A+New+Energy+Optimal+Control+Scheme+for+a+Separately+Excited+DC+Motor+Based+Incremental+Motion+Drive

8/2/2019 A+New+Energy+Optimal+Control+Scheme+for+a+Separately+Excited+DC+Motor+Based+Incremental+Motion

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International Journal of Automation and Computing 6(3), August 2009, 267-276

DOI: 10.1007/s11633-009-0267-4

A New Energy Optimal Control Scheme for a Separately

Excited DC Motor Based Incremental Motion Drive

Milan A. Sheta1 Vivek Agarwal2, Paluri S. V. Nataraj11Systems and Control Engineering, Indian Institute of Technology Bombay, India

2Department of Electrical Engineering, Indian Institute of Technology Bombay, India

Abstract: This paper considers minimization of resistive and frictional power dissipation in a separately excited DC motor basedincremental motion drive (IMD). The drive is required to displace a given, fixed load through a definite angle in specified time, withminimum energy dissipation in the motor windings and minimum frictional losses. Accordingly, an energy optimal (EO) controlstrategy is proposed in which the motor is first accelerated to track a specific speed profile for a pre-determined optimal time period.Thereafter, both armature and field power supplies are disconnected, and the motor decelerates and comes to a halt at the desireddisplacement point in the desired total displacement time. The optimal time period for the initial acceleration phase is computed sothat the motor stores just enough energy to decelerate to the final position at the specified displacement time. The parameters, suchas the moment of inertia and coefficient of friction, which depend on the load and other external conditions, have been obtained usingsystem identification method. Comparison with earlier control techniques is included. The results show that the proposed EO control

strategy results in significant reduction of energy losses compared to the existing ones.

Keywords: Energy optimal control, speed profile, incremental motion drive (IMD).

1 Introduction

Certain applications such as robots, automatic machinetools, and electric cranes require that a load is movedfrom one location to another, where it is held for a cer-tain time duration, before the next motion command is is-sued. An electric drive that facilitates the repeated step-ping of the load through incremental steps at specified fre-quency or time intervals is called an incremental motiondrive (IMD)[1,2].

Any motor can be used in an IMD application. How-ever, the performance of the IMD will be determined bythe type of motor used. DC motors are a strong candi-date for an IMD due to their simple control requirements,precise control over a wide bandwidth, and fast responsefeatures. Among the DC motors, a permanent magnet DC(PMDC) motor may be considered for an IMD applicationdue to its high power density, high efficiency, and bettercontrollability[3]. However, the magnetic field in a PMDCmotor cannot be varied, ruling out the operation in thefield weakening mode. This puts a serious limitation on theuse of a PMDC motor in an IMD. A series or shunt con-

nected DC motor can be used, but has limitations becauseindependent control of the armature and field circuit is notpossible. Therefore, a separately excited DC motor is byfar the most eligible candidate for an IMD application.

In the last few decades, AC motors have also become pop-ular for use in electric drives due to the availability of highpower inverters and powerful processors (e.g., digital signalprocessors), the latter being a requirement for the usuallycomplex and non-linear control of the AC drives. In fact, itis due to this control complexity that the popular AC mo-tor control methods like the vector control try to emulatethe DC motor control characteristics. Among the AC mo-

Manuscript received February 18, 2008; revised December 24, 2008

*Corresponding author. E-mail address: [email protected]

tors, permanent magnet synchronous motor (PMSM) drivesare popular due to their high power to weight ratio, largetorque to inertia ratio and high efficiency. However, oneof the major disadvantages of a PMSM is the presence oftorque ripple[4]. This ripple may particularly be a matterof concern in applications requiring precise control. Fur-ther, a PMSM with damper windings consumes more en-ergy (i.e., more losses) during the transient, start-up phase

as the permanent magnets exert a braking torque

[5]

. An-other problem with a PMSM is the possibility of its per-manent magnets getting de-magnetized during an electricfault or in due course of their life. This will hamper theperformance of the PMSM.

In view of the drawbacks discussed above in addition tothe higher cost and losses in the power stage and highercontrol complexity, the AC drives are usually preferred forhigh power applications only. For low power rating applica-tions, including the battery operated autonomous systems(e.g., robots), a DC motor drive is still considered a betteralternative[6,7].

This paper is concerned with a DC motor based IMD.As stated previously, among the DC motors, a separately

excited DC motor is a highly suitable configuration for anIMD, in which the efficiency can be enhanced and bettercontrollability can be achieved with the help of variablefield flux, which can be independently adjusted. If the mo-tor is connected to a constant load, then for achieving anoverall high efficiency, an appropriate combination of thearmature and field power can be used. Thus, the motorcan be operated with an optimized reduced power[6,8,9].

Minimization of the energy losses not only improves theefficiency of the drive but also the performance of the drive.For example, it causes an indirect reduction in the arma-ture current. From a practical point of view, this has severaladvantages. The reduced armature current eliminates thepossibility of saturation in the power amplifiers and convert-


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268 International Journal of Automation and Computing 6(3), August 2009

ers, thereby inhibiting nonlinear effects in the systems[10].A reduced armature current also leads to reduced lossesin the control circuit. Further, a reduction in the arma-ture current also yields a reduction in the I2R losses andmagnetic flux to the necessary minimum levels[11]. This inturn results in a reduction of the core losses and stray-load

losses[12]

. Therefore, a motor with lower power rating canbe used for a given application.

