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Abstract

Focusing on exploring optimal control method of high-speed train, this research explicitly introduces a fuzzy predictive control framework to enhance safety, comfort, and energy-efficiency. The dynamic equation of the high-speed train is modeled as a cascade of cars connected by spring-like couplers and then represented by Takagi–Sugeno fuzzy model. Based on model predictive control approach, an effort is made on the design of fuzzy predictive controller such that each car tracks the target speed, and every coupler is stabilized to the equilibrium state as well as optimizing the performance function value at every time instant with regard to accelerating and braking constraints. Sufficient conditions for the existence of corresponding controller optimization problem are given in a set of linear matrix inequalities. Some simulations are performed to show the effectiveness of the model and the algorithm.

Keywords

High-speed train optimal operation fuzzy model predictive control linear matrix inequality

Introduction

Background and literature review

Since the world’s first high-speed railway, Japan Tokaido Shinkansen, was put into commercial operation in 1964, high-speed railway systems have attracted much attention as they can provide greater transport capacity, significantly faster speeds, and outstanding features of punctuality when compared to aircraft and auto vehicle. Nowadays, high-speed railways have gone through a rapid development in Europe, Russia, Japan, China, and many other developing countries. As the high-speed railway networks are becoming more and more complex, many new challenging technical and commercial issues such as reduction of energy consumption, safety, and comfort have been brought up. With the breakthrough in computer control and communication technology, the automatic train control (ATC) system is equipped to automatically supervise the train speed to follow a desired trajectory which also makes it possible to operate the train in a safe, energy-efficient, and comfortable way.^1–3

As shown in the literature, the dynamic models of the high-speed train can be divided into two categories: the single-mass point model and the cascade mass point model. Some researchers modeled the whole train as a single point mass and applied the Pontryagin maximum principle to prove that the optimal sequence of train driving strategy should consist of four phases including maximum acceleration, cruising, coasting, and maximum braking.^4–6 Moreover, various algorithms were proposed to determine the optimal switching strategy among different phases considering the real-world operation environments such as track gradient and speed limits, commercial punctuality constraints, and the uncertainty of some performance parameters.^7–9

Although the single-mass-point model makes the controller design much convenient, it overlooks the in-train dynamics which play essential roles when considering the operation safety for a long train. Research based on the cascade-mass-point model, therefore, has become a new trend in recent years. Gruber and Bayoumi¹⁰ investigated the longitudinal dynamics of multi-locomotive powered train and approximated it as a nonlinear mass–spring dashpot model which was interconnected via couplers at the first time. They demonstrated that this method could efficiently minimize the coupler forces which resulted in safer operation and increased traveling speeds. With the cascade-mass-point model, Zhuan and Xia¹¹ proposed an optimal open-loop controller approach and employed the Lagrange multiplier method to minimize the coupler forces between connected cars. With the similar model as Zhuan and Xia,¹¹ Chou and Xia¹² introduced a linear quadratic regulation algorithm to optimize in-train forces as well as energy consumption. Yang and Sun¹³ extended the cascade-mass-point model to describe the dynamic features of high-speed train, and they formulated a mixed $H_{2} / H_{\infty}$ controller to handle the convex optimization problem. Simulation results indicated that this method had excellent flexibility in trade-off between command tracking and disturbance attenuation. Li et al.¹⁴ investigated the sampled-data-based cruise controller and converted the sampling period into a bounded time-varying delay. The stability for the proposed controller was validated, and a prescribed $H_{\infty}$ disturbance attenuation level with regard to the wind gust was ensured. Furthermore, in the study by Li et al.,¹⁵ a new model with uncertain disturbances and parameters was analyzed and the sufficient condition was proved.

The cascade-mass-point model is believed to be much more accurate than the single-mass one; it thus imposes a higher computational complexity in solving the optimization problem. Besides, as the practical train model is nonlinear, scholars insist on finding possible methods to simplify the cascade mass point model while maintaining its merits and advantages. Most researchers choose to linearize the dynamical model to facilitate the controller design by which way some vital factors may be ignored and certain operational constraints are not concerned.^14–16 However, with the increasing train speed, the deviations caused by linearization become more and more significant so that it should be carefully contemplated in control design. With development of the fuzzy logic control in recent decades,^17,18 a lot of researchers begin to take the fuzzy control methods and optimization techniques as alternatives to control problems associated with high-speed train.^19–21

Proposed approach

As shown in the literature, the majority of studies mainly investigate the optimal high-speed train control according to the current information, and future behaviors during the whole travel are usually not taken into consideration. Therefore, these approaches cannot accomplish an overall optimization for the high-speed train movement due to the uncertainty during a long trip. Since model predictive control (MPC) approach is computed repeatedly under certain cost function, this method can adapt to unfavorable factors such as model inaccuracies, operation constraints, and unanticipated changes in system parameters.^22–26 Moreover, it has been proved that Takagi–Sugeno (T-S) fuzzy models can approximate any smooth nonlinear system to any accuracy within a compact set.^27,28 Therefore, this characteristic makes it suitable for dealing with the deviations caused by just linearizing the high-speed train model.

Motivated by above discussions, we use the T-S fuzzy model to represent the nonlinear train dynamics and convert the controller design to a solution of an optimization problem with MPC approach. This article is aimed at offering the following contributions to the growing body of work on optimal train control methods:

To better depict the in-train dynamics, the high-speed train is abstracted as a cascade of cars connected by spring-like couplers. Different to most high-speed train controllers which linearize the train dynamical model to facilitate the controller design, the T-S fuzzy model is employed to approximate the nonlinear dynamic model. The dynamic model is captured by a set of fuzzy rules which characterize local correlation in the state space, so that the deviations caused by linearization could be effectively alleviated. Furthermore, the fuzzy controller is given the fuzzy parallel distributed compensation form.

For the train operation problem, the fuzzy predictive approach uses the current train dynamic state, train model, and operational limits to calculate future changes in the control variable such that train performance criteria are optimized sequentially in infinite-time horizon subject to accelerating or braking constraints. A set of linear matrix inequalities (LMIs) is given as sufficient conditions for the existence of the proposed fuzzy predictive controller (FPC) as well as guaranteeing the various railway operational performances such as safety, energy consumption, riding comfort, and velocity tracking.

