any speed and power. Simulation results using the. MATLAB software package for a'specific six-pulse con- trolled rectifier fed pmdc motor are presented for two.
INTELLIGENT CONTROL STRATEGIES FOR PERMANENT MAGNET DC MOTOR DRIVES Michael E. Fisher Arindam Ghosh Adel M. Sharaf Dept. of Mathematics Dept. of Electrical Engg. Dept. of Electrical Engg. IIT Kanpur Universjty of New Brunswick University of Western Australia Nedlands, W.A., Australia Kanpur, U.P., India Fredericton, N.B., Canada Abstract - This paper presents three effective speed regulation schemes for permanent magnet dc motor drives with a nonlinear load. A feedforward/feedback control strategy is designed for each scheme to ensure effective, high performance tracking of reference speed trajectories. The structure of the feedforward cqmponent is the same for each scheme and uses only the online reference speed. All the control strategies utilize the output speed error and its derivative as feedback d a m p ing signals. The different feedback controllers used are proportional plus derivative, multi-zonal rule based proportional plus derivative and a fuzzy logic based scheme. The digital simulation and control validation results are presented for both set point tracking and set point plus ramp tracking. 1. INTRODUCTION
High performance permanent magnet dc (pnidc) motor drives are used for a multitude of industrial applications such as in process control, guided vehicles, paper and steel mills and mining and smelting plants. Precise, fast, effective speed reference tracking with niinimuni overshoot/undershoot and small steady state error are essential control objectives of such a drive system. Conventional control strategies are of a fixed structure, fixed parameter design [1,2] hence the tuning and optimization of these controllers is a challenging and difficult task, particularly under varying load conditions, parameter changes, abnormal modes of operation etc. Attempts to overcome such limitations [3-51 U* ing adaptive and variable structure control have had limited success due to complexity, requiring of estimation stages, model sturcture changes due to discontinuous drive mode of operation, parameter variations, load excursions and noisy feedback speed and current signals. New Artiiicial Intelligence technologies [6-91 such as rule based, expert systems, fuzzy logic and artificial neural networks (ANN) started emerging during the last decade and promise to simplify and enhance the roboustness of speed/position control designs for pnidc motor drives. This paper presents three differelit speed regulation schemes for a pmdc motor drive with a nonlinear load. Each of the three control schemes consists of a comnion
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feedforward component and different feedback components, the latter based on the output speed error and its derivative. The inclusion of the feedforward term reduces the tracking problem to a nonlinear regulator problem thus simplifying the structure of the various feedback control components. The feedforward component used is a function only of the reference speed and its derivative and differs from the more popular types of controllers based on the notion of feedback linearization [lo] in which the feedforward term is a function of the motor speed and its derivatives. The three different feedback controllers used are proportional plus derivative (PD), multi-zonal Tule based PD and a fuzzy logic controller with gain adjustment. Siniulation results b i n g the MATLAB software package for a specific six-pulse converter fed prndc motor are presented for two arbitrarily chosen reference speed trajectories, one with set point changes and the other with ramp changes to the reference speed. The dynamic performances of the three controllers are compared under parametric variations, load changes, abnormal load Fxcursions and noisy speed feedback.
2. PMDG MOTOR M'ODEL AND CONTROL FORMULATION The pmdc motor model is shown in Fig. 1 where the delay term is associated with the six-pulse converter firing circuit. The system equations are given by: Load Equations: T L = K,+K,w+K2w2 TM = KTI
T M - T L = .JW+Bw
(1)
(2)
(3)
where w is the speed of the motor, TMand TLare respectively the motor and the load torques, I is the motor armature current, J and B are the motor inertia and damping constaats, KT is the torque constant and we have assumed a general purpose quadratic load model. Electrical Equations:
360
E V-E
=
KEW
= RI+Li
( 1) (5)
-
where E is the motor emf, I/ is the control input given by V = 1 . 3 5 V cos ~ ~ a,a begin the flring angle of the sixpulse rectifier bridge, K E is the back emf constant, R and L are the resistance and inductance of the armature circuit. It is assumed that the rectifier firing angle ( a ) is limited between 2" and 88".
