A Driverless Control System for the Beijing Yizhuang Metro Line

Transcription

1 A Driverless Control System for the Beijing Yizhuang Metro Line ANDREAS LINDGREN Master s Degree Project Stockholm, Sweden April 211 XR-EE-RT 211:7

2 Abstract To improve the traffic situation in Beijing, large investsments are made in the field of infrastructure. An extra focus is put on railway, and this thesis aims at developing a driverless control system, which could be implemented in the Beijing Yizhuang metro line. With given train models, controllers are designed using PID and MPC technology. Furthermore, two methods for calculating velocity reference curves are developed. An additional feedback controller is used to ensure that the requirements on stopping accuracy are satisfied. Simulations are performed where the different controller and reference curve configurations are compared. All requirements on the control system are satisfied, but the performance varies with different configurations. The MPC controller gives the best performance in terms of energy consumption and timing, whereas the PID controller gives a somewhat higher passenger comfort. Summarized, the two controller types give similar results and could both be implemented in the train, but if equally practical to implement, the MPC controller is preferred. 1

3 Notations Symbol Explanation Unit a,t Acceleration of the traction model m/s 2 τ t Time constant of the traction model s τ d,t Time delay of the traction model s a,b Acceleration of the braking model m/s 2 τ b Time constant of the braking model s τ d,b Time delay of the braking model s a t Maximum allowed acceleration m/s 2 a b Maximum allowed deceleration m/s 2 D Distance between two stations m s b Braking distance m t des Desired runtime from one station to the next s t ref Runtime of reference curve s t run Actual runtime from one station to the next s v AT P Velocity restriction imposed by the ATP system km/h v des Desired velocity km/h v r Velocity reference curve km/h vy X Reference curve using X technology, t des = Y km/h i Gradient a g Acceleration caused by gradient m/s 2 s Stopping accuracy m J Passenger comfort m/s 3 E Energy consumption MJ H Timing - T Sampling time s N x System output horizon for MPC controller - N u Control signal horizon for MPC controller - Q v Weight on system output error - Q u Weight on control input error - 2

4 Contents 1 Introduction Background Previous Work Objective Outline Delimitations Theory MPC Observer System Models Traction Model Braking Model Gradient Velocity Curve Theory Velocity Restrictions Braking Distance Desired Speed Approaching the Station Conflicts with Velocity Restrictions Method 1: Direct Implementation (DI) Method 2: MPC Comparison Two Trains

5 5 System Analysis and Control Design System Analysis Signals and Disturbances Fixed Signal Limitations Requirements on the Control System Control Design PID Controller MPC Controller Controller Addons Stopping Accuracy Controller Robustness Summary Simulation Standards for Comparison Simulink Model Stopping Accuracy and Addons Measurement Noise PID - MPC Comparison PID Controller MPC Controller Running Time versus Energy Consumption Summary Conclusion Feasability Future Work References 51 4

6 1 Introduction This thesis is written at the Beijing Jiaotong University in Beijing, China. The main objective is to develop a driverless control system, which could be used in the newly opened Yizhuang metro line in the southern parts of Beijing. Mathematical models of the trains are given, and focus of the control system is put on passenger comfort, timing and energy consumption. 1.1 Background To improve the traffic situation in Beijing, large investments are being made in the field of infrastructure. A special focus is put on railway, and at present, several new metro lines are being constructed in and around the city [1]. A light rail line between the two areas of Songjiazhuang and Yizhuang has already been constructed and is running from the beginning of 211. At present, a driverless system based on PID technique is used, but since the comfort level has been reported not to be high enough, this thesis aims to investigate whether it is possible to design a control system for Automatic Train Operation (ATO), which gives a higher comfort level using more advanced control methods. The ATO system should calculate a reference velocity curve, which except for comfort also considers the timing and energy consumption of the train and then runs it according to this curve. The system should also consider velocity restrictions imposed by the Automatic Train Protection (ATP) system as well as other static velocity restrictions along the track. It should be noted that the system in this project is driverless in the sense that the train velocity is not controlled by a human being, although a human hand is needed to press a button to confirm that all passengers have entered the train and that the ATO system can start the ride to the next station. Before initiating the writing of this thesis, field experiments have been performed to collect relevant data, and train models have been derived, describing traction and braking dynamics respectively. All models contain nonlinearities and time delays. 1.2 Previous Work In [2], the development of automated train control systems in China is summarized for the case of high speed railways, of which much can be applied to a metro system as well. Previous papers have designed ATO systems by using for example fuzzy control [3], genetic methods [4], [5] or neural networks [6], [7]. Fuzzy logics have also been used to design stopping accuracy controllers [8]. Furthermore, previous papers have in depth analyzed the effects coasting has on passenger comfort and energy consumption [5], [9]. Analyses of using splines to create velocity reference curves and a more in depth analysis of passenger comfort were made in [1]. Methods for optimizing reference trajectories and minimizing energy consumption are analyzed in [11] and [12]. 5