The rest of this article is organized as follows. Section “Problem formulation” first makes a detailed description for the nonlinear dynamic model of the high-speed train and then formulates the problem as a T-S fuzzy model. In section “FPC of high-speed train,” the FPC of high-speed train is designed. In section “Numerical simulations,” some numerical experiments are conducted to demonstrate the effectiveness of the introduced approaches. Finally, a conclusion is given in section “Conclusion.”

Notation

The symbol ☆ in a matrix denotes an entry that can be deduced from the symmetry of the matrix.

diag{·} represents a block-diagonal matrix.

Problem formulation

The dynamic model of high-speed train

As shown in Figure 1, we consider a train with n cars connected with elastic spring-like couplers. According to Newton’s law, the longitudinal dynamics for each car can be established in the following equation

m_{i} {\overset{\cdot}{v}}_{i} = u_{i} + f_{i - 1} - f_{i} - F_{i}^{R}, i = 1, 2, \dots, n

(1)

where $m_{i}$ is the mass of the ith car, $u_{i}$ is the traction or braking efforts for the ith car, $f_{i}$ denotes the coupler force between the $i th$ and the $(i + 1) th$ car, and $F_{i}^{R}$ is the general running resistance posed on the ith car affected by aerodynamic drag and rolling resistance $M_{i}^{R}$ , track gradient resistance $L_{i}^{R}$ , and line curvature resistance $C_{i}^{R}$

F_{i}^{R} = M_{i}^{R} + L_{i}^{R} + C_{i}^{R}

(2)

Figure 1.

Longitudinal dynamics of high-speed train.

The running resistance $M_{i}^{R}$ consists of rolling resistance and air resistance. The former item acts as a linear function of the adhesion and the wheel rims, while the latter one quadratically increases as a function of the train speed. Based on Davis’²⁹ equation, the running resistance is approximated as

M_{i}^{R} = m_{i} (c_{0} + c_{1} v_{i} + c_{2} v_{i}^{2})

(3)

where $c_{0}$ , $c_{1}$ , and $c_{2}$ are determined by the physical properties of the high-speed train.

Concerning line resistance and curve resistance,³⁰ the values of $L_{i}^{R}$ and $C_{i}^{R}$ are given by equations (4) and (5), respectively

L_{i}^{R} = m_{i} g \sin θ

(4)

where g is the gravity constant g = 9.81 N/kg and $θ$ is the gradient angle

C_{i}^{R} = m_{i} g \frac{700}{R}

(5)

where R is the track curve radius.

In this article, for the controller design of the high-speed train, we assumed that each car in the high-speed train has a separate control command. Then, the dynamic model of the high-speed train can be obtained from equations (1) to (5) that

{\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = v_{i} - v_{i + 1}, & i = 1, 2, \dots, n \\ m_{1} {\overset{\cdot}{v}}_{1} (t) = u_{1} - f_{1} - (c_{0} + c_{1} v_{1}) m_{1} - c_{2} m_{1} v_{1}^{2} - m_{1} g \sin θ - m_{1} g \frac{700}{R} \\ m_{i} {\overset{\cdot}{v}}_{i} (t) = u_{i} + f_{i - 1} - f_{i} - (c_{0} + c_{1} v_{i}) m_{i} - c_{2} m_{i} v_{i}^{2} - m_{i} g \sin θ - m_{i} g \frac{700}{R}, & i = 2, 3, \dots, n - 1 \\ m_{n} {\overset{\cdot}{v}}_{n} (t) = u_{n} + f_{n - 1} - (c_{0} + c_{1} v_{n}) m_{n} - c_{2} m_{n} v_{n}^{2} - m_{n} g \sin θ - m_{n} g \frac{700}{R} \end{matrix}

(6)

where $m_{i}$ is the ith car’s mass; $x_{i}$ is the relative coupler displacement between the two connected cars i and $i + 1$ ; $v_{i}$ and $u_{i}$ are the velocity and effort of the ith car, respectively; and $f_{i}$ is the in-train force between the ith car and the $(i + 1) th$ car. As mentioned in Chou and Xia¹² and Li et al.,¹⁴ here we also assume that the interactive force is a linear function of the coupler’s relative deformation between two adjoining cars, which can be established as follows

f_{i} (x) = k x_{i}, i = 1, 2, \dots, n - 1

(7)

where k > 0 is the stiffness coefficient. As $x_{i}$ represents the relative elastic deformation of the ith coupler corresponding to the original length, it could be positive or negative.

In this article, we mainly consider the rolling mechanical resistance and air resistance as leading factors for the running resistance, and the track geometry factors in the form of equations (4) and (5) are not considered. Besides, we suppose that the rolling resistance exerts on each car, while the air resistance acts only on the front car. Thus, the dynamic motion in the form of equation (6) can be simplified and transformed to the following form to facilitate the FPC design

{\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = v_{i} - v_{i + 1}, & i = 1, 2, \dots, n \\ m_{1} {\overset{\cdot}{v}}_{1} (t) = u_{1} - k x_{1} - (c_{0} + c_{1} v_{1}) m_{1} - c_{2} (\sum_{i = 1}^{n} m_{i}) v_{1}^{2} \\ m_{i} {\overset{\cdot}{v}}_{i} (t) = u_{i} + k x_{i - 1} - k x_{i} - (c_{0} + c_{1} v_{i}) m_{i}, & i = 2, 3, \dots, n - 1 \\ m_{n} {\overset{\cdot}{v}}_{n} (t) = u_{n} + k x_{n - 1} - (c_{0} + c_{1} v_{n}) m_{n} \end{matrix}

(8)

We assume that the desired train speed is $\bar{v}$ , and each coupler’s elastic deformation is $\bar{x}$ which is equal to zero at the equilibrium state. Then, the control effort is calculated as

{\begin{matrix} {\bar{u}}_{1} = (c_{0} + c_{1} \bar{v}) m_{1} + c_{2} (\sum_{i = 1}^{n} m_{i}) {\bar{v}}^{2} \\ {\bar{u}}_{i} = (c_{0} + c_{1} \bar{v}) m_{i}, & i = 2, 3, \dots, n - 1 \end{matrix}

(9)

Let ${\hat{x}}_{i} = x_{i} - \bar{x}$ , ${\hat{v}}_{i} = v_{i} - \bar{v}$ , ${\hat{u}}_{i} = u_{i} - {\bar{u}}_{i}$ , and choosing $x (t) = [{\hat{x}}_{1} (t), {\hat{x}}_{2} (t), \dots, {\hat{x}}_{n - 1} (t), {\hat{v}}_{1} (t), {\hat{v}}_{2} (t), \dots, {\hat{v}}_{n} (t)]^{T}$ as the state variable, and $u (t) = {[{\hat{u}}_{1} (t) / m_{1}, {\hat{u}}_{2} (t) / m_{2}, \dots, {\hat{u}}_{n} (t) / m_{n}]}^{T}$ as the decision vector, the corresponding error dynamic equation is formulated as

