The Optimal Control of a Train

Transcription

1 Annals of Operations Research 98, 65-87, Kluwer Academic Publishers. Manufactured in The Netherlands. The Optimal Control of a Train PHIL HOWLETT p.howlett@unisa.edu.au 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 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 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.

2 66 HOWLETT 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 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.

3 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] 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.

4 68 HOWLETT 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 and with dx du = Fix, u, u) (2) at 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) <Q<Fix,O,p) for all X [0, X]. We also assume that the partial derivatives df df df, and ax dv du exist and are continuous with df -T-ix,v,u)>0 (3) au for all ix, v,u) [0, X] x (0, V] x U. The cost of the journey is measured by the fuel consumption J= / f[uit)\dt, (4) Jo where /:[ q, p] i-> 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 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

5 TRAIN CONTROL ; 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 ;?} CE 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 p- 1,/?} CR,

6 70 HOWLETT where p and q are fixed positive integers.* For each fixed sequence [ukjr\]k=o.\ n of control settings and each partition 0 = /o^ri ^ ^ WI ' (5) ofthe positive r-axis there is a corresponding control function M : (0, /n+i)» V defined essentially by for t 6 (tk, tk+\) and each ^ = 0, 1,..., n. We denote the corresponding strategy by (6) 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 <-- <;c«+i (9) ofthe positive ;c-axis, where the points [xk]k=\.2 n are known as the switching points with XQ the starting point and Xn^\ the stopping point. We write T^.^i = Ar* and ^^+1 = Ajjt and also write V^ = v(tk) for the speed at the switching time /; If the cost rate of each control is defined by a function / : t/ -^ M then the cost of the strategy 5 is given by f "*' f[u{t)] dt = (10) k=0 Note that 7(5) depends continuously on the switching times ^i. r2r,,'«we can now state the problem precisely. Problem 2.3. For each fixed sequence {MJH-I1A=O,I n of control settings find the switching times [tk]k=\.2,..,n that minimise J{S) subject to the initial condition VQ = 0, the final condition V^+i = 0 and the constraints tn+\ ^ T and Xn+i ^ X. Remark 2.4. Howlett and Pudney [14] have shown that any sequence of positive measurable control can be approximated as closely as we please by a sequence of power-coast pairs. Thus for a sufficiently long control sequence [p, 0, /?, 0 p, 0, q) on nonsteep track it is obvious that the strategy with the best switching points will be close to ** The condition that df/du > 0 can be replaced in this case by the weaker condition that Fix, u, uj) < Fix, V, uj) whenever u i, ii2 U and «i < U2.

7 TRAIN CONTROL 71 the minimum cost strategy for continuous control. More information about the interpretation of solutions to the discrete control problem can be obtained from the book by Howlett and Pudney [14] Existence of a solution for the discrete control problem For the moment we consider the speed v and the control u as functions of position X [0, X]. The equations of motion can be written as a single differential equation v-^ = Fix,V,uix)). ' (11) dx Assume that the solution v = Vp{x) to the initial value problem v^ = Fix,v,p) (12) dx for the region J: ^ 0 with ii(o) = 0 exists and is unique and that the solution v = if_^ {x) to the final value problem, V = Fix,v,-q) (13) for the region x ^ X with v(x) = 0 exists and is unique. In this case the conditions Fix, 0, -q) < Fix, 0, p) for all x e [0, X] and Fix, v, -q) < Fix, v, p) for all ix, v) G [0, X] X (0, V) imply there is a uniquely defined point Xf, (0, X) with Vpixf,) = v^gixfj). It is now easy to see that there is a maximum speed strategy on [0, X] with an initia! phase of maximum power using u^ix) = p for x e [0, x^] and a final phase of maximum braking using u^ix) = q for x e [xb, X]. The speed V = Vm{x) defined by satisfies the boundary conditions v^ig t(x) = 1 0 Vm(x) dx ifo<jc<jcfc, if Xb K X ^ X = o.u then the maximum speed strategy is feasible. Now let r be a fixed positive integer and consider the set Tr of feasible strategies of the form {p, 0, /?, 0 p, 0, -q} with switching points 0 = = X. The set J^r is non empty since it contains the maximum speed strategy (the maximum speed strategy has.ri =.^2 = = ^2r = xi,) and so Jinf= inf 7(5) ^0 5JP;