In incremental motion control, the driving motor mostlyremains in the dynamic state thus consuming a largeamount of energy[1214]. Therefore, the resistive and fric-tional losses greatly depend on the speed profile throughwhich the motor is rotated. An elegant energy optimal (EO)control scheme was proposed by Trzynadlowski[12] to min-imize the energy losses in the IMD. In this scheme, usingthe calculus of variations, an optimal parabolic or trape-zoidal speed profile is generated, which minimizes the en-ergy losses. The energy losses corresponding to this opti-mized speed profile are further minimized with respect tothe field current and displacement time. One of the draw-

backs of this scheme is that if the obtained optimal value offield current is higher than the specified allowable maximumvalue, then it is set at its maximum value, i.e., it is assumedthat the energy losses are minimum at the maximum fieldcurrent value. Actually, it may not be true because anyvalue in the given range of field current may correspond tominimum losses. While optimizing the losses with respectto the displacement time, if the obtained displacement timevalue is less than the specified maximum value, then the ob-tained value itself is used. In applications where the loadmust stop exactly at the specified displacement time, thismay not be applicable. Another drawback of this schemeis that only resistive losses are minimized with respect to

the speed profile, while frictional losses are not considered.It is shown in a subsequent section of this paper that iffrictional losses are also considered, better performance interms of losses can be achieved. The disadvantages, dis-cussed above, limit the application of the scheme proposedby Trzynadlowski[12].

This paper presents a new EO control scheme by tak-ing into account the frictional losses for the minimizationof overall energy losses and optimal use of kinetic energystored in the motor. The considerations are limited to con-stant load-torque and possible angular displacement withrespect to specified displacement time and maximum al-lowable speed. Both the resistive and frictional losses areconsidered. In the proposed scheme, the motor is first accel-

erated as per a specific speed profile up to a pre-computedmaximum speed for a pre-computed intermediate displace-ment time. Subsequently, both armature and field powersupplies are disconnected. The motor decelerates and comesto a halt exactly at the specified displacement time. Thespecific EO speed profile through which the motor must berotated during the initial acceleration phase is obtained us-ing the theory of calculus of variations. The parameterssuch as the coefficient of friction, moment of inertia, etc.,which depend on the type of load and the external condi-tions in which the drive is required to work, are obtainedusing system identification method, which is both easy andaccurate. It is assumed that the load and external environ-

mental conditions, for which the system identification hasbeen done, do not vary significantly during an operationcycle.

A modification to Trzynadlowskis scheme[12] is also in-cluded, which takes into account the frictional losses. It isshown that this modification improves the performance of

the drive as compared to the original scheme.The rest of this paper is organized as follows. The ba-

sic concept behind the proposed scheme and the determi-nation of the EO speed profile are discussed in Section 2.Section 3 discusses the implementation of the proposedcontrol scheme. Section 4 presents the details of the ear-lier EO control[12], modified to take into account the fric-tional losses. Discussion and comparison of various controlschemes is included in Section 5. Experimental results cor-responding to the proposed scheme and comparison withthe simulation results are included in Section 6. Finally,the main conclusions of this work are given in Section 7.

2 The proposed energy optimal control

scheme

In the proposed scheme, the motor is accelerated up to aspeed (t

d ) for a pre-computed optimal time period [0, td ]

with a pre-computed EO speed profile ((t)) to minimizethe total losses (see Fig. 1 (a)). (t) can be determinedby minimizing the total losses with respect to the speed(t) by using the theory of calculus of variation along withthe given constraints[15]. However, the field current is stillan unknown quantity. To determine an optimal value ofthe field current, the total losses corresponding to (t) areminimized with respect to the field current.

(t) is implemented by varying the armature voltage

Ea(t), while field current If is set at its optimal value thatdepends on the pre-computed disconnecting speed (t

d ),

optimal intermediate displacement time td , electrical andmechanical parameters of the motor like armature and fieldresistances, inductance of the armature circuit, moment ofinertia, load torque, viscous friction, etc. The power sup-plies are disconnected at time instant t

d , thereby forcing

armature current ia(t) and If to zero. Therefore, after td ,the electrical losses are zero.

The kinetic energy stored in the motor up to instant td

takes the motor to its final position. As the stored energyis spent, the motor decelerates and comes to a halt at thedesired displacement d, in the desired displacement timetd as shown in Fig.1 (a). The time interval [td , td] depends

on the motor speed (td ), at which the supplies are dis-connected. It also depends on the motor time constant,connected load torque, and displacement

dd

.The analysis to get the overall EO speed profile ((t)

and dec(t)) is done in two steps.The first step includes determination of

dd

and td for

(td ) [min, max]. The second step includes the deter-

mination of (t) for the disconnecting speed (td ). (t)

is determined such that it minimizes the total losses byconsidering it as an isoperimetric problem of calculus ofvariation with integral constraint (

od ), initial condition

((0) = 0), and final condition ((td ) = (t

d )). These

two steps are described in details next.