\overset{\cdot}{x} (t) = \bar{A} x (t) + \bar{B} u (t)

(10)

where

\bar{A} = [\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}], A_{12} = {[\begin{array}{l} 1 & - 1 & 0 & 0 & \dots \\ 0 & 1 & - 1 & 0 & \dots \\ 0 & \dots & 0 & 1 & - 1 \end{array}]}_{(n - 1) \times n}, A_{21} = {[\begin{array}{l} - \frac{k}{m_{1}} & 0 & 0 & \dots & 0 \\ \frac{k}{m_{2}} & - \frac{k}{m_{2}} & 0 & \dots & 0 \\ 0 & \dots & 0 & \frac{k}{m_{n - 1}} & - \frac{k}{m_{n - 1}} \\ 0 & \dots & 0 & 0 & \frac{k}{m_{n}} \end{array}]}_{n \times (n - 1)}

A_{22} = {[\begin{matrix} - c_{1} - c_{2} \frac{(\sum_{i = 1}^{n} m_{i}) ({\hat{v}}_{1} + 2 \bar{v})}{m_{i}} & 0 & 0 & \dots & 0 \\ 0 & - c_{1} & 0 & \dots & 0 \\ 0 & \dots & 0 & 0 & - c_{1} \end{matrix}]}_{n \times n}

A_{11} = 0_{(n - 1) \times (n - 1)}, \bar{B} = [0_{n \times (n - 1)}, I_{n \times n}]^{T}

In order to apply the MPC framework, we choose a proper sampling period $T_{s}$ and use the zero-order hold method to discrete the continuous time-domain state-space equation (10)

x (k + 1) = Ax (k) + Bu (k)

(11)

where $A = e^{\bar{A} T_{s}}$ and $B = \int_{0}^{T_{s}} e^{\bar{A} τ} d τ \bar{B}$ .

T-S fuzzy model for the high-speed train

To approximate the nonlinear model of the high-speed train, the T-S fuzzy model is adopted to combine some linear models to form an overall single model through nonlinear membership functions. Here, we choose the speed deviations from the target speed profiles as the premise variables. According to the fuzzy model proposed by TS,²⁷ the high-speed train T-S fuzzy model is described by fuzzy IF-THEN rules, and the lth rule is presented as follows:

Fuzzy rule l: IF ${\hat{v}}_{1} (k)$ is $M_{l 1}$ and ⋯ and ${\hat{v}}_{n} (k)$ is $M_{\ln}$ , THEN

x (k + 1) = A_{l} x (k) + B_{l} u (k), l = 1, \dots, r

(12)

Here, r is the number of model rules, $M_{li} (i = 1, 2, \dots, n)$ are the fuzzy sets, $x (k)$ is the state vector and $u (k)$ is the input vector both of which are defined in equation (11), and $A_{l}$ and $B_{l}$ are the constant matrices. Given a pair of $(x (k), u (k))$ , the T-S fuzzy model of high-speed train is written as

x (k + 1) = \sum_{l = 1}^{r} h_{l} (z (k)) (A_{l} x (k) + B_{l} u (k)) l = 1, \dots, r

(13)

where

\begin{array}{l} z (k) = [{\hat{v}}_{1} (k), …, {\hat{v}}_{n} (k)], h_{l} (z (k)) = \frac{w_{l} (z (k))}{\sum_{l = 1}^{r} w_{l} (z (k))}, \\ w_{l} (z (k)) = \prod_{i = 1}^{n} M_{l i} ({\hat{v}}_{i} (k)) \end{array}

The term $M_{li} ({\hat{v}}_{i} (k))$ is the grade of membership functions of ${\hat{v}}_{i} (k)$ in $M_{li}$ . It should be clarified that since

{\begin{matrix} \sum_{l = 1}^{r} w_{l} (z (k)) > 0, \\ w_{l} (z (k)) \geq 0, & l = 1, 2, \dots, r \end{matrix}

for all k, we have

{\begin{matrix} \sum_{l = 1}^{r} h_{l} (z (k)) = 1, \\ h_{l} (z (k)) \geq 0, & l = 1, 2, \dots, r \end{matrix}

FPC of high-speed train

In this article, we mainly concern the safety, energy consumption, riding comfort, and punctuality performances of high-speed train. Therefore, the following objective function is introduced

J = \sum_{t = t_{0}}^{T_{e}} (\sum_{i = 1}^{n - 1} α f_{i}^{2} + \sum_{i = 1}^{n} β {\bar{u}}_{i}^{2} + \sum_{i = 1}^{n} η {\hat{v}}_{i}^{2})

(14)

where $t_{0}$ and $T_{e}$ are related to the start time and end time of the optimizing period, respectively, and $α$ , $β$ , and $η$ denote the weight factors of different objectives.

Remark 1

In the objective function (14), the relative coupler deviations are defined as indicators of safety and riding comfort. Generally speaking, the integrand $u_{i} (t) v_{i} (t)$ represents the exported tractive power of train i at time t. Since our attention is focused on the energy consumption associated with the control strategy tending to the equilibrium state, we use the second part to express the energy-efficiency performance. The third term represents the error of tracking the desired speed profile, and it has close relationship with punctuality.

