Speed control of a field controlled D.C. traction motor

0005-109S/81/040627-04 $02.00/0 Pergamon Press Ltd t ' 1981 International Federation of Automatic Control

Automatica, Vol. 17. No. 4, pp. 627-630, 1981 Printed in Great Britain

Brief Paper Speed Control of a Field Controlled D.C. Traction Motor* M. PACHTERt Key Wor~--D.C. traction motor; minimum-time control; microprocessor control; computer control

Abstract--An optimal (minimal-time) feedback control law is synthesized for speed control of a field controlled D.C. traction motor in a battery-driven vehicle application.

Rf

1. Introduction

,,)

ef

±

AN EXPERIMENTAL electric vehicle (Welz and Van Niekerk, 1979) is considered, which has a drive train that consists of a D.C. electric traction motor driving the input shaft, a clutch and a gearbox connected to the output shaft. We assume that the gearbox is disengaged (via the clutch) and address ourselves to the specific problem of controlling the motor so as to synchronize the speed of the input (motor) and output shafts prior to gear engagement. Motor control could be implemented by means of armature current control with a conventional PI (proportional integral) controller, since in this case the electric motor could be considered a linear first-order control system. It was felt, however, that it would be desirable to 'field control' the motor, i.e. higher speeds are then achieved by the so-called 'field weakening', since then less current (and energy) would be involved in the procedure. It is thus our aim in this paper to synthesize a controller, for field control of the motor speed, for gearbox synchronization. Chow and Kokotovic (1978) present a method for the approximation of feedback controllers which is based on a two-time-scale approach (and which is applicable to electrical machine models). Indeed, considering a reduced order system (in Section 3) proves to be helpful; however, a complete analysis by means of Pontryagin's maximum principle which takes into account the control and state constraints is possible here. Specifically, in Section 2 we model the inherently nonlinear control system associated with field control of the traction motor, and in Section 3 we synthesize the field current (minimum-time) control law for a simplified first-order model which arises when the dynamics of the field circuit are neglected. In Section 4 we then present the synthesis of the feedback (minimum-time) control law for the second-order model, while concluding remarks on implementation are made in Section 5.

Lf

J

-2"

~"

Rx

FIG. 1. The field and armature circuits. The equations of motion are i,R:, = V - e

(1)

e=Kritm

(2)

T, = kli$i o

(3)

do)

J --~-- = T.

(4)

di/ es = RsJS + Ls--~-.

(5)

and .

Here, t Is] is the independent time variable and the motor constant (in the correct units) is KI=0.098. In addition, we have the following technological constraints

2. The field control model of the D.C. traction motor

Consider the D.C. electric traction motor shown in Fig. 1. The battery voltage, V=144 volts ; e IV] is the back electromotive force of the motor; e~- I-V1 is the field circuit voltage; io and i.r [A] are the armature circuit and field circuit currents, respectively; the armature circuit and field circuit resistances are, Rx=0.12fl and R i = 1 0 f l , respectively; the field circuit inductance, L$--1.4 Henry (the armature circuit inductance of the traction motor is ,~,6 x 10-3H and it has been neglected) and T=[Nm], co[s-l], and J = 0 . 4 [kgm 2] are the motor torque, speed and moment of inertia, respectively.

i° _~220

(6)

210
(7)

which means that at any instant the motor speed must be below 6100r.p.m. and above 2000r.p.m. 2.34 = i$ ~ 7.5

(8)

O ~ e f ~ 144.

(9)

and *Received II October 1980; revised 4 August 1980; revised 15 December 1980. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor P. Dorato. This work was partially supported by a grant from Control Data. *National Research Institute for Mathematical Sciences of the CSIR, P.O. Box 395, Pretoria, South Africa.

Finally, the five forward gears of the gearbox have the gear ratios ai= 5.034:1, 2.769:1, 1.644:1, 1:1 and 0.793:1 for the first, second, third, fourth and fifth gears, respectively. Since the vehicle inertia is large in comparison with the inertia of the motor, we conclude that synchronization for a gear shift, say for a shift from first gear to second gear, implies that the motor must be slowed down so that the 627