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M. A. Sheta et al. / A New Energy Optimal Control Scheme for a Separately Excited DC Motor 269

(a)

(b)

Fig. 1 (a) EO speed profile of an IMD using proposed technique.

td is the time instant at which power supplies are disconnected,

after which the motor decelerates and comes to a halt exactly at

time instanttd; (b) EO speed profiles at different disconnecting

speeds

Step 1. The motor torque equation is

J.

+f + Tl = kmifia (1)

where km is the motor constant, ia, if are the armature andthe field currents, J is the equivalent mass moment of iner-tia at the motor shaft, f is the coefficient of viscous friction,and Tl is the constant load torque. After td [0, td], ia andif are zero because the power supplies are disconnected attd . Using this, dec(t) for (td ) [min, max] and t td

can be determined by solving (1) as

dec(t) = Tl

f+

(td ) + T

l

f

e

(tt

d

) (2)

where = J/f. Substituting dec(t) = 0 at t = td in (2)and solving for td yields

td = td + ln

Tlf

(td ) + Tl

f

. (3)

The corresponding displacement required during the inter-val 0 to t

d is

od = d dd (4)

which is the area covered by the speed versus time curve,

from 0 to td

as shown in Fig.1 (a). In (4), d

d can be

determined by integrating (2) over the time interval [ td , td]and is given by

dd

=

f

Tl + f (td )

1 e

tdd

Tl tdd

. (5)

The constants (td

), d

d, and td

determined from theabove analysis are used as the final condition, integral con-straint, and integral limit, respectively, in that order.

Step 2. To determine (t) and the optimal value offield current, If for t [0, td ], consider the energy lossessupplied by the source

W(t,,.

, if) =

td

0

(2f + i2aRa + i2fRf)dt (6)

where Ra and Rf represent the armature and the field cir-cuit resistances, respectively. Using (1) and (6), we get

W(t,,.

, if) =td

0

2f +

J

.

+f + Tlkmif

2Ra + i

2fRf

dt. (7)

The EO speed profile can now be determined by minimiz-ing (7) within the constraints

od , (0), and (t

d ). This

can be posed as an isoperimetric problem as shown below.Isoperimetric problem formulation. Consider the

problem of determining a curve that minimizes a given func-tion W(t,,

., if) with respect to within an integral con-

straint and the given initial and final values[15]:

minimize W(t,,.

, if) =

td

0

f(t,,.

, if)dt (8)

with respect to (t), with initial and final conditions (0) =0, (t

d ) [min, max] subject to the constraint

od =

td

0

g(t, .

, if)dt (9)

where

f(t,,.

, if) = 2f +

J

. +f + Tl

kmif

2Ra + i

2fRf (10)

g(t,,.

, if) = . (11)

In order that (t) be the solution of (7), it is necessary

that it should be an extremal of the following integraltd

0

(f(t,,.

, if) + g(t,,.

, if))dt (12)

for a certain constant , called the Lagrange multiplier.To determine (t) for the extremal solution of (11), thenecessary condition is[3]

(f + g) +

d

dt

.

(f + g)

= 0. (13)

Equation (13) can be simplified into the following form:

m2

..

m1

1 = 0 (14)


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where m2 = R

aJ2, m1 = f + R

af2, 1 = (2R

af Tl + )/2,

and R

a = Ra/(k2mi

2f). In time-domain, the solution of (14)

is obtained as

(t) =1(cosh nt 1)+

. (0)nm2 sinh nt

m22n(15)

where 1 and . (0) are obtained by simultaneous solutionof (16) and (17), given below:

. (0) =

m22n(td ) + 1(1 cosh ntd )

nm2 sinh ntd(16)

1 =n

od m2

2n+

. (0)m2(1 cosh ntd )

sinh ntd ntd

(17)

where n =

m1/m2.By taking the Laplace transform of (15), the transfer

function for a step input Vi is given by

(s)s

Vi =

.(0)Vi

m2s +1Vi

m2(s2 2n) . (18)

Several values are possible for (td ) [min, max] as

shown in Fig. 1 (b). Therefore, interval [min, max] is di-vided into a set of equidistant points. EO speed profiles(i (t)) are determined at each point, and the correspond-ing losses are compared. This comparison gives the overallEO speed profile (t), the optimal supply disconnectingspeed ods(td ) and the corresponding overall optimal valueof the field current, If

. A systematic way to determinethe above unknowns is given below.

Algorithm 1. The procedure to determine ods(t

d )

[max, min], I

f [Ifmin, Ifmax] and (t) is as follows:

Step 1. Divide the interval [min, max] into a set of

p equidistant points given by mi, i = 1, 2, , p and seti = 1.

Step 2. If i > p, then go to Step 6, else determine td

and od from (3) and (4) for (td) = mi.

Step 3. Determine i (t) from (15) for the initial andfinal conditions (0) = 0, (t

d ) = mi and integral con-

straint od .

Set the motors field current as an unknown variable,which is required to be optimized.