Letting $x (k + j | k)$ be the predicted state of the high-speed train at time $k + j$ and $u (k + j | k)$ be the future control at time $k + j$ , we consider the infinite-horizon MPC problem with a quadratic objective as Cuzzola et al.²² According to the error dynamical equation of high-speed train (11), above-mentioned performance cost function can be mathematically transformed into an equivalent one as

J_{0}^{\infty} (k) = \sum_{j = 0}^{\infty} {[\begin{matrix} x (k + j | k) \\ u (k + j | k) \end{matrix}]}^{T} [\begin{matrix} Q & 0 \\ 0 & R \end{matrix}] [\begin{matrix} x (k + j | k) \\ u (k + j | k) \end{matrix}]

(15)

where

\begin{matrix} Q = [\begin{matrix} diag \overset{(n - 1)}{\overset{︷}{{α k^{2}, \dots, α k^{2}}}} & 0 \\ 0 & diag \underset{n}{\underset{︸}{{η, \dots, η}}} \end{matrix}], \\ R = diag \underset{n}{\underset{︸}{{β, \dots, β}}} \end{matrix}

At sample time k, construct a quadratic function $V (x (k)) = x (k)^{T} P (k) x (k)$ , where $P (k)$ is a common positive definite matrix. To guarantee the stability condition for FPC of high-speed train, we impose following design requirement

\begin{array}{l} V (x (k + j + 1 | k)) - V (x (k + j | k)) \\ \leq - [x {(k + j | k)}^{T} Q x (k + j | k) + u {(k + j | k)}^{T} R u (k + j | k)] \end{array}

(16)

By summing equation (16) from $j = 0$ to $\infty$ , we obtain the following formula

V (x (k + \infty | k)) - V (x (k | k)) \leq - \sum_{j = 0}^{\infty} [x {(k + j | k)}^{T} Q x (k + j | k) + u {(k + j | k)}^{T} R u (k + j | k)]

For the object function (15), it is reasonable to state that $V (x (k + \infty | k)) = 0$ as $x (k + \infty | k) = 0$ . Obviously, the design requirement (16) can be converted to

J_{0}^{\infty} \leq x (k | k)^{T} P (k) x (k | k)

where $x (k | k)$ is the initial state.

Furthermore, the equivalent formulation of the optimal control problem is obtained as follows

x (k | k)^{T} P (k) x (k | k) \leq γ

(17)

where $γ$ is a nonnegative coefficient to be minimized.

FPC of high-speed train

In this subsection, we will detail the FPC for optimal control of high-speed train. We use the fuzzy rule–based state-feedback control law and consider the following fuzzy parallel distributed compensation form:

Rule l: IF ${\hat{v}}_{1} (k)$ is $M_{l 1}$ and ⋯ and ${\hat{v}}_{n} (k)$ is $M_{\ln}$ , THEN

u_{l} (k) = F_{l} x (k), l = 1, \dots, r

where $x (k)$ is the input of the controller for the high-speed train and $F_{l}$ is the gain matrix of the state-feedback controller. Hence, the control input can be written as

u (k) = \sum_{l = 1}^{r} h_{l} (z (k)) F_{l} x (k)

(18)

According to the fuzzy dynamic model of high-speed train (13) and fuzzy state-feedback law (18), the closed-loop system is represented by

x (k + 1) = \sum_{l = 1}^{r} \sum_{m = 1}^{r} h_{l} (z (k)) h_{m} (z (k)) [A_{l} x (k) + B_{l} F_{m} x (k)]

(19)

We first introduce the following lemma which is useful for further proof.

Lemma 1

For any real matrices $X_{l}$ and $Y_{l}$ for $1 \leq l \leq r$ , and $S > 0$ with appropriate dimensions, the following inequalities hold³¹

\begin{matrix} 2 \sum_{l = 1}^{r} \sum_{m = 1}^{r} h_{l} h_{m} X_{l}^{T} S Y_{m} \leq \sum_{l = 1}^{r} h_{l} (X_{l}^{T} S X_{l} + Y_{l}^{T} S Y_{l}) \\ 2 \sum_{l = 1}^{r} \sum_{m = 1}^{r} \sum_{q = 1}^{r} \sum_{w = 1}^{r} h_{l} h_{m} h_{q} h_{w} X_{lm}^{T} S Y_{qw} \\ \leq \sum_{l = 1}^{r} \sum_{m = 1}^{r} h_{l} h_{m} (X_{lm}^{T} S X_{lm} + Y_{lm}^{Y} S Y_{lm}) \end{matrix}

where $h_{l} \geq 0, \sum_{l = 1}^{r} h_{l} = 1$ .

Based on the Lyapunov stability theory, the following theorem will provide a sufficient condition for the existence of the FPC for high-speed train which stabilizes the overall system and minimizes the upper bound on the performance criteria defined in equation (15).

Theorem 1

Consider the FPC of high-speed train in equation (13) with a properly designed fuzzy state-feedback control strategy in equation (19). For a given pair of positive definite matrices ${Q, R}$ , if there exist matrices $H (k) > 0, G (k)$ and $Y_{l} (k)$ for $l = 1, 2, \dots, r$ with appropriate dimensions such that the following LMI optimization problem is feasible

min_{γ, H (k), Y_{i} (k)} γ

(20)

subject to the following constraints

[\begin{matrix} H (k) - G (k) - G {(k)}^{T} & ⋆ & ⋆ & ⋆ \\ {(A_{l} G (k) + B_{l} Y_{l})}^{T} & - H (k) & ⋆ & ⋆ \\ Y_{l}^{T} & 0 & - γ R^{- 1} & ⋆ \\ {(Q^{1 / 2} G (k))}^{T} & 0 & 0 & - γ I \end{matrix}] < 0, l = 1, 2, \dots, r

(21)

[\begin{matrix} H (k) - G (k) - G {(k)}^{T} & ⋆ & ⋆ \\ (\frac{A_{l} G (k) + B_{l} Y_{m} + A_{m} G (k) + B_{m} Y_{l}}{2}) & - H (k) & ⋆ \\ (Q^{1 / 2} G (k)) & 0 & - γ I \end{matrix}] < 0, 1 \leq l < m \leq r

(22)

[\begin{matrix} - 1 & x {(k | k)}^{T} \\ x (k | k) & - H (k) \end{matrix}] \leq 0

(23)

then the control gain $F_{l} (k) = Y_{l} (k) G (k)^{- 1}$ is obtained to guarantee each car tracks the predefined velocity profile, every coupler is stabilized to the equilibrium state, and meanwhile the upper bound of performance objective function (15) is minimized.