7 1.3 Objective This project aims to design a driverless control system specifically adapted to the Beijing Yizhuang Metro Line. The system should make it possible to move the train from one station to another in a specified amount of time while obeying velocity restrictions and at the same time fulfilling requirements on passenger comfort and stopping accuracy, where stopping accuracy is measured as the distance between the calculated stopping position and the actual stopping position of the train. In addition, focus should be put on making the controllers energy efficient, which in this case means avoiding frequent accelerations. Given the desired runtime, distance between the stations, velocity restrictions along the track and data on train acceleration and deceleration, methods for calculating suitable reference velocity curves should be developed. A reference curve will inform the train at which velocity to run at the current location, thus taking it to its destination without violating the velocity restrictions, nor exceeding the specified requirements on passenger comfort and timing. Controllers should be designed to track the velocity reference curves, and Matlab and Simulink will be used for simulations. Disturbances such as rail gradient, model deviations and measurement noise are introduced, and it is a part of this project to make the impact of these disturbances as small as possible. PID controllers should be designed and their performance is to be compared to MPC controllers, being a more advanced control method. MPC will be implemented using the built in MPC controller block in Simulink. Since the train exhibits different behavior for traction and coasting, controllers need to be designed for each driving mode. Furthermore, since it is of utmost importance that the requirement on stopping accuracy is satisfied, it needs to be analyzed whether switching to a different kind of controller is necessary as the train is approaching the station or if the PID/MPC controller can be used all throughout the journey. Except for the train velocity, this controller would probably also need the distance to the station as input. 1.4 Outline Section 2 gives general information on theory used in later sections. This information is not necessary to understand the thesis, but will be of use if the results of the thesis are to be implemented in a real system. Section 3 defines the train models, used later when creating the controllers. This section also defines the function converting the gradient along the track to the acceleration it imposes on the train. Section 4 describes the methods used for creating the reference curves, which are later used as input to the controllers. Section 5 gives basic information about the system. Signals and disturbances are defined, as well as the requirements on the control system. Controllers are designed, as well as a number of controller addons. Section 6 defines methods of comparing the designed controllers and reference 6

8 curves. All simulations are performed, and the result is discussed. Section 7 summarizes the results and gives recommendations and possibilities for future work. 1.5 Delimitations In a real railway system, the track is normally divided into blocks, thus preventing the trains from colliding [13], Chapter 2. These blocks will not be considered in this thesis, but instead only the velocity restrictions will prevent the train from moving. It should be noted that this could easily be converted to a block system, as the velocity restriction could be set to zero at a certain position. In this project, different controllers are used during the same simulation. In reality, this might cause bumps in the control signal, but this effect is considered to be outside the scope of this thesis. Furthermore, the effects wind resistance or a curved track have on the train are seen as small enough no to affect the system behavior and are therefore not considered in this thesis. 2 Theory General information on theory used in later sections is given. The information is not needed to understand the calculations performed in other parts of this thesis, but will be of use if the system is to be implemented in a real system. 2.1 MPC The basic idea of Model Predictive Control, MPC, [14] is to use the control input u to optimize the future response of the system. Given information about the current state, an objective function is set up with weights on the states and the output, and based on this function, an optimal control input sequence of finite length is calculated. Only the optimal value obtained for the current time, being the first value in the sequence, is implemented and the system is then allowed to evolve one sample. After this, new measurements are collected and the optimization is repeated. An MPC controller needs access to all the states of the system. If there are states which can not be measured, an observer needs to be designed. An important advantage of using MPC is the ability to impose constraints on the states as well as on the input and the output in the optimization process by using constrained optimization methods. Unlike in the early days of MPC, there are today very powerful methods of solving this kind of optimization problems numerically, which makes it possible to use MPC online for a large number of systems. All calculations made in the optimization process are based on a discrete time 7

9 representation of the system. Given the state space matrices A, B and C of the continuous system and using the zero order hold method with sample time T, the discretized system [15], page 11, is represented as where Λ and Γ are given as x((k + 1)T ) = Λx(kT ) + Γu(kT ) (1) y(kt ) = Cx(kT ) Λ = e AT Γ = T e As dsb A general MPC formulation is where min u f(x, u) (2) f(x, u) = N x (x i x ref,i ) T Q x (x i x ref,i ) + (3) i=1 N u 1 i= (u i u ref,i ) T Q u (u i u ref,i ) in which i = is equal to the present sampling time. Q x and Q u are weights on the errors. At the same time, the following constraints should be satisfied u min u i u max i [, N u 1] (4) y min Cx i y max i [1, N x ] It should be noted that the output error y y ref with corresponding weight Q y is easily included by setting Q x = C T Q y C (5) The values N x and N u determine how many samples into the future the controller should calculate the state and the control input. Input values after sample N u have to be set by the control designer and are often set to the last value of the control sequence. Long horizons give a more stable system, but also a slower controller since the optimization problem grows with growing horizons. N x should be chosen long enough to cover the transient behavior of the system and N u is often chosen much smaller than N x. Reference values need to be set by the control designer. Often, u ref,i = i [, N u 1] and choosing x ref,i = C y ref i [, N x ] ensures that the states are moving in a direction which gives the desired output y ref. C here denotes the pseudo inverse of C. Setting x = x i+1.. x i+nx, x ref = x ref,i+1.. x ref,i+nx u = u i. u i+nu 1, u ref = u ref,i. u ref,i+nu 1 8