Step 4. Optimal value of the field current If for thecorresponding mi and

i (t) can be determined by usingEulers explicit numerical integration method. For this pur-pose, let T be the sampling time. Then, (8) can be writtenin the following discrete form

W(nT) = T

f(nT, (nT),.

(nT), if, )

+

+ W((n 1)T) (19)

where W( 0 ) = 0, nT {0, T, 2T, , mT} andmT = td . The objective is to determine I

f [Ifmin, Ifmax], such that W(mT) is minimum. This canbe done as follows:

Step 4.1 Divide the interval [Ifmin, Ifmax] into a set ofq equidistant points given by Ifr , r = 1, 2, , q and setr = 1.

Step 4.2 If r > q, then go to Step 4.4, else determineWr = W(mT) for Ifr and mi.

Step 4.3Set r = r + 1 and go to Step 4.2.

Step 4.4 Set Ifi {If1, If2, , Ifq } corresponding toWi = min{W1, W2, , Wq}.

Step 5. Set i = i + 1 and go to Step 2.Step 6. Determine the overall optimal value of

field current If {I

f1,I

f2,, , I

fp,},

ods(td ) {m1, m2, m3, , mp} and td corresponding to

W

= min{W

1 , W

2 , W

3 , , W

p }.Step 7. From (15), determine (t) corresponding to

If and

ods(td).

3 Implementation of the proposed con-

trol scheme

After determination of the overall EO speed profile(t), the motor needs to track this profile up to the op-timal intermediate displacement time td . Because of themoment of inertia, viscous friction, load torque, armatureresistance, and inductance, the motor has a certain set-tling time and steady-state error that causes deviation from

(t), if the motor is fed with the exact voltage equivalentof (t) as shown in Fig. 2, where VEOS (t) corresponds tovoltage equivalent of (t), and TM(s) is the motor trans-fer function. Therefore, it is necessary to improve the set-tling time and to make the steady-state error zero, whichis done with the help of a proportional-integral-derivative(PID) controller as discussed later in this section.

Fig. 2 Deviation from the desired EO speed profile due to set-

tling time and steady-state error of the separately excited DC

motor

After improving the motor transient and steady-state re-sponse, the motor is fed with the armature voltage equiv-alent to (t). This variation is achieved at the outputof the tracking controller from the constant voltage sup-ply as shown in Fig. 3. Note that both the armature andfield supplies are disconnected at time instant td . Deter-mination of the PID controller CPID(s) and the trackingcontroller CT(s) is shown next. It must be pointed outthat any other controller that makes the steady-state error

zero and improves the transient response can also be usedinstead of the PID controller.

Fig. 3 Block diagram of the control system used for implement-

ing and testing the proposed scheme


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3.1 Determination of PID controller

CCCPIDPIDPID(sss)

To track the overall EO speed profile, the drive must havea lower settling time than the given displacement time andzero steady-state error. This subsection discusses the pole

cancelation technique to determine the PID parameters.Let the transfer function of the PID controller be givenby

CPID (s) = kp(1 +1

is+ ds) (20)

where kp is the proportional gain, and i, d are the integraland derivative time constants, respectively.

The PID constants are determined such that the actualpoles of the motor transfer function TM(s) are canceled outby proper selection of i and d, i.e., the second-order sys-tem is reduced to a first-order system. The proportionalgain kp is chosen to improve the settling time.

Let the motor transfer function be given by

TM(s) = (s)Ea(s)

=

KMJLas2 + (RaJ + Laf)s + (Raf + KMKB)

(21)

where Ra and La are the resistance and inductance respec-tively of the armature circuit, and KB = KM = kmIf. Tocancel the poles of TM(s), i and d should be chosen asbelow:

i =RaJ + Laf

Raf + KMKB, d =

JLaRaJ + Laf

. (22)

3.2 Determination of tracking controller

CCCTTT(sss)

After i and d are determined, TMPID(s) (see Fig.3) isgiven by

TMPID(s) =(s)

VEOS (s)=

KMkpi(Raf + KMKB)s + KMkp

. (23)

In (23), kp is an adjustable parameter that can be usedto track (t) satisfactorily.

The overall transfer function of the control system shownin Fig. 3, is given by

(s)

Vi(s) = CT(s)TMPID(s). (24)

Since the effect of the poles of TMPID(s) on CT(s) isinsignificant, TMPID(s) can be taken as unity. By takingLaplace transform of(t), CT(s) =

(s)/Vi(s) is deter-mined. Therefore, from (24), we have

CT(s) =(s)

Vi(s)=

(s)

Vi(s). (25)

Equation (25) indicates that the speed of the motor, us-ing this control scheme, can be approximately tracked asper the overall EO speed profile (t). It must be pointedout that if switch S is kept ON continuously (see Fig. 3),

even beyond td

, the system will get unstable. Both the

controllers (CT(s) and CPID (s)) must be reset during theinterval [N td , N td] (N = 1, 2, ), before the commence-ment of the next cycle.

4 EO control[12], modified to account for

frictional losses

In this section, the analysis of Trzynadlowskis scheme[12]

is expanded to take into account the frictional losses, whichare not considered in the original scheme. Accordingly, theEO speed profile of the drive is obtained by considering thefrictional plus armature and field resistive losses.