628

Brief Paper

motor speed is decreased from to0 to ml. Here mo is the motor speed prior to the gear change and mt =~Oo7i where 7i A =cq+l/~ i, i=1, 2, 3, 4 and for i=1 here )h=2.769/5.034; similarly, for a shift from second to third gear, from third to fourth gear and from fourth to fifth gear, the motor deceleration speed ratios 1/7 i are 1.684, 1.644 and 1.261, respectively. Furthermore, engagement of a lower gear entails accelerating the motor in conformity with the motor speed ratios given above. In order to achieve a synchronized and smooth gearbox design it was felt that minimum-time performance would be ideal in this situation. Although energy considerations are of paramount importance in this application, the actual energy expenditure for specifically synchronizing the gearbox is very small because in this mode of operation the load is not applied to the motor, viz. the motor and gearbox are disengaged (declutched). Therefore a minimum-energy performance criterion is irrelevant. Hence we pose the following optimal control problem of motor speed synchronization for a gear shift. Given the motor speed ~oo prior to the gear change, find an optimal control (an optimal field voltage e: time history) which will transfer the motor speed from o% to ~ol in minimum time, subject to the dynamical constraints (1)-(5), and to the technological constraints (6)-(9); here to I is related to coo according to the corresponding gear shift motor speed ratio yl where the index i (1 < i < 4 ) specifies the gear shift in question. It is convenient to employ the following notation A

A

A

A .

second gear xl =XoTt, i.e. xt =x0/1.818) subject to the state constraint (13), the state/control constraint ux>=1200

(17)

2.34 < u < 7.5.

(18)

and the control constraint

First consider the right-hand side of equation (16) (see Fig. 2). Obviously, given x, then if the control (the field current) u < ( b / a ) ( 1 / x ) the motor accelerates and if u > ( b / a ) l l / x ) the motor decelerates.

I

2,34

b

1200

b~

17,5

FIG. 2. The motor acceleration capability as a function of the field current.

A

x=co, (and Xo=O~o, x~ =~Ol), y=~y, u = e y , A 2 '~ a=K:/(R~J), b=K:V/(R~J),

A r=

L:/R:, C & I/L:.

In addition, the following holds for the specific system parameters given in Section 2

Upon substituting io from (1), e from (2) and Tm from (3) into (4) we then have the following differential equation Yc= - a x y 2 + b y

(I0)

for our specific motor the numerical values are a = 0 . 2 and b =300. Equation (5) is now rewritten

b ->1200

and

a

for all possible x given by (13) so that we can always accelerate the motor b

1500

ax

c 7.5

(20)

x

(11)

and here r=0.14 and c=5/7. In addition, because of (1), (2) and (6) we obtain the state constraint x y > 1200

(19)

ax

- 1 9= --y+cu

b -->2.34

(12)

for all possible x given by (13) so that we can always decelerate the motor. Conditions (19) and (20) are basically a controllability statement b 750 1200 . . . . < for all x 2ax

x

(21)

x

so that the state/control constraint (17) is active as long as

and (7)-(9) are now 210
(13)

2.34< y__<7.5

(14)

1200 x < 2-.~-= 513

(22)

and thereafter, for x>513, u<2.34. Next we note that if x ( t ) is a (time-)optimal trajectory then

and 0
(15)

x ( t ) must be a monotonically increasing

or decreasing function

3. T h e f i r s t - o r d e r model

In view of the specific system parameters given in Section 2 it is apparent that the synchronization control process is of the order of magnitude of one second; on the other hand the time constant of the field circuit [see equation (11)] is z =0.14s. Hence in this section we shall neglect the dynamics of the field circuit and consider the field current as o u r control variable. Specifically, we then have the optimal control problem given below (having renamed the field current y as u). The nonlinear (bilinear) control system Yc= - a x u 2 + bu

(23)

this depends on whether we have an acceleration or deceleration sequence. Therefore the state constraint (13) can be replaced by the simpler requirement 210
(24)

Since the constraint (17) involves the control variable explicitly we now apply the Pontryagin maximum principle g i v e n in Theorem II.1 of Hestenes (1966), i.e. we write the Hamiltonian

(16) A

H=-~kaxu

is steered in minimum time from the initial state Xo t o the final state x 1 (for example, for a shift from the first to the

2

+ ~kbu + # u x

where the (not identically zero) continuous costate variable

Brief Paper

629 Y

satisfies @= @ a u 2 - # u ,

@(0)=@o

x y = 1200

(25)

..............

(26)

............

THE FEASIBLE STATE SPACE REGION

and the piecewise continuous multiplier/z(t)~_0; if/a>0, (and if 2.34
~

5,71

A

D ,]

y=7,5

I y = 2,34

[then the constraint (17) is active and it determines u], and if # = 0 , (and if ux> 1200) then u = arg max [@(- axu ~ + bu)].

210

(27)

I

.513

640

=

FIG .3. The feasible state space region.

