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Higher Order G(s) and Time Delay
CHE 361(LHO_05)
Higher Order Transfer Function Models
Linear dynamic models of chemical processes often represent many simple (1st & 2nd order)
transfer functions in series.
MultiTanks = 1,2,3,4,5,6 Equal Sized Tanks in Series
1.2
1
T'out
0.8
0.6
0.4
0.2
0
0
10
20
30
Time , [min]
40
50
MultiTanks = 1,2,3,4,6 Tanks with same Total Volume
1.2
1
T'out
0.8
0.6
0.4
0.2
0
0
10
20
30
Time , [min]
40
For these higher order G(s) with larger number of poles, initial values of higher order time
derivatives are equal to zero and you observe “S”-shaped curves. Often high order systems are
approximated by lower-order systems with time delay (see Sec. 6.3.1 Skogestad’s “Half Rule” in
SEMD3).
Skog_1_2_Gfirst.slx
1st/2nd order plus time delay
FOPTD/SOPTD approximations
of 4th order transfer function
1
1
1
1
8s+1
s+1
0.2s+1
0.05s+1
GA1
GA2
GA3
GA4
Tin Step
1
Mux
8.5s+1
Time Delay1
GB1
1
1
8s+1
1.1s+1
GC1
GC2
Skog_1_2
Time Delay2
Mux
50
Page 2 of 4
Skogestad Approx. of 4th order by FOPTD or SOPTD
1
Skogestad Approx. of 4th order by FOPTD or SOPTD
0.15
0.8
G4(s)
FOPTD
SOPTD
0.1
T'out
T'out
0.6
G4(s)
FOPTD
SOPTD
0.4
0.05
0.2
0
0
10
20
Time , [min]
30
40
0
0
0.5
1
Time , [min]
1.5
2
Two additional examples of Skogestad’s method are shown in Examples 6.4 and 6.5 in SEMD3.
In Example 6.4, Skogestad’s method is better than using a Taylor Series.
G6.4 ( s ) =
K ( −0.1s + 1)
( 5s + 1)( 3s + 1)( 0.5s + 1)
In Example 6.5 two models using Skogestad’s method are compared: a FOPTD model and a
SOPTD model. The SOPTD model is a better approximation, as expected.
G6.5 ( s ) =
K (1 − s ) e − s
(12s + 1)( 3s + 1)( 0.2s + 1)( 0.05s + 1)
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True or Pure Time Delay as Part of Transfer Function Model
“Pure” time delay can arise from delays in fluid in a pipe as discussed on pages 96-97 SEMD3.
G ( s ) = e −θ s Here θ is the time delay.
Time delays in process model cause problems with feedback control loops because the effect of the
manipulated variable changes are not seen in the output immediately. This can cause unstable
control loops and will be studied further in CHE 461. For analysis purposes, it is often useful to
approximate the e−θ s factor as a ratio of polynomials, the Pade family of approximations. Here we
use first and second order versions.
e −θ s ≈
1−
θs
2 first order Pade'
θs
1+
2
and
≈
1−
θ s θ 2s2
+
2
12 second order Pade'
θ s θ 2s2
1+ +
2
12
Time Delay: Pure, 1st, 2nd Pade Approximations
1
0.8
T'out
0.6
0.4
Pure
Pade-1
Pade-2
0.2
0
-0.2
0
10
20
Time , [min]
30
40
Time Delay: Pure, 1st, 2nd Pade Approximations
0.4
0.3
Pure
Pade-1
Pade-2
Compare Simulink results to Figure 6.7b
on right from SEMD3 page 98.
T'out
0.2
0.1
0
-0.1
0
1
2
3
Time , [min]
4
5
Page 4 of 4
Speed-up Effect of LHP Zero (simplest = overdamped 2nd-order Step Response)
So … a LHP zero “speeds-up the response” of a process (Fig. 6.3 SEMD3) -but Why? and How?
Simple example: poles at -0.2 and -1 with a zero at -0.5, thus
τ 1 = 5, τ z = 2, τ 2 = 1 then for a gain of 1:
K (τ s + 1)
2s + 1
2s + 1
= 2
= 2 2 z
(5s + 1)(1s + 1) 5s + 6s + 1 τ s + 2τζ s + 1
1
2s
Gc ( s ) = Ga ( s ) + Gb ( s ) = 2
+ 2
5s + 6 s + 1 5s + 6 s + 1
Gc ( s) =
"The presence of a zero changes the shape (coefficients) but not the speed (exponentials)."
−t
−t
⎛ τ ⎞
⎛ τ ⎞
y0/2 (t )
= 1 − ⎜ 1 ⎟ e τ1 + ⎜ 2 ⎟ eτ 2 for simple second-order overdamped response
KM
⎝ τ1 − τ 2 ⎠
⎝ τ1 − τ 2 ⎠
−t
5
−t
1
= 1 − 1.25e + 0.25e = 1 − 0.4598 + 0.0017 = 0.5419 at t = 5
⎡ ⎛ y0/2
⎢ d ⎜ KM
and 2 ⎢ ⎝
⎢ dt
⎢⎣
⎞⎤
−t
−t
⎟⎥
⎠ ⎥ = 2 ⎡0.25e 5 − 0.25e 1 ⎤ = 0.5(0.36788) − 0.5(0.00674) = 0.1805 at t = 5
⎢
⎥
⎥
⎣
⎦
⎥⎦
−t
⎛ τ −τ ⎞
⎛ τ −τ
y1/2 (t )
= 