10 the objective function in Equation (3) can be rewritten as (x x ref ) T Qx (x x ref ) + (u u ref ) T Qu (u u ref ) where Q x and Q u are matrices of appropriate size with Q x and Q u respectively on the diagonal and zeros elsewhere. In order to make the optimization problem in Equation (2) a function of u only, x needs to be rewritten as a function of u, which can be done by using Equation (1) repeatedly. This yields x = Λ Λ 2. Λ Nx 1 Λ Nx x + Γ ΛΓ Γ Λ Nx 2 Γ Λ Nx 3 Γ Γ Λ Nx 1 Γ Λ Nx 2 Γ ΛΓ Γ u = Āx + Bu where x is the given current state. Furthermore, the constraints in Equation (4) on the output y can now be rewritten as u min u u max y min Cx = C(Āx + Bu) y max where C is a diagonal matrix with C on the diagonal and zeros elsewhere. Now having written the states and constraints as a function of u only, the MPC formulation as described in Equation (2) can be rewritten as a Quadratic Program, which is useful due to the powerful methods available for solving this kind of problem. In Matlab, this can be done with the function quadprog. min u u T Hu + h T u such that Lu b (6) As mentioned, the first of the calculated outputs u is used and the system is allowed to evolve one sample. After this, new values of H, h, L and b are calculated and the process is repeated. Weights and control horizons will have to be adjusted in order to obtain a desired system behavior. 2.2 Observer Sometimes the controller requires information about all the states in the system. In practice, this is often not possible and the unmeasurable states need to be estimated. This can be done using an observer. Given the system x = Ax + Bu y = Cx where the matrices A, B and C are known but there are states, which can not be measured. If ˆx is used to approximate x and y C ˆx is used to measure how well ˆx approximates x, the observer can be defined as ˆx = Aˆx + Bu + K(y C ˆx) where K is a vector of length equal to the number of states in x. Defining the estimation error as x = x ˆx 9

11 the dynamics of the observer can be calculated as x = ẋ ˆx = (A KC) x This means the eigenvalues of A KC determine how fast the observer error x goes to zero. According to [16], Equation 9.22, it is true that x(t) c x() e at if the real part of all eigenvalues of A KC is smaller than a. c is a constant. In this thesis, one of the states can be measured, and designing an observer for it is therefore not necessary. Instead, a reduced observer could be designed, as described in [15], page System Models To be able to design adequate controllers, train models are necessary. Field experiments have been performed [17], [18] and the behavior of the train has been analyzed. It is clear that the train exhibits different behavior during traction and braking, and therefore two models have been designed. Both contain nonlinearities and time delays. The train is controlled by a throttle which can take values u [ 1, 1]. During the field experiments, a constant throttle value was applied and the velocity was measured. Using this data, the model parameters were identified and the process was then repeated for different throttle values. All models take the throttle value u as input and give the train acceleration a as output. In the real system, only the velocity is measurable and an integration is therefore added to the models to obtain the train velocity v. It should also be noted that the time delay used in the model represents the largest possible throttle change, namely from u = to u = 1 or u = to u = 1. Other throttle changes will induce smaller time delays, but these will not be considered in this thesis since a controller designed to deal with long time delays can also deal with shorter ones. 3.1 Traction Model The traction model is used when the train is accelerating, this being the case when the throttle u (, 1]. Field experiments show that the model parameters do not change for different throttle values. Except for the time delay, the train exhibits approximately linear traction behavior for small velocities. At velocities v > 4 km/h the train begins to accelerate more slowly, which is a result from train power limitations. This behavior can be modeled as a decrease in the throttle value and is added as a nonlinear factor to the linear model. The complete traction transfer function from throttle value u to velocity v is given as G t (s) = G u a,t 1 s G delay,t f(v) (7) = a,t τ t s s e sτ d,t f(v), 1

12 where a,t =.913 m/s 2, τ t =.8 s, τ d,t = 1.2 s, and f(v) = { 1 v 4 km/h.2v 2.45v v > 4 km/h (8) It should be noted that f(v) < 1 v (4, 8], where 8 km/h is the highest velocity allowed on the track. The linear model G u a,t 1/s is given on state space form as ẋ(t) = Ax(t) + Bu(t) (9) y(t) = Cx(t), where x = [x 1, x 2 ] T = [v, a] T and [ ] [ 1 A = B = 1/τ t ] a,t /τ t C = [ 1 ] 3.2 Braking Model The braking model is to be used when the train is decelerating, this being the case when the throttle u [ 1, ). In contrast to the case of traction, the braking model parameters do change for different throttle values, although the change is relatively small. This change is added as an extra, slightly nonlinear, factor f(u) to the linear model. The braking model exhibits no change in behavior for increasing velocity and is given as G b (s) = G u a,b 1 s G delay,b f(u) (1) = a,b τ b s s e sτ d,b f(u), where a,b = 1. m/s 2, τ b = 1.2 s, τ d,b = 1.5 s, and f(u) =.1898u u 1 u < (11) It should be noted that f(u) < 1 u [ 1, ). The linear model G u a,b 1/s is given on state space form as ẋ(t) = Ax(t) + Bu(t) (12) y(t) = Cx(t), where x = [x 1, x 2 ] T = [v, a] T and [ ] [ 1 A = B = 1/τ b a,b /τ b ] C = [ 1 ] 3.3 Gradient The slope of the railway, known as the gradient, influences the acceleration of the train, and can be seen as a disturbance to the system. In railway systems, the gradient i is normally known and can be described as a function of the 11

