Robust PI protective tracking control of decentralized‐power trains with model uncertainties against over‐speed and signal passed at danger

Funding information National Natural Science Foundation of China, Grant/Award Numbers: U1834211, 62073027; National Research Center of Railway Intelligent Transport System Engineering Technology Founding, Grant/Award Number: RITS2019KF03; Natural Science Foundation of Beijing Municipality, Grant/Award Number: 4192046; Signal and Communication Research Institute, China Academy of Railway Sciences Corporation Limited, Grant/Award Number: 2020HT04 Abstract Controlling the movements of trains to desired target speed and distance without breaking through safety regions is of primary importance in practical applications for safety reasons. In the classical train control and protection framework, automatic train operation regulates the speed and distance with respect to tracking desired ones under the supervision of automatic train protection, an independently operating subsystem, to prevent the phenomenon of over-speed and signal passed at danger. This motivates to develop an integrated control scheme combining functions of control and protection, achieving protective tracking control against over-speed and signal passed at danger doubtlessly. Meanwhile, computationally inexpensive control structure is desired for practical applications due to limited computing resource provided by on-board computer on trains. In this paper, a robust control with PI structure and protective tracking is proposed for decentralized-power trains regardless of model uncertainties. Specifically, the circumvent problem of over-speed and signal passed at danger is formulated as prescribed performance control. It is proved rigorously that the proposed approach results in stable closed-loop system, and finally, comparative simulation results are given to demonstrate the effectiveness and advantages.


INTRODUCTION
In the last decades, control design for various kinds of trains' automatic train operation (ATO) has been greatly advanced and developed to improve the performance due to high requirements of multiple objectives, such as energy-saving, riding comfort, and so on [1]. The major function of ATO system is to drive the train automatically with respect to tracking target speed curve versus distance, based on preallocated timetable information obtained from the trackside. While, automatic train protection (ATP) system takes charge of continually checking that the speed of a train is compatible with the permitted speed allowed by trackside signals and the position of a train is compatible with the position authorized by moving authority unit, including automatic stop at certain signal aspects, if not, ATP activates a timely emergency brake to slow down or stop the train.
In the fields of advanced control design, there are numerous reported method done by worldwide scholars. To name a few, intelligent methods including fuzzy systems and neural networks are designed for ATO system in [2][3][4][5][6][7][8], adaptive methods for ATO system are developed in [9][10][11][12][13][14], optimal train control algorithms are reported in [15][16][17][18][19][20], cooperative control and scheduling methods are given in [21][22][23][24][25][26][27][28]. Despite these well established theoretical results, there are still two important issues from both aspects of practical applications and theoretical analysis can be considered for further improvements: (i) due to the limited computing sources that can be provided by on-board computers equipped on trains, how to design a computationally inexpensive yet efficient control algorithm for practical applications in the presence of model uncertainties? (ii) the violations of permitted speed and position allowed by trackside signals and moving authority unit bring in unsafe factors, and a naturally raised question is, how to design a proper control algorithm combining the functions of "tracking" and "protection" in an integrated one, and thus avoid unexpectedly triggered emergency braking implemented by ATP doubtlessly? The aim of this work is to give a systematic solution to dispose these issues.
Some previous efforts have been made to keep the states or tracking errors in some predefined allowed regions. An adaptive control that provides an arbitrarily good transient and steady-state response is designed in [29] for a class of minimum phase linear system, which is capable of forcing tracking error to be confined by pre-specified constant after a pre-specified period of time and ensuring overshoot to be bounded by pre-specified upper bound, containing a LTI compensator and switching mechanism. Inspired by this success, a tracking control with prescribed transient behaviour, later named as funnel control [30,31], is proposed [32], which releases the conditions that controlled plant is of relative degree one and extends to a class of nonlinear systems. The idea of funnel control is to construct an adaptive gain which exhibits large values in the condition that error variable is close-enough to the funnel boundary, and theoretical results have been reported for nonlinear systems with relative degree one [33,34], relative degree two [30], and high-order nonlinear systems by backstepping methodology [35]. To further present a simplified scheme circumventing the complexity problem of backstepping, a prescribed performance control (PPC) methodology is proposed to also achieve the funnel control result by incorporating the allowed performance function into the original error variable, which has been employed to neural approximation-based linearisable MIMO nonlinear systems [36], neural approximation-based strict feedback systems [37], MIMO affine nonlinear systems [38]. In order to further simplify the control structure, an approximation-free low-complexity PPC is proposed in [39] because of that approximating the "ideal controller" by existing neural networks or fuzzy systems is hard task in practical applications.
In this paper, we propose a robust PI tracking control for a class of decentralized-power trains incorporating the target speed versus position tracking and over-speed and signal passed at danger protection simultaneously. To better mimic the dynamic property of decentralized-power trains, a multiplemass model is adopted, with n cars connected by n − 1 couplers. With comparison to existing literatures, the developed controller exhibits the following twofold newly significant features.
1. Since ATP subsystem in practical applications is generally assumed to be capable of guaranteeing the safe concerns of trains independently, methods reported in existing literatures do not consider the "protection" boundaries consequently, thus, unexpected emergency brakes are ineluctable with 100% certainty when implementing control schemes in . Our previous work [40] presents the first attempt to address control design with protection constraints, which is developed based on a simple singlemass model. Alternatively, the proposed robust PI protective control in this paper, partially inspired by low-complexity PPC [39] and ATO control with protection constraints [40], is on the basis of multiple-mass model reflecting the decentralized-power property of currently operated trains in most railway lines, and capable of circumventing the problems of over-speed and signal passed at danger with absolute certainty. 2. Approximation and identification based methods are actually computationally expensive for most industrial applications and trains are no exception, because of the reliabilityoriented aim, not computing performance-oriented, is desired for on-board computers. In this sense, although wellestablished and proved theoretical results are obtained, the methods in [29][30][31][32][33][34][35][36][37][38] are not applicable to the control of trains due to limited computing resource. Meanwhile, the approximation-free low-complexity PPC [39] ensures satisfactory tracking performance if and only if monotonically decreasing boundary functions are adopted in transforming the tracking error variables, while the safety boundaries against over-speed and signal passed at danger for trains are generally sectionalized constants. In order to improve the closed-loop convergence rate and ultimately steady tracking performance without monotonically decreasing boundary functions, this paper addresses the tracking control problem with PI structure without requiring the prior accurate information of dynamic model, yielding a robust scheme against model uncertainties.
The rest of this paper is organized as follows. Some necessary and formulated problem are given in Section 2. Section 3 presents the main results of this paper, including equivalence analysis of safe protection and prescribed performance, detailed control design steps and rigorous closed-loop stability analysis. Comparative simulation results are shown in Section 4 to show the effectiveness and advantages of the proposed control by implementing to Beijing subway Yizhuang line, containing fourteen stations and thus thirteen operating intervals. Section 5 ends this paper by conclusions.

