Proportional-Integral Plus Bang-Bang Control of DC Servo Motors with PWM Drives

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PROPORTIONAL-INTEGRAL PLUS BANG-BANG CONTROL OF DC SERVO MOTORS WITH PWM DRIVES 11 Hong Suh*, Seung Ho Hwang* and Zeungnam Bien** 'Technical Center, Daewoo H eavy Industries Limited, 6 Manseog-Dong, Dong-ku, Incheon, Korea "Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, P.O. Box 150, Chongyangni, Seoul, Korea

Abstract. A speed controller with proportional-integra1(PI)-plus bang bang action is proposed for dc servomotors with transistorized pulse width modulated drives. The controller employs the PI-action when the magnitude of the error between the reference signal and the speed output signal is smaller than some prescribed value. Otherwise, the controller produces the maximum allowable control signal with the integrator reset. The servo motor system with the proposed speed controller is analyzed in both continuous and discrete time domain and then a method for controller parameter design is provided. To show that the performance of the controller with PI-plus bang bang action is improved compared to that of conventional PI-acti.on controllers, some experimental resu1ts of a design example are illustrated. Keywords. Proportional plus integral action; proportional-integral plus bang-bang ac tion; d.c. servo motor control; transistorized PWM drives; settling time-and-overshoots. INTRODUCTION

In this paper, to overcome the difficulties . due to the integral wind up of the speed controller with the PI action, a speed controller with PI plus bang bang action as shown in Fig. 3 is proposed. In this proposed configuration, controller employs the proportional-integral action when the magnitude of the error is smaller than a prescribed value. Otherwise, the controller generates the maximum permissible control signal with the integrator being forced to reset. A design method of the analog and digital controllers based on a mathematical analysis will be given. The design, which is experimentally verified, proves that the speed controller with PI-plus bang bang action renders a fast settling time property with small overshoots regardless of the magnitude of the reference input signal by not only preventing the integral wind up, but also utilizing the maximum ratings of hardware components such as power transistor and motor armature windings.

There have been a number of research works (Sen and MacDonald, 1978, Krishnan and Ramaswami, 1974, 1976, Taft and Slate, 1979) for the analysis and synthesis of the speed controller for separately excited DC servo motors. Most of the proposed design methods are aimed at a control system with the properties of small overshoot and fast settling time to a step input change. Traditionally, current controllers of the P(proportiona1) - or PI (proportional-integral) type are used (Krishnan and Ramaswami, 1974) in cascade with a device of limiting the armature current of the motor to a prescribed maximum allowable value. Also the current feedback is frequently used to suppress the armature current (Bailey, 1981). At any rate, the design in these cases means to establish a rule of determining the parameters such as PIgains for a simplified first order open-loop speed control system (Krishnan and Ramaswami, 1976, Bailey, 1981).

In the sequel, the following notational tions will be used

Even if it practically exists in the current reference input node (see, Fig. 1), however, the saturation type non1inearity is not taken into account in these conventional analysis and design of the speed controller. As a consequence, the so-called integra1-wind-up phenomenon is not effectively controlled, causing large overshoots and/or lengthy settling time (Takahashi, et al., 1972) .

B J

L Ra Kt Kb T TL Kc KcF Kv KvF Ts ia va vb r y e=r - y ei q o

The integral wind-up phenomenon, causing an actuator saturation and large overshoots, was discussed in detail by Phelan (1977), who furthur suggested as a possible solution to turn off the integral action when the integral term excess some prescribed value. More recently, Krike1is (1980) proposed another type of 'intelligent' integrator shown in Fig. 2 to avoid the integral wind up. In the Krike1is' intelligent integrator in Fig. 2, the parameters ~ and H are supposed to be the designer's choices. But, unfortunately, it was found that such a freedom vanishes when the integrator is to be used for the purpose of designing a speed servo system of DC servo motors.