.
It is worth noting that the feedforward term alone can never satuwte the control input if the reference trajectory can be tracked in the absence of error (this is a necessary but not sufficient condition for stability). This is in contrast to an approach based on the so called "exact feedback linearization" (lo] in which the feedforward part of the control may saturate if the actual speed differs from the reference speed, leaving no control left for the error feedback.
w
I
I
Fig. 1 Block diagram of the pmdc motor di-ive. Combining (1) to (3) and then diffrentiating we obtain the following equations
1 =
1' -KT p
k+ (B
+ K1)w 4-K2w2 + KO]
Fig. 2 Feedforzuald/feedback control stmcture.
(6)
3. FEEDBACK CONTROLLER DESIGNS AND SIMULATIONS
Using (6) and (7) to eliminate I and I from (5) gives the pmdc motor dynamic model as
where 01 to cy6 are positive scalar constants which depend on the motor parameters and are given by
JL KT'
0 1
=
-
a5
=
2LK2 -
KT '
a2
=
RJ + L(B + K i ) KT
I
RKO ag= -.
KT
Let w, be the reference speed and e = w, - w be the speed error. The control V is now broken in two parts
In this section, three different feedback control designs of V b are presented, increasing in complexity from a simple PD controller to a fuzzy logic based controller. Specific design details are given for a 230V, 125HP,1150 rpm pmdc motor to illustrate the approach, although the techniques are applicable to pmdc motors of almost any size. The motor and load parameters are given in the Appendix. In the simulation studies, two different reference trajectories are chosen to-illustrate the results. The first, which is shown in Fig. 3(a), is a staircase type function used to investigate how the controllers respond to set point changes. The second, shown in Fig. 3(h), is a combination of both ramp and step functions. These will be referred to as trajectory 1 and 2 respectively.
3,1 The PD Controller where V, is the feedforward component of the controller defined by
and fi is the feedback componet which is function of the tracking speed error and its derivative. The controller structure is shown in Fig. 2. Combining (8j, (9) and (lo), we obtain the feedback control equation in terms of the tracking error as
The purpose of the feedback is to force the nonlinear error equation (11) to zero. This is equivalent to choosing the coefficients of e and t in this equation in such a way that closed-loop poles can be placed in a desired way. This can only be achieved through a PD controller as it influences the coefficients of e and 1 directly. The introduction of &I integral control action will increase the order of the error equation needlessly. The feedback is then given by
vb = Kpe -b Kd& The problem of designing a suitable controller for the tracking problem has now been reduced to that of designing a suitable feedback controller V, for the nonlinear regulator problem defined by (11).
(12:
where K p and Kd are gains to be chosen. Substutinl (12) in (11) and rearranging we get ale 4-
[Kd
+ a2 + hy5wr)t + + a4(wr -t-w ) + a.+,]e
+[Kp
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a3
=o
(I:
Since the coefficient CY,are all positive, and also w cannot be negative, the sufficient conditions for the error equation to be stable are that the gains K p and K d are positive and K p satisfies (for the extreme conditions of w = w, = 0 and ijr < 0)
the rated value, thus remaining well within the acceptable range.
It is easy to satisfy the above condition and any value of K p > 0 will guarantee stability. 0
I .
0
2
4
6
8
10
tirnc(secs)
Fig.
01 0
4 Speed
tracking along trajectory-1 controller.