Proof

Considering the closed-loop system in equation (19), the stability constraint (16) yields that

\begin{matrix} {[\sum_{l = 1}^{r} \sum_{m = 1}^{r} h_{l} (z (k + j | k)) h_{m} (z (k + j | k)) (A_{l} x (k + j | k) + B_{l} F_{m} x (k + j | k))]}^{T} \cdot P (k) \cdot \\ [\sum_{q = 1}^{r} \sum_{w = 1}^{r} h_{q} (z (k + j | k)) h_{w} (z (k + j | k)) (A_{q} x (k + j | k) + B_{q} F_{w} x (k + j | k))] - x (k + j | k)^{T} P (k) x (k + j | k) \\ + x (k + j | k)^{T} Qx (k + j | k) + [\sum_{l = 1}^{r} h_{l} (z (k + j | k)) F_{l} x (k + j | k))] R [\sum_{m = 1}^{r} h_{m} (z (k + j | k)) F_{m} x (k + j | k))] \leq 0 \end{matrix}

Let $h_{l} = h_{l} (z (k + j | k))$ . Above condition becomes

\begin{matrix} \sum_{l = 1}^{r} \sum_{m = 1}^{r} \sum_{q = 1}^{r} \sum_{w = 1}^{r} h_{l} h_{m} h_{q} h_{w} x (k + j | k)^{T} [(A_{l} + B_{l} F_{m})^{T} P (k) (A_{q} + B_{q} F_{w}) - P (k)] x (k + j | k) \\ + x (k + j | k)^{T} Qx (k + j | k) + \sum_{l = 1}^{r} \sum_{m = 1}^{r} h_{l} h_{m} x (k + j | k)^{T} F_{l}^{T} R F_{m} x (k + j | k) \leq 0 \end{matrix}

By Lemma 1, above inequality can be represented as

\sum_{l = 1}^{r} \sum_{m = 1}^{r} h_{l} h_{m} x (k + j | k)^{T} [(A_{l} + B_{l} F_{m})^{T} P (k) (A_{l} + B_{l} F_{m}) - P (k) + Q + F_{l}^{T} R F_{l}] x (k + j | k) \leq 0

(24)

Noting that

\begin{matrix} \sum_{l = 1}^{r} \sum_{m = 1}^{r} h_{l} h_{m} [(A_{l} + B_{l} F_{m})^{T} P (k) (A_{l} + B_{l} F_{m}) - P (k) + Q + F_{l}^{T} R F_{l}] \\ = \sum_{l = 1}^{r} h_{l}^{2} [(A_{l} + B_{l} F_{l})^{T} P (k) (A_{l} + B_{l} F_{l}) - P (k) + Q + F_{l}^{T} R F_{l}] + \\ 2 \sum_{l = 1}^{r - 1} \sum_{m = l}^{r} h_{l} h_{m} [{(\frac{A_{l} + B_{l} F_{m} + A_{m} + B_{m} F_{l}}{2})}^{T} P (k) (\frac{A_{l} + B_{l} F_{m} + A_{m} + B_{m} F_{l}}{2}) - P (k) + Q] \end{matrix}

it is obvious that the FPC is stable if we guarantee the following inequalities

(A_{l} + B_{l} F_{l})^{T} P (k) (A_{l} + B_{l} F_{l}) - P (k) + Q + F_{l}^{T} R F_{l} < 0, l = 1, 2, \dots, r

(25)

{(\frac{A_{l} + B_{l} F_{m} + A_{m} + B_{m} F_{l}}{2})}^{T} P (k) (\frac{A_{l} + B_{l} F_{m} + A_{m} + B_{m} F_{l}}{2}) - P (k) + Q \leq 0, 1 \leq l < m \leq r

(26)

However, according to the inequality $- G (k)^{T} H (k)^{- 1} G (k) \leq H (k) - G (k)^{T} - G (k)$ , where $G (k)$ is a non-singular matrix and $H (k) > 0$ , pre- and post-multiplying both sides of equation (21) by $diag {G (k)^{- T}, I, I, I}$ and its transpose, inequality (21) can be described by

[\begin{matrix} - H^{- 1} (k) & ★ & ★ & ★ \\ {(A_{l} + B_{l} Y_{l} G {(k)}^{- 1})}^{T} & - H (k) & ★ & ★ \\ Y_{l} G {(k)}^{- 1} & 0 & - γ R^{- 1} & 0 \\ Q^{1 / 2} & 0 & 0 & - γ I \end{matrix}] < 0

Since $F_{l} (k) = Y_{l} (k) G (k)^{- 1}$ , using Schur complement, one can find that above inequality is equivalent to

\begin{array}{l} - H {(k)}^{- 1} + {(A_{l} + B_{l} F_{l})}^{T} H {(k)}^{- 1} (A_{l} + B_{l} F l) \\ + F_{l}^{T} R F_{l} γ^{- 1} + Q γ^{- 1} < 0 \end{array}

Let $H (k)^{- 1} = γ^{- 1} P (k)$ , above inequality can be transformed to equation (25).

Using similar proof technique, inequality (26) can be obtained from equation (22). Thus, the FPC of high-speed train is stable.

Pre- and post-multiplying both sides of equation (23) by $diag {I, H (k)^{- T}}$ and its transpose, we have

[\begin{matrix} - 1 & x {(k | k)}^{T} P {(k)}^{T} γ^{- 1} \\ γ^{- 1} P (k) x (k | k) & - γ^{- 1} P (k) \end{matrix}] \leq 0

By Schur complement, we obtain

x (k | k)^{T} P (k) x (k | k) \leq γ

Thus, the upper bound of the cost function is minimized, which indicates that control performance in equation (17) can be ensured through this approach. The proof is complete.

Remark 2

Theorem 1 gives a sufficient condition for choosing proper control gain $F (k)$ to optimize high-speed train operation in infinite-time horizon. The conditions in Theorem 1 take the term of LMIs, which can be determined by solving a convex problem using MATLAB LMI toolbox. Consequently, the proposed fuzzy controller can meet the demands for optimal performance.

T-S fuzzy model for the high-speed train with input constraints

In practical movement, it is common to consider the constraints on the control inputs $u_{i} (k)$ on each car. More specifically, $u_{i} (k)$ needs to satisfy $u_{imin} (k) \leq u_{i} (k) \leq u_{imax} (k)$ for some constants $u_{imin} (k)$ and $u_{imax} (k)$ , which is closely related to the traction or brake characteristics. As energy-efficiency and comfort are two main issues to be concerned in this article, the following constraint on ${\hat{u}}_{i} (k + j | k)$ in the following inequality is studied

| {\hat{u}}_{i} (k + j | k) | \leq δ u_{imax}, i = 1, 2, \dots, n, j > 0

(27)

To facilitate the FPC design for high-speed train with regard to the input constraints (27), we first introduce the stable invariant ellipsoid properties.