2 . 3 4 ~u_~'L5

Consider the optimal control problem of accelerating the motor from Xo to x~>xo. Let us first assume that @>0. For x_~513, if we assume that (17) is not active ~ = 0 ) and therefore equation (27) applies then we obtain a u which in view of (21) and (22) violates the constraint (17): a contradiction. Hence, for x_~513 we have that ~ > 0 so that (17) is active and u * = 1200/x. If x ~_513 then (17) must not be active because otherwise (18) would be violated. Therefore for x>513, /~=0 and (27) is applicable, which implies that u* = 2.34. If, however, we assume that for some time t, @(t)<0, then for p = 0 [i.e. equation (27) is then applicable] we have, in view of Fig. 2, that u*> b/ax, which means that we decelerate and in view of (23) we then know that this control cannot be optimal: a contradiction. Therefore, if @(t)<0 it must be that # > 0 throughout, i.e. u*=12OO/x. Now, assuming that 2.34 < u*< 7.5, we see that (26) is applicable and in view of this we have that

I

work we need consider the maximum principle only in the interior of ABCDE, i.e. the maximum principle without state constraints. Then the Hamiltonian is

H=@l(--axy2 +by)+ @2(--! y+cu) and the costate vector (@~,@2) satisfies

(30)

~l =@lay 2

~z=2@laxy+@21--@lb,

@2(tl)=0

(31)

T

and the optimal field circuit voltage is u* = arg sup (@,u). The singular case [@z(t)=0 for O~_t~_h] cannot arise, since then also ~b: = 0 which in view of equations (31) would imply

i.e.: a contradiction. Hence, when solving the optimal control problem of accelerating the motor the costate variable @(t)>0 for all t, 0 < t <-tl, [x~ = x(h )], and therefore the optimal control law is

•

, (12001x if x ~ 5 1 3 u (x)=~2.34 if x > 5 1 3

so that b xY=~a

(28)

and in addition it is readily verifiable that /z(t)>0 in the regime x<513. Similarly, the optimal decelerating control law is u*(x)=7.5 [then @(t)<0 for all O
2@laxy--@lb=O

(29)

It should be noted that in both the acceleration and deceleration cases (i.e. a shift to a lower and higher gear, respectively) the feedback control laws, viz. equations (28) and (29), are independent of the actual magnitude of the required change of speed, i.e. the actual gear engaged; in order to implement these feedback control laws it is necessary to measure the system state, i.e. the motor shaft speed (x).

which is not possible since then the state constraint (12) is violated. Now assume that @2(0 cannot change sign and that hence in the interior of ABCDE we have u* = 0 and u* = 144 when the motor is accelerated and decelerated, respecthcly. (Similar to the reduced order model we thus assume that @2 > 0 in the process of acceleration and @2< 0 in the process of deceleration.) Then the optimal flow field when the motor is accelerated (i.e. u* = 0 ) can be constructed, as shown in Fig. 4. The broken line y = 1500/x is the locus of d x / d y = 0 in the phase plane. Specifically the optimal accelerating control law in ABCDE'F is

u*(x,y)=o

(32)

4. The second-order model We have the following optimal control problem: it is required to steer the nonlinear control system (10) and (11) in a short time from the initial state x0, y(0) to the final state xt, subject to the state constraints (12)--(14) and the control (the field circuit voltage) constraint (15). The state space of the control problem is given in Fig. 3. In view of the specific system parameters given in Section 2 it is readily verifiable that the state constraint ABC in Fig. 3 is a feasible (acceleration) trajectory. Similarly, the state constraint DE is a feasible (deceleration) trajectory. Moreover, in view of the analysis in Section 3 we conclude that these trajectories satisfy Pontryagin's maximum principle for trajectories lying on tbe boundary of the set of state constraints, namely those of Theorem 22 of Pontryagin and co-workers (1965). Thus, in view of Theorem 25 in the same

I\

E E'

,

I 210

.. V= 7,5

BI''--~-_ ', .... 513

"~------ xy :,~o0 I- x Y - ' 2 ° °

640

=

FIG. 4. The optimal fiow field when the motor is accelerated.

630

Brief Paper

except on the boundary BC, where •

1

u (x,y)= 2.34-- = 23.4

(33)

and on the boundary AB where 1200 1 1 dy u*(x,y)= ---~ x zc c dt

1200 1 x zc

1 72000 c x2 Y

12000(1_42y x

\

5x J

(34)

whereas if the initial state is in the small region EE'F of the feasible state space region then it is not possible to increase the speed of the motor, i.e. initial states which are in the region EE'F are not accelerable. The line E'F which delimits this region is an optimal trajectory [u*(x,y)=O] which is terminated at the point F. In Fig. 5 the flow field for the case when the motor is decelerated (i.e. u * = 144) is shown. The optimal decelerating