t
2
4
6
8
10
time(secs)
* 0
2
4
6 time(secs)
8
10
Fig. 3 Refrence trajectories chosen for simulation studies: ( a ) trajectory-1, (b) trajectory-2
Fig. 5 Tracking error for trajectory-2 with P D w ntrol ler. 3.2 The Multi-zonal Rule Based PD Controller An improvement in the tracking performance can be obtained if the gains of the controller of (12) are varied according to the size of the excursion vector
For the specified pmdc motor, the error euqation (13) Re= ( e , & ) is stable for all positive values of K p and K d . These gains are chosen in a sinilar way as discussed in [ll] In particular if the error and its derivative are relatively and they are given by small, then the gains can be chosen to achieve critical or near critical damping so as to reduce overshoot whereas Kp=10 K d = 3 for larger errors the major consideration in choosing the gains is to obtain a fast response. Fig. 6 depicts which results in a slightly underdamped system. In- the structure of a %zone PD controller for the specified creasing the gains does not improve the response to set pmdc motor, where the gains K p and Kd depend on the point changes due to control saturation. zones defined by the excursion vector &. The values of Fig 4 shows the speed response of the PD controller the gains chosen for the three zones are shown in this while tracking trajectory-1. This figure demonstrates figure. Fig. 7 shows the speed response of the multi-zonal how effective even a simple PD feedback is in responding to changes in the reference speed. Fig. 5 shows the PD controller in tracking trajectory-1 and only a slight tracking error resulting from the tracking of trajectory- difference from the performance of !he PD controller 2. The maximum tracking error of approximately 0.02 can be observed (see Fig. 4). Fig. 8 shows the trackpu occurs due to the discontinuous change in the slope ing error while tracking trajectory-2. It can be seen of the reference trajectory and is rapidly damped. It that there is some improvement, both in the magnitude was observed during the simulations that the armature of the errors and the time taken to damp ,out the ercurrent values were always between 30% and 110% of rors, over the P h n t r o l l e r . Overall, the multi-zonal
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P D controller performs slightly better than the PD controller, which is to be expected.
2kiangular membership functions pA(e) and p ~ ( k of ) e and b respectively are assigned to each of these 7 fuzzy sets. For the specified pmdc motor, the same 7 fuzzy sets were chosen for e and t (ie., pA(e) = p ~ ( t )and ) these are shown in Fig. 9. Fig. 10 shows the fuzzy sets for the output function, U ,together with its membership function.
NL
NM
NS
-0.3
-0.1 -0.02
t"
Zt
PS
PM
I'L
0
0.02
0.1
0.3'
e,e Fig. 6 Three-zone rule based P D gains.
y-qL
0
0 . 8_ - b-
__..-
0.4
I\ 8
' 0
Fig. 9 Fuzzy sets and membership functions for inputs.
. --c
0
2
4
6
8
IO
U c
timc(sccs)
Fig. 7 Speed tracking along trajectory-1 with multi-zonal PD controller.
T"
'
0
-0.7
-0.4
-0.2
0
0.2
0.4
0.7
Fig. 10 Fuzzy sets and membership junctiom for output.
A fuzzy control rule decision table is then defined for all the 49 corresponding fuzzy rules with the control actions chosen from the 7 alternatives as for the input data. The fuzzy control decision table is defined in Table l. The weight or truth value associated with the ith fuzzy control rule:
2
4
h
8
IO
-
if e is Ai and t is Bi then
U
is C;
is given by
liinc(sccs) Fig. 8 Tmcking error f o r trajectory-2 with multi-zonal PD controller.
3.3 The f i z z y Logic Based Controller The fuzzy logic controller adopted for generating Vb is of a standard type based on the input variables e and i. [12]. The fuzzification strategy converts the input data into the fuzzy sets:
NL: Negative Large NS: Negative Small PS: Positive Small PL: Positive Large
NM: Negative Medium ZE: Zero PM: Positive Medium
The crisp value of the output V, is then obtained using the so called maximum criterion method [12]
where U , is the output value for which the output nienibership function attains a niaxinium value. Fig. 11 shows the speed response of the fuzzy logic controller while tracking trajectory-1. A comparison with Fig. 7 shows that the fuzzy logic controller and multizonal PD controller behave in an almost identical
Table 1: Fuzzy control rules
fashion. Fig. 12 shows the tracking error for the reference trajectory-2 and again, the performance is similar to that of the multi-zonal P D controller.
4" 0.8
0.4
0
2
4
6
8
10
ti me(secs)
Fig. 11 Speed tracking along to trajectory-1 with fuzz9 controller.