The EO speed profile is determined by minimizing (7)with respect to (t) for the time period 0 to td. The de-sired EO speed profile FL(t) should satisfy the integralconstraint

d =

td0

FL(t)dt (26)

which is subject to the initial and final conditions FL(0) =

0 and

FL(td) = 0, respectively.

FL(t) can be determinedby substituting (td) = 0, od = d, and td = td in (15),(16), and (17).

After determining FL(t), the optimal value of the fieldcurrent IFL is calculated by minimizing the total energylosses for FL(t) with respect to the field current.

For implementation of this modified control scheme, thesame control system, as shown in Fig.3, is used; the onlydifference being that now both the controllers remain activefrom 0 to td. CPID (s) is determined for I

FL in a similarmanner as discussed in Section 3, while CT(s) is determinedby taking the Laplace transform of FL(t) for a step inputVi(s). CT(s) is given by

CT(s) =

FL(s)Vi(s)

. (27)

5 Discussion and comparison of various

control schemes

To highlight and study the improvements obtained withthe proposed technique over earlier techniques, the follow-ing control schemes were implemented:

1) EO control[12];2) EO control[12], modified to take into account frictional

losses;3) Proposed EO control scheme.In addition to these, a conventional control scheme, the

state feedback technique, is also implemented, which doesnot involve any optimization. The idea is to show the im-portance of optimization, without which the drive incurshuge losses.

A separately excited DC motor with the following specifi-cations was considered: Prated = 250W, Ea(rated) = 250 V,Ra = 46, Rf = 1000, Ifmin = 0.07A, Ifmax = 0.15A,min = 180 rad/s, max = 320 rad/s. A given load of0.1239Nm (Tl) is required to be displaced by 565.48 rad(d), i.e., 90 rotations in 3 s (td).

Complex and time-consuming computations can be donefor the determination of the other unknowns like equiv-alent moment of inertia, frictional coefficient, etc. How-

ever, this may not give any additional advantage since the


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motor performance significantly depends on the environ-ment in which the drive is required to work. The perfor-mance also depends on the type of load connected to thedrive. The easiest and accurate way is to identify the sys-tem (system identification) in the same environment anddetermine the unknowns. For the given system, a second-

order system was identified, and the following parameterswere obtained: J = 0.00081 kgm2, f = 0.00021734 Nms.,KB = KM = 5.8104 If Nm/A

2, La is neglected, where KBand KM are the back electromotive force (emf) and torqueconstants, respectively, of the motor.

The comparison of losses, armature current variation, po-sition, and speed profiles corresponding to these schemes,are included later in this section.

5.1 EO control[12]

The earlier, EO control scheme[12] is discussed in thissubsection. The control system shown in Fig.3 remainsthe same. The only difference is that both the PID and the

tracking controllers are operated from 0 to td. Similarly, themotor armature and field supplies also remain ON during[0, td], i.e., the power supplies are no longer switched OFFat td , unlike the case with the proposed scheme.

The procedure to determine the EO speed profile andtracking controller CT(s) for the given motor specificationsis given below:

Step 1. For a given d, max, and td, check the inequalitygiven by

d >2

3maxtd. (28)

For the drive specifications given, (28) is not satisfied.Therefore, parabolic EO profile is applicable and can be

determined.Step 2. Determine optimal values of the field current

If(p) and displacement time t

d(p) from the equations given

below[12]:

If(p) =

2

RaRf

Tl

km

2 14(29)

td(p) =

12

JdTl

2 14. (30)

Step 3. If If(p) ifmax and/or t

d(p) td, then set theircorresponding values to maximum.

In the present case, If(p) = 0.0804A and is less thanifmax (0.15 A). Therefore, the inequality is not satisfied.However, td(p) is obtained as 3.5786s, and it satisfies theinequality, so td(p) is taken as 3 s.

Step 4. Using the information obtained in Step 3, deter-mine the parabolic EO speed profile (p)(t) from the equa-tion given below:

(p)(t) =6d

(td(p)

)3(td(p) t t

2). (31)

Equation (31) gives

(p)(t) = 125.637(3t t

2

). (32)

Step 5. By taking Laplace transform of (32) for a givenstep input Vi(s) (0100 V), CT(s) is given by

CT(s) =(s)

Vi(s)=

3.77s2 2.5123s

s3. (33)

It may be noted that the EO speed profile given by (32)

is the profile that minimizes only the resistive losses in thearmature and field windings. The frictional losses are notminimized.

To determine the PID controller parameters, consider themotor transfer function calculated for If(p) = 0.0804 A.

(s)

Ea(s)=

12.5

s + 6.133. (34)

Using (22), d and i are obtained as 0 and 0.1631, respec-tively. The proportional gain, kp, is an adjustable param-eter that determines the deviation from the desired speedprofile. For satisfactory tracking of the actual speed profile,kp, is taken as 12 so that the settling time turns out equalto the actual settling time of the motor as it is very lowcompared to the given displacement time. Now, the systemis able to track the desired profile, and the motor is fed withthe output of the tracking controller as shown in Fig. 3.