13 θ F grad 1 mg i Figure 1: Force imposed on the train by gravity when gradient i train position, i(s). In railway systems, the gradient is often given in parts per thousand, as shown in Figure 1. The force which gravity imposes on the train when the gradient i is denoted F grad. The mass of the train and the gravity acceleration are denoted m and g = 9.81 m/s 2 respectively. Noting that tan θ = i/1 = θ = tan 1 (i/1) and that F grad = mg sin θ gives ( ) F grad = mg sin tan 1 i i = mg 1 i According to Newton s second law F grad = ma g, where a g is the acceleration the gradient imposes on the train. The final expression for this acceleration as a function of the current gradient i can thus be written gi a g (i) = (13) i Velocity Curve The main objective of this master thesis is to create a system, which will move the train from one station to another. An important part of this system is a reference velocity curve, which informs the controller at which velocity the train should be running at the current train position s. Given speed restrictions, distance between the stations and the desired runtime, a desired velocity is calculated for each sample. To create a velocity reference curve which accelerates the train to the desired velocity in a smooth way, two methods for imitating a train are developed. In other words, a reference curve is created offline, and the outcome can then be used as input to the controller when simulating the system. If the system is to be implemented in a real train, the velocity reference curve needs to be recreated several times during the journey, since the conditions might change en route. 4.1 Theory A velocity reference curve should take the train from an initial velocity of v = at position s = to the next station located at s = D, at all times ensuring the current train speed v(s) < v AT P (s), where v AT P (s) is the speed restriction at the present position. Furthermore, the train velocity should be adapted such 12

14 that the complete journey takes t des seconds. D, v AT P (s), t des together with the maximum allowed train acceleration and deceleration a t, a b are all given. The maximum acceleration is set to a t = 1. m/s 2 and maximum deceleration a b =.7 m/s 2, although the deceleration is allowed to reach a b = 1. m/s 2 in the case of emergency braking. Note that a b is seen as a positive constant all throughout this thesis. Since the reference curve is created using a computer, it is discretized to v r (k), which means it needs to be interpolated if implemented in a real train. Using a fixed sample time T =.2 s, the curve is built up one sample at a time in the time domain. At each sample v r (k) it is decided where the next sample v r (k + 1) should be located relative to v r (k). The distance the train has moved in the last sample is calculated as d = T v r (k) and the total distance traveled is updated at each sample as s(k) = s(k 1) + d Velocity Restrictions On a level above the Automatic Train Operation (ATO) system, the Automatic Train Protection (ATP) system ensures that the train runs in a secure way. This means the train brakes are automatically applied if the train velocity exceeds the current velocity restrictions, or if the train gets too close to the train in front. In the case of a train approaching a train in front, the velocity restriction could be an continuously decreasing curve, which makes the train make full stop at a specified safety distance from the train in front. Furthermore, there are static restrictions along the track, which do not change regardless of the circumstances. An example of a static restriction curve is seen in Figure 2. In this project, the ATP curve is seen as equal to the static curve, and they are therefore both defined v AT P. 8 Velocity [km/h] Distance [m] Figure 2: Example of velocity restriction curve v AT P (s) Braking Distance Since sudden changes in velocity have a very negative effect on passenger comfort, it is important to avoid hard braking. When designing the velocity curve, it is therefore important for the system to look ahead for changes in velocity restrictions, thus ensuring that sudden decreases in velocity are discovered in time. Here, the braking distance of the train plays a vital role, and there are several ways of calculating the distance the train needs to come to a full stop [19]. In this thesis, a relatively simple method will be used. As described in 13

