Annals of Operations Research 98, 65-87, 2000 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.
The Optimal Control of a Train PHIL HOWLETT
[email protected] Centre for Industrial and Applicable Mathematics, University of South Australia, Adelaide, SA 5000, Australia
Abstract We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with die sel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn-Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem. Keywords; train control, optimal control, discrete control, optimal switching times
1.
Introduction
We consider the problem of determining an optimal driving strategy for a train. We assume that the joumey must be completed within a given time and seek a strategy that minimises fuel consumption. Our main results are established using a generalised equation of motion and, as such, are new. The derivations and many of the specific results are also new. We introduce the discussion by presenting a brief review of the significant papers in this area and illustrate our remarks by considering a typical train control problem. 7.7. The significant milestones Although the train control problem was studied in early works by Ichikawa [20] in 1968, Kokotovic and Singh [22] in 1972 and by Milroy [23] in 1980 the first comprehensive analysis on a flat track was presented by Asnis et al. [1] in 1985. Asnis et al. assumed that the applied acceleration was the control variable; that the control was a continuous control with uniform bounds; and that the cost associated with a particular strategy was the mechanical energy consumed by the train.
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Although continuous control is available on some trains it is normally not possible to directly control acceleration and it is probably not reasonable to assume that the acceleration is a uniformly bounded control. Asnis et al. used the Pontryagin principle to find necessary conditions on an optimal control strategy and showed that these conditions could be applied to detennine an optimal driving strategy. A similar solution was discovered independently by Howlett [9] in 1984 but for various reasons was not published externally [10] until 1990. While the paper by Asnis et al. was a more elegant treatment the paper by Howlett showed that after the optimal control sequence had been determined by the Pontryagin principle a simplified problem could be formulated as a finite dimensional constrained optimisation in which the variables are the unknown switching times. This idea was the basis for subsequent solutions of the train control problem in the case where only discrete control is allowed. The next significant development was the observation by Benjamin et al. [2] in 1989 that the control mechanism on a typical diesel-electric locomotive is a throttle that can take only a finite number of positions. Each position determines a constant rate of fuel supply to the diesel motor and thereby detemiines a constant level of power supply to the wheels. Benjamin et al. also observed that except at very small speeds the acceleration of the train is inversely proportional to the speed. The theoretical basis for the optimal control of a typical diesel-electric locomotive was developed during an extensive railway research program begun in 1982 by the Scheduling and Control Group (SCG) at the University of South Australia. This research program is described in a recent book by Howlett and Pudney [14]. The significant theoretical papers produced by the SCG on the discrete control problem include papers by Cheng and Howlett [4,5], Pudney and Howlett [24], Howlett et al. [11-13], Howlett [15] and Howlett and Cheng [16]. The Ph.D. thesis by Cheng [7] is also a significant work and contains many additional details including a comprehensive collection of realistic examples. Several other useful results were detailed by Howlett and Pudney [14] in their 1995 book. They showed that the optimal control problem for a train with a distributed mass on a track with continuously varying gradient can be replaced by an equivalent problem for a point mass train and that any strategy of continuous control can be approximated as closely as we please by a strategy with discrete control. They also showed that where speedholding is possible it must be optimal. In the case of continuous control the first comprehensive analysis on a track with continuously varying gradient was given by Khmelnitsky [21] in 1994 who formulated the problem using kinetic energy as the primary dependent state variable. Khmelnitsky showed that the predominant speedholding mode must be interrupted on steep uphill sections' by phases of maximum power and on steep downhill sections^ by phases of coasting. These results are consistent with earlier work by Howlett et al. using discrete An uphill section is said to be steep if the speed cannot be maintained at or above the desired level using maximum power. A downhill section is said to be steep if the speed cannot be kept at or below the desired speed by coasting.
TRAIN CONTROL
6*1
control. Khmelnitsky also showed that the continuous control problem can be solved when speed restrictions are imposed. The corresponding problem for discrete control is extremely difficult because it is no longer possible to follow an arbitrary smooth speed limit precisely. This problem has been solved only recently by Cheng et al. [6]. 7.2. Some new results In this paper we formulate and solve a generalised train control problem using a general equation of motion. In the first place we consider the problem with continuous control and in the second place we study the problem when only discrete control is available. In the case of continuous control we will present an argument that is essentially equivalent to the argument presented by Khmelnitsky. We will however give a more usual formulation^ with speed as the primary dependent state variable. To begin we follow Khmelnitsky and show that the Pontryagin principle can be applied to find necessary conditions on a solution. However we will also take some time to establish new formulae that show how the solution of the continuous control problem can be related directly to the solution of the corresponding discrete control problem. These results are new. We also present new results relating to the calculation of optimal switching points. ln the case of discrete control we show that the known results can be extended to apply in more general circumstances. We pay particular attention to the derivation of the key equations that provide a basis for calculation of the strategies of optimal type. We present two new methods for the derivation of these equations. We show that an argument of restricted variation first used by Howlett for flat track [19] can also be used in this more general situation and that a rather obscure continuity principle for the Hamiltonian function [26] can be used to derive the key equations for optimal switching. L3. Other relevant research The problem of finding optimal driving strategies for a solar powered racing car is closely related to the above train control problems. We will refer to a paper by Howlett et al. [17] which considered optimal driving strategies on flat track and a paper by Howlett and Pudney [18] which considered the problem on undulating road. An altemative formulation of the solar car problem as a problem of shortest path by Gates and Westcott [8] is less relevant to our current discussion but is nevertheless an interesting paper. 2.