8

2809

conven-

Friction coefficient Moment of inertia Total inductance of armature and external reactor Armature resistance Torque constant Back EMF constant Output torque Load torque Proportional gain of current controller Current feedback gain Proportional gain of speed controller Speed feedback gain Sampling time Armature current Armature voltage Back EMF voltage Speed reference input voltage Speed feedback voltage Speed error signal voltage Current reference input voltage Integration of error signal e Zero of speed PI controller Zero of current PI controller

11 Hong Suh, Seung Ho Hwang and Zeungnam Bien

2810

Per unit overshoot 6 peak overshoot value-steady state value ~ steady state value Voltage magnitude of switching Output speed Laplace transform of ei Laplace transform of ia Laplace transform of va Laplace transform of Maximum allowable voltage of ei Maximum allowable current Maximum allowable speed Speed reference input magnitude Incremental speed input magnitude

n w Ei la Va 11

~ax

Imax Ilmax R

R

DC SERVOMOTOR DYNAMICS WITH PI-CURRENT CONTROLLER

v

a

vb+R i a a

motor

ANALOG SPEED CONTROLLER WITH PI PLUS BANG BANG ACTION Consider the simplified open loop speed control system (2) shown in Fig. 1 whose dynamics is given by

In the design of current controller of the P or PI type (Sen and MacDonald, 1978), it is customarilly assumed that the magnitude of the inherent motor armature inductance is sufficiently small and thus is neglected. However, in case of the transistorized pulse width modulated dc servomotor drives (Taft and Slate, 1979), the armature in ductance is made highly increased not to be neglected by connecting a reactor (an external inductor) in series with the motor armature winding so that the magnitude of the current ripple is to be reduced. Thus the design method of the current controller proposed by Sen and MacDonald (1978) and Taft and Slate (1979) may not directly apply for the motor with transistorized PWM drives. To be specific, consider the servo described by

It is remarked that the maximum current reference voltage ~x should be limited to a certain value, which lies in practice between 6 and 10 volts. A rationale is that ~ax is the output signal of an operational amplifier whose absolute magnitude of the supply voltages is normally less than 15 V and that it should not be greater than the maximum allowable motor current. Thus the open loop speed control system with current controller of the PItype should include the saturation type nonlinearity in the current reference input node as shown in Fig. 1.

system

w=

-pw + Ku,

For the system (3), let us apply the controller with PI plus bang-bang action shown in Fig . 3. Let y(t)

i KvFw(t)

(4)

e(t)

~

(5 )

and q(t) t;,.

r(t) - y(t)

r

J~ dt +

Then the control law can be written as __ { KEv(e(t)+ aq(t)), for le(t)1 ~ n,

di +L __a_ dt

BW

+ TL'

u(t)

(1)

and

The PI-current controller is given by

max and q(t) = O,for e(t)

i

> >

n,

(7)

-n .

In (4) and (7), KvF is the positive real constant which represents the gain of the speed to voltage transducer, and Kv, a and n are the real constants to be determined. Recall that our design problem is to find Kv, a and n such that the system response after controller switching not only remains in an n band but also has the small overshoot values. In the following is given a sufficient condition that y(t), after controller switching, remains in an n band.

where e

(6)

e(t) dt .

0

-Emax and q(t)= 0, for e(t) T

(3)

i current reference command,

Theorem 1. Let Kv and a be chosen such that characteristic equation of the system

the

Now, if the inequalities given by

(Ra + Kcc K F) 2 »

LoK K F' f,J CC

and

hold, then the motor with current controller of the PI-type can be approximated as a first order system as follows (2)

where P and K are constants defined by

P i B/ J,

has two negative real roots -a and -b satisfying b > 2a

(9)

and further the per unit overshoot 0 of Xl (t) in (8) is less than one. Then if the ine qualit y (10)

Proportional-Int egral plus Bang-bang Control holds, the output response y(t) of the system (3) with the controller (7) asymptotically tracks the reference step input r(t). The proof of Theorem 1 is somewhat lengthy will be elaborated in the presentation.

and

It is noted from Theorem 1 that, though the inequality condition in (10) seems very restrictive, there exists n satisfying the inequality condition in (10) for all practical systems. This can be shown as follows. If

(ll)

a < 2b

and la-al is sufficiently small so as to make the per unit overshoot 0 be small. Choose n such that n is sufficiently small to obtain fast settling time response, while satisfying

Step 3.