15 [2], assuming the deceleration a b is constant and using the well known laws of physics v = at and s = at 2 /2, the braking distance s b, needed to bring the train from the current velocity v to a full stop, can be calculated as s b = v 2 2(a b + a g ) + (t r + t s +.5t b )v, (14) where t r and t s are the reaction times of the system and the brakes respectively, and t b is the time it takes for the brakes of the whole train to reach full braking capacity. In this paper t R = t r + t s +.5t b = 2.5 s. a g is the acceleration caused by the gradient, as derived in Equation (13), and is here seen as small enough not to be considered when creating the velocity curve. That is a g =. Calculating s b at each sample, the system will look for changes in velocity restrictions at s = s pres + s b, where s pres is the present location Desired Speed In order for the train to reach the station at s = D and t = t des, the desired average speed of the train needs to be calculated. In a real system, the train accelerates until it reaches a certain velocity, which it keeps for some time. This is followed by coasting where the throttle is set to zero and the train slows down only because of resistance due to mechanical reasons and wind. As the train approaches the station, the braking part is initiated and the train is brought to a full stop. A simplified approximation of this behavior is to assume constant initial acceleration, which brings the train to a constant speed v des. Resistance is neglected which means the velocity is kept constant until the train approaches the station, where a constant deceleration brings the train to a full stop. See Figure 3. This model means the velocity reference curve is fixed to v r = v des throughout the journey, except for at the end when it is set to v r (s) = s [D s b, D] v [m/s] v des a t a b t t t b t des t [s] Figure 3: Simplified train movement in the time domain Using this model with given D, t des, a t and a b, the movement can be described as D = a tt 2 t 2 + v des(t des t t t b ) + a bt 2 b 2 where t t = v des, t b = v des a t a b This means the desired velocity v des can be calculated as v des = t des 2 a (tdes ) 2 2 a Da, a = a ta b (15) a t + a b 14

16 A more flexible approach is calculating a desired reference speed at each sample. By doing this, it is possible to compensate for driving too slow in the beginning by driving faster in the later part of the journey. The most natural choice of reference speed is v des (s) = D s b s (16) t des t b t Worth noting is that Equation (16) excludes the last part of the journey, where D s b s D. This is due to the fact that timing becomes less important as the train approaches the station and other methods for calculating the reference velocity are used. This will be further examined in Section Approaching the Station As the train is approaching the station, timing becomes much less important than distance accuracy. For example, the train stopping 5 cm away from the platform doors would impose a greater problem to the passengers than if it arrives 1 seconds late. Therefore, the speed reference in Equation (16) does not take the last part of the journey into account. Instead, using constant deceleration, it is known that v = at and s = v dt = at 2 /2, which by eliminating t gives the distance dependent velocity v(s) = 2as Noting that the deceleration takes place at s (s b, D] and that the velocity should be decreasing with growing distance, the curve is desribed as v(s) = 2a(D s), D s b < s D Furthermore, using the initial value v(s b ) = v, the deceleration which will make the train come to a full stop at s = D is calculated as v 2 D s a = 2(D s b ) = v(s) = v D s b and the complete expression for desired velocity can now be written v des (s) = D s b s t des t b t v D s D s b s D s b D s b < s D (17) where v = v des (D s b ) Conflicts with Velocity Restrictions The expression for desired velocity derived in Section assumes there are no obstacles for the train to run at the desired speed. In reality there are always speed restrictions (see Section 4.1.1), and since these are known when creating the curve, they should be taken into account. If they are not, the train will drive at the desired speed until it reaches the speed restrictions. Having passed the restrictions, it has to accelerate to a relatively high speed in order to keep 15

17 the time table. This is of course a possible solution, but not optimal from a passenger comfort point of view. To deal with this problem, the desired velocity v des is calculated according to Equation (16) at each sample and the system supposes this velocity will be constant all throughout the journey. It is then compared to the ATP curve v AT P (s) for s [s, D s b ] where s is the current position. The sections where the restrictions imposed by the ATP curve interfere with the calculated reference curve are denoted S such that v des (s) > v AT P (s) s S. Unless S, the train needs to compensate for this by driving faster where s / S. Figure 4 gives a schematic picture of this situation. v new v 1 v new v 2... v[m/s] s d 1 r 1 d 2 r 2... D s b s[m] Figure 4: Velocity restrictions along the route Here, r i S and v i is the speed restriction. v new is the velocity, at which the train must be driving in order to keep the time schedule. Setting up the equations yields ds t des t b = v AT P (s) + ds which gives s S v new = t des t b s S s / S ds s / S ds v(s) v new (18) In this model, the train is supposed to follow the speed restrictions exactly, which is never the case in reality. A real train will need some time to accelerate and decelerate to the new velocities, but as an approximation this model is good enough. The velocity added to the initial desired velocity is calculated as v extra = v new v des 4.2 Method 1: Direct Implementation (DI) The name Direct Implementation stems from the fact that the difference between the present velocity v and the desired velocity v des has direct impact on the acceleration of the velocity reference curve v r. v des is calculated at each sample according to Equation (16) and Equation (18), and, once again using the laws of physics together with appropriate initial conditions, the acceleration of the curve can be calculated as a = v2 des v(k)2 2s b where v(k) is the velocity in the last sample and s b is the braking distance as calculated in Equation (14). This means the system is able to take the train to the desired velocity within the present braking distance. 16

