Mohamed M. M. Negm, Fahad N. Al-Ghnnam, T.M. Nasab, Sabry A. Leithy

Technical College at Dammam, Dammam, 31472, Saudi Arabia

Abstract:

Speed control of a separately excited dc motor using digital PID controller is proposed in this paper. The optimalPID controller gains are obtained off-line using the root-locus technique with satisfying the requiredperformance. Digital computer simulation results are made for continuous system and discrete system. Theseresults indicate the difference between the optimal PID controller gains incorporated with both of the twosystems with satisfying the required performance. Based on these results, a digital PID controller is synthesizedand implemented on-line. Effects of the microprocessors execution time, sampling time and the resolution of theanalog to digital - digital to analog, ADDA, converter card used with the controlled system are investigated. Therobustness of the optimal PID controller is demonstrated, and comparison between the experimental results andsimulation results is depicted. The recommendation and conclusion of the theoretical and experimental studiesare illustrated in this paper.

1. Introduction

In contrast to the continuous-time system whoseoperation is described or modeled by a set ofdifferential equations, the discrete-time system is onewhose operation is described by a set of differenceequations. The transform method employed in theanalysis of linear time-invariant continuous-timesystem is the Laplace transform, in a similar manner,the transform used in the analysis of linear time-invariant discrete-time is the Z-transform [1, 2]. In thispaper, the continuous-time controlled system and thediscrete-time controlled system comprise theseparately excited dc motor and the PID controller.This is to control the speed of the dc motor. Therequired performance of the controlled system isachieved by selecting the PID controller gains usingthe root-locus technique. Furthermore, the digital PIDcontroller is synthesized and implemented on-line.Effects of the sampling time , microprocessorsexecution time, PID controllers gains, and resolutionof the ADDA converter card used with the controlledsystem are investigated. Comparison between theexperimental results and the simulation results are alsodemonstrated. In addition the recommendation andconclusion are given.

2. Continuous-Time System

The dynamic equation of a separately excited dcmotor is given by [6]

a a

aa

b

a aa

d i t

dtR

Li t K

Lt

Lv t

( )( ) ( ) ( )= - - -w

1 (1)

d t

dtKJ

i tB

Jt

JT tm a L

ww

( )( ) ( ) ( )= - - -

1 (2)

where

w ( )t : angular velocity, rad/sec

av t( ) : armature voltage, V

ai t( ) : armature current, A

aR : armature circuit resistance, W

aL : armature circuit inductance, H

bK : back emf constant, V/rad/sec

mK : motor constant, N.m./AJ : total moment of inertia , kg.m2

B : viscous friction, N.m./rad/sec

LT t( ) : constant load torque, N.m.

3. Discrete-Time System

To illustrate the idea of the discrete-time system,consider the digital control system shown in Fig.1a.The digital computer performs the compensationfunction within the system. The interface at the inputof the computer is an analog to digital (A/D)converter, and is required to convert the error signalwhich is a continuous-time signal into a form that canbe readily processed by the computer. At the computeroutput a digital to analog (D/A) converter is requiredto convert the controlled binary signals of thecomputer into a form necessary to drive the dc motor.

Suppose that the A/D converter, the digital computer,and the D/A converter are to replace an analog, orcontinuous-time, proportional - integral - derivative

(PID) controller, such that the digital control systemresponse has essentially the same characteristics as theanalog system.

The analog controller output is given by

u t K e t K e t d t Kde t

d tuP I

t

D o( ) ( ) ( )( )

= + + +0

(3)

where e t( ) , which is the difference between thereference signal, r(t), and the measured signal, b(t),is the error or controller input signal, while u t( ) is thecontroller output signal, and PK , IK and DK areconstant gains determined by the design process.The initial condition is denoted byou .Since the digital computer can be programmed tomultiply, add, and integrate numerically, the controllerequation can be realized using the digital computer.For this reason, the rectangular rule of numericalintegration, illustrated in Fig.1b, will be employed.The first-order linear difference equation of Eq.(3), isgiven by

u kT K e kT K x kTP I( ) ( ) ( )= +

+ - - +D oKT

e kT e k T u( ( ) ( ) )1 (4)

where

x kT x k T T e kT( ) [ ( ) ] ( )= - +1 (5)

and T denotes the sampling period of the samplerused, and x t( ) is the numerical integral of e t( ) . It iseasy to derive the pulse transfer function,cG z( ) , ofthe digital PID-controller using Eq.s (4)-(5) , andconsidering zero initial conditions.

c pI DG z

U zE z

KK T

z

K zT

( )( )( ) ( )

( )= = +

-+

--

-

1 1

1 1 (6)

Fig.2, represents the closed-loop direct digital control,DDC, system in which the transfer function of theseparately excited dc motor is given by:

ma

m

a a b mG s

s

V sK

R L s J s B K K( )

( )( ) ( )( )

= =+ + +

W (7)

and that of the 3-phase bridge converter is,

aa

aG sV s

U sA( )

( )( )

= = (8)

while for the zero order hold,

ho

TsG s

es

( )=- -1

(9)

Finally, the transducer transfer function is,

H s At( )= (10)

Let

G s G s G s G sho a m( ) ( ) ( ) ( )= (11)

The closed-loop pulse transfer function is

G zz

R zG z G z

G z GH zc

c( )

( )( )

( ) ( )( ) ( )

= =+

W1

(12)

where

G z( ) is the z-transform of G s( ) .

