Sos, Fopz, Sopz
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Transcript of Sos, Fopz, Sopz
SOS โ General Form
โข General second order system
Standard form
SRSaunders - WSU - ChE 441 141
๐2
๐2๐ฆ(๐ก)
๐๐ก2+ ๐1
๐๐ฆ(๐ก)
๐๐ก+ ๐๐๐ฆ(๐ก) = ๐๐ข(๐ก)
๐2๐2๐ฆ(๐ก)
๐๐ก2+ 2๐๐
๐๐ฆ(๐ก)
๐๐ก+ ๐ฆ(๐ก) = ๐พ๐ข(๐ก)
๐ =๐2
๐๐, 2๐๐ =
๐1
๐๐, ๐พ =
๐
๐๐
๐2๐ 2๐ ๐ + 2๐๐๐ ๐ ๐ + ๐ ๐ = ๐พ๐(๐ )
๐ ๐ =๐พ
๐2๐ 2 + 2๐๐๐ + 1๐(๐ )
SOS Parameters
โข Three key parameters
Gain K
Natural period ฯ
Damping coefficient ฮถ (Zeta)
โข Existence and nature of oscillation are
characterized by ฮถ and ฯ
โข Poles?
SRSaunders - WSU - ChE 441 142
๐1,2 =โ2๐๐ ยฑ 4๐2๐2 โ 4๐2
2๐2=
โ๐ ยฑ ๐2 โ 1
๐
Lets look at 3 test cases and their
response to a unit stepโฆ
K 10 10 10
ฯ2 40 42.25 13
2ฯฮถ 25 13 3
SRSaunders - WSU - ChE 441 143
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
time
y(t)
K=10,t2=40, 2tz=25
K=10,t2=42.25, 2tz=13
K=10,t2=40, 2tz=3
Which is :
Underdamped?
Overdamped?
Critically Damped?
SOS Dynamics
Damping Coefficient ฮถ
โข Damping coefficient characterizes qualitative
process response
โข Case 1: 0 < ฮถ < 1
Poles are real or imaginary?
Displays oscillation
Has OVERSHOOT
โข progression past the final value,
โข followed by a return to the
โข steady state
Underdamped
SRSaunders - WSU - ChE 441 144
0 20 40 60 80 1000
5
10
15
time
y(t)
K=10,t2=40, 2tz=3
SOS Dynamics
Damping Coefficient ฮถ
โข Case 2: ฮถ = 1
Poles are real or
imaginary?
Fastest approach to
final value w/o
overshoot
Critically damped
โข Case 3: ฮถ > 1
Poles are real or
imaginary?
Slower response than
case 2
Overdamped system
SRSaunders - WSU - ChE 441 145
0 20 40 60 80 1000
2
4
6
8
10
time
y(t)
Overdamped
Critically Damped
SOS Dynamics
Damping Coefficient ฮถ
โข Case 4: ฮถ = 0
โข Pole are real or imaginary?
โข Oscillatory response
โข with no damping
โข Frequency of oscillation :
โข 1/ฯ => period = ฯ
โข Case 5: ฮถ < 0
โข UNSTABLE
SRSaunders - WSU - ChE 441 146
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
30
time
y(t)
Oscillatory No Damping
Unstable
FOS vs. SOS
โข Step response comparison
SOS (w/ no zeros) always more
sluggish than FOS
SOS has an โSโ shape response
TOS is even more sluggish
System FOS SOS
Final Value AK AK
Initial Value 0 0
Initial
SLOPE
Finite, non-
zero0
SRSaunders - WSU - ChE 441 147
Underdamped SOS
โข Rise time
Time to the first crossing of the final steady state
value
SRSaunders - WSU - ChE 441 148
tr=12
๐ก๐ =๐
๐ฝ๐ โ ๐
๐ฝ = 1 โ ๐2
๐ = tanโ1๐ฝ
๐A=1, K=10, ฯ2=40, 2ฯฮถ=3
Underdamped SOS
โข Period
Time between successive oscillation peaks
SRSaunders - WSU - ChE 441 149
๐ =2๐๐
๐ฝ๐๐๐๐๐๐ ๐๐ ๐๐ ๐๐๐๐๐๐ก๐๐๐
๐ =๐ฝ
๐๐๐๐๐๐ข๐๐๐๐ฆ ๐๐ ๐๐ ๐๐๐๐๐๐ก๐๐๐
A=1, K=10, ฯ2=40, 2ฯฮถ=3
Underdamped SOS
โข Decay Ratio
A measure of the rate of oscillation decay
โข Overshoot
SRSaunders - WSU - ChE 441 150
a1=4.64
a2=1.0๐ท๐ =๐2
๐1= ๐
โ2๐๐๐ฝ
A=1, K=10, ฯ2=40, 2ฯฮถ=3๐๐ = ๐ด๐
โ๐๐
1โ๐2
Underdamped SOS
โข Settling Time โ time at which the output
enters (and remains within) a percentage of
the final value
Often 90%, 95% or 99% settling time
SRSaunders - WSU - ChE 441 151
t90%=49
t95%=69
t99%=123
A=1, K=10, ฯ2=40, 2ฯฮถ=3
FOS - Lead-lag Systems
โข System with a โproperโ transfer function
Gain K
Zero -1/๐๐
Pole -1/ฯ
Lead-to-lag ratio ฯ = ๐๐๐
โข Lead arises from the zero, lag from the pole.