Here, it is remarked that the maximum magnitude of reference signal should be limited as ~ax .::.10 V, since we usually implement the controller by the OP Amp's for signal amplification.

then there always exists n satisfying (10). Note that the inequality (11) can be rewritten as E

max

+

(12)

K v

P

28 11

DIGITAL SPEED CONTROLLER WITH PI PLUS BANG BANG ACTION Given the continuous-time system in (3), it is not difficult to derive the discretized version described by the difference equation

Also note that in practical motor systems, w(k+l) = exp(-pTs). (k)+K(l-exp(-pTs»u(k)/p, K E

n max

max

<

(13)

(14)

lu(k)1 '::'Emax·

P

since the maximum allowable current command Emax should be large enough for the steady state value of the motor output speed response to be rated speed Wmax' Thus from (13), it follows that

Consider the digital PI plus bang-bang control law given by Kv (e (k)+ q (k» ,

E and q(k)=O, for e(k) > n, max

u(k) < K

n

< K

!

-Emax and q(k)=O, for e(k)

vF max E

max

+

<

(15)

- n,

where

E

vF P

for le (k) 1 .::. n,

max

k

K v

q (k) ~

This implies that for all the practical there exist n statisfying the condition Theorem 1.

systems, (10) of

To propose a rule of thumb for the deisgn of parameters ~, a and n , it should be noted that as ~ and a gets larger, the speed response becomes faster if we design the speed controller only for the idealized open speed control s ystem in (3). However, there are some practical limitations such as the dynamics of the current controller and current feedback filter which have been neglected. Hence in designing speed controller, we should choose Kv and a such that all those dynamics do not seriously affect the closed loop system performance. This implies that the zero of the speed controller should be located far from the zero of the current control system in the right hand side of the zero of the current control system. For this reason, it is proposed that a be in the range of

E e (i).

i=O

As in the analog speed controller with PI plug bang bang action, the design problem to be solved here is to find Ky, a , and n such that the output response after controller swit ching is completed lie within an n band. It is noted that the digital controller output does not change during the sampling time interval. It is further noted that a limit cycle oscillation can occur when n is smaller than the incremental value ~y ~y ~

KvF( w(k+l) - w(k»

= KvF(l-exp(-pTs» (K.Ema/P-w(k».

(16)

When the viscous friction coefficient B is very small and thus p in (2) is also very small, then (16) can be approximated as (17)

And the gain parameter ~ is chosen such that the two negative real poles are located at -a and -b in the s-plane while satisfying the condition (9), and -a is located nearby - a to have small overshoot. These are summarized in the following,

to

Step 1.

Choose

Step 2.

Choose ~ such that two poles -a and -b satisf y

such that

<

a

<

t

negative

It can be shown that, if ~y < n , a similar result of Theorem 1 may apply for the digital controller synthesis except that (10) is replaced with the inequality given by

<

real

n

<

E

max

/K. v

(18)

2812

Il Hong Suh, Seung Ho Uwang and Zeungnam Bien EXPERIMENTAL RESULTS

REFERENCES

To evaluate the performances of the speed controllers (A) (B)

with PI-action, and with the proposed PI plus bang-bang action,

some experiments are conducted for a dc servomotor with parameters in table 1. To determine the design parameters Kc'~ and KcF of the current controller, let the maximum allowable current and maximum current reference signal ~ax be 6 Amperes ~ and 6 Volts, respectively. Then KcF' Kc and are determined to be I, 26 and 256, respectively. In this case, the transfer function in (2) is given by rl (s)

185

Ei(s)

s + 0.2

(19)

Now to compare the performances of the two types of the speed controllers (A) and (B), let KvF be 0.05, which implies that the rated output speed rl max is 120 rad/sec for 6 V maximum speed reference input voltage ~ax' Since the zero of the current control system is located at -256, let a be equal to 50 by the design step 1. Then by choosing the per unit overshoot 6 of xl(t) of the system in (8) to be 0.10, we can obtain that Kv is equal to 30. From these gain parameters, the characteristic equation of the system (8) given by

