Controls Lecture 2

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ME4053
Controls Lecture 2
Control of a DC Motor
Dr. Ferri
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2nd-order DC Motor Model
where
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Proportional Control (P-control)
Q
R
E
M
Kp
error
actuator
with P-control, m(t) Kp e(t)
open-loop transfer function
closed-loop transfer function
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Closed-loop poles roots of denominator of
cl-transfer function
Compare characteristic equation with standard form
See that
and
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Root Locus
Imag
Real
-1/Tm
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Root Locus
Imag
Closed-loop pole locations
K 0
Real
-1/Tm
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Root Locus
Imag
K 0.1/Tm
Real
-1/2Tm
-1/Tm
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Root Locus
Imag
K 0.2/Tm
Real
-1/2Tm
-1/Tm
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Root Locus
Imag
K 0.25/Tm
Real
-1/2Tm
-1/Tm
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Root Locus
Imag
K 0.4/Tm
Real
-1/Tm
-1/2Tm
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Root Locus
Imag
K 0.5/Tm
45O
Real
-1/Tm
-1/2Tm
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Root Locus
Imag
Real
-1/Tm
-1/2Tm
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Unit Step Response, dependence on proportional
gain
Tm 1 sec
q
Time
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Time-domain performance specifications
unit step response
Mp max overshoot
q
/- 2 of qss
t
ts , settling time
tr , rise time
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Settling time
(usually conservative)
Max overshoot
Mp
z
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Performance Specifications
5 max percent overshoot, Mp 0.05
But, Mp depends only on z
Mp
z
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ts 4t
2nd performance specification ts 2.5 sec
What is the time constant of a second-order
underdamped system?
for impulsive input, r(t) d(t), R1
-1
Time constant corresponds to
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Specifying z and ts (or z and t) completely
determines the cl poles
Imag
see that specifying z and t locates cl pole
wd
f
z sin(f)
wn
Real
-zwn -1/t
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Root Locus with P-control
Imag
line of constant z
desired cl pole locations
-1/Tm
Real
line of constant t
-1/2Tm
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Root Locus with P-control
Imag
See that we can satisfy the z (or Mp )
requirement, but not the time-constant (or ts)
requirement with P-control alone
line of constant z
desired cl pole locations
-1/Tm
Real
line of constant t
-1/2Tm
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Proportional plus Derivative Control (PD-control)
Q
R
E
M
error
actuator
with PD-control,
.
high e is corrected with Kd
e
t
high e is corrected with Kp
PD anticipates large future error
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Open-loop transfer function
Closed-loop transfer function
See that
and
PD design procedure
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PD Root Locus
Zero at
OL TF
Imag
zd
Desired pole location
-1/td
Real
z
-1/Tm
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PD Root Locus
Zero at
OL TF
Imag
zd
Desired pole location
-1/td
Real
z
-1/Tm
Move zero left or right to make the root
locus pass through the desired cl pole location
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PD Root Locus
Zero at
OL TF
Imag
zd
-1/td
Real
z
-1/Tm
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See that with PD-Control, we can attain both the
desired z and the desired ts, but the relation
between z and Mp is now different. Why?
Closed-loop transfer function
This is NOT the standard 2nd-order form because
of the zero in the numerator
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Actual percent-overshoot vs z for a PD-controller
Note that, even though the Kp and Kd gains put
the cl-poles in the correct location, the
overshoot is higher, and gets worse as the
damping ratio gets larger.
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Implementation issues of PD-control
Pure derivative is not a good idea because of
noise. Instead of
use
Approximates a pure derivative at low frequencies
(below p rad/s) then levels off so that
high-frequency noise is not amplified
20db/dec
90O
Phase
Mag
0O
Log frequency
Log frequency
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Effect of p
u
Objective is for y to be the derivative of u
y
.
.
What if u is a noisy sinewave? u sin(10t)n,
u 10cos(10t)n
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With p 5, the output looks sinusoidal, but the
magnitude is way off
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With p 25, the output looks like a reasonable
approx of the derivative of u
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With p 250, the noise has been intensified too
much
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The problem is more difficult when the signal
that you want to measure has several
frequencies, like
u sin(10t) 0.5cos(20t) - 0.5sin(30t)
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Now, corrupt the signal with some noise
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PD-Control Step Response
See that as p increases, the overshoot goes
down, meaning that the controller looks more
like a true PD controller
However, as p increases, cl-system is more
sensitive to noise
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Effect of Derivative Modification
Effective controller is
This is the form of a lead-compensator. In
general, when people say they are using
PD-control, they are really using
Lead-Compensation.
Design option place the lead-compensator zero
right On top of the plant pole put the new pole
whereever needed to get the desired root-locus
controller pole/zero
original plant poles
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desired cl-pole locations
One advantage of this design option is that the
cl-transfer function no longer has a zero. Thus,
the attained overshoot is exactly the one
predicted by the formula
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Proportional plus Derivative plus Integral
Control (PID-control)
Q
R
E
M
error
actuator
with PID-control,
Low but nonzero steady-state error
e
t
As time grows, integral of error increases,
causing m(t) to grow
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Open-loop transfer function
Closed-loop transfer function
Compare with the following third-order form
Choose s, then equate coefficients of powers of s
in the denominators of each expression for Gcl
to get 3 equations for Kp, Kd, and Ki
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How do we choose s?
Imag
Desired cl pole locations
-zwn
Real
-s
s nzwn
Choose n to be around 5 so that system appears to
have 2nd-order characteristics
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PID Gains
n your choice
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Experimental Setup
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Linear Simulink Model
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Nonlinear Simulink Model
X
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Dead Zone
Saturation
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Dead Zone
Positive when input gt VH
1 or -1
Negative when input lt VL
VH
VL
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MATLAB Dead Zone
Experimental Dead Zone
out
out
Lower Limit
Lower Limit
in
in
Upper Limit
Upper Limit
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Alternative Nonlinear Simulink Model
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If your Simulink model runs
very slowly. . .

• Try using the alternative dead zone model
• Try using a different ODE solver
• ODE45
• ODE23
• ODE113
• If all else fails Make a small change in your
  gains

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Guidance for Controls 2 Reports

• Discuss the results of the qualitative
  investigation of the P, I and D controllers
• Compare the linear and nonlinear SIMULINK models
  with experimental results
• When you compare with a theoretical result you
  must show where the theoretical result comes from
  i.e . the equation or the model.
• Data must support your conclusions
• Make your own sketch of the experiment