Lemma 2

Consider the closed-loop control system for high-speed train in the form of equation (19). At sampling time k, if there exist $G (k) > 0$ , $Y_{l} (k)$ , and $F_{l} (k) = Y_{l} (k) G (k)^{- 1}$ such that the optimization problem defined in Theorem 1 is feasible. Thus, $ξ = {x | x^{T} H^{- 1} x \leq 1}$ is an invariant ellipsoid for the predicted states of high-speed train.

Proof

Following equation (16), we have

V (x (k + j + 1 | k)) - V (x (k + j | k)) \leq 0, j > 0

which is equivalent to

\begin{array}{l} x {(k + j + 1 | k)}^{T} P (k) x (k + j + 1 | k) \\ \leq x {(k + j | k)}^{T} P (k) x (k + j | k), j > 0 \end{array}

Then, along equation (23), the following can be obtained

\begin{array}{l} x {(k + j + 1 | k)}^{T} P (k) x (k + j + 1 | k) \\ \leq x {(k + j | k)}^{T} P (k) x (k + j | k) \leq γ, j > 0 \end{array}

Let $P (k) = γ H^{- 1}$ and consider the fact that $x (k + i | k) \to 0$ as $i \to \infty$ ; therefore, the proof is completed.

Based on Lemma 2, the following theorem points out the specific condition under which the cost function in equation (15) is optimized with respect to the control input constraints.

Theorem 2

Consider the FPC of high-speed train in equation (13) with a properly designed fuzzy state-feedback control strategy in equation (19) and control input constraint in equation (27). For a given pair of positive definite matrices ${Q, R}$ , if there exist matrices $H (k) > 0, G (k)$ , $Y_{l} (k)$ , and $Ξ$ for $l = 1, 2, \dots, r$ with appropriate dimensions such that the LMI optimization problem in equation (20) is feasible and subject to equations (21)–(23) and the following inequality

[\begin{matrix} - Ξ & ★ \\ Y_{l}^{T} & H (k) - G {(k)}^{T} - G (k) \end{matrix}] \leq 0

(28)

e_{i}^{T} Ξ e_{i} \leq δ u_{\max}^{2}, l = 1, 2, \dots, r, i = 1, 2, \dots, n

where $e_{i}$ is the ith column of the identity matrix. Then, the control gain $F_{l} (k) = Y_{l} (k) G (k)^{- 1}$ is obtained to ensure each car tracks the desired velocity profile, every coupler is stabilized to the equilibrium state as well as the upper bound of performance objective function (15) is being minimized.

Proof

We only need to prove that constraint (27) is ensured by equation (28). Considering the control input constraint on each car of high-speed train at sampling time k in equation (27), we have that

\begin{matrix} max_{j > 0} | u_{i} (k + j | k {) |}^{2} = max_{j > 0} | (F_{l} x (k + j | k {))}_{i} |^{2} \\ \leq max_{w \in ξ} {‖ {(F_{l} w)}_{i} ‖}^{2} \\ \begin{matrix} \leq ‖ {(F_{l} H^{1 / 2})}_{i} ‖_{2}^{2} ‖ {(H^{- 1 / 2} w)}_{i} ‖_{2}^{2} \\ \leq (F_{l} {HF}_{l}^{T})_{ii}, i = 1, 2, \dots, n, l = 1, 2, \dots, r \end{matrix} \end{matrix}

By Schur complement, above inequality is true if the following inequality holds

[\begin{matrix} - Ξ & ★ \\ F_{l}^{T} & - H {(k)}^{- 1} \end{matrix}] \leq 0

(29)

e_{i}^{T} Ξ e_{i} \leq δ u_{\max}^{2}, i = 1, 2, \dots, n, l = 1, 2, \dots, r

where $e_{i}$ is the ith column of the identity matrix. Pre- and post-multiplying both sides of equation (29) by $diag {I, G (k)^{T}}$ and its transpose, inequality (29) can be represented by

\begin{matrix} [\begin{matrix} - Ξ & ★ \\ Y_{l}^{T} & - G {(k)}^{T} H {(k)}^{- 1} G (k) \end{matrix}] \leq 0 \\ e_{i}^{T} Ξ e_{i} \leq δ u_{\max}^{2}, i = 1, 2, \dots, n, l = 1, 2, \dots, r \end{matrix}

According to the inequality $- G (k)^{T} H (k)^{- 1} G (k) \leq H (k) - G (k)^{T} - G (k)$ , where $G (k)$ is a non-singular matrix and $H (k) > 0$ , inequality (28) can be obtained. Thus, the proof is complete.

Remark 3

Theorem 2 gives a sufficient condition for selecting proper control gain $F (k)$ to optimize high-speed train operation with input constraints. Once the high-speed train T-S fuzzy model is established according to certain fuzzy rule, the conditions in Theorem 2 can be obtained with the help of MATLAB LMI toolbox. At each sampling step k, the controller first measures current state $x (k)$ . Given the input constraint $δ u_{\max}$ , one can obtain the control gain $F (k) = Y (k) G (k)^{- 1}$ according to Theorem 2. Then, calculating the control output and applying it to the train dynamic model in (11), we can get the next state $x (k + 1)$ . By this way, the optimal output is computed repeatedly under the train performance function until the train reaches the destination. Since LMI can be solved efficiently, the FPC is able to cope with the needs of on-line control of high-speed train. Consequently, the proposed approach can meet the demands for practical application and satisfy the real time requirement.

Numerical simulations

To test the effectiveness of the FPC for the high-speed train, numerical experiments will be implemented in this section where analysis on the three weights will also be detailed. All the simulation studies are conducted based on Japanese Shinkansen high-speed train with 10 cars,¹³ and the parameters are given in Table 1.

Table 1.

The simulation parameters of high-speed train.

Parameters	Value	Unit
$m_{1}, m_{2}, \dots, m_{10}$	$80 \times 10^{3}$	$kg$
k	$80 \times 10^{3}$	$N / m$
$c_{0}$	0.01176	$N / kg$
$c_{1}$	$7.7616 \times 10^{- 4}$	$N s / m kg$
$c_{2}$	$1.6 \times 10^{- 5}$	$N s^{2} / m^{2} kg$

With the consideration of both theoretical and practical factors, the membership functions for the speed-tracking error are chosen as shown in Figure 2.

Figure 2.

The membership functions of speed-tracking error.