8 72 HOWLETT is well defined. We need to show the existence of a strategy S e Tr with J{S) = Jinf. Let {5'''}j=i.2,... be a sequence of feasible strategies with switching points such that JTO 5; X] ^ ^.1^2/- We can choose a subsequence {/(s)}i=i,2,... such that, for each /: = 0, 1 sequence of switching points 2r - - ], the converges, as 5 -> oo, to a limit x^. Clearly 0 = io ^ -^1 ^ ^ -V2r ^ X.2r+\ = X. If we define 4 = /(-^t) as the corresponding switching time then the strategy 5 = [[p\ (0, fl)], [0; (fl, fa)] [p\ ihr-l, hr-x)], [O; ihr-xjlr)]^ [-<?; (f2r, f2r-fl)]) is ajninimum cost strategy in the set Tr- It is obvious from equation (10) that as s» oo. 3. The general solution procedure for continuous control In this section we will show how the Pontryagin principle [3] can be applied to find an optimal strategy for the continuous control problem and indicate how the key equations can be derived. The Hamiltonian is given by and the adjoint equations are //(JC, v,a,^,u) = -/(«) -\-av -\- ^Fix, v,u) (14) da dh 3F and ; d^ 9ff o^f dt dv dv for (x, v) [0, X] X (0, V] and (a, ;S) Z, c R^ ^here L is an appropriate closed rectangle. Let u = u{t) for f G [a, b] be a given smooth control strategy^ and suppose that the initial conditions xia) = x^, via) = Va and the final conditions a{b) = at,, pib) = pt} are known. Under these circumstances we can solve the state and adjoint ^ We refer here to a control strategy that is smooth on some subinterval [a, b] C [0, T]. We will ultimately piece together a succession of smooth strategies to form a piecewise smooth control strategy.

9 TRAIN CONTROL 73 equations to find uniquely defined state functions jc(f), v(t) and adjoint functions a(t),. We will often use the abbreviated notation K(t) = K(x(t), v(t), a(t), m> "(0) when we evaluate a function /:: [0, X] x (0, V] x L x t/ M- R along the unique trajectory determined by «(/). In particular we write and it is easy to see that Hit) = -f[u{t)]+av{t)- ^F{t) (17) H'(t) = r-/[m(0] + ^(0^(0l"'(0. (18) The Pontryagin principle tells us that the control u{t) is an optimal control at the point (JC, u,a, ^) only if u(0 e (/is the value of the control that maximises the Hamiitonian H{x,v,oc,^,u). Since there arefivedifferent cases to consider. We have L ^ > /'(H)/f => uit) = p; 2. ^ = /'(u)/ J>0=^u(Oe[0,p]: 3. 0<p<f'(u)/' =>uit) = Q; 4. ^ = 0 =^ u(t) [-q, 0]; and 5. /3 <O=^u(t) = -q. ^ -/'(«)+ ^ ^ (19) Bu au Notice that in all cases [ + ^^{x, V,tt(O)l«'(O = 0 (20) and hence the Hamiitonian H(t) is necessarily constant along any portion ofthe optimal trajectory generated by a smooth control. In fact the Pontryagin principle states that the Hamiitonian is constant along the entire optimal trajectory even if the control is not smooth. Therefore we have ^ = -f[uit)] + ait)vit) + mno (21) for some constant /i K. It is often useful to integrate the adjoint equations along the optimal trajectory. Using equation (21) the first adjoint equation can be rewritten as,

