ & = - \left[ {c_{\upsilon } + c_{a} (\dot{x}_{d} + x_{2} )} \right]z_{2} + c_{a} \sigma_{1} z_{2} + c_{\upsilon } \sigma_{1} + c_{a} \sigma_{1} (2\dot{x}_{d} - \sigma_{1} ). \\ \end{aligned} $$
(9)
Substituting Eq. (9) into Eq.(8), we have
$$ \begin{aligned} \dot{z}_{2} { = }\frac{1}{m}\left[ {\tilde{\theta }^{T} \phi_{d} + \tilde{m}\left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right) + c_{\upsilon } \sigma_{1} + c_{a} \sigma_{1} (2\dot{x}_{d} - \sigma_{1} )} \right] \\ - \frac{1}{m}\left[ {c_{\upsilon } - c_{a} \sigma_{1} + c_{a} (\dot{x}_{d} + x_{2} )} \right]z_{2} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}z_{2} + \frac{1}{m}u_{s} . \\ \end{aligned} $$
Since \( M_{1} \ge \left| {\sigma_{1} } \right| \) can be designed, if we choose \( M_{1} \le \left( {\underline{c}_{\upsilon } /\bar{c}_{a} } \right) \), then the term \( - c_{a} \sigma_{1} \) can be dominated by \( c_{\upsilon } \), i.e., \( c_{\upsilon } - c_{a} \sigma_{1} \ge 0 \). Letting \( \varepsilon_{2} \triangleq c_{\upsilon } - c_{a} \sigma_{1} + c_{a} (\dot{x}_{d} + x_{2} ) \), because parameter \( c_{a} \), and desired and actual train speed \( \dot{x}_{d} \), \( x_{2} \) are all positive-definite, \( \varepsilon_{2} \) is positive-definite. Then, the derivative \( \dot{z}_{2} \) can be forward expressed as
$$ \dot{z}_{2} { = }\frac{1}{m}\left[ {\tilde{\theta }^{T} \phi_{d} + \tilde{m}\left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right) + c_{\upsilon } \sigma_{1} + c_{a} \sigma_{1} (2\dot{x}_{d} - \sigma_{1} )} \right] - \varepsilon_{2} z_{2} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}z_{2} + \frac{1}{m}u_{s} . $$
(10)
The residual term \( - \varepsilon_{2} z_{2} \) acts as a damping term to help stabilizing the dynamic \( \dot{z}_{2} \), so the robust control \( u_{s} \) needs to suppress \( \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}z_{2} \) and model mismatch to render \( z_{2} \) converge or at least, bounded. By considering the saturation problem, the robust control is designed as sign
$$ u_{s} = - \sigma_{2} (z_{2} ), $$
(11)
$$ - \sigma_{2} \left( {z_{2} } \right) = \left\{ {\begin{array}{*{20}c} {\iota_{2} z{}_{2},} & {\text{if}\,\left| {z_{2} } \right| < L_{2} } \\ {\text{sign}\left( {z_{2} } \right)M_{2} } & {\left| {z_{2} } \right| \ge L_{2} } \\ \end{array} } \right. $$
(12)
Noting this is the last step, thus \( \sigma_{2} (z_{2} ) \) only needs to be continuous. Thus, the overall of the control input is given by
$$ \left\{ {\begin{array}{*{20}l} {u = u_{a} + u_{s} ,} \hfill \\ {u_{a} = \tilde{\theta }^{T} \phi_{d} + \hat{m}\left( {\ddot{x}_{d} + \frac{{\partial \sigma_{1} }}{{\partial z_{1} }}\sigma_{1} } \right),} \hfill \\ {u_{s} = - \sigma_{2} (z_{2} ).} \hfill \\ \end{array} } \right. $$
(13)

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:
  1. i)

    The closed-loop system is global stable.

     
  2. (ii)

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

     
$$ \Omega _{d}^{{\prime }} = \left\{ {e(t) \in {\mathbb{R}}^{2} \left| {\left| e \right| \le \frac{h}{{k_{2} }}} \right.} \right\}, $$
where, \( h,k_{2} \) are positive constants, which will be specified in the proof analysis.

3 Comparative Experiments

3.1 System Setup and Implementation

Comparative experiments are constructed in the form of numerical simulation to demonstrate the effectiveness of the proposed saturated adaptive robust control. The train dynamics under study is stemmed from 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 \( \bar{B} = \bar{B}_{1} + \bar{B}_{2} \).
Table 1

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

\( {\text{kN}} \)

From design and empirical record, the prior region of train dynamics is defined as
$$ \Theta \in \left[ {\left[ {450,1,0.02,0.0005} \right]^{T} ,\left[ {500,20,0.2,0.005} \right]^{T} } \right]. $$

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

where \( u_{a} \) can be easily calculated on-line from Eq. (5). It can be verified that these parameters satisfy the selection guidance of Theorem 1. Apparently, the saturation function \( \sigma_{1} ,\sigma_{2} \) are also determined, as shown in Fig. 1.
Fig. 1

Saturation function designed for \( z_{1} ,z_{2} \)

3.2 Comparable Results

The experiment is constructed to operate the train travelling through a distance of 28.565 km, and the whole process is design to consist of three typical mode: acceleration, cruising and braking in turn. The desired location and velocity are given in Fig. 2. Some 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:
Fig. 2

Desired location and velocity trajectory

Fig. 3

Time-varying resistance coefficients

  1. (i)

    The contraction adaptive control given in [11] is used, but the input is truncated when the control effort reaches the actuator limit.

     
  2. (ii)

    The saturated adaptive robust control given in Theorem 1 is used.

     
Both the cases are running with the same configurations. The tracking errors and control input under Case i and Case ii are given in Figs. 4 and 5, and the comparison of transient response and steady-state tracking performance is given in Fig. 6. Under Case (i), the control input is truncated at the place marked by ellipse as shown in Fig. 4b, and the tracking error is enlarged consequently as shown in Fig. 6. Under Case (ii), the control input generated by the proposed saturated adaptive robust control are always within the capacity limit of actuator as shown in Fig. 5b, and compensates the model dynamics effectively, so the closed-loop dynamics is working within the normal region \( \Omega _{C} \) for almost all the time as shown in Fig. 5a, and superior transient and steady-state tracking performance is achieved as shown in Fig. 6.
Fig. 4

Tracking errors and control input under Case (i)

Fig. 5

Tracking errors and control input under Case (ii)

Fig. 6

Transient response comparison

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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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.BeijingChina

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