Formulation of the generalised train control problem
A train travels from one station to the next along a smooth track with non-zero gradient. The journey must be completed within a given time and it is desirable to minimise the fuel consumption. ^ Our fonnulation is more usual but not necessarily better.
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2.J. The generalised train model We consider a control problem with r € [0, 7] c ]R as the independent variable and state variables x = xit) e [0. X] c ]R and u = vit) € [0, V] c E satisfying a simple autonomous system of differential equations in the form dx and du — = Fix, u, u) at
(2)
with x(0) = 0,
xiT) = X and u(0) = viT) = 0
for ix, v) € S = [0, X] X {0, V] C E^ and « 6 f/ c [-q, p] C R where ;? and ^ are positive real numbers. The variable w is the control variable. For the train control problem we interpret / as time, x as position, and v as speed. We assume that Fix, v, u) is well defined and continuous for all (jt, u,«) e [0, X] x (0, V] x U with F(x,O,-q) R is a continuous function which is strictly increasing on [0, p] and differentiable on (0, p). We also assume that / ( « ) = 0 for M e [—q, 0]. Remark 2.L Use of a generalised equation of motion (2) will allow us to extend the established theory to more realistic models that could incorporate such things as the effects of track curvature. 2.2. The vehicle control problem with continuous control In some modem diesel-electric locomotives continuous control is available. In this case the control variable u can take any value in some bounded interval [—q, /?]. We assume
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;
69
that each value u e [0, p] of the control variable determines a constant rate of fuel supply. Although braking is more complex the complexities relate mainly to engineering and safety issues and it is reasonable to assume that each value u € [—q, 0] determines a constant negative acceleration. It is also pertinent to observe that the time spent braking is relatively small. We assume that no fuel is consumed during braking. We wish to find a bounded and measurable control function that minimises fuel consumption but allows the train to arrive at the destination on time. We use the model described in subsection 2.1 and assume that the control variable u e V = [-q, p] c E, Remark 2.2. To state the problem precisely at this stage and in this general form is difficult and somewhat pointless. In general terms we can say that it is necessary to establish a suitable function space for the set of control functions and to show that each control function from this set determines a unique speed profile for the train. It is also necessary to show that there is at least one feasible strategy and that there exists a control function from within the specified set that minimises the cost of fuel. We refer to a paper by Howlett and Pudney [18] where a similar problem is considered in detail and a complete proof of existence is given. • i 2.3. The vehicle control problem with discrete control In this case the control variable u can take only a finite number of pre-determined values. This is the typical situation in a diesel-electric locomotive where there are a finite number of distinct traction control settings M G {0, 1
;?} C E
and each setting determines a constant rate of fuel supply. It is reasonable to assume that there are a finite number of brake settings ' u€{-q,-q
+l
-1}CR
and that each setting determines a constant negative acceleration. We assume that no fuel is consumed during braking. As long as the track is not steep the train will be controlled using a finite sequence of traction phases and a final brake phase. During each phase the control setting is constant. For each sequence of controls there are many different strategies, each one determined by different switching points. For each strategy there is a uniquely defined speed profile detennined by the equations of motion. We will say that a strategy is feasible if the distance and time constraints are satisfied. We wish to find a feasible strategy that minimises fuel consumption. We use the model described in subsection 2.1 and assume that the control variable
ueU ^[-q,-q
+1
0
p - 1,/?} CR,
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where p and q are fixed positive integers.* For each fixed sequence [ukJr\]k=o.\ n of control settings and each partition 0 = / o ^ r i ^ ••• ^
WI
'
(5)
ofthe positive r-axis there is a corresponding control function M : (0, /n+i) —»• V defined essentially by
(6) for t 6 (tk, tk+\) and each ^ = 0, 1 , . . . , n. We denote the corresponding strategy by
and use (
«)
(8)
to denote the collection of all strategies using the given sequence {«A-htl*=o.i....,n of control settings. The value u^+i is the control setting on the interval {t^, ^A+I). The times [tk]k=\.i n are known as the switching times with ^o the starting time and tn^\ the stopping time. We write X](_ — x(tk) to generate a corresponding partition O = J : O < ^ I < - - •