E

D

Z-..7-

I

I

210

513

L

y = 2,34

'

x j : 1200

I

640

x y = 1500 ~

X

FIG. 5. The oPtimal flow field when the motor is decelerated.

control in ABCDE, is now u * ( x , y ) = 144

(35)

except on the boundary ED, where 1

u*(x,y)=7.5--=75

(36)

TC

The validity of our assumption from above that ~b2(t)>0 in the optimal acceleration process and ~,2(t)<0 in the optimal deceleration process is now evident in view of the flow fields in Figs. 4 and 5. Indeed if for example there would exist a time O < t < t l such that 1 ~ 2 ( t ) < 0 in the optimal acceleration process, this would imply that we should employ at time t the feedback control of Fig. 5 in Fig. 4; which in view of the flow fields in Figs. 4 and 5 is indeed non-optimal. F u r t h e r m o r e , Figs. 4 and 5 provide a regular synthesis for the optimal control, viz. we have a global optimal feedback control (and accordingly a global solution for the Hamilton-Jacobi equation). This in turn (Boltyanskii, 1971) implies that the optimal control synthesized herein via Pontryagin's necessary conditions also satisfies the sufficiency conditions for optimality. Note that minimum time optimal acceleration of the motor encompasses an initial deceleration of the motor for those initial states which are above the line x y = 1500 and similarly, minimum time optimal deceleration of the motor encompasses an initial acceleration of the motor for those initial states which are below the line x y = 1500. Finally, we note that in this second-order model also, the feedback control laws synthesized here, viz. (32)-(34) for accelerating the motor, and (35) and (36) for decelerating the motor do

not depend on the actual magnitude of the required change of speed, i.e. the actual gear engaged. In order to implement these feedback control laws it is necessary to measure the system state, i.e. the motor shaft speed (x) and the field circuit current (y). 5. Conclusion I have in this paper synthesized feedback control laws for field control of the speed of a D.C. traction motor; specifically the (declutched) motor is accelerated tin ABCDE'F) or decelerated (in ABCDE) in minimum time to synchronize the speed of the input (motor) and output shafts prior to the engagement of a new gear. The optimal control laws are given by equations (28) and (29) for a simplified (reduced order) model and by equations (32}--(34) and, (35) and (36) for the second-order model. These simple feedback control laws can be readily implemented on a microprocessor whose input is based on measurements of the system state, i.e. the motor speed and the field current. The only additional information which must be supplied to the microprocessor is whether an acceleration or deceleration sequence is to be performed, i.e. whether a shift to a lower or higher gear is required. In the first case equations (32)-(34) would apply. The control process terminates (at tl) when the ratio Xo/X(ti )corresponds to the gear change i in question, i.e. to 1/7~ (for example, for a change from first to second gear Xo/X(t I )= 1/71= 1.818). This synchronization procedure is basically a mechanization of the 'double declutching' and 'neutral pause' driving techniques employed when changing gear in a car without synchromesh gearbox. We also remark that in this specific application of motor speed control for synchronization, the standard method of dealing with field controlled D.C. motors failed, i.e. it was not possible to apply linearization (where the model of the motor is a first-order linear system a n d the situation is similar to that which occurs when armature control is employed) and subsequent use of linear compensation techniques; in fact it became necessary to resort to the nonlinear analysis of this paper. We finally remark that this work could be extended to include the design, by methods of linear-quadratic theory, of a (minimum-energy) speed-hold controller. In this mode of operation the load is applied to the motor and one would then obviously augment the model to include the vehicle inertia, road friction, etc. and one would then linearize around the prespecified nominal motor speed. Acknowledgeraent--I was presented Barnard van Zyl from the National Research Institute of the CS1R, whose gratefully acknowledged. In addition, Professor D. H. Jacobson who read paper.

with this problem by Electrical Engineering cooperation and help is I would like to thank an earlier draft of this

References Boltyanskii, V. E. (1971). Mathematical Methods of Optimal Control. Holt, Reinboldt, Winston Publishers, New York, Section 12. Chow. J. H. and P. V. Kokotovic (1978). Two-time-scale feedback design of a class of nonlinear systems. IEEE Trans. Aut. Control AC-23, 438. Hestenes, M. R. (1966). Calculus Of Variations and Optimal Control Theory. John Wiley, New York, p. 347. Pontryagin, L. S., V. G. Boltyanski, R. V. Gamkrilidze and E. F. Mishcenko (1965). The Mathematical Theory of Optimal Processes, Interseience, New York. Welz, J. J. and H. R. Van Niekerk (1979). The use of a gearbox for efficiency control in the drive system of a battery vehicle. Paper presented at the Electric Transport Symposium, 10-11 April 1979, CSIR Conference Centre, Pretoria.