Problem Formulation
The objective of this work is to design a state feedback robust PI control  =  (, ) ∈ ℝ n to ensure the following targets: (i) for the given coupled reference position and speed vectors  r (t ) and  r (t ) = r (t ), the controlled trains can track the reference trajectories vectors with the tracking errors  −  r and  −  r tunable characterized by control parameters explicitly and can be adjusted to as small as possible, therein,  r = [p 1,r , p 2,r , … , p n,r ] T ∈ ℝ n , p 2,r = p 1,r + l 1 , ⋯, p n,r = p 1,r + ∑ n−1 i=1 l i ,  r = [v r , v r , … , v r ] T ∈ ℝ n , p 1,r and v r are obtained from off-line optimization [43,44]. In this study, the length of cars is omitted since no dynamics is introduced by constant lengths. (ii) considering the safe manipulation and overspeed protection of trains, the error variables vectors  −  r and  −  r are kept between specified regions characterized by upper and lower boundaries invariably. And (iii) all the resulted closed-loop signals are guaranteed to be bounded.

Assumptions and lemmas
In this paper, the following assumptions and lemmas are used throughout the whole work.

Assumption 1.
The reference position and speed trajectories vectors  r and  r are bounded versus time and  1 functions with r =  r .

Assumption 2.
Depending on diversified speed sensors and location techniques, the state variables of a controlled train, including position  and speed , are available for measurements and feedback to design feasible control.