18 In case an ATP restriction lower than the present velocity is found within the braking distance s b, this is treated as a special case and the deceleration is then described as a = v2 AT P v(k)2 2(s AT P s) where v AT P < v(k) is the restricted velocity and s AT P is the position where the ATP restriction begins. This ensures that the deceleration is large enough to brake the train down to a velocity not exceeding the restriction before arriving at s = s AT P. Furthermore, in order not to exceed the possible acceleration of the real train, thus making the curve impossible to implement, a is restricted to a b a a t. Additionally the change of acceleration from sample to sample is restricted to a T/3, where T is the sample time. This means it would take 3 seconds for the train to change from no acceleration to full acceleration, which is seen as normal behavior since no real train driver would change the throttle value from to 1 in one sample, corresponding to.2 seconds. The next sample can now be calculated as v(k + 1) = v(k) + at. If v(k + 1) turns out to be larger than the allowed velocity at the present situation, v(k +1) is set to v(k), which makes the train drive at constant speed. This process is repeated until s = D s b, where the braking algorithm described in Section takes over. An example of two velocity curves v r created using the DI method is seen in Figure 5. The curves correspond to desired running times t des = 22 s and t des = 25 s respectively. Note that an additional safety margin of 5 km/h is added to the ATP curve in order to avoid the train velocity to exceed the velocity restrictions in the case of slight overshoots in the simulation. The extra curve in the lower left corner of the distance-velocity plot shows the additional speed added in order to compensate for the future speed restriction at 1 s 15, as discussed in Section The two curves end at s = D = 3 m in the space domain and at t = 23 s and t = 252 s respectively in the time domain. The latter stopping 2 seconds too late it expected since stopping accuracy is more important than timing, as discussed in Section The fact that the curve corresponding to t des = 22 s stops at t = 23 s indicates that the desired time is too short and that the problem is close to being unsolvable. This can also be seen in the figure, where the curve goes into final braking in a very abrupt way. 4.3 Method 2: MPC As suggested in [21], MPC theory can be used for designing an ATO control algorithm. In this thesis, a modified MPC controller is used to create a reference curve from s = to s = D. For each sample, a desired velocity is calculated according to Equation (16) and Equation (18), and this velocity is used as reference for the controller. In case an ATP restriction lower than the present velocity is found within the braking distance s b, the ATP restriction is set as reference. As the train reaches s = D s b, the MPC controller is no longer used, and the braking algorithm described in Section is used instead. As described in Section 2.1, one of the main advantages of using MPC control is that restrictions can easily be imposed on the control signal as well as on the system output. In this case, output is restricted to the highest possible 17

19 8 Velocity [km/h] 6 4 v ATP 2 v r, 22 v r, Distance [m] Velocity [km/h] v r, 22 v r, Time [s] Figure 5: Two velocity reference curves created using the DI method. velocity allowed by the ATP system at the current location, and the control signal, being the throttle controlling the acceleration of the train, is restricted to u 1. Furthermore, just as in the case of DI, u < T/3 to prevent the acceleration of the train to change too rapidly, since this would impose a high level of discomfort for the passengers. Technically, this restriction is implemented in the controller by adding u ET/3 + u to Equation (6). u is here the control signal, u the control signal in the last sample and E is a lower triangular matrix of size N u N u where all the elements not equal to zero are set to unity. Another advantage of using MPC is that the train models described in Section 3 can be a directly integrated part of the controller, thus making it possible to give the velocity reference curve a behavior more similar to the real train compared to the DI method. Since MPC uses a discretized state space model, it is problematic to implement the time delay present in the original train model. Replacing the delay with a Padé approximation yields e sτ d 1 sτ d/2 1 + sτ d /2 which gives a minimum phase system with an additional pole and zero. This system turns out to be hard to control and does not give a smooth curve. Because of this, the delay is disregarded when creating the velocity curve and must instead be compensated for when doing the simulations. The same applies to the nonlinear elements of the models, which are also disregarded when creating the reference curve. Furthermore, it turns out not to be practical to switch between the two train models for traction and braking when the train should accelerate or decelerate, since this makes the curve unstable and often makes the optimization problem unsolvable. Instead, only one model is used for both traction and braking. Since the final braking to a full stop is controlled by the 18

20 algorithm described in Section 4.1.4, the main task of the reference curve is to make the train accelerate. Because of this, together with the fact that the models for traction and braking are quite similar when disregarding time delays and nonlinear elements, the traction model described in Equation (9) is used. An example of a two velocity reference curves created using the MPC method is seen in Figure 6. Just as in the case of DI, they correspond to the desired 8 Velocity [km/h] v ATP v r, 22 v r, Distance [m] Velocity [km/h] v r, 22 v r, Time [s] Figure 6: Two velocity reference curves created using the MPC method. running times t des = 22 s and t des = 25 s respectively, and an additional safety margin has been added to ensure that the train does not violate the velocity restrictions when simulating the system. The extra curve in the lower left corner of the distance-velocity plot shows the additional speed added in order to compensate for the future speed restriction at 1 s 15. Both curves end at s = D = 3 m in the space domain and t = 225 s and t = 251 s respectively in the time domain. Note worthing is that the curve corresponding to t des = 22 s brakes off rather abruptly as the final braking begins. This indicates that the problem is close to being unsolvable, which is further proven by the fact that the train arrives 5 seconds late. Control horizons are chosen to N x = 8 and N u = 1 since 8 samples, corresponding to 8T = 16 s, clearly cover the model time constant τ t =.8 s and thus all transient behavior. N u is chosen smaller than N x, as described in Section 2.1. The error weights defined in Equation (3) and Equation (5) are set to Q u = 3 and Q v = 1 respectively. The weight Q u is chosen large to ensure that the control signal is kept low and to make sure it does not reach its limits too often, since this does not correspond to the behavior of a real driver. 19

