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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.

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 [

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 [

In the most existing literature of the high-speed train, the resistance coefficients of train were often assumed to be constant [

Model predictive control cannot only deal with multiobjective constraint problem, but its dynamic response is fast [

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:

The rest of this paper is arranged as following. In Section

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.

Figure

The multiple mass-point model structure of high-speed train.

The multiple mass-point dynamic equation of a train can be described as

It can be seen from

Assume that when the velocity of the high-speed train reaches the desired velocity, the current state of the train is the equilibrium state. The velocity of a high-speed train at equilibrium state is denoted as

So we define the error displacement variable as

Choosing

To be exact,

Then the above continuous time-domain state-space equation is discretized by the zero-order hold method with sampling period

As shown in Figure

Adaptive model predictive control scheme.

Cruise control of high-speed train must track the desired velocity profile quickly so that the train arrives at its destination on time. With the development of high-speed railway, energy-saving driving and safe driving are of much concern. So, the optimization objective function we set up includes energy consumption, velocity tracking, and the relative displacements between neighboring cars. In this paper, the control input is used to express energy consumption.

Consequently, the optimization objective function can be established as follows:

As a practical system, in order to ensure the safe and efficient operation of high-speed trains, some specific constraints must be satisfied as follows.

First, the traction and brake forces are bounded because of the nature physical characteristics of the traction motor. Second, the maximum allowable speed of high-speed trains is affected not only by line conditions and operating conditions, but also by their physical characteristics. Third, coupler force must be manipulated to vary in an acceptable range in order to ensure the train’s run safety. In this paper, the coupler deformation is used to represent intrain forces characteristic. Then, the constraints can be illustrated by the following inequality:

where

For (

Design an estimated system for (

The actual high-speed trains state and the estimated system state can be rewritten into another forms

Define a cost function for the estimated error

In the MPC framework,

Suppose that the output of the estimated system is given by

The optimization objective function can be written as

In order to minimize objective function (

Each

To facilitate the MPC design, constraints (

The relative displacements constraints between neighboring cars are included in constraints of the state variable x(k). Designing a matrix

Consequently, the relative displacements constraints

Combining constraints (

The proposed adaptive MPC algorithm can be summarized as follows:

A simulation study on a high-speed train is presented to demonstrate the effectiveness of the proposed adaptive MPC algorithm. The simulation in this paper is to solve the optimization problem of model prediction with quadprog toolbox of MATLAB simulation software version 2016b under the system environment of Windows 10 operating system. The parameters of the train model are from the CRH-3 high speed train in China, which are given in Table

Parameters of the CRH-3 high-speed train.

Symbol | Value | Unit |
---|---|---|

| 47.5 | t |

| | |

| | |

| | |

| | |

In order to evaluate the performance of the controller, the desired velocity curve including accelerating, decelerating, velocity step increase, velocity step decrease, and constantvelocity stages, the speed command of high-speed train is given in Table

Speed Command of high-speed train.

Phase | Time(s) | Velocity(m/s) |
---|---|---|

acceleration | 0 | 0 |

cruise | 100 | 40 |

acceleration | 400 | 40-70 |

cruise | 450 | 70 |

deceleration | 850 | 70-50 |

cruise | 900 | 50 |

Figure

Speed curves for each car of high-speed train in the cruise phases.

Estimated output errors.

The curves for the tracking and braking forces of each car in the cruise phases are plotted in Figure

The force output of each car.

The curves of relative spring displacements between the two neighboring cars are plotted in Figure

The coupler deformation between neighboring cars.

The norm of estimation errors is defined by

Norm of estimation errors.

The variation of estimated parameters.

In this subsection, we further discuss the performance of the proposed adaptive model predictive controller in terms of superiority and computation efficiency.

In order to verify the superiority of the method proposed in this paper, we make a simulation comparison with the method in literature [

Comparison of adaptive control system with nonadaptive control system in predicted output velocity and the desired velocity error.

Adaptive control system

Nonadaptive control system

The coupler force of each car is plotted in Figure

Comparison of adaptive control system with nonadaptive control system in predicted output velocity and the desired velocity error.

Adaptive control system

Non adaptive control system

In this subsection, by considering the different prediction horizons, this paper considers the computation complexity of the adaptive model predictive control with different prediction horizons. Within different prediction horizons, the adaptive MPC and the traditional MPC are respectively implemented on (

Computation time of adaptive MPC and traditional MPC.

Algorithm | | | |
---|---|---|---|

Adaptive MPC | 9.12s | 10.45s | 11.96s |

Traditional MPC | 10.08s | 11.06s | 13.29s |

In this paper, the optimal cruise control of high-speed trains with time-varying air resistance coefficients and control constraints is investigated. The control objective is accurate speed tracking control with minimum energy consumption and safe relative displacement between two neighbored cars. First, a multiple-mass-point model of high-speed trains is built. By considering multiple constraints and performance metrics, an adaptive MPC method is proposed to design the cruise control controller. In order to improve the accuracy of the method, a dynamic estimated system model of high-speed trains with time-varying parameters is proposed. Also, an adaptive updating law for estimated system parameters by the Lyapunov stability theory is designed. Then the optimization objective and operation constraints are analyzed in detail. In addition, the cruising control problem is transformed into a constrained finite-time optimal control problem with aquadratic objective function, which can be uniformly solved by a quadratic programming approach. Using the method in this paper, the high-speed trains track the desired speed quickly and precisely, and the relative spring displacement between the two neighbored cars is stable at the equilibrium state. Performance of the closed-loop system is substantiated by simulation results.

The simulation result data used to support the findings of this study have been deposited in the

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The research work is supported by National Nature Science Foundation of China (Grant Nos. 61772558, 61672537, 61873353, 61672539).

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_{∞}cruise controller design for high speed train