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Abstract: ME4053Controls Lecture 1Modeling and Identification of a DC MotorDr. Ferri Announcements!• Vibrations reports are next week• You will be assigned to do one of the following

ME4053
Controls Lecture 1
Modeling and Identification of a DC Motor
Dr. Ferri
Announcements!
• Vibrations reports are next week
• You will be assigned to do one of the following
– Presentation on the week 1 ( 2DoF) lab at 10 am Weds April 14th
– Extended abstract on the the week 1 (2DoF) lab due by 4 pm on Weds
April 14th
– Presentation on the week 2 (FreeFree Beam) lab at 10 am Friday,
April 16th
– Extended abstract on the the week 2 (FreeFree Beam) lab due by 4 pm
on Friday, April 16th
• Assignments will available on the website on Friday
• Room assignments will be posted
• Deposit extended abstracts in the bin outside the lab
• Email electronic copies of presentations and abstracts to
[email protected]
ME4053
Controls Lab 1
Modeling and Identification of a
Brushless DC Motor
Controls Lab 2
Control of a Brushless DC Motor
DC Motor Model
Ra La
ia
eo(t) eb
q
b
JL
Kirchoff Voltage Law (KVL):
Backelectromotive force (emf)
Torquecurrent constant
Moment balance on load: JL T
bw
Laplace Transforms
Laplace Transforms
combine algebraically, or…
Motor can be represented in the form of
a blockdiagram showing “internal feedback”
KVL
Torquecurrent relation
Moment summation
Back EMF
R C
G
H
Letting and
or
In many cases, La is very small, allowing reduction to 1storder system:
or
where
Motor time constant Motor gain
See how time constant and motor gain depend on physical parameters
Step response:
Inverse Laplace Transform
wss= Vin Km
w
w = 0.632 wss
t/Tm = 1 t/Tm
Km = 1 2% settling time
Vin =4
Vin =3
w
Vin =2
Vin =1
t/Tm t/Tm = 4
Deadzone
wss
Km
Vin
Negative deadzone Positive deadzone
System Identification, Continued
motor
A
B
Note phase lag
Steadystate harmonic excitation, harmonic response
Mag Phase Lag
3dB
Due to nonlinearity, we use a biased sinusoid as input
thigh
Vhigh
Vlow
Dt
A
tinp
Form Bode Plot
3dB point
wc =“corner frequency”
20dB/dec
45O phase lag
Phase Lead
(Note: plot shows
phase lead)
wc
SIMULINK : Measurement Model
SIMULINK : Simulation Model
Review of Feedback Control…
System ID was done using the motor/flywheel speed, but
for servocontrol, we first need to obtain the Transfer Function
from Voltage to Rotation Angle
Recall = transfer function from applied
voltage to motor speed
What about position?
= transfer function from applied
voltage to motor angular position
Change system performance through use of feedback control
commanded
output
angle
angle
R E M
motor Q
controller
error actuator plant
signal
Unity feedback
H=1
(includes unit conversion)
Proportional Control (Pcontrol)
R E M Q
Kp
error actuator
openloop transfer function:
R C
G
H
Closedloop transfer function:
Closedloop poles = roots of denominator of cltransfer function
at K = 0, roots are at s = 0, and s = 1/Tm
as K >0, roots move together along the real axis
at K = 1/(4Tm), repeated roots at s = 1/(2Tm)
for K > 1/(4Tm), roots become complex conjugates
Root Locus
Imag
Real
1/Tm
Root Locus
Imag
Closedloop pole locations
K=0
Real
1/Tm
Root Locus
Imag
K = 0.1/Tm
Real
1/Tm 1/2Tm
Root Locus
Imag
K = 0.2/Tm
Real
1/Tm 1/2Tm
Root Locus
Imag
K = 0.25/Tm
Real
1/Tm 1/2Tm
Root Locus
Imag
K = 0.4/Tm
Real
1/Tm
1/2Tm
Root Locus
Imag
K = 0.5/Tm
45O
Real
1/Tm
1/2Tm
Root Locus
Imag
Real
1/Tm
1/2Tm
Compare characteristic equation with standard form
See that
and
as Kp wn
but z
From z, determine KP
Unit Step Response, dependence on proportional gain
Tm = 1 sec
q
Time
Timedomain performance specifications
unit step response
Mp = max % overshoot
q
=/ 2% of qss
t
tr , rise time ts , settling time
Settling time: (usually conservative)
Max % overshoot:
For a second
order system
Mp in standard form,
there is a 1to1
relationship
between Mp and z
z
Root Locus
Imag
In reality, the system will
go unstable as the gain is
increased, due to the presence
of unmodeled highfrequency
poles
Real
1/Tm
possible unmodeled
dynamics
1/2Tm
Pole Placement
Root locus control design is based on the principle of Pole Placement. The
question is, if we can place the closedloop poles anywhere we want, where
should we place them?
for impulsive input, r(t) = d(t), R=1
1
Closedloop time constant corresponds to
Pole Placement
Root locus control design is based on the principle of Pole Placement. The
question is, if we can place the closedloop poles anywhere we want, where
should we place them?
z = sin(f)
Imag Radial lines are lines of constant z
Vertical lines are lines of constant t
wd
Horizontal lines are lines of constant wd
f
Circles are lines of constant wn
wn
Real
zwn = 1/t
Root Locus with Pcontrol
Imag
line of constant z
desired cl
pole locations
1/Tm
Real
line of constant t
1/2Tm
Root Locus with Pcontrol
Imag
See that we can satisfy
line of constant z the z (or Mp ) requirement,
but not the timeconstant
desired cl (or ts) requirement with
pole locations Pcontrol alone
1/Tm
Real
line of constant t
1/2Tm
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