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Last day:
(1) Identify individual entries in a Control Loop Diagram
(2) Sketch Bode Plots by hand (when we could have used a computer
program to generate sketches).
How might this be useful?
• Can more clearly see how individual terms affect system performance and system output.
• Easier to understand why different changes in the control loop might be used compensate for specific shortcomings and specific terms in the system.
1
• Defining stability
• Interpreting stability from the magnitude and phase plots
2
Stability
Stability - definition
“A stable system is a system which generates a bounded output when given a bounded input.”
• E.g. a sine wave input results in a sine wave output that doesn’t grow unbounded until the system finally “breaks”.
• Resonance is still acceptable (the output can become very large, as long as it does not become unbounded)
• In analyzing systems, we are MOST concerned about stability in Feedback Loops. Let's have a closer look at how a Feedback Loop might become unstable by looking at a Feedback Loop Diagram.
3
A closer look at Feedback Loop diagram for simplification:
Calculate the output
Value using Algebra:
Y = G * Error
= G * (X-FY)
= GX - GFY
Y * (1+GF)=G X
Y/X= G/(1+GF)
GF
G
X
YH
1
G(s)
F(s)
+
-
X(s) Y(s)
Motor and amplifier behaviour
Sensor behaviour
Error
Where:
G= forward transfer function
GF = loop transfer function
This is the transfer function for a Feedback Loop!
4
Aside – general ru les for S imp lifying System B lock Diagrams
• You can often derive the overall transfer function of a system using just the System Block Diagrams
• The final result may have just a single block containing all the original control, process and feedback blocks
• Simplify the diagram by combining functions together at the different nodes.
5
Image from:http://www.indianshout.com/control-systems-study-material-block-diagram-reduction/2152
Aside: Rules for simplifying block diagrams:
This is a feedback loop
simplification. We just
solved for this
relationship on Slide 4!
6
This is the ultimate simplification of a feedback system, into a
single step
X(s) Y(s)
)()(1
)(
sFsG
sG
7
G(s)
F(s)
+
-
X(s) Y(s)
Motor and amplifier behaviour
Sensor behaviour
Error
For feedback loops, putting the feedback loop into this form is
useful to identify something important about stability of the
system.
X(s) Y(s)
)()(1
)(
sFsG
sG
GF
G
X
YH
1
overall transfer function
Major Question: What happens when GF gets to be a
value of -1?
● H gets bigger and bigger, becoming an unstable
system.
• GF can become -1 depending on the gain and
phase of the system. This is the whole point of
stability analysis, to determine the gain and phase
when the system becomes unstable..8
Stability analysis usually assumes that the feedback element F(s) =1
(unity gain) , in order to simplify the analysis.
• Most of the time, stab ility ana lysis in contro l theory assumes that the feedback ga in is a constant value of 1. This simplifies analysis to just looking at the true open-loop gain G(s):
X(s) Y(s)
)(1
)(
sG
sG
G
G
X
YH
1
overall transfer function
X(s) G(s)
1
+-
Y(s)
Therefore, H becomes unstable if G ever
becomes a value of -1
G is called the open-loop gain since G is the
gain in the system if the feedback loop is
removed.9
Stability comes from avoiding G(s)=-1
• G(s) = -1 when magnitude plot = 1 = 0 dB, and phase plot* = -180 degrees
• A stable system is one where the open-loop gain is less than 1 when the open-loop phase
angle is -180 degrees
• An unstable system is one where the open-loop gain is greater than 1 at the same time that the
open-loop phase is -180 degrees (since this guarantees that at some point, the value G(s)
would be “equal” to -1**)
Aside:
( * the word “phase” is from the fact that “s”, H(s), and G(s) are all really complex number (with real
and imaginary components), which can be represented with a magnitude and phase value )
( **actually, this is only the “real” part of G(s) which equals -1, since G(s) is a complex # with real
and imaginary parts, but really that doesn’t matter in this analysis, it still guarantees that the
system will become unstable at some point )
10
For clarification: Why analyze the open-loop response instead of the
closed-loop response directly?
X(s) Y(s)
)(1
)(
sG
sG
Why not analyze this entire expression
H(s) instead of just analyzing G(s)?
Because G(s) remains bounded and stable even as the entire expression H(s)
becomes unstable. That is, the open-loop system can remain stable, but as soon as
it is used in a feedback loop there is a chance that the entire system will inherently
become unstable!
We can use Bode plots and straight-line asymptotes to examine G(s) even when the
system becomes unstable, but we don’t have any tools to accurately analyze H(s) as
the entire expression becomes unbounded and unstable.
H(s)
You check stability by looking at the gain and phase margin of the open-loop system
12
Plot from lecture notes, 8B, pg. 12
• Gain Margin- the amount of gain that you could add to the system before reaching a gain of 1 = 0 dB. Measured at the phase
crossing frequency (phase = +/-1800)
• Phase Margin- the amount of phase you could add/subtract to the system before
reaching +/-1800. Measured at the gain crossing frequency (gain = 1 = 0dB)
wgain_cross= gain crossing (frequency at which gain =
0dB, and where the phase margin is measured)
wphase_cross= phase crossing (frequency at which
phase = +/- 180deg, and where gain margin is measured)
wgain_cross w phase_cross
Magnitude plot
Phase plot
-180
0
0db
What can you do with the gain and phase margin?