10 74 HOWLETT and if we define,b)= ^,^is)ds (23) J,^, Fis) dx for t e[a,b] then it follows that '""r^i±/t^^<" (24) for all t [a, b]. By substituting the expression (24) into the equation (21) we can find a similar expression for ^(r) for all t e [a, i*]. This expression can also be obtained by direct integration along the optimal trajectory although the details are a httle tricky. Such integrations can be used to obtain key equations that determine the optimal switching points. We will consider this more closely in the next section. 4. A typical train control problem with continuous control For a typical train control problem we can write Fix, V, u) = ^ ^ + Biu) + six) - riv), (25) where A > 0 is a constant, B(w) = 0 for 0 ^ M ^ p and BiU]) < Biu2) < 0 for U\ < U2 < 0, gix) is the component of gravitational acceleration in the direction of motion and riv) is the frictional resistance per unit mass opposing the motion. The equations of motion become and dv _ Afiu) dt V and the Hamiltonian is given by djc -^ = v (26) + Biu) + gix) - riv) ill), V, a, y5, M) = -/(«) -\-av + m -^^^-^ + Biu) + gix) ~ r{v) \. (28) The adjoint equations are and da = ~^g'ix). (29) db VAfdA 1 (30)

11 TRAIN CONTROL 75 We maximise the Hamiltonian function by writing it in the form Hiu) = I - \]fiu) + ^Biu) + (31) \ V ) and note that there are five different cases to consider. We have 1. ^ > vja =^ u = p, 2. ^ = v/a^u [0,p], 3. 0 < ^ < v/a =^ M = 0, 4. ^ = 0 => u e [ q, 0], and If the condition A^^v. (32) is maintained over a non-trivial time interval we will show that the corresponding control mode is a speedholding mode. Although the speedholding control is a singular control it is nevertheless the key to the entire optimal strategy Speedholding with continuous control If the condition (32) holds for all t e [a, b] then by differentiating both sides and rearranging we obtain where (piv) = vr(v). However we also have from which it follows that d«.,_ vg'ix) d f gix) I ^^^^ dt '^ A dr I A Qt = for all / [a,b'\. Therefore we deduce that A (35) ip'iv) = C. (36) If we assume that the graph y = (piv) is convex then it follows that i; = V is constant and that yg{x) -\-(piv) fiu) = ^^^/ ^^ \ (37)

12 76 HOWLETT Since the Hamiltonian is constant along an optimal strategy we have H{t) = fx for some constant /^ e R and all r [o, b]. From (28) it follows that ^ (38) where ^(u) = v^r'iv). On fiat track it has been shown that the optimal strategy consists of an initial phase of maximum power with u = p followed by a speedholding phase^ with u [0, p], a coasting phase with «= 0 and a final brake phase with u = ~q. The speed at which braking begins is related to the holding speed by a simple formula. On a track where the gradient varies continuously the most significant difference is that there may be sections of track which are so steep that speedholding is not possible. If gix) < r{v) - ^ ^ (39) then the track is so steeply uphill that the desired speed V cannot be held even under maximum power whereas if g(x)>riv) (40) then the track is so steeply downhill that the desired speed V cannot be held even if no power is applied. On steep uphill sections we will show that maximum power must be used and on steep downhill sections we will show that no power is used. We will also show how to calculate the precise switching points when a steep section of track is to be negotiated. If there are no steep sections then the optimal joumey has the same form as it does on fiat track Calculation of switching points for steep sections of track We make the following crucial observation. If the train is in speedholding mode before reaching a steep uphill section and wishes to return to speedholding mode after traversing the steep section, then the switch from speedhold to maximium power must occur before the steep section is reached. The switch back to speedhold must occur after the traverse of the steep section has been completed. We have the following result. Proposition 4.1. Let V be the desired holding speed and let for all J: G ixt, Xc). Suppose there is some interval [Xa, x^] with ix^, Xc) c [x^, AV] and On a long joumey the speedholding phase becomes the dominant phase and the holding speed is approximately the total distance divided by the total time.