5.2 EO control[12] modified to take into ac-

count frictional losses

From Section 4, FL(t) is obtained as

FL(t) = 332.7083(cosh(1.5047t 1))+

325.4990(sinh(1.5047t)). (35)

CT(s) is calculated from (35) for a given Vi(s) and is givenby

CT(s) =

FL(s)

Vi(s) =

4.889s 7.492

s2 2.245 . (36)

Further, the total losses given by (7) are minimized withrespect to the field current for obtained FL(t). The vari-ation of the energy losses with respect to the field currentis shown in Fig.4 using dotted line. IFL = 0.0945 A givesminimum losses and is taken as the optimal field current.Using the value ofIFL, d, and i, for the PID controller, arecalculated from (22) as 0 and 0.1185, respectively. kp = 12is valid in this case also.

Fig. 4 Energy losses versus field current for the IMD


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5.3 Proposed EO control scheme

This subsection presents the obtained parameters of thetracking and PID controllers for the proposed scheme. Fromthe proposed algorithm, (t), ods(td), I

f , and td aredetermined for the given drive specifications as follows:

The interval [min, max] is divided into a set of eightequidistant points. For each point, intermediate displace-ment time td and EO sp eed profile are calculated. Further,the total losses are minimized with respect to the field cur-rent for every EO speed profile.

Fig. 4 shows the variation of the energy losses with fieldcurrent for each disconnecting speed. From the figure, itis obvious that minimum losses occur at the disconnect-ing speed, ods(td ) = 320 rad/s, optimal field currentIf = 0.1205 A, and optimal, intermediate disconnectingtime instant, t

d = 1.3395 s. The overall EO speed profile,

(t) is given by

(t) = 399.0148(cosh(1.8995t 1))+

391.4802(sinh(1.8995t)). (37)

By taking the Laplace transform of (37), CT(s) is obtainedas

CT(s) =(s)

Vi(s)=

7.436s 14.4

s2 3.608. (38)

Likewise, d and i, for PID controller, are calculatedfrom (22) as 0 and 0.0731, respectively, and kp is set at 12as before.

5.4 State feed-back control

As an example of a conventional control technique usedfor applications such as electric drives, state feedback orthe pole placement method[16] is considered in this section,

and disadvantages of arbitrary design of this control schemeare presented. No optimization is involved here. Fig. 5shows the general block diagram of the state feedback con-trol scheme for the DC motor. KM and KB are functionsof the field current.

Fig. 5 General block diagram of the state feedback control sys-

tem

To design the state feedback controller for position con-trol, the field current If is arbitrarily taken as 0.1 A lyingbetween Ifmin and Ifmax. From the chosen value of If,motor systems state-space representation is given by

.

X= AX + Bu

and

Y = CX (39)

where

X = d

, u = Ea, Y = d

and the matrices A, B, and C are computed by using thegiven parameters

A =

0 1

0 9.34

, B =

0

15.6

, C =

1 0

.

(40)Using (39) and (40), the corresponding transfer function isgiven by

TSFB(s) =d(s)

Ea(s)=

15.6

s2 + 9.34s. (41)

From the transfer function (41), it is obvious that the sys-tem is unstable. The objective is to determine the statefeedback gain matrix K = [k1 k2], which stabilizes the sys-tem and gives the desired transient response. Since thegiven displacement time is 3 s, the system must have a set-tling time equal to the given displacement time td. In thegiven case, the inductance of the armature circuit is ne-glected as it is very less. Therefore, the armature currentcannot be considered as a state vector. However, in gen-eral, if La is not negligible, then armature current must beconsidered as a state vector. In this case, DC motor sys-tem becomes a third-order system, and it needs to stabilize

three poles. For that case, let the new characteristic equa-tion, which gives the desired response with state feedbackcontrol be

(s2 + 2nts + 2nt)(s + P) = 0. (42)

In (42), and nt are so determined that the characteris-tic equation gives the desired transient response, providedthat P is such that it does not affect the transient responseobtained with the help of and nt. P is generally takenas 10nt.

For the control law implementation, is assumed to be0.8 and for the desired settling time, nt is obtained as1.6667 (as the armature current state vector is not present

in the given case, the second term in the characteristic equa-tion, i.e., s + P is ignored). Therefore, the poles (0, 9.34)of the original characteristic equation (denominator of (41))must be placed at 1.333j, i.e., the original characteristicequation must be modified as s2 + 2.667s + 2.778 = 0. Af-ter inserting the state feedback gain matrix in the originalsystem (represented by (39)), the new state matrix, AFBis given by AFB = A BK. For the desired transient re-sponse, K must be determined such that AFB has the abovedesired characteristic equation. Accordingly, K is obtainedas K = [0.1780 0.4277].

For zero steady-state error, a gain constant, kg = 0.1927is inserted after the desired reference input, d, as shown in

Fig. 5.


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5.5 Summary of comparison of energy

losses, armature current variation, po-

sition and speed profiles for various

control schemes

Fig.6 shows the position profiles of the drive obtained

with various control schemes discussed in this section. Itis seen that all these profiles have the same settling time(3 s) and steady-state value (565.48 rad), even though thevariation of position of the motor with respect to time isdifferent.