Typical errors in these parameters should not therefore exceed 10%. Simulation studies were carried out with parameter variations up to 10% and virtually no change in tracking performance was detected. The parameters J and B are, however, susceptible to larger inaccuracies and hence simulations were conducted with variations of up to 50% in this parameters. When tracking set point changes in the reference speed, with J and B increased by up to 50%, the results showed only a slight degradation in performance of the P D qnd the multi-zonal PD controllers and effectively no change in performance of the fuzzy controller. With decreases in J and B of up to 50% the performance of all the three controllers was virtually unaffected to set point changes. When tracking the reference trajectory-2, the performence of all controllers was significantly affected, with tracking errors Iarger when J and B were increased than when J and B were decreased. Figs. 13 to 15 show the errors incurred by the three controllers when tracking this trajectory with J; and B at 150% of their original vahes. These figures show that the magnitude of the tracking errors has increased in all cases compared with when J and B are exact, with the PD controller giving the largest errors and the fuzzy controller giving the smallest errors. More significant is the inability of the P D controller and, to a lesser extent, the multizonal P D controller to damp out the errors on the ramp following part of the trajectory which are successfully damped using the fuzgy controller.
t" -0.04
I
L"-
0
2
4
-0.w 0
2
4
6
8
10
timc(secs)
6 time(sccs)
8
10
Fig. 13 Tracking error with P D controller for increase in J and B.
Fig. 12 Tracking error f o r trajectory-2 with fuzzy controller. 0.02 4. SIMULATIONS UNDER PARAMETRIC
A
0
AND LOAD VARIATIONS The controllers described in Section 3 were designed for the measured motor and load parameters defined in the Appendix. The robustness of the different controllers is assessed in this section by examining their ability to track under parameter and load variations as wel) as in the presence of noisy feedback signal. The model parameters R, L, K E and KT can be estimated to a reasonable accuracy for a given motor.
In
-0.02 -0.04
t v I
0
b
4
-2
6
8
IO
tiiiie(secs)
Fig. 14 Tracking error with multi-zonal P D controller for increase in J and B .
364
-_
.
As can be seen, the noise significantly affected the set point tracking of the multi-zonal PD controller, but had only a small effect on the fuzzy controller. The performance of the PD controller is even worse than the multi-zonal P D controller and is not shown here. As U was increased to 0.01, the tracking performance of all the three controllers suffered and they all failed to eliminate significant steady state errors which were typically between 2% and 5%. This however, cannot be considered a severe limitation as the sensors used for critical drive systems are fairly accurate. -0.04 I 0 2 4 6 8 10
timc(sccs) Fig. 15 Tracking error with fuzzy controller for increase in J a n d B. Significant load changes are to be expected during a motor's operating cycle. There may be many occasions when the motor is required to operate at less than full load for a significant length of time, whereas overload conditions would occur with less frequency and only for shorter periods of time. Simulations were undertaken with varying load conditions to test the robustness of these controllers. As an illustration, results are shown for the case when the load coefficients KO,Kj and Kz are changed, after 1 s, to 50% of their nominal values. Fig. 16 shows the tracking error with the PD controller for the reference trajectory-2 and one can see that the controller fails to completely eradicate the error on the ramp part of the trajectory leaving a constant error of approximately 0.01 pu, altough the error is damped out as the reference speed becomes constant. Fig. 17 shows the corresponding result using the fuzzy logic controller. It can be seen that when tracking the ramp section of the trajectory, there is a small error offset, but the r e sults are a significant improvement over those of the PD controller. The multi-zonal P D controller proved to be slightly inferior to the fuzzy logic controller under this condition, but better than the P D controller. It is to be noted that all the three controllers performed satisfactorily for this test while tracking trajectory-1 and the results are not shown here. Another set of simulations tested the ability of the controllers to respond to severe overload conditions. In these simulations the load was doubled for a short period of time with the motor operating a t 90% of its rated speed. The three controllers responded with the motor speed virtually unchanged during the overload period while at the same time the current almost doubled. To investigate how these controllers perform in the presence of inaccurate sensors, the final set of simulations were carried out. To model sensor inaccuracy, zero-mean white gaussian noise with standard deviation U and proportional to the size of w was added to the feedback signal. Simulations were carried out for values of U up to 0.01. Figs. 18 and 19 show the results for set point tracking with the multi-zonal PD controller and the fuzzy controller respectively when CT = 0.005.