13 TRAIN CONTROL T' for alljc e [Xa, Xb) U (Xc, x^]. Suppose also that we can find a feasible strategy R with VR(xb) = VRixd) = V, VR(X) = V forjc [Xa,Xb] and VR(X) < V for JC (xi,,xd) and with HR(JC) = pfor X e (x^, x^) and a feasible strategy S with vs(xa) = vs(xc) = V, Vs{x) > V forjc e (jc^, jc^) and VR(X) = V for ;C e [jc^, JC^], and with us(x) = p for JC (Xa,Xc). If g'(jc^) < 0 and g'(jcc) > Othcn neither R nor 5 is optimal. Remark 4.2. Although it is often helpful to think of JC as the independent variable and to think of V and t as dependent variables we will nevertheless continue to use the notation JC = jc(a) = Xa and v = v(a) = Va when / = a. Remark 4.3. The above proposition shows that it is not optimal to switch maximum power on at the start of the steep section and also shows that it is not optimal to switch maximum power off at the end of the steep section. ' Remark 4.4. The assumptions that ^'(JC^) < 0 and g'ixc) > 0 are reasonable since the track is only steep at speed V between x^ and Xc and hence g(x) is decreasing at x^, and increasing at jc^. Proof of proposition 4.1. Consider the strategy R and suppose that it is part of an optimal strategy. Define rjr = A^R/VR and note that r}r(b) = r)r(d) = 1 with t]r{t) > 1 for all r G (b, d). From the basic state and adjoint equations it can be seen that and if we use the expression d/ VR \_ VR then it follows that By differentiating again we can show that and since we also have <P"{VR) ( I VR VR^ J VR^ J d/ VR FR{b) = 0

14 78 HOWLETT Differentiating once more shows us that (P'{VR)\ Q( \_ VR \ VR VR J VR- \ (Xt where the additional terms are multiples of FR, drjr/dt and d'^t]r/dt^. Since it follows that dfr dt FR Now it follows that r)rit) < T]Rib) fort>b and sufficiently close to b. This contradicts the fact that T)R{t) > 1 for all / e {b, d). Hence R is not optimal. A similar argument can be applied to S with the derivatives evaluated at x^. The next result is a key equation that will allow us to determine the precise position ofthe switching points. If we use the same notation as we used in proposition 4.1 then we need to find an interval [.ico. -'^i] with Xa < Xo < Xf^ < x^ < Xj < x^ and a strategy Q with M(0 = p and r?(r) > 1 for r (a, r) and with rjia) = r/(t) = 1. We can integrate the adjoint equation for a as we did in the general case to obtain where p(, T,)^ I L ^ J Jt=a V{_t} Ij(5)g'[jc(5)]d5 {41} However we also know that I [] \ (42) from which it follows easily that ^^^'^^ t=o L ^^K (43) By using the expression (35) and the equation (43) we have (44)

15 TRAIN CONTROL and if (41) is used to eliminate a we obtain the key equation where /x = i^(v)/a. If we define e^'"'] = jj"' ^ + r[u(/)]] de^"''>, (45) Au ^+riv) (46) and use integration by parts then the key equation can be written in the more compact form I Q^^''^dE^[vit)] = O. (47) Jt= Remark 4.5. The key equation (47) can be used to find the precise switching points. ChooseCT (a, b) and solve the state equations forward in time with ii^ = V' to find T (c, d) with I;(T) = V. Now evaluate the integral /(u, r) in (47) over the interval (CT, T). If Her, r) 7«^ 0 choose a again and repeat the calculation. Remark 4.6. The switching points can also be found in the following more standard way. Choose a e ia, b) and solve the state equations forward in time with u^ = V to find T ic, d) with ii(r) = V. Now solve the adjoint equations backward in time with ^j. = V/A to find ^(a). \f ^ia) ^ V/A choose a again and repeat the calculation. Remark 4.7. The same equation, with M = 0 for / (CT, T), and the same procedure can be used to determine the precise switching points for a coasting phase when speedholding is interrupted by a steep downhill segment Calculation ofthe point at which braking begins If we assume that speedholding finishes at r = a and that we coast to t = b where braking begins then we can use the adjoint equations to relate the speed U = vib) at which braking begins to the holding speed V = via). For t e [a, b\ we define ' and we define the effective speed V{t) on the interval la, b] by the formula = e^*'-^' + / de^*^-'*. (49) Vit) vit) Js=tvis) In each ofthe above definitions x = xit) and v = vit) are the solutions in the region t [a,b]to the differential equations ^ = V (50) dr