Equivalence of protective safe operation and prescribed performance
A schematic diagram showing the relationship of tracking operation and safe protection is given in Figure 1, from where it is observed the permitted speed and authorized position boundaries are characterized by the supervised speed curve versus position (the red solid line, denoted as  b ), including ceiling speed, braking curve, and stopping point. In the meantime, a delay-free operation lower boundary (the blue dot-dash line, denoted as  l ) is used to avoid large excursion from operating plan and disruption. While, the green solid line (denoted as  t ) outputs the target position and speed curves, defining FIGURE 1 Illustration of safe boundaries for train control the reference trajectories to drive a train to track. During the whole running process, it is desired that the actual running curve  of trains track  t with zero or small constant error and as the allowed error boundaries. In consequence, the safe protection control of trains against over-speed and signal passed at danger can be formulated as prescribed performance control problem with proper defined speed and position tracking errors.

Robust PI prescribed performance control design
Step I-a: Define the position tracking error vector as We make use of a nonlinear mapping function to perform the error transformation, with which the original position error vector  p is converted into the new coordinate  p . Therein, the p takes charges of receiving the moving authority (MA) related information via ground-side equipments, that is,  is always kept under upper bound  r +  p ,  = [1, … , 1] T to circumvent the signal passed at danger phenomenon.
Step I-b: Supposing that the initial position of all cars of a train  0 =  (0) are located in the MA-permissible regions, that is, and  0 r ∈ ℝ n , respectively. Design the intermediate virtual controller as: with k int,1,i ∈ ℝ + , ∀i = 1, … , n, respectively. It is noticed that an initial presupposed condition is required to design the feasible intermediate virtual control. Such an initial condition is easy to satisfy in practical applications since a train is not allowed to break through the boundary of MA grimly and the initial states are also confined in such boundary.
Step II-a: Define the speed tracking error vector as In a same manner as Step I-a, we use to perform the error transformation, with which the original position error vector  v is converted into the new coordinate  v . Therein, the v takes charges of receiving the speed restriction related information from automatic train protection (ATP) via ground-side equipment, that is,  is always kept under to circumvent the over-speed phenomenon. It is noticed that the precondition of over-speed free is that no signal passed at danger phenomenon happens, which has actually been guarantee in previous steps by constraining  as Step II-b: Supposing that the initial speed of all cars of a train  0 =  (0) are located in the speed restriction-permissible regions, that is, (,  r ) ∈ ℝ n , respectively. Design the control input as: with k int,2,i ∈ ℝ + , ∀i = 1, … , n, respectively. It is also noticed that an initial presupposed condition is required to design the feasible control input. Such an initial condition is also easy to satisfy in practical applications since a train is not allowed to break through the boundary of speed restriction grimly and the initial states are also confined in such boundary.
Remark 1. By utilizing the PPC method, to be specific, the non- to perform the error transformation, the constrained control problem to circumvent the over-speed and signal passed at danger becomes a unconstrained one by incorporating these constraint boundaries and original error variables into new coordinates  p and  v . In this sense, the following control design problem can be solved without considering the nondifferentiable points caused by hard constraints raised by overspeed and signal passed at danger, no extra techniques, such as hyperbolic switching control [46], pure hyperbolic functionbased control [47], are required and thus the control design procedure is simplified immensely. Meanwhile, though model uncertainties exist directly in the considered model (1), it is noticed that the control design procedures are independent of dealing with uncertainties using adaptive method [10], neural approximation-based control [11], disturbance observer-based method [48], and so on. In this sense, under proper and easilysatisfied practical assumptions, the proposed control method is model-free against model uncertainties even with over-speed and signal passed at danger constraints. Moreover, the proposed control is also applicable in the presence of external disturbances by modifying theoretical analysis parts and keeping the control structure unchanged, which is discussed in the following Section 3.4 in details.

Theorem 1. Consider a class of decentralized-power trains with model uncertainties (1) obeying Assumptions 1 and 2. Supposing that a decentralized-power train is initially located satisfying the conditions
being the ith elements of and (,  r ) ∈ ℝ n , respectively. The proposed robust PI control (2) and (4), shown in Figure 2, is capable of guaranteeing the solve of formulated problems in Section 2 regardless of the model uncertainties.