Fig. 6 Variation of position with respect to time for various con-

trol schemes (Note that the initial starting time, the desired dis-

placement and the total displacement time td are the same for

all the profiles, irrespective of the control scheme used.)

Fig. 7 compares the speed profiles obtained with various

techniques. It is seen that in state feedback, the speed pro-file exceeds max, which may be unacceptable. While inthe case of original EO control scheme[12] and modified EOcontrol scheme[12], to maintain the symmetry of the speedprofile, armature current in the reverse direction is required.This kind of braking is not required in the proposed tech-nique as shown in Fig. 8.

Fig. 7 Variation of speed with respect to time for various con-

trol schemes (Note that the initial starting time instant and the

final specified displacement time td are the same for all control

schemes.)

Fig. 9. shows that the conventional state feedback con-trol, EO control[12], and EO control[12] (modified), incur en-ergy losses that are approximately 2.23, 1.61, and 1.53 timeshigher than in the proposed scheme. Because of very highlosses in the case of state feedback control, as compared toother control schemes, the maximum value of armature cur-

rent is approximately 2.5 times, i.e., high armature currentrated motor is required if state feedback control is appliedfor the same displacement angle and time.

Fig. 8 Variation of the armature current with respect to time

for various control schemes

Fig. 9 Comparison of losses incurred by the IMD

In the proposed control scheme, the supplies are discon-

nected after time instant 1.3395s. Therefore, the lossesremain constant after time instant 1.3395 s. After discon-necting both power supplies, the stored kinetic energy isdissipated as frictional losses, and the motor comes to ahalt at the desired displacement time td and displacementangle d.

6 Experimental results

To validate the theory and simulation results, a labo-ratory prototype of the incremental motion DC drive wasbuilt. An MOSFET (IRFP450) based buck-boost type DC-to-DC converter was used to power the drive. Intels micro-

controller 8052 was used for controlling the power converter.


9/10


The microcontroller generates variable duty cycle pulsescorresponding to the EO speed profile, which are fed intothe gate of the MOSFET through an isolation cum driverIC, Agilents HPCL3120. This makes the power convertersoutput voltage (which supplies the armature of the DC mo-tor) vary in accordance with the desired EO speed profile.

Fig. 10 shows the photograph of the experimental setup.

Fig. 10 Photograph of the experimental set up

The speed sensor used for the experiment is a perma-nent magnet tacho generator that generates 1 V for a motorspeed of 17.16 rad/s. In the experimental system, overall 87rotations were achieved with a total energy loss of 56 Ws.Fig.11 shows a comparison of the simulation results andthe experimental results. Figs. 11 (a) and (b) show the EOarmature voltage profiles obtained with Matlab simulationand with experimental prototype respectively. Figs. 11 (c)and (d) show the resulting speed profiles. It is important to

note that in the armature voltage profile obtained experi-mentally (see Fig. 11 (b)), the voltage is not zero even afterdisconnecting the supplies at the disconnecting speed. Thisis because the motor is still in rotation, and the presence ofresidual magnet field causes the motor to work as a gener-ator until the machine comes to a halt. However, this doesnot affect the operation of the IMD since the armature ter-minals are open (supply is disconnected). Experimental re-sults show good agreement with the theoretical/simulationresults.

(a)

(b)

(c)

(d)

Fig. 11 Comparison of experimental results with simulation re-

sults. (a) EO armature voltage profile obtained with Matlab;

(b) EO armature voltage profile obtained with the experimental

prototype; (c) EO speed profile obtained with Matlab (A scaling

of 1:17.16 has been used for comparison with experimental out-

put); (d) EO speed profile obtained with experimental prototype

(The speed sensor used, generates 1 V for 17.16 rad/s.)


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7 Conclusions

A new EO control scheme for a separately excited DCmotor based IMD has been proposed and verified throughcomputer simulations and experiments. The scheme mini-mizes not only the resistive losses in the armature and field

windings, but also the frictional losses. The minimization ofthe energy dissipation in the drive is achieved by minimizingthe losses with respect to the sp eed profile and the field cur-rent and by using the kinetic energy stored in the system.The performance of the proposed scheme, with respect tothe total energy losses, has been compared with other EOcontrol schemes, and its superiority has been established.Also, the importance of optimization in the control of elec-tric drives has been highlighted.

The merits of the proposed scheme can be summarizedas below:

1) The energy losses being low, the armature power re-quirement decreases, which implies that a lower rating mo-tor can be used for the same application.

2) On account of low energy losses, the thermal dissipa-tion requirements are less stringent.

3) Braking current is not required because the motornaturally comes to a halt (zero speed), unlike most otherschemes.

One of the drawbacks of the proposed scheme is the as-sumption that the load and the surrounding environmentalconditions do not change. This restricts its applications.However, in this paper, the major objective was to demon-strate a new scheme. In the future, it is proposed to con-sider conditions involving varying load and environmentalconditions. This would require real-time system identifica-tion, for which a digital signal processor (DSP) or a fieldprogrammable gate array (FPGA) will have to be employed.This work would be reported in a future paper.