21 4.4 Comparison Looking at the figures for DI and MPC separately, it might be hard to tell them apart. In Figure 7 the curves corresponding to t des = 22 s are put in the same graph and the difference is more distinct. 8 Velocity [km/h] v ATP v r, DI v r, MPC Distance [m] 8 Velocity [km/h] v r, DI v r, MPC Time [s] Figure 7: Comparison of DI and MPC curves. t des = 22 s. In terms of acceleration, the MPC curve tends to accelerate more quickly and then keep the same velocity, whereas the DI curve accelerates more slowly but during a longer period of time. The slower acceleration of the DI curve causes it to be slower and to arrive at s = D approximately five seconds later than the MPC curve. This problem is only present when the desired running time is short. For longer running times, the timing of the two curves is close to identical, as shown in Figure 5 and Figure 6. Furthermore, the MPC method gives a smoother curve, which might turn out to make a difference in terms of passenger comfort. In terms of efficiency, the DI curve takes practically no time to calculate, whereas the MPC curve needs approximately.15 s per sample on a relatively fast PC, which means a simulated runtime of 3 s = 3/T = 15 samples requires 23 s of calculation time. This is due to the fact that optimization problems must be solved for each sample in the MPC curve, which is not the case for DI. If the system is to be implemented in a real train, where the curve needs to be recalculated several times during the ride, this aspect has to be considered. 2

22 4.5 Two Trains The discussion in the previous sections assume that only the speed restrictions limit the train movement. In reality, an important restriction is the presence of other trains in front. To ensure a safety distance is kept between the current train, called train 1, and the train in front, called train 2, the ATP system will force the train 1 to brake if it gets too close to train 2. The models derived earlier can easily be adapted to fit to this behavior by letting the ATP curve go to zero with constant deceleration, thus forcing the train to brake. The three variables s det, s brake and s stop are introduced, corresponding to the locations where the ATP system informs train 1 about train 2, where train 1 starts braking and where train 1 should stand still respectively. Furthermore, the variable t wait is added to the discussion, corresponding to the time train 1 waits at s = s stop. As the location s = s det is reached, train 1 can start preparing for the braking. That is, increase the speed in order to avoid heavy acceleration at the end. The methods described in Section 4.1 to calculate the desired velocity can be used with the modification of adding t wait to Equation (18) as follows s / S ds v new = t des t b t wait s S ds v(s) (19) where the set S was defined in Section An example of a two train situation is shown in Figure 8, where the DI method has been used with desired runtime t des = 3 s. The second train is detected at s = s det = 3 m, which explains why the first train starts accelerating there. Note the waiting time t wait = 2 s, visible in the time domain of the figure. Velocity [km/h] v ATP v r Distance [m] 8 v r Velocity [km/h] Time [s] Figure 8: DI curve for a two train system. s det = 3 m 21

23 5 System Analysis and Control Design In this section, the complete system and all signals in and out from it are described and analyzed. Furthermore, limitations and disturbances are introduced and handled. Requirements for the control system are set up regarding stability, robustness and accuracy, and controllers are designed. Additionally, the built in Simulink MPC controller is described and configured. 5.1 System Analysis A schematic picture of the complete system is shown in Figure 9. The signal s is the current location of the train, which is obtained by integrating the current velocity v. The function f 1 maps the current location to a reference speed v r by using one of the methods of creating a velocity curve discussed in Section 4. The gradient at the current position is denoted i and f 2 is the function described in Equation (13), which converts the gradient to the acceleration a g it imposes on the train. n denotes the measurement noise, which will be further discussed in Section f 2 i s v r e u a v a g 1 f 1 F G s n Figure 9: Schematic picture of the complete system As described earlier, the system exhibits different behavior for traction and braking, and the train model G does therefore change for different control inputs u. Furthermore, adding the nonlinearities described in Section 3, the block G in Figure 9 can be illustrated as shown in Figure 1. f t v u G t a f b G b Figure 1: Extended schematic picture of train model G As seen, a switch directs the control input u to the traction model for u and 22

24 to the braking model for u <. The functions f t (v) and f b (u) correspond to the nonlinearities of the system which are defined in Equation (8) and Equation (11) respectively. The models G t and G b are defined as G t = G u a,t G delay,t from Equation (7) and G b = G u a,b G delay,b from Equation (1) respectively. Since there are two train models, one controller should be designed for each model, and the block F in Figure 9 can be expanded according to Figure 11. e F t u F b Figure 11: Expanded schematic picture of controller F Signals and Disturbances The reference curve v r (s), created by using one of the methods described in Section 4, is the input to the system. In reality, the train velocity v is the only measurable output from the system, although in this thesis, the acceleration a is the output from the system G. The velocity v, the distance s traveled since t = and the jerk j are also used and are obtained directly in the simulation as v = adt s = vdt (2) j = da dt If these variables are to be used in the real train, observers need to be designed, for example by using the method described in Section 2.2. The measurement noise n and the gradient i are seen as disturbances. The gradient i does not exceed 3, and the measurement noise n is modeled as white Gaussian noise with deviation.1 m/s, giving variance σ =.1 2. Both the variance and the gradient are based on data from the real system Fixed Signal Limitations The train is designed to take throttle values in the interval [-1, 1], which means the control input has a fixed limitation of u 1. Furthermore, the Yizhuang metro line has an upper velocity restriction of 8 km/h which means v 8 km/h at all times. 23