Therefore, the fuzzy dynamic model of high-speed train (13) can be transformed to

x (k + 1) = \sum_{l = 1}^{8} h_{l} ({\hat{v}}_{1} (k)) (A_{l} x (k) + B_{l} u (k))

(30)

where

\begin{matrix} h_{1} ({\hat{v}}_{1} (k)) = {\begin{matrix} \frac{- {\hat{v}}_{1} (k) - 20}{20}, & - 40 \leq {\hat{v}}_{1} (k) < - 20 \\ 1, & {\hat{v}}_{1} (k) < - 40 \end{matrix} h_{2} ({\hat{v}}_{1} (k)) = {\begin{matrix} \frac{{\hat{v}}_{1} (k) + 40}{20}, & - 40 \leq {\hat{v}}_{1} (k) < - 20 \\ 0, & {\hat{v}}_{1} (k) < - 40 \end{matrix} \\ h_{3} ({\hat{v}}_{1} (k)) = {\begin{matrix} \frac{- {\hat{v}}_{1} (k),}{20} & - 20 \leq {\hat{v}}_{1} (k) \leq 0 \\ 0, & {\hat{v}}_{1} (k) > 0 \end{matrix} h_{4} ({\hat{v}}_{1} (k)) = {\begin{matrix} \frac{{\hat{v}}_{1} (k) + 20}{20}, & - 20 \leq {\hat{v}}_{1} (k) \leq 0 \\ 0, & {\hat{v}}_{1} (k) < - 20 \end{matrix} \\ h_{5} ({\hat{v}}_{1} (k)) = {\begin{matrix} \frac{20 - {\hat{v}}_{1} (k),}{20} & 0 \leq {\hat{v}}_{1} (k) \leq 20 \\ 0, & {\hat{v}}_{1} (k) > 20 \end{matrix} h_{6} ({\hat{v}}_{1} (k)) = {\begin{matrix} \frac{{\hat{v}}_{1} (k)}{20}, & 0 \leq {\hat{v}}_{1} (k) \leq 20 \\ 0, & {\hat{v}}_{1} (k) < 0 \end{matrix} \\ h_{7} ({\hat{v}}_{1} (k)) = {\begin{matrix} \frac{40 - {\hat{v}}_{1} (k),}{20} & 20 \leq {\hat{v}}_{1} (k) \leq 40 \\ 0, & {\hat{v}}_{1} (k) > 40 \end{matrix} h_{8} ({\hat{v}}_{1} (k)) = {\begin{matrix} \frac{{\hat{v}}_{1} (k) - 20,}{20} & 20 \leq {\hat{v}}_{1} (k) \leq 40 \\ 1, & {\hat{v}}_{1} (k) > 40 \end{matrix} \end{matrix}

In addition, $A_{i}$ and $B_{i}$ can be easily determined with respect to equations (11) and (30), which are not mentioned here to keep the article concise.

Example 1

In the first example, we set the sampling period $T_{s}$ to be 0.5 s, and related weights are $α = 2 \times 10^{- 7}$ , $β = 200$ , and $η = 0.5$ . Meanwhile, the control force constraint for each train is $δ u_{imax} \leq 0.4, i = 1, 2, \dots, 10$ . According to conditions (21)–(23) in Theorem 1 and condition (28) in Theorem 2, the fuzzy predictive control gain F is calculated as $F_{l} (k) = Y_{l} (k) G (k)^{- 1}$ . With the help of MATLAB LMI toolbox, gain F is solved efficiently and the minimum performance value is $γ^{*} = 223, 971.8$ . As the dimensions for gain F are very large, it is not given here for the reason to make this article concise.

Suppose that the high-speed train is cruising at a initial speed of 180 km/h during the time interval [0 s, 100 s], that is, $v_{1} = v_{2} = \dots = v_{10} = \bar{v} = 180$ km/h, and the initial coupler displacement between adjacent cars is $x_{1} = x_{2} = \dots = x_{9} = \bar{x} = 0$ . Then, at the time t = 101 s, the target speed raises to 200 km/h, and each car of high-speed train begins to accelerate to track the reference speed. As depicted in Figure 3(a), the speed curves for each car are described in solid line and the dotted line indicates the desired speed. We can find out that every car is able to track the target speed $\bar{v} = 200 km / h$ perfectly after t = 573 s and the acceleration phase lasts nearly 473 s. It is easy to calculate that the mean acceleration rate is approximately 0.042 m/s² which can greatly improve the safety and comfort of the high-speed train movement. It can be inferred from above results that although maximum traction force is usually adopted as an energy-efficient driving strategy, it may not be the optimal choice when other factors such as speed tracking and in-train forces are contemplated as a whole.

Figure 3.

Optimal curves for each car under the fuzzy predictive controller with a higher desired speed: (a) Car speed, (b) Relative displacement of couplers, and (c) Traction effort.

Figure 3(b) and (c) depicts the variation curves of relative coupler deformations and the control signals exerted on each car. When the train starts to accelerate, both the deformations of spring-like coupler and the traction efforts remain positive. In order to provide necessary traction power, both the displacements and traction efforts go up remarkably at the beginning of acceleration phase. As the speed approaches the desired one, they follow a downward trend to balance different counting factors. Finally, the relative coupler deformations reach zero point which is considered to be the equilibrium state after t = 573 s. This trend is in accordance with speed-tracking error.

Example 2

In this example, a decreasing scenario is elaborated to verify whether the FPC can track a lower reference speed. The control parameters are the same to those used in Example 4.1 unless otherwise noted.

Suppose the desired speed $\bar{v}$ is 210 km/h at the time interval [0 s, 100 s] and it decreases to 180 km/h at the time 100 s. With MATLAB LMI toolbox, we can obtain the minimum performance value of $J_{0}^{\infty}$ as $γ^{*} = 835, 588.4$ . Under the FPC, the speed curves, deviations of the relative coupler displacements, and brake forces for each car of the high-speed train under this circumstance are plotted in Figure 4(a)–(c), respectively. At first, we set the cruising speed to be 210 km/h, that is, $v_{1} = v_{2} = \dots = v_{10} = \bar{v} = 210 km / h$ , and the initial relative coupler deformation between the two connected cars is $x_{1} = x_{2} = \dots = x_{9} = \bar{x} = 0$ .

Figure 4.

Optimal curves for each car under the fuzzy predictive controller with a lower desired speed: (a) Car speed, (b) Relative displacement of couplers, and (c) Brake effort.

