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PD Servo
Author:
John Bratcher
PD Servo is a linear
math model of a proportional/psuedo derivative
control system. This means that the feed forward
compensation is proportional and the feedback
compensation is both proportional and differential.
This is most common in electric motor servos
with tachometer as well as potentiometer feedback.
This model, being linear, applies mostly to
small signal stability analysis. Large command
inputs will usually saturate current limits,
which would greatly affect overall performance.
Also, non linearity's such as coulombic and
static friction as well as backlash are not
included, which can also significantly influence
performance. This program is useful as a first
cut design/analysis of a servo system, which
will later be enhanced with nonlinear simulation/analysis.
To
use PD Servo first initialize all the physical parameters taking care to
use consistent units. I edit the file using the edit option of the main
keys of MtrxCal. Then define the period and frequency ranges and their
increments. This will most likely take several iterations to get the best
combination. Then switch to matrix keys and execute the program. If all
parameters are properly defined, a graph of the gain will appear. Tap the
screen and the phase plot will appear. Tap again and the step response
will be plotted. Finally, after another tap, a list of computed
performance parameters will appear such as the natural frequency,
resonance frequency, damping ratio, time constant and settling time.
Note: the parameter definitions are not present
in the executed program.
They are only included here for definition.
I've used metric units for consistency, but
English units could also be used provided appropriate
conversion factors are included.
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kin=1; %input
gain (outside closed loop)
kp=500; %feed forward proportional
gain
kt=.01; %motor torque constant
[NM/Amp]
kbemf=.01; %motor back emf
constant [V-sec]
rm=1; %motor winding resistance
[Ohms]
kfb=1; %proportional feedback
gain
kd=.1; %feeback rate (derivative)
gain
jm=10^-6; %motor rotor moments
of inertia [NM-sec^2]
jf=.01; %load moments of
inertia [NM-sec^2]
kdm=10^-6; %motor rate damping
coefficient [NM-sec]
kdf=.1; %load rate damping
coefficient [NM-sec]
n=150; %motor to load gear
ratio
t0=0; %plot start time [sec]
tf=2; %plot stop time [sec]
ti=.02; %plot time increment
[sec]
f0=-1; %plot start frequency
exponent, 10^f0 [Hz]
ff=2; %plot stop frequency
exponent, 10^ff [Hz]
fn=100; %number of logarithmically
spaced points from f0 to ff
f=logspace(f0,ff,fn);
w=1j*2*pi.*f;
t=[t0:ti:tf];
kdfe=kdf+n^2*kdm;
jfe=jf+n^2*jm;
k1=kin*kp*kt/(jfe*rm);
k2=kdfe/jfe+n*kbemf*kt/(jfe*rm)+kd/jfe;
k3=kfb*kp*kt/(jfe*rm);
wn=sqrt(k3);
damp=k2/(2*wn)
fn=wn/(2*pi)
del=sqrt(1-damp^2);
fr=fn*del
tau=1/(damp*wn)
ts=4*tau
g=k1./(w.^2+k2*w+k3);
semilogx(f,20*log10(abs(g)));
title('Freq Resp: Magnitude')
xlabel('Frequency Hz')
ylabel('Mag dB')
pause;
clf
semilogx(f,angle(g)*57.3);
title('Freq Resp: Phase')
xlabel('Frequency Hz')
ylabel('Phase degs')
pause;
clf
pos=1-exp(-damp*wn.*t)./del.*sin(wn*del.*t+atan(del/damp));
plot(t,pos);
title('Step Response')
xlabel('time sec')
ylabel('position degs') |
Fig. 1
Fig. 2
Fig. 3
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