Pressure Control System:Root Locus Plotting
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Outline
Pressure System Background Previous Work: Transfer Function Root Locus Theory Modeling Results Conclusions
Background: Schematic Diagram
Background: Block Diagram
Manipulated Variable
M System C
Controlled Variable
% Motor Speed Pressure in cm-H2O
Background: SSOC
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 10 20 30 40 50 60 70 80 90 100
% Motor
Operating Range: 30 – 70%
Ou
tpu
t: 1.3 –3.8 cm
H2O
Background: FOPDT Parameters
K = 0.061 cm of H2O / %
to = 0.25 sec = 0.4 sec
Feedback Control
Root Locus Theory
Where is the damping ratio.
Decay Ratio =
Effects of
Overdamped = monotonic and stable
Underdamped = oscillatory and stable
Undamped = sustained oscillations
Unstable = growing oscillations
Run-away = monotonic and unstable
Four Possible Responses
Stable, Monotonic
Unstable, Oscillatory Unstable, Monotonic
Stable, Oscillatory
Root Locus Parameters
KCD = highest value of KC that provides monotonic and stable output (critically damped)
KQD = value of KC that produces one-
quarter oscillating decay KCU = highest value of KC that causes
oscillatory and stable output (marginally stable)
Root Locus Parameters
KCD is found where the roots go from real to complex
KQD is found where the ratio of the imaginary root to the real root is 4.5
KCU is found where the real portion of the root goes from negative to positive.
Parameter Locations
ROOT LOCUS PLOT
-10
-8
-6
-4
-2
0
2
4
6
8
10
-6 -5 -4 -3 -2 -1 0 1
REAL AXIS
IMA
GIN
AR
Y A
XIS
KCD
KCUKQD
Dr Henry’s Suggestion:Put the VALUES of the Kc’sfor the different choices you have on this graph
Locations ROOT LOCUS PLOT
-10
-8
-6
-4
-2
0
2
4
6
8
10
-6 -5 -4 -3 -2 -1 0 1
REAL AXIS
IMA
GIN
AR
Y A
XIS
DR=0
DR=1/500
DR=1/10
DR=1/4
DR=1/1
Results from Root Locus
Ultimate Kcu= 37
Quarter Decay Kc = 24.
Critically Damping Kc = 0.1
Underdamped 0.1< Kc < 37
Overdamped 0 < Kc < 0.1
*all units are % / cm H2O
Conclusions
For Kc needed
Overdamped 0 < Kc < 0.1
Critically Damped Kc = 0.1
Underdamped 0.1< Kc < 36.9
Quarter Decay Kc = 24.2
*all units are % / cm H2O
Questions?