From Figure 4, we can observe that as the reference speed changes to 180 km/h at the time $t = 101 s$ , each car begins to brake to track the predefined speed. At this time, the offsets of each car’s coupler remain negative. Therefore, the braking force is generated in the longitudinal direction which are consistent with real life. It is worth to note that instead of using constant maximum brake force, the brake force keeps decreasing until the train reaches the desired speed which guarantees the safety and comfort of the operation of the high-speed train. Under the FPC, both the relative spring displacements and speed of each car are eventually kept at the equilibrium state from 781 s.

From what has been discussed above, it can be concluded that the FPC can be adapted to the operational constraints and allows the train to track predefined speed profile perfectly.

Performance of the FPC with different parameters

In this example, we give the sensitivity analysis on the three weights $α$ , $β$ , and $η$ , which allow the proposed controller to make some adjustments with in-train force minimization, energy consumption reduction, and velocity tracking improvement.

For the convenience of energy consumption analysis in different control strategies, we use $E = \int_{t_{0}}^{T_{f}} | u_{i} v_{i} | dt$ to approximate the energy consumed during the optimizing interval. First, we change the $β$ from 200 to 8000 and fix other two parameters as in Example 4.1, which implies that controller is tuned to be an energy-efficiency-emphasized one. Consequently, energy consumption decreases from 35,459 to 35,327 MJ, and the maximum traction effort is reduced accordingly which is shown in Figure 5(c). By contrast, as plotted in Figure 5(a), the mean speed is lower during the whole trip with a larger $β$ , and the acceleration phase undergoes nearly 540 s compared to 423 s in Example 1, which results in worse speed-tracking results. The difference between maximum coupler deviation becomes more distinctive in this case. Moreover, Figure 5(b) indicates that the maximum in-train forces also increase a little accordingly. Thus, it is impossible to make any improvement on energy performance without making the two other performance indicators worse off, and we should coordinate the three parameters regarding to the desired performance in practice.

Figure 5.

Sensitivity of different $β$ : (a) Mean speed, (b) Max relative displacement of couplers, and (c) Max traction effort.

To check what follows when attention is paid on better velocity tracking results, we carry out the following simulation to examine the effects of velocity tracking weight $η$ on the overall performance. We augment $η$ from 0.5 to 10, and the other parameters are the same as that presented in above example unless otherwise noted. From Figure 6(a) we can see that it costs as soon as 440 s to catch the desired speed. Similar to the results in Figure 5(b), improvement of velocity tracking inevitably leads to greater coupler displacements and more traction efforts as plotted in Figure 6(b) and (c), respectively. This phenomenon can be explained by the reason that more efforts are exerted on each car to prompt the velocity in a shorter time, thereby leading to the fact that the maximum traction effort has a slight rise in this case. In addition, the energy consumption also endures a little increase to obtain a better velocity tracking outcomes, 35,454 MJ in this case compared to 35,327 MJ in the former case.

Figure 6.

Simulation results with different $η$ : (a) Mean speed, (b) Max relative displacement of couplers, and (c) Max traction effort.

Similar effects for in-train forces can be found by doing some adjustments with $α$ . It is easy to conclude that the preferable performance is achieved at the expense of other two aspects’ deterioration. In practical application, we should make some trade-off to harmonize all the related performances.

Comparative analysis

To better illustrate the performance of the proposed method, we implement another set of experiments to compare the performance of the proposed FPC with Zhang and Zhuan’s method (abbreviated as Zhang).¹⁶ The optimization horizon $N_{p}$ and the control horizon $N_{c}$ are set to be 6 and 4, respectively. The control parameters remain the same as those used in Example 1 unless otherwise noted. The comparisons of mean speed, maximum coupler displacement, and maximum output of controller generated by FPC and Zhang are shown in Figure 7(a)–(c).

Figure 7.

Comparisons between FPC and Zhang: (a) Mean speed, (b) Max relative displacement of couplers, and (c) Max traction effort.

These simulation results indicate that above two methods have very close train mean speed, but Zhang has greater maximum traction effort and relative coupler displacement than FPC. Especially, the maximum extension of coupler varies drastically in Zhang, which may bring more risk to the operational safety. As both FPC and Zhang adopt the MPC concept to solve the train optimal problem, it is safe to conclude that using T-S fuzzy model to describe the nonlinear train model can better control the in-train forces than just linearizing the train model in Zhang. The detailed performance values of the two methods are shown in Table 2, including energy consumption (EC), maximum relative coupler deformations (MRCD), and time to reach the equilibrium state (TRES).

Table 2.

Performance comparison.

Method	EC (MJ)	MRCD (m)	TRES (s)
FPC	35,459	0.0848	473
Zhang	35,808	0.1180	468

EC: energy consumption; MRCD: maximum relative coupler deformations; TRES: time to reach the equilibrium state.

It is shown that FPC and Zhang do not have much discrepancy in TRES ability, but FPC has great advantages in energy-efficiency and longitudinal in-train forces. Thus, these results demonstrate the improved overall performance of the proposed approach.

Conclusion

In this article, we discuss the problem of fuzzy predictive optimal control of high-speed train. We use the T-S fuzzy model to represent the nonlinear train dynamics and convert the controller design to a solution of an optimization problem with MPC approach. A set of LMIs is given as sufficient conditions for the existence of the proposed FPC subject to input constraints. With the introduced method, each car tracks the desired velocity profile, every coupler is stabilized to the equilibrium state, and meanwhile the safety, comfort, energy-efficiency, and speed-tracking ability are guaranteed. The effectiveness and applicability of the proposed method have been illustrated by numerical examples. Furthermore, as a typical networked control system, the optimal control of high-speed train with imperfect communication links can be an interesting research topic which remains to be evaluated in my future research work.

Footnotes

Academic Editor: Anand Thite

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper was partially supported by Beijing Laboratory of Urban Rail Transit, Beijing Key Laboratory of Urban Rail Transit Automation and Control, and by Beijing Jiaotong University Technology Funding Project under grant 2016JBM005.

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Optimal operation of high-speed train based on fuzzy model predictive control

Abstract

Keywords

Introduction

Background and literature review

Proposed approach

Notation

Problem formulation

The dynamic model of high-speed train

T-S fuzzy model for the high-speed train

FPC of high-speed train

Remark 1

FPC of high-speed train

Lemma 1

Theorem 1

Proof

Remark 2

T-S fuzzy model for the high-speed train with input constraints

Lemma 2

Proof

Theorem 2

Proof

Remark 3

Numerical simulations

Example 1

Example 2

Performance of the FPC with different parameters

Comparative analysis

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References