GH z( ) is the z-transform of G s H s( ) ( ) .

4. Simulation Results and Comments

The MATLAB software program is used to simulatethe performance of the controlled system in the caseof continuous-time and discrete-time.A step response is carried out to demonstrate effects ofthe sampling time and the values of the controllergains PK , IK and DK in cases of continuous-timeand discrete-time. Figs.49 indicate the digitalcomputer simulation results for continuous-timeand discrete-time. Fig.4a, Fig.5a and Fig.6a,illustrate the root-locus of the continuous-time systemcontrolled by proportional (P), proportional-derivative(PD), and proportional - integral - derivative (PID)controllers , respectively, with controller gains

PK = 1.1203, DK = 0.0026,IK = 47.4399. WhileFig. 4b, Fig. 5b and Fig. 6b demonstrate thecorresponding Bode plots. From these figures, thecontinuous-time controlled system is stable when thephase margin is set at maximum value as with the PIDcontroller. Furthermore, Fig. 7a, depicts the stepresponse of the continuous-time controlled systemwith P, PD, and PID controllers. Fig. 7b indicatesthe effect of changing the armature resistance on themotor speed with PID controller, while Fig.7cdemonstrates effect of changing the load torque,which is considered as input to the controlled system,on the regulated speed, with PID controller gains

PK = 1.1203,DK = 0.0026,IK = 47.4399.In comparison with the continuous P-controller, Fig.8,indicates the corresponding root-locus and the stepresponse for discrete-time system with differentsampling periods and PK = 0.049. This figureindicates the maximum overshoot is increased andthe settling time is increased when the sampling

time increased. It is noticed also that the digitalproportional controller gain be reduced to attaingood step response. In addition, a compromisebetween the value of the sampling period and thecontrollers gain should be taken into consideration.Fig.9, illustrates the step response of the discrete-timesystem controlled by the digital PID controller withdifferent sampling time and PK = 0.02,

DK = 0.0001,IK = 0.15. In comparison with the continuous PID controller ofFig. 7a, the digital controller gains are decreased andthat increases maximum overshoot and reduces risetime and settling time due to the effect of the samplingperiod. In this case the controlled system may beunstable with increasing the sampling time.

5. Experimental Results

The experimental results are obtained from theexperimental set-up indicated in Fig.3, where a DataComputer Package with 12-bit AD-DA adapter is usedwith sampling time T=10 msec. The dc shunt motorused in the experimental set up has the followingspecifications:

2-pole, av = 110 V, ai =23 A, w =183.2 rad/sec,

LT = 8.91 N.m. , aR =1.6 W, aL =0.016 H,

mK = 0.53 N.m./A, bK =0.53 V/rad/sec,J = 0.52 kg.m2, B = 0.043 N.m./rad/sec.

Fig.10, indicates the experimental results whereFig.10a, illustrates the speed response of the dc motorwhich corresponds to the change of load torque fromno-load to full load. This figure is obtained under theeffect of the P-controller with gain PK = 0.4. Effectof changing the load torque from no-load to fullload on the motor speed with PID-controller isdepicted in Fig. 10b. The gains of this controller are:

PK = 0.5, IK = 0.05,DK = 0.01.

6. Conclusion

Digital PID controller is implemented to control thespeed of a separately excited dc motor. The optimalPID controller gains are obtained off-line using theroot-locus technique to satisfy the requiredperformance of the controlled system. The digital PIDcontroller is synthesized and implemented on-line.Effects of the sampling time, PID controllers gains,microprocessors execution time, and resolution of theADDA converter card are investigated. Comparisonbetween the performance of the controlled continuous-time system and the controlled discrete-time system ismade using simulation and experimental works.

7. References

[1] Hale, F.J., " Introduction to control system analysis and design" , Prentice-Hall, Inc., 1988.

[2] Phillips C.L., and Nagle, H.T., " Digital Control system -Analysis and Design" , Prentice- Hall, Inc., 1990.

[3] Money, S. A., " Practical microprocessor inter- facing ", Collins P. Book, 1987.

[4] Rashid, M. H.," Power electronics - circuits, devices and applications", Prentice - Hall, Inc., 1993.

[5] Webb, J. and Greshock, K.," Industrial Control Electronics", Macmillan Publishing Company, 1993.

[6] Awad, A. S., Mohamed, E. A., Negm, M. M. and Said, A. I., " Speed control of DC motor drives based on efficient utilization of energy and optimal performance ", IEE, CIRED 97, Con-ference Publication No. 438 , 1997, pp. 5. 22.1- 22.5.

(a)

(b)

Fig. 1 (a) Digital control system (b) Rectangular rule of numerical integration

Main Page