16 80 HOWLETT and dv = gix)-riv) (51) at with the given boundary conditions. First we note that //(/) = fx is constant on the interval [a, b] and by evaluation ai t = a that /x = i/f(v)/a as before. Following our calculations in the general case we see that ah - a(a) e^'"-''* = ^ / ^. (52) Jt=a vit) If we evaluate the Hamiitonian at f = a and t = bwe get and and equation (52) becomes u am = -i^ + ' ^ (53) «6 = ^ (54) In the case of a flat track with g(x) = 0 for all x we have V(t, fr) = 0 for all / and Via) = V and the above equation takes the simplified form U = 5. The restricted variation argument for the general problem with discrete control We consider the general vehicle control problem with discrete control described in section 2.3 and use the same notation. This problem was solved by Howlett and Cbeng [16] who found a rather elegant form for the key equations that determine necessary conditions on a strategy of optimal type. However, the simplicity of the key equations was somewhat obscured by the calculation of some complicated derivatives in the Kuhn- Tucker equations. One reason for the difficulty of these derivatives is that for each A' = 0, 1,..., n the quantities ) and -^ foth < k depend on the solution of the state equations in the interval (tk.tt+i)- This solution depends, in tum, on the initial condition and the rate of change dvk 9^ for /i < k

17 TRAIN CONTROL 81 V y=vit) Figure 1. Variation of t/^-i with fixed. of the initial condition with respect to the variable ^/,+i = Xk+\ Xh- Thus these variables depend in a recursive way on each of the previous switching points. An interesting idea [19] is that instead of considering the simultaneous variation of all switching points we may be able to obtain the necessary conditions for optimality by considering a more restricted variation of the switching points. We might thereby require only a simplified form of the full recursion. Indeed, a little consideration should convince us that the time constraint r^+i ^ T and the distance constraint Xn+\ ^ X could be preserved with a variation of any three successive switching points. This is illustrated in figure 1 where only the switching times 4_], tk and tk+] are changed and they are changed in such a way that the total distance travelled is unchanged. It is clear that the distance travelled is represented by the area under the curve which is clearly preserved in the variation. In fact the only switching points to change arex^-i, x^ andxk-\^\. From now on it will be convenient to regard x as the independent variable with t = tix) and V = vix) as the dependent variables. Consider a differential variation of successive switching points.tfc_],xft andjc^+j for an arbitrarily chosen integer/: (l,n) in a feasible strategy of optimal type. A necessary condition for optimality is essentially that the differential increment of cost is non-negative when the differential increments of distance and time are respectively non-negative and non-positive. Thus, when we have the distance constraint 0, (57) and the time constraint = dtt_] -I- dzk + + (58)

18 82 HOWLETT we must also have the cost minimisation condition d/'** = /(Mi_i)dri_i + f{uk)drk + /(Wi+Odr^+i + fiuk+2)drk+2 > 0. (59) In terms of the Lagrangean differential dj^<*> = d7(*> + ;rad^^'=* - Pkdx^^\ (60) where ;ri e R and pt R are non-negative multipliers the necessary conditions for a local constrained minimum are given by and = 0 (61) ;rid t*'-/>,dt(^' = O. (62) By calculating the relevant partial derivatives we can express this equation in the general form A(d i_i,d^a,d i+,) = O, (63) where A e (E^, R) is a linear mapping and where the equation (63) is satisfied for all feasible differential increments (d^t_i, d^^, d^^+i). To establish the precise form of equation (63) we need the following additional notation. We write Fs^\{x, v) = F(jc, u, H^+I) and define \ df I (w dx for each x [^j,xi.i.i]. We write V^ix) = 'Dsix,x,+i) and V, - VsiXs) for convenience. We also let and define the effective speed Vsix) at each point x [Xs, Xs+\] by the formula 1 1 ^ TT e H / (65) VAX) vix) J^^, viw) We write V, = V^Xs)- The basic recursive relations derived by Howlett and Cheng [16] are simple enough and are given by and (67)