-0.04 0
2
4
8
6
10
-
time(secs) Fig. 16 Tracking error with PD controller f o r a decrease in load.
1"
-0.04 I 0
2
4
+ 6
8
10
time(secs) Fig. 17 Tracking error with fuzzy controller for a decrease i n load.
5. CONCLUSIONS In this paper three effective speed regulation schemes for pmdc motor drives with nonlinear load are presented. Each scheme utilizes a feedforward/feedback control strategy to ensure effective, high performance tracking of the reference speed trajectory. The structure of the feedforward component is the same for each scheme and is computed online from the reference speed. The inclusion of the feedforward term reduces the tracking problem to a nonlinear regulator problem thus simplifying the structure of the various feedb;!, L control components. All the control strategies utiliz< the output speed error and its derivative as feedback damping signals.
365
Designs of the three feedback controllers are presented for a 230 V, 125 HP, 1150 rpm pmdc motor, although the techniques are applicable to pmdc motor of almost any speed and power. Simulation results using the MATLAB software package for a‘specific six-pulse controlled rectifier fed pmdc motor are presented for two arbitrarily chosen trajectories. All three controllers exihibit excellent tracking performance when the motor and load parameters are exactly known. With the introduction of parameter and load variations, all controllers exhibited good robustness characteristics with the fuzzy logic controller giving the best response of all. The fuzzy logic controller also copes best with a noisy feedback signal which does not unduly affect the tracking performance of the controllers unless the noise to signal ratio is too large.
[3] Y.Y. Hsu and W.C. Chan, “Optimal variable structure controller for dc motor speed control”, Proc. IEE, Pt. B, vol. 131, pp. 223-237, 1984. [4] S. Weerasoorya and M.A. El-SharGwi, “Adaptive
tracking control for high performance dc drives”, IEEE Trans. Energy. Convr., vol. 4, pp. 502508, 1989.
[5]M.A. El-Sharkawi and S. Weerasoorya, “Develop ment and implementation of self-tuning tracking controllers for dc motors”, I E E E Trans. Energy. Convr., vol. 5, pp. 122-128, 1990. [ti] E.H.
Mamdani, “Application of fuzzy logic to a p proximate reasoning using linguistic I E E E Trans. Computers, vol. 26, 1191, 1977.
[7]C.P. Pappis and E.H. Mamdani, “A fuzzy logic controller for a traffic junction” , I E E E T9-am. Syst.
0.8
0.4
103-113, 1985. 0
c
0
2
4
6
8
logic controller for dc motor drives”, Proc. Int. Conf. Appl. Industry, IASTED’92, 1992.
10
time(secs.1 Fig. 18 Speed tracking with multi-zonal PD controller under sensory noise.
[lo] J.-J.E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood-Cliffs, 1991. [ll] M.E. Fisher, A. Ghosh and A.M. Sharaf, “An optimal proportion& plus integral set point controller for a permanent magnet dc motor”, Control - Theory and Advanced Technology, CTAT, Vol. 10, No. 4, Pt. 3, pp. 1431-1444, 1995.
1121 C.L. Chen, ‘Tuzzy logic in control systems: fuzzy logic controller”, part 11, I E E E Trans. Syst. M a n €d Cybern. vol. 20, pp. 419-435, 1990.
0‘
0
2
4
6
8
c
10
time(secs)
Fig. 19 Speed tracking with fuzzy controller under s e n s o 3 noise.
REFERENCES [l] B.A. White, R.T. Lipczynski and A.R. Daniels, “A simple digital control scheme for a dc motor”, Proc. IEE, Pt. B, vol. 130, pp. 143-147, 1983. [2] J. Zhang and T.H. Baston, “Robustness enhancement of dc drives with a smooth optimal sliding mode control”, I E E E Trans. Ind. Appl., vol. 27, pp. 686-693, 1990.
APPENDIX The parameters of the pmdc motor chosen for tlie simulation studies are: Rated Power = 125 Rated Volatge VLL= 230 V, Rated Speed = 1150 rpm, R = 0.0125 R, J = 3 Nnis2, K E = 1.91 VS, KO = 300, Kz = 0.022
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