SRSaunders - WSU - ChE 441 152
๐บ ๐ =๐พ(๐๐๐ + 1)
๐๐ + 1
FOS โ Lead-Lag Systems
โข Partial Fraction Expansion
โข General Form
SRSaunders - WSU - ChE 441 153
(๐๐๐ + 1)
๐๐ + 1= ๐ด +
๐ต
๐๐ + 1
๐๐๐ + 1 = ๐ด ๐๐ + 1 + ๐ต
๐ 1: ๐๐๐ = ๐ด๐๐
๐ด =๐๐
๐= ๐
๐ 0: 1 = ๐ด + ๐ต๐ต = 1 โ ๐ด = 1 โ ๐
๐บ ๐ =๐พ(๐๐๐ + 1)
๐๐ + 1= ๐พ๐ +
๐พ 1 โ ๐
๐๐ + 1
Lead-Lag Step Response
โข Observations
For very small t (t->0, i.e., use initial value
theorem) y(0)=KฯA
โข Discontinuous jump in the output signal
โข Effect of the zero
For very large t (t->โ, i.e., use final value
theorem) y(โ)=KA
โข Effect of lag term
Behavior of y(t) is a big function of ฯ
SRSaunders - WSU - ChE 441 154
๐ ๐ =๐พ ๐๐๐ + 1
๐๐ + 1๐ ๐ =
๐พ๐๐ด
๐ +
๐พ 1 โ ๐ ๐ด
๐ (๐๐ + 1)
Lead-Lag Systems โ Effect of ฯ
โข Case 1: 0 < ๐๐ < ฯ (0 < ฯ < 1)
Discontinuous Jump @ moment of
step
๐๐-> 0, becomes more of a โpureโ
FOS and lag dominates
SRSaunders - WSU - ChE 441 155
A=1, K=5, ๐๐=0.5, ฯ=1y(t
)
0
Lead-Lag Systems โ Effect of ฯ
โข Case 2: ๐๐ = ฯ (ฯ = 1)
Pole-zero cancellation
Pure gain system G(s)=K
Discontinuous jump @
moment of step
SRSaunders - WSU - ChE 441 156
A=1, K=5, ๐๐=0.5, ฯ=0.5
y(t
)
0
Lead-Lag Systems โ Effect of ฯ
โข Case 3: ๐๐ > ฯ (ฯ > 1)
Overshoot
Lead dominates
Discontinuous jump @
moment of step
SRSaunders - WSU - ChE 441 157
A=1, K=5, ๐๐=1, ฯ=0.5
y(t
)
0
Lead-Lag Systems โ Effect of ฯ
โข Case 4: ๐๐ < 0 < ฯ (ฯ < 0)
Inverse response
Initial move away from the SS Value
SRSaunders - WSU - ChE 441 158
A=1, K=5, ๐๐=-1, ฯ=0.5
y(t
)
t0
0
FOS In Parallel
๐บ1 ๐ =๐พ1
๐1๐ + 1
๐บ2 ๐ =๐พ2
๐2๐ + 1
๐ ๐ = ๐1 ๐ + ๐2 ๐
= ๐บ1 ๐ ๐ ๐ + ๐บ2 ๐ ๐ ๐
= ๐ ๐ ๐บ1 ๐ + ๐บ2 ๐
= ๐ ๐ ๐บ(๐ )
๐บ ๐ = ๐บ1 ๐ + ๐บ2 ๐ =๐พ1
๐1๐ + 1+
๐พ2
๐2๐ + 1
SOS with
Zeroes!