Stability analysis and proof of Theorem 1
To proceed the proof of Theorem 1, we first introduce the following two denotations: and with⊗ denoting a multiplier exporting ℝ n×1⊗ ℝ n×1 → ℝ n×1 by internal multiply operation using Therein, the p and v are also known as the position prescribed performance function (PPF) and speed PPF, respectively. With respect to the new coordinates Θ p and Θ v , the new closed-loop dynamics of a decentralized-power train can be rewritten aṡ can be rewritten in a following compact form: What follows, the proof of Theorem 1 would be completed by three steps.
Step A: Let us denote Λ Θ = Λ Θ,p × Λ Θ,v with which is nonempty and open vector set. According to the principles mentioned in the above design steps, the initial conditions are and (,  r ) ∈ ℝ n , respectively. These initial conditions are equivalent to , containing terms of constants and functions vectors with respect to  and, is a smooth functions vector versus , and time. As the receiving on-board terminals of information from MA and ATP, ( p ) ∈ ℝ n and ( v ) ∈ ℝ n are also ensured to be smooth functions vectors. According to Assumption 1,  r , r =  r and r are smooth functions vectors. Further, the components of proposed intermediate virtual control (,  r ) and  (,  r , ,  r ) are also smooth functions vectors over Λ Θ . Based on these observations and Lemma 2, we can reach the conclusion that an one and only maximal solution Θ ∶ [0,  ) → Λ Θ ofΘ = Φ(Θ p , Θ v , t ) over [0,  ) guaranteeing Θ(t ) ∈ Λ Θ , ∀t ∈ [0,  ), to be specific, Θ p ∈ Λ Θ,p 1 and Θ v ∈ Λ Θ,v are all true for all t ∈ [0,  ).
Step B-I: Choose the positive-definite Lyapunov function V p = 1 4  T p  p , and deduce the time derivative of V p along with the closed-loop dynamic equations (7a), we can obtaiṅ It has been proved in the above Step A that Θ p ∈ Λ Θ,p is true for all t ∈ [0,  ), meanwhile, p > 0 is the indispensable condition to avoid the singularity problem of nonlinear transformation function Based on these facts, each element of vector is guaranteed to be strictly positive with Let us denote It is not hard to obtain one sufficient condition to guarantee the negative property ofV p is |s p,i | > h i, † 1 min{k pro,1,1 ,…,k pro,1,n } for all i = 1, … , n, this means with s 0 p,i = s p,i (0) denoting the initial value of s p,i (t ). Noting the proposition of intermediate virtual controller (,  r ) = − pro,1  p (t ) −  int,1 ∫ t 0  p (s)ds is a smoothly defined function vector with respect to  p and is ensured to be bounded over t ∈ [0,  ) based on extreme value theorem. Recalling  = Θ v⊗ ( v ) +  r + (,  r ) and Θ v ∈ Λ Θ,v , one ensures  is bounded over t ∈ [0,  ) using Assumption 1. The inverse function vector of which is equivalent to Based on these arguments, the time derivativė is bounded vector over t ∈ [0,  ) in line with abovementioned analysis.
Step B-II: Choose the positive-definite Lyapunov function and deduce the time derivative of V v along with the closed-loop dynamic equations (7b), we can obtaiṅ It has been proved in the above Step A that Θ v ∈ Λ Θ,v is true for all t ∈ [0,  ), at the same time, v > 0 is the indispensable condition to avoid the singularity problem of nonlinear transformation function By this facts, it is known that each element of vector is ensured to be strictly positive with Let  2 being a sumaration of following terms It is not hard to obtain one sufficient condition to guarantee the negative property ofV v is |s v,i | > for all t ∈ [0,  ), i = 1, … , n, with s 0 v,i = s v,i (0) denoting the initial value of s v,i (t ). The inverse function vector of versus Θ v can be obtained as herein, the ith element can be written as In conclusion, one obtains which is equivalent to Finally, it is not hard to conclude that is also bounded for all t ∈ [0,  ) steadily.
Step C: From above proof steps, one knows Θ L p < Θ p < Θ U which is nonempty and compact sets vector with In this sense, one knows Λ ⋆ Θ ⊂ Λ Θ . Suppose that  < +∞, one can find a time moment  ⋆ ∈ [0,  ) satisfying Θ( ⋆ ) ∉ Λ ⋆ Θ , it is an obvious contradiction with foregoing conclusions. Therefore,  is unbounded, that is,  = +∞. Finally, it is concluded that The proof completes.
Remark 2. From the above proof, it is known that a train is well controlled without violating potential signal passed at danger via guaranteeing −1 ⋅ p <  p < 1 ⋅ p , therein, p is the receiving terminal of tolerable bound which is bigger than signal passed at danger bound certainly. In this sense, the circumvention of signal passed at danger is well guaranteed. Based on the definition and U being the minimum and maximum numerical values of respectively. It is known circumvention of over-speed, equivalent to the constrained problem of  −  r , can be guaranteed in the situation that a train is not very far from the target position at the begin running time, or a train will running aggressively to follow target position. In practice, the initial position and speed of trains are generally relative zero, and target distance-to-go curves, containing position versus time and speed versus time, are obtained using offline optimization with zero initial and terminal values. In this sense, the proposed scheme is applicable to whole process operation control of trains among stations with relatively small calculative values of L and U .
Remark 3. For practical decentralized-power trains, not all cars are powered-cars and trailer-cars only provide braking control, i.e. negative input of u i if ith car is trailer one. To cope with this issue without destroying the continuous control signals, we design a nonlinear gain − tanh(u g ⋅ u i (t )) + 1 2 for ith trailer-car with u g being positive constant. Speaking mathematically,