• Te lls you how much proportiona l gain (k) you can add to the system (to make the system operate faster) without becoming unstable.
• Tells you how you might select and design a controller to compensate for aspects of the system (e.g. PID) to increase stability or improve performance
13
14
Changing the Performance by using
Controllers/Compensators
• Controllers alter the Bode Plots of the overall system
• PID controls revisited
• Improving performance with Controllers / Compensators
Some Terminology
• Controllers/Compensators
– An operation introduced into a control loop to make it operate in a desired way (rise time, fall time, stability, etc).
• Bandwidth
– The range of frequencies over which a system “operates”.
– Sometimes defined as the range of frequencies over which the magnitudes exceeds some arbitrary threshold (e.g. gain of 1, magnitude of -40dB, magnitude of 0.05, etc).
– In control theory, often defined as a drop of -3dB drop from the maximum value (a drop of 1/sqrt(2)).
15
16Images from: http://rf-filter-circuits.blogspot.com/
Bode plot of pole (aka “lowpassfilter”)
(bandwidth = FC– 0 = FC)
BandpassFilter (bandwidth = FH– FL)
BandpassFilter with narrow band
(bandwidth = FH– FL)
Aside: Why is -3 dB drop used?
-3dB = 20 log(0.7071...) = 20 log (1 /sqrt(2))
A -3 dB drop is equal to a drop of 1/sqrt(2) in the
magnitude of the signal. This means thesquareof the
amplitude drops by a factor of 2 at the -3dB point. This is
also called the half-power point (“power” is usually the
magnitude squared)... Don’t worry, not mentioned again in
this course)
FC= corner frequency, FH= high cutoff frequency, FL= low cutoff frequency
Bode Plots – bandwidth measurements
Controllers/compensators change the Bode Plot (and therefore the
performance) of the system
17
10
0
-10
-20
dB
(Compensated System ) = (Controller )* (Uncompensated)
The bode plot for the final Compensated System is the same as putting the
Uncompensated and Controller bode plots in cascade series (and then combining multiple
bode plots as we have done in previous examples).
Controller
PID Controllers Revisited
18
In the in-class demo (arm moving back and
forth when connected to a potentiometer),
turning one of the three tuning knobs
changed the constant values Kp, Ki, and Kd
(i.e. We were changing the constants for
proportional, derivative and integral gain in
the transfer function)
sKs
KKsC D
IP )(
PID Controllers
• Looks like this has 2 zeros (z1 and z2), proportional gain, and one integrator (s in denominator)
• Can use this controller to generate phase compensation as well as a higher operating frequency
bandwidth
Proportional Gain– increases speed of system
Derivative Gain- Acts like the damping /friction term ;
Used to get reduce overshoot.
Integral Gain- reduces steady-state error (error
value accumulates over time, changes inputs)
s
zszsk
s
KsKsKsK
s
KKsC IPD
DI
P
))(()( 21
2
19
PID Controller - Bode plots
Image from: http://courses.ece.ubc.ca/360/lectures/Lecture24.pdf
Amplitude boost at low frequency –
reduces steady-state error
Phase margin increase at high
frequencies, higher bandwidth
Higher gain at high
frequencies might lead to
problems.
20
s
zszsk
s
KsKsKsK
s
KKsC IPD
DI
P
))(()( 21
2
Use PID or other types of contro llers to get a “more desirable” output
1. How to get faster response?
– Increase proportional gain
– Increase the system bandwidth (frequency response)
2. How to reduce overshoot?
– Increase damping (e.g. Increase derivative gain)
3. How to eliminate steady-state error
– Increase integral gain
21
1. How to get faster response?In general:
wider bandwidth = faster response
higher gain = faster response
22
System T4 has a higher bandwidth than
System T3 on the magnitude Bode Plot
(frequency domain)
Figure: Response of two second-order systems
(fromDorf,Beiser)
Therefore, System T4 has a faster rise
time than System T3 (time domain).
May or may not result in higher
overshoot, oscillations.
frequency
time
Time Response to a step-input of a system
with increasing proportional gain Kp.
Note that as the gain increases:
•rise time is faster
•overshoot and oscillations are higher
•steady-state error is reduced (gets closer to
desired final value of 1)
All of these were seen in the PID demo done
in-class as well -
http://blog.analogmachine.org/2012/02/04/pid-control-demonstra
tion/
An increase in the proportional gain shifts the
Bode Plot vertically upwards (this may also
increase system bandwidth as well)23
(fromDorf,Beiser)
1. How to get faster response?In general:
wider bandwidth = faster response
higher gain = faster response
2. How to reduce overshoot
Increase damping / increase derivative gain
24
Image from: http://www.industrialheating.com/Articles/Column/bdcffcf0ddbb7010VgnVCM100000f932a8c0____
Increasing KD
Time Response to a step-input of a system
with increasing derivative gain KD.
Note that as KDincreases, system
overshoot and oscillation decreases and
disappears; higher KDvalues can severely
reduce response time (a very slow-reacting
system).
3. How to reduce steady-state error
Increase integral gain
25
Example from: http://www.stanford.edu/~boyd/ee102/int-ctrl.pdf
If KIis chosen correctly, the steady-state
error reduces to zero in a reasonable
time.
If KItoo small, results in slow asymptotic
approach to zero (may take a long time)
If KIis too large, oscillatory response, or
even instability may result.
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