SRSaunders - WSU - ChE 441 159
=๐พ1(๐2๐ + 1) + ๐พ2(๐1๐ + 1)
(๐1๐ + 1)(๐2๐ + 1)
=(๐พ1 + ๐พ2)
(๐พ1๐2 + ๐พ2๐1)๐พ1 + ๐พ2
๐ + 1
(๐1๐ + 1)(๐2๐ + 1)
=(๐พ1๐2 + ๐พ2๐1)๐ +(๐พ1 +๐พ2)
(๐1๐ + 1)(๐2๐ + 1)=
๐พ1๐2๐ + ๐พ1 + ๐พ2๐1๐ + ๐พ2
(๐1๐ + 1)(๐2๐ + 1)
G2
G1
U(s) Y(s)+
+
SOS with Zeroes
โข 2-pole, 1 zero system:
โข Output step response:
SRSaunders - WSU - ChE 441 160
๐บ ๐ =๐พ ๐๐๐ + 1
๐1๐ + 1 ๐2๐ + 1
๐ ๐ =๐พ ๐๐๐ + 1
๐1๐ + 1 ๐2๐ + 1
๐ด
๐= ๐พ๐ด
๐ต
๐ +
๐ถ
๐1๐ + 1+
๐ท
๐2๐ + 1
๐ต = 1
๐ถ = โ๐1 ๐1 โ ๐๐
๐1 โ ๐2
๐ท =๐2 ๐2 โ ๐๐
๐1 โ ๐2
๐ฆ ๐ก = ๐ด๐พ 1 โ๐1 โ ๐๐
๐1 โ ๐2๐
โ๐ก๐1 +
๐1 โ ๐๐
๐1 โ ๐2๐
โ๐ก๐2
SOS with Zeroes โ Step Response
A=1, K=10, ๐1=5, ๐2=10
SRSaunders - WSU - ChE 441 161
>
SOS โ Case Evaluations
โข Let ฯ1
< ฯ2, ฯ
a> 0
Unless otherwise noted
โข Case 1: ฯa
> ฯ2
Overshoot
โข Case 2: ฯa
= ฯ1
or ฯa
= ฯ2
Pole-zero cancellation
Yields a FOS
SRSaunders - WSU - ChE 441 162
๐บ ๐ =๐พ ๐๐๐ + 1
๐1๐ + 1 ๐2๐ + 1=
๐พ
๐๐๐ + 1
>
SOS โ Case Evaluations
โข Case 3: 0 < ฯa
< ฯ2
Resembles a FOS until ฯa
<< ฯ1
โข Case 4: ฯa
< 0
Always displays inverse response
SRSaunders - WSU - ChE 441 163
Case Summary โ FOS ฯa
Values Key Observations
0 < ฯa < ฯJump at t=0 toward
y(โ)
ฯa = ฯ
Pure gain system
(pole-zero
cancellation)
ฯa > ฯ Overshoot
ฯa < 0 < ฯ Inverse response
SRSaunders - WSU - ChE 441 164
โข Discontinuous jump @ t=0 for all cases
โข Let ฯ1
< ฯ2, ฯ
a> 0
โข Unless otherwise noted
โข No discontinuous jump @ t=0
Case Summary โ SOS ฯa
Values Key Observations
0 < ฯa < ฯ2 Similar to FOS
ฯa = ฯ1 or ฯ2
FOS
(pole-zero cancellation)
ฯa > ฯ2 Overshoot
ฯa < 0 Inverse response
SRSaunders - WSU - ChE 441 165
Case Summary โ SOS ฮถ
Values Key Observations
ฮถ < 0 Unstable
ฮถ = 0Underdamped
oscillates forever
0 < ฮถ < 1Overshoot and
underdamped
ฮถ = 1 Critically damped
ฮถ > 1Overdamped โ
sluggish
SRSaunders - WSU - ChE 441 166
Inverse Respone
โข When a process output initially moves in a
direction opposite to its steady state value
followed by a return to steady state
Effect of (at least) two opposing processes are
different timescales
Occurs when ฯa
< 0 in single-zero systems
y(t) crosses the zero axis (in deviation variables)
in response to a step input
โข Where does this show up?
SRSaunders - WSU - ChE 441 167
A Two Timescale Exercise
โข Given the following block diagram and
transfer functions, calculate G(s)
SRSaunders - WSU - ChE 441 168
๐บ1 ๐ =5
10๐ + 1๐บ2 ๐ =
โ1
1๐ + 1
๐บ ๐ = ๐บ1 ๐ + ๐บ2 ๐
=5
10๐ + 1โ
1
๐ + 1
=5 ๐ + 1 โ (10๐ + 1)
(10๐ + 1)(๐ + 1)
=4 โ
54
๐ + 1
10๐ 2 + 11๐ + 1
G2
G1
U(s) Y(s)+
+
FOS in Parallel
โข Two FOS in Parallel
โข Let:
|K1|> |K
2|
K1
and K2
be opposite signs
ฯ1
> ฯ2
(G2
is faster than G1)
โข Consequences
Fast process => Initial response
Slow Process => final response (due to higher
gain)
SRSaunders - WSU - ChE 441 169
๐บ(๐ ) =(๐พ1 + ๐พ2)
(๐พ1๐2 + ๐พ2๐1)๐ ๐พ1 + ๐พ2
๐ + 1
(๐1๐ + 1)(๐2๐ + 1)
Real Inverse Response
โข Drum Boiler
Used for steam generation
SRSaunders - WSU - ChE 441 170
Heat
Source
Steam
Cold Water