19 TRAIN CONTROL 83 for r < s. Howlett and Cheng also show that = 77^ (68) and 3^^ J 1_ -1 + V. V, (69) for r < s. We will write fs = /(«s) and Afs = fs+i fs- Although the algebra is still somewhat tedious it is now possible to calculate the required partial derivatives. In making these calculations one must remember that only Xk^\, x^ and Xk+] are varied. Because of the restricted variation the extensive recursion and excessive complication of the original derivation [7,16] are avoided. We can now substitute these expressions into the necessary condition (63) and equate the coefficients of df^+i, d^^ and d^^t-i to zero. From the coefficient of d^^^+i we have = Ttic. (70) From the coefficient of d^k and by elimination of 7Tk using equation (70) we get -A/, e^^, From the coefficient of d^k-\ with rrr^ once again eliminated using equations (70) and (71) we obtain Af, Q, Vk Qk-i T =0. (72) If these conditions are applied to all values of k e (\,n) we obtain the necessary conditions for a strategy of optimal type. In particular we note that if k is replaced by /: 1 in equation (71) and the resulting equation is compared with equation (72) we can see that PJ^_I = pj and hence pk = P for some p K with p > 0 and all k. By replacing k by /c 1 in equation (70) and comparing the new equation with the original it is now quite easy to see that +p, QA-H r fk-\ J The conditions (70) and (73) were found recently by Cheng [7] using a more complicated version of the above argument involving a full variation.

20 84 HOWLETT 6. The key equations for a typical train control problem with discrete control For the typical train control problem with discrete control where Afiu) Fix, V, u) = ^^-^ -^ Biu) + gix) - riv) (74) the key equations take a simplified form. In this case equation (71) becomes ^ (75) yk+\ By using the equation = r*" mm^ ^ - r[vix)]\ [ ] \ ^^' (76) and applying an appropriate integration by parts the key equation can be written in the equivalent form fjxi =0. (77) It is interesting to compare this equation with the equation (47) for optimal switching in the continuous control problem. 7. The continuity principle Once again we consider the general vehicle control problem with discrete control described in section 2.1. We form a Hamiltonian function H -.U^ x U -*- Rby setting Hix,v,a,fi,u) = -fiu)-\-av + fifix,v,u), (78) where the adjoint variables a ait) e R and ;3 = fiit) e R satisfy the adjoint differential equations (15) and (16). It is easily seen that there is a sequence {Mit+i}i=o,i n e R with forr itk. tk+i). Therefore (79) and hence -fk+i -\-cxv + fifk+iix,v) = ij.k+i (80), v)

21 I TRAIN CONTROL 85 on this interval. If we rewrite equation (15) in the form -^Jki,,) I (82) dt Fk+]ix,v) Fk-i.\ix,v) dx then we can integrate equation (82) from tk to tk+i to deduce that '*+'\ (83) where we have used the notation ak = aitk) to denote the value of a at time tk. If we assume that the Hamiltonian is continuous at tk and if we write fik = ^ (h) to denote the value of ^ at time tk then, by equating the left and right hand limits of Hit) at tk we have and hence -fk + akvk + ^kfkixk, Vk) = -/t+i + auvk + ^kfk+xixk. Vk) (84) ^ ^ '. (85), Vk) - Fkixk, Vk) 1 - Qk V) The continuity ofthe Hamiltonian also implies that there is some /^ G M with and hence f (86) for all t (tk, tk+i) and each k = 0,\,...,n. Evaluating the Hamiltonian by taking the right hand limit at tt, and using equation (85) gives and by substituting in equation (83) and simplifying we obtain Ajk ^ Jk+l I M l-qk Vk Vk ~"'^+'* =M (87) On the other hand we note that evaluation of the Hamiltonian by taking the left hand limit at tk gives /i- + oftvt - - BiiPkixh, Vt) = ju. (89) and by rearranging and using equations (85) and (88) we have _ A + /^ Qk AA ^ A + ^ Qk ijk \ jk-n I r- _ I foci The equations (88) and (90) are the key equations given by Cheng [7].