Further discussions on external disturbances
In practical situations, external disturbances are inevitable due to the changes of running environments and various kinds of  (1) can be rewritten as follows: where the denotations are exactly the same as (1), while the extra term (1) represents the external disturbance vector with (1) = diag (1, … , 1) ⏟⎴⏟⎴⏟ n times ∈ ℝ n×n and  = [d 1 , … , d n ] T ∈ ℝ n×n , therein, d i , i = 1, … , n denotes the external disturbance imposed on the ith car. Now, we will discuss the feasibility of main which satisfies the assumptions, i.e. locally Lipschitz, continuous, locally integrable, mentioned in Lemma 2. Meanwhile, the extreme value theorem is applicable to  ′ directly in above theoretical analysis. In this sense, the main results in Theorem 1 and corresponding closed-loop analysis are applicable to this case directly without modifying any control structure and theoretical analysis.
Case 2 : (1) is piecewise-time defined bounded constants or functions vector. In this case, a proper assumption is needed to circumvent the self-Zeno behaviour of piecewise (1), that is, there are finite number, denoted as  s , of switching constants t i, j with i = 1, … , n being the sequential number of car and j = 1, 2, … being the times of switching in the j th external disturbance channel. By this proper assumption, it is easy to know that (1) can be reconstructed as right-continuous function vector as which is a subset of [0,  ). Until now, the remaining analysis procedures of Theorem 1 can be applied to this case without any modifications.

COMPARATIVE SIMULATION RESULTS
To verify the effectiveness and advantages of proposed control scheme, we applied the robust PI protective tracking control (2) and (4), as given in Algorithm 1, to the whole operation process of Beijing subway Yizhuang line, shown in Figure 3. The target speed profiles versus distance and gradient profiles in the whole line are shown in Figures 4(a) and 5, and line information and resistance parameters are given in Tables 1 and 2 respectively. The controlled train contains eight cars, with first, third, fifth, seventh and eighth ones being powered-cars and thus second, fourth, and sixth ones being trailer-cars, and the parameters u g for nonlinear gains in trailer-cars are all chosen as 200.
In order to present unprejudiced comparisons, two existing controllers are utilized in the simulations: (i) proportional control (labelled as Prop. control for short) with pc (,  r ) = − pc pro,1  p (t ) and  pc (,  r , ,  r ) = − pc pro,2  v (t ), and (ii) prescribed performance control in [39] (labelled as Pre-per. control for short in the Figures) with pp (,  r ) = − pp pro,1  p (t ) and  pp (,  r , ,  r ) = − pp pro,2  v (t ). For unprejudiced comparison, the gain matrices of above two controllers are set equal  Based on the above results and chosen PPFs, one knows that both these algorithms guarantee the circumvention of violating the PPFs by choosing proper control parameters. In order to illustrate the protective tracking control under such situation, we