25 5.2 Requirements on the Control System Except for the restrictions mentioned in Section 5.1.2, there are a number of additional restrictions regarding velocity and passenger comfort, which the controllers have to take into consideration. Naturally v < v AT P at all times. The maximum acceleration and deceleration used when creating the velocity reference curves are defined as a t and a b respectively. When designing the controllers and simulating the systems, accelerations and decelerations up to 1 m/s 2 are accepted though. Furthermore, the jerk j defined as the change of acceleration per time unit is very important for measuring passenger comfort and should not exceed.6 m/s 3 as mentioned in [22], chapter 3.5. Another very important factor when designing the controller is the stopping accuracy, defined as s = s final D. This parameter shows how close to the desired stopping position D the train actually stops. The requirement on stopping accuracy has been set to s.3 m, and it is of great importance that this requirement is satisfied, thus making it possible for the passengers to enter and exit the train. Except for stopping accuracy, timing, defined as t = t run t des, is also very important, but no specific requirements are set in this thesis. The controller should be stable and should be robust enough to handle model deviations, where the model parameters are allowed to differ up to ±15 % from their original value. Additionally, the controller should function also in the presence of measurement noise n. All requirements on the control system are summarized in Table 1. Table 1: Requirements on the control system Parameter Requirement Velocity v v AT P 8 km/h Throttle u 1. Acceleration a 1. m/s 2 Jerk j.6 m/s 3 Stopping accuracy s.3 m Stability The system should be stable Robustness Parameter deviations up to ±15 % 5.3 Control Design In this section, a number of controllers are designed and the outcome is compared. Since MPC technology was used to create velocity reference curves in Section 4.3, an MPC controller is designed and used to track the curve. The result is compared to a simple PID controller designed using specifications in the frequency domain. Worth noting is that two controllers are to be designed for both the MPC and PID case respectively, since the train models differ for traction and braking. Additionally, a feedback controller is designed, which is to be used in the final braking part to ensure that the requirement on stopping accuracy is satisfied, and a number of addons are designed to further improve the performance of the already designed controllers. 24

26 5.3.1 PID Controller Before designing a PID controller, the basic restrictions of the system need to be defined. In the case of the train models defined in Section 3, the largest constraint is the time delay. Intuitively, the control system can not be faster than the time delay, since this would mean the system G does not have time to react before a new control command is issued. As discussed in [15], this means the bandwidth ω B of the control system is restricted by the time delay τ d as ω B < 1 τ d (21) The open loop of the system is denoted G = GF. Noting that ω c ω B, where ω c is the crossover frequency defined as G (iω c ) = 1 = db, the restriction is easy to implement in the controller. Except for the time delay, the constraint on the control output u 1 also poses a great constraint on the speed of the controller, since a fast controller requires a large control signal, which might not be possible in this system. Using Bode plots is a very straight forward method of designing a PID controller, and will therefore be the method used here. Basic theory is found in [15], Chapter 5. Controller for Braking Model A controller is to be designed for the model defined in Section 3.2, although not including the nonlinear part. That is, the model is defined as G b = a,b τ b s s e sτ d,b To get an idea of the behavior of the system, a P controller is chosen as F = K, where K = 1. This yields the open loop Bode plot shown in Figure 12. The phase margin ϕ m is defined as the distance between the phase curve and 18 at the frequency ω = ω c. Since instability occurs as G (iω) encircles 1 according to the Nyquist criterion, and since this occurs when G (iω) > 1 and arg G (iω) = 18, ϕ m could be seen as a measurement of how much the phase curve could be displaced before instability occurs. As seen, due to the time delay, the phase curve is decreasing rapidly for growing frequencies, making the system highly unstable in that region. Choosing the crossover frequency ω c =.5/τ d,b =.33 rad/s gives the phase margin ϕ m = 4. By using the relation e sτ = e iωτ = e iϕm the largest time delay τ allowed can be calculated to τ = 2.9 s, which is more than the actual time delay of the system. In the real system, overshoots make the train velocity follow a wavy pattern, which means the acceleration keeps changing sign. It is difficult for people to brace themselves to sign changing acceleration, and for the sake of passenger comfort, overshoots should therefore be avoided. To achieve this, the phase margin needs to be raised further, for which a differentiating lead link F lead is introduced as F lead = τ Ds + 1 βτ D s + 1 As seen, the parameter β decides how close to a pure differentiating link F lead should be, and setting β = gives a pure PD controller. To see how much the 25