22 86 HOWLETT References 11 ] I. A. Asnis, A. V. Dmitruk and N.P. Osmolovskii. Solution of the problem of the energetically optimal control of the motion of a train by the maximum principle, U.S.S.R. Comput. Math. Math. Phys. 25(6) (1985) 37^W. [2] B.R. Benjamin, I.P. Milroy and P.J. Pudney. Energy-efficient operation of long-haul trains, in: Proceedings of the 4th Intemational Heavy Haul Railway Conference, leaust. (Brisbane, 1989) pp [3] L. Ccsari, Optimisation - Theory and Applications (Springer. 1983). [4] J. Cheng and P.G. Howlett, Application of critical velocities to the minimisation of fuel consumption in the control of trains, Automatica 28(1) (1992) [5] J. Cheng and P.G. Howlen, A note on the calculation of optimal strategies for the minimisation of fuel consumption in the control of trains, IEEE Transactions on Automatic Control 38(i 1) (1993) [6] J. Cheng, Y. Davydova, P.G. Howlett and P.J. Pudney, Optimal driving strategies for a train journey with non-zero track gradient and speed limits, IMA Joumal of Mathematics Applied in Business and Industry 10(1999) [7] J. Cheng, Analysis of optimal driving strategies for train control problems, Ph.D. thesis. University of South Australia (1997). [8] DJ. Gates and M.R. Westcon. Solar cars and variational problems equivalent to shortest paths. SIAM Joumal on Control and Optimization 34(2) (1996) [9] P.G. Howlett, The optimal control of a train. Study Leave Repott, School of Mathematics. University ofsouth Australia (1984). [10] P.G. Howlett, An optimal strategy for the control of a train, J. Aust. Math. Soc. Ser. B 31 (1990) [11] P.G. Howlett, PJ. Pudney and B.R. Benjamin. Determination of optimal driving strategies for the control of a train, in: Computational Techniques and Applications; CTAC 91, eds. B.J. Noye, B.R. Benjamin and L.H. Colgan, Computational Math. Group, Division of Applied Math., Aust. Math. Soc. (1992)pp [12] P.G. Howlett, LP. Milroy and P.J. Pudney, Energy-efficient train control. Control Engineering Practice 2(2) (1994) [13] P.G. Howlett, J. Cheng, and P.J. Pudney. Optimal strategies for energy-efficient train control, in: Control Problems in Industry, eds. 1. Lasiecka and B. Morton. Progress in Systems and Control Theory (Birkhauser, 1995) pp [14] P.G. Howlett and P.J. Pudney, Energy-Efficient Train Control, Advances in Industrial Control (Springer. London. 1995). [15] P.G. Howlett, Optimal strategies for the control of a train. Automatica 32(4) (1996) [16] P.G. Howlett and J. Cheng, Optimal driving strategies for a train on a track with continuously varying gradient, J. Aust. Math. Soc. Ser. B 38 (1997) 388^10. [17] P.G. Howlett, RJ. Pudney, D. Gates and T. Tamopolskaya, Optimal driving strategy for a solar car on a level road, IMA Joumal of Mathematics Applied in Business and Industry 8 (1997) [18] P.G. Howlett and P.J. Pudney, An optimal driving strategy for a solar powered car on an undulating road. Dynamics of Continuous, Discrete and Impulsive Systems 4 (1998) [19] P.G. Howlett, A restricted variation argument to derive necessary conditions for the optimal control of a train, in: Progress in Optimization 11; Contributions from Australasia., eds. X.Q. Yang, A.I. Mees, M.E. Fisher and L.S. Jennings (Kluwer) to appear. [20] K. Ichikawa, Application of optimization theory for bounded state variable problems to the operation of a train, Bulletin of Japanese Society of Mech. Eng. 11(47) (1968) [21] E. Khmelnitsky, On an optimal control problem of train operation. Report for the Faculty of Engineering, Department of Industrial Engineering, Tel-Aviv University (1994).

23 TRAIN CONTROL [22] P. Kokotovic and G. Singh. Minimum-energy control of a traction motor. IEEE Trans, on Autom^ic Control 17(1) (1972) I [23] I.P. Milroy, Aspects of automatic train control, Ph.D. thesis, Loughborough University (1980). [24] P.J. Pudney and P.G. Howlett. Optimal driving strategies for a train journey with speed limits. J. Aust. Math. Soc. Set. B 36 (1994) [25] P.J. Pudney, P.G. Howlett, B.R. Benjamin and I.P. Milroy, LP., Modelling the operational performance of large systems; A railway example, in: Computational Techniques and Applications: CTAC 95, eds. R.L. May and A.K. Easton (World Scientific, 1996) pp [26] K.L. Teo, C.J. Goh and K.H. Wong, A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics 55 (Longman Scientific and Technical, Longman. UK, 1991).

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