Journal of Advanced Transportation

Volume 2019, Article ID 7261726, 11 pages

https://doi.org/10.1155/2019/7261726

## Adaptive Model Predictive Control for Cruise Control of High-Speed Trains with Time-Varying Parameters

^{1}School of Information Science and Engineering, Central South University, Changsha 410000, China^{2}Hunan Engineering Laboratory of Rail Vehicles Braking Technology, Changsha 410000, China^{3}Changsha University of Science and Technology, Changsha 410000, China

Correspondence should be addressed to Yingze Yang; nc.ude.usc@ezgniygnay

Received 14 December 2018; Revised 28 March 2019; Accepted 2 April 2019; Published 2 May 2019

Guest Editor: Belen M. Batista

Copyright © 2019 Xiaokang Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The cruise control of high-speed trains is challenging due to the presence of time-varying air resistance coefficients and control constrains. Because the resistance coefficients for high-speed trains are not accurately known and will change with the actual operating environment, the precision of high speed train model is lower. In order to ensure the safe and effective operation of the train, the operating conditions of the train must meet the safety constraints. The most traditional cruise control methods are PID control, model predictive control, and so on, in which the high-speed train model is identified offline. However, the traditional methods typically suffer from performance degradations in the presence of time-varying resistance coefficients. In this paper, an adaptive model predictive control (MPC) method is proposed for cruise control of high-speed trains with time-varying resistance coefficients. The adaptive MPC is designed by combining an adaptive updating law for estimated parameters and a multiply constrained MPC for the estimated system. It is proved theoretically that, with the proposed adaptive MPC, the high-speed trains track the desired speed with ultimately bounded tracking errors, while the estimated parameters are bounded and the relative spring displacement between the two neighboring cars is stable at the equilibrium state. Simulations results validate that proposed method is better than the traditional model predictive control.

#### 1. Introduction

In recent years, the high-speed railway transportation has played a more and more important role in modern society. High-speed train has many more advantages such as high speed, large volume, and safe and comfortable environment than traditional railway traffic. With the speed of the high-speed trains rising, it is extremely difficult for human drivers to guarantee the safety of the operation of high-speed trains. In order to ensure the safe and effective operation of high-speed trains, the automatic train control (ATC) system is proposed, which is used to monitor, control, and adjust the train operations to guarantee safety, punctuality, and comfort [1–3].

One of the demanding control problems associated with the automatic train control (ATC) is cruise control problem in which the speed of the train is automatically controlled to follow a desired trajectory. The methods proposed for cruise control of high-speed trains which are developed based on a motion model obtained from Newton’s second law can be classified into two categories. One is to model the whole train that consists of multiple cars as a single point mass [4, 5], while the interaction force between the two cars of the train is ignored. Considering the relative movement between the two cars, the other one is to construct the high-speed train model by a cascade of masses connected by flexible couplers, which provides much more accuracy in characterizing the dynamics of the high-speed trains [6, 7].

In the most existing literature of the high-speed train, the resistance coefficients of train were often assumed to be constant [8–10]. However, for a high-speed train, the aerodynamic resistance will change large, when the train is traveling at high speed. So, the dynamic motion model of the train is a time-varying model dependent on the operating conditions. The robust adaptive tracking control method was derived for a multiple-mass-points high-speed train dynamics model with unknown and time varying resistance coefficients [11]. Besides, the robust output feedback cruise control is developed for speed tracking with the unknown parameters [12]. Moreover, some more complex train operating conditions are considered in paper [13], and an adaptive controller is developed to deal with the problem of the uncertainty of the air resistance coefficients of the piecewise model. However, the studies of the robust adaptive controller neglected the state constraints and control input constraints. The safe speed and the saturation characteristics of traction and braking units are very important for online operation of high-speed train. The model predictive control has an advantage that fully considers the input and state constrains of the system [14, 15]. However, the influence of time-varying air resistance parameters on the system model is neglected in [15], resulting in low system model precision.