Bailey, S.J. (1981). Servomotor-amplifier choice is key to better position loop response. Control Engineering, (28), 63-68. ---Krikelis, N.J. (1980) . State feedback integral control with 'Intelligent integrators'. Int. J. Control, (32), 465-473. Krishnan, T. and Ramaswami, B. (1974) . A fastresponse dc motor speed control system. IEEE Trans. Ind. Appl., (10). Krishnan, T. and Ramaswami, B. (1973). Speed control of dc motor using thyristor dual converter. IEEE Trans. Indust . Electr. Contr. Instrum., (23) . Phelan, R.M. (1977). Automatic Control Systems. Comell University Press, London. Sen, P.C. and MacDonald, M.L. (1978). Thyristorized dc drives with regenerative braking and speed reversal. IEEE Trans. Indust. Electr. Contr. Instrum., (25). Taft, C.K. and Slate, E.V. (1979). Pulsewidth modulated dc Control: a parameter variation study with current loop analysis. IEEE Trans. Indust. Electr. Contr. Instrum., (26). Takahashi, Y., Rabins, M.J. and Auslander, D.M. (1972) . Control and Dynamic Systems, AddisonWesley, London. TABLE 1.

Item D(s) = s2 + (0.2 + 9 . 25Kv )s + 9 . 25Kv a

Specifications of a DC Servomotor

Value

Unit

(20) Rated Power

has two real roots -65, and -212, which satisfy the condition in (9). Thus by design step 3, n is chosen to be 0.2. To evaluate the performances of the two types of speed controllers (A) and (B), we apply 50 mVolts square wave signal as 'small' reference speed input signal, and apply 1.6 Volts square wave signal as 'large' reference speed input signal. The speed output response of the system with controllers (A) and (B) with respect to the small signal input and large signal input are shown in Fig. 4 and in Fig. 5, respectively. We can observe from Fig . 4 that the speed responses of the system with controller (A) and (B) are identical as we expected, and from Fig . 5, the speed response of the system with controller (B) is better than that of the system with controller (A) in the sense that the speed response shown in Fig. 5-b has smaller overshoots and faster settling time than that of the speed response in Fig. 5-a. Here, it should be noted that with the same Kv and a, the speed response of the system with controller (B) ocillates as shown in Fig. 6, when n reduces to zero which violate the condition in (10). CONCLUSIONS For DC servo motor systems with the transistorized PWM driver applied to a large inductive load, a novel type of speed controller using PI action plus bang-bang action is suggested, analyzed and its design rule is given. It was further shown via comparative experimental study that the proposed speed controller was better than the speed controller only with PI-action. The proposed controller may be used for many industrial process control which are often modelled as first order systems with saturation nonlinearity.

400 7. 2 36

Kg.cm2

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V a max I

max

rl

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Watt Kg.cm 2 /Rad . Sec

rl

6666

2 2 Kg.cm /sec .A

0.7

Volt/rad/sec

90

Volt

5

Amp

120

Rad/sec

28 13

Proportional-Integral plus Bang- bang Control

SPEED

(RPM)

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I

CURRENT REF. INPUT NODE Fi).;Url· 1.

5

Simpl i fied OpE>n -Loop Speed Cont ral Systl:'m

-+...L.L...I.....L.+.:-'--'-'-........,.Ioo~-'-....L...~-_TIME (mM<:)

E

'J:~"-,----T",_--,OU..

Fi gu r e 4.

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SPEED

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of Krtkelis '

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320 (b)

REF .

240

TJ

160 REF. +

80 U(I)

BANG - BANG DRIVER -, : Emax

OUTPUT (SPEED)

-'1]:

100

I.

300

~ Figu r e ').

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SPEED

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Spe~d

RrsponsE'

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(RPM)

100

_-f'-.l..-L-I.......L........l.:.....L.......L-..L..-.L......J'--I.---'---'----_

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Figure 6.

T I ME (msec)

Oscillat ing Response of Large Si gnal Input with

n=O Volt.