References

[1] C. K. Lai, K. K. Shyu. A Novel Motor Drive Designfor Incremental Motion System via Sliding-mode ControlMethod. IEEE Transactions on Industrial Electronics, vol.52, no. 2, pp. 499507, 2005.

[2] J. Tal, S. Kahne. Control and Component Selection for In-cremental Motion Systems. Automatica, vol. 9, pp. 501507, 1973.

[3] C. C. Chan, R. Zhang, K. T. Chau, J. Z. Jiang. OptimalEfficiency Control of PM Hybrid Motor Drives for Elec-trical Vehicles. In Proceedings of the 28th Annual IEEEPower Electronics Specialists Conference, vol. 1, pp. 363

368, 1997.[4] J. Holtz, L. Springob. Identification and Compensation ofTorque Ripple in High Precision Permanent Magnet MotorDrives. IEEE Transactions on Industrial Electronics, vol.43, no. 2, pp. 309320, 1996.

[5] P. Pillay, R. Krishnan. Modeling, Simulation, and Anal-ysis of Permanent-magnet Motor Drives Part 1: ThePermanent-magnet Synchronous Motor Drive. IEEE Trans-actions on Industry Applications, vol. 25, no. 2, pp. 274279, 1989.

[6] G. Zhang, A. Schmidhofer, A. Schmid. Efficiency Optimisa-tion at DC Drives for Small Electrical Vehicles. In Proceed-ings of IEEE International Conference on Industrial Tech-nology, IEEE Press, vol. 2, pp. 11501155, 2003.

[7] E. E. EI-kholy, S. S. Shokralla, A. H. Morsi, S. A. EI-Absawy. Improved Performance of Rolling Mill Drives Us-ing Hybrid Fuzzy-PI Controller. In Proceedings of the 5th

International Conference on Power Electronics and DriveSystems, IEEE Press, vol. 2, pp. 10101015, 2003.

[8] T. Egami, J. Wang, T. Tsuchiya. Efficiency OptimizedSpeed Control System Synthesis Method Based on Im-proved Optimal Regulator Theory Application to Sep-arately Excited DC Motor System. IEEE Transactions onIndustrial Electronics, vol. IE-32, no. 4, pp. 372380, 1985.

[9] A. Kusko, D. Galler. Control Means for Minimization ofLosses in AC and DC Motor Drives. IEEE Transactions onIndustry Applications, vol. IA-19, no. 4, pp. 561570, 1983.

[10] S. Murtuza, N. Narasimhamurthi, S. T. Dilodovico. DCServo Control-academic and Real. In Proceedings of theAmerican Control Conference, IEEE Press, vol. 3, pp. 20362037, 1995.

[11] R. N. Danbury. Servomechanisms for Incremental Motion:Power Dissipation Considerations. Mechatronics, vol. 4, no.1, pp. 2536, 1994.

[12] A. M. Trzynadlowski. Energy Optimization of a CertainClass of Incremental Motion DC Drives. IEEE Transactionson Industrial Electronics, vol. 35, no. 1, pp. 6066, 1988.

[13] F. J. Zheng, G. Cook. Energy Optimal Control for SteelRolling. IEEE Transactions on Industrial Electronics, vol.

IE-32, no. 4, pp. 388392, 1985.[14] J. C. Brierley, R. E. Colyer, A. M. Trzynadlowski. The

SOAR Method for Computer Aided Design of Energy-optimal Positioning DC Drive Systems. In Proceedings ofIndustry Applications Society Annual Meeting, IEEE Press,vol. 1, pp. 464467, 1989.

[15] E. R. Pinch. Optimal Control and the Calculus of Varia-tions, Oxford University Press Inc., New York, USA, 1993.

[16] I. J. Nagrath, M. Gopal. Control Systems Engineering, NewAge International Ltd., New Delhi, India, 1989.

Milan A. Sheta received the bachelor degree in electricalengineering from Vyavasayi Vidya Pratishthan Engineering Col-lege, Rajkot, Gujarat, India in 2002, and the M. Tech. degree insystems and control engineering at the Indian Institute of Tech-

nology Bombay, India, in 2005.

Vivek Agarwal received the bachelordegree in physics from St. Stephens col-lege, Delhi University, USA, the Master de-gree in electrical Engineering from the In-dian Institute of Science, India, and thePh. D. degree in the Depterment of Electri-cal and Computer Engineering, Universityof Victoria, Canada in 1994. He worked forStatpower Technologies, Burnaby, Canada,as a research engineer. In 1995, he joined

the Department of Electrical Engineering, Indian Institute ofTechnology, Bombay, India, where he is currently a professor.He is a senior member of IEEE, fellow of IETE, and a life mem-ber of ISTE.

His research interests include power electronics, modeling andsimulation of new power converter configurations, intelligentand hybrid control of power electronic systems, power quality,EMI/EMC, and conditioning of energy from non-conventionalenergy sources.

Paluri S. V. Nataraj received thePh. D. degree in process dynamics and con-trol from Indian Institute of TechnologyMadras, India in 1987. He is a faculty of thesystems and control engineering at IndianInstitute of Technology, Bombay, India.

His research interests include robust con-trol, process automation, nonlinear systemanalysis and control, and reliable comput-ing.

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