Model predictive control cannot only deal with multiobjective constraint problem, but its dynamic response is fast [16]. The goal of cruise control of high-speed trains is to track the desired target speed quickly and accurately, so, model predictive control is very suitable for the high-speed train cruise controller. According to the principle of model predictive control, it is known that model predictive control requires a prediction model with high precision. The accuracy of the prediction model determines the performance of the controller. Therefore, the starting point of this paper is how to improve the accuracy of the dynamics model of high-speed. In this paper, a multibody model of the high-speed train with time-varying air resistance coefficients and control constraints is considered. The train dynamics model set up in this paper contains time-varying resistance coefficients, and the relative movements among the connected cars of the train are considered, so the dynamics model in this paper is more accurate than that in paper [15]. This paper designs an adaptive model predictive control for cruise control of high-speed trains. Based on Lyapunov’s stability theory, an adaptive updating law is given for estimated system model parameters. The closed-loop system is capable of tracking the desired speed, and the relative spring displacements between the two neighbored cars are stable at the equilibrium state.

Compared to the existing work, the proposed adaptive MPC not only solves the cruise control problem with time-varying resistance coefficients but also ensures the train operations within the range of safety constraints. The model complexity is equivalent to the traditional model. The main contributions in this paper can be summarized as follows:

By considering the time-varying air resistance coefficients, this paper firstly constructs a linear multiple-mass-points dynamical model of high-speed trains with time-varying resistance coefficients.

In order to improve the accuracy of the constructed model, an estimated system is proposed based on the proposed model. Then, an adaptive updating law is designed for time-varying parameters of estimated system. Based on the estimated system model, an adaptive model predictive control framework is introduced and the estimated system model is used as the prediction model. Based on the estimated system prediction model, the control problem can be formulated as an objective optimization problem with multiple constraints.

According to the practical requirement, the objective optimization problem with multiple constraints is transformed into the linear quadratic programming problem, which determines the optimal cruise control for the high-speed train with time-varying resistance coefficients and control constraints to improve the safety and energy efficiency of the operation of the train.

The rest of this paper is arranged as following. In Section 2, the linear dynamical model of high-speed train with time-varying parameters is introduced. In Section 3, an adaptive model predictive controller is developed for the train with time-varying parameters to speed tracking. In Section 4, a simulation study is presented to show the performance of the proposed method. Finally, some conclusions are given in Section 5.

#### 2. Dynamic Model of High-Speed Train

In this section, the nonlinear multiple mass point dynamic model of high-speed trains with time-varying resistance coefficients is established by analyzing their dynamical characteristics. It is difficult to design the system controller because of the complex characteristics of the nonlinear model. So, the linear error dynamic model of high-speed train is constructed around the equilibrium point.

##### 2.1. The Dynamic Model of the High-Speed Train

Figure 1 presents the multiple mass-point model structure for high-speed train. A high-speed train contains cars; these cars are connected by flexible couplers. In high-speed train operation, the flexible couplers play an important role in the connected cars and transmit interaction force between two connected cars. As the couplers between two adjacent cars are not perfectly rigid, the function of couplers can be described by a spring model such that the coupler force is a function of the relative displacement between two connected cars.where is the stiffness coefficient, which is positive. The running resistance of high-speed train consists of the rolling mechanical resistance and aerodynamic drag, which is commonly expressed aswhere is the car velocity and , , and are the resistance coefficients and they are bounded time-varying parameters. The parameters change near the nominal value [17]; they are can be described as , , and . , , and and , , and are the nominal value of the resistance coefficients and the deviation value near the nominal value, respectively, where the deviation value is assumed to be time-varying and bounded, i.e., , , and . defines the train’s rolling resistance component, defines the train’s linear resistance coefficient, and defines the train’s nonlinear resistance coefficient. represents the rolling mechanical resistance, and denotes the aerodynamic drag.