• Controls Lecture 1


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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

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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 (Free-Free Beam) lab at 10 am Friday,
April 16th
–  Extended abstract on the the week 2 (Free-Free 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

•  E-mail 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):
Back-electromotive force (emf)
Torque-current constant
Moment balance on load: JL T
bw
Laplace Transforms
Laplace Transforms
combine algebraically, or…
Motor can be represented in the form of
a block-diagram showing “internal feedback”
KVL
Torque-current relation
Moment summation
Back EMF
R C
G
H
Letting and
or
In many cases, La is very small, allowing reduction to 1st-order 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
Steady-state 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 servo-control, 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 (P-control)
R E M Q

Kp
error actuator
open-loop transfer function:
R C
G
H
Closed-loop transfer function:
Closed-loop poles = roots of denominator of cl-transfer 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
Closed-loop 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
Time-domain 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 1-to-1
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 high-frequency
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 closed-loop poles anywhere we want, where
should we place them?
for impulsive input, r(t) = d(t), R=1
-1
Closed-loop 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 closed-loop 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 P-control
Imag
line of constant z
desired cl
pole locations
-1/Tm
Real
line of constant t
-1/2Tm
Root Locus with P-control
Imag
See that we can satisfy
line of constant z the z (or Mp ) requirement,
but not the time-constant
desired cl (or ts) requirement with
pole locations P-control alone
-1/Tm
Real
line of constant t
-1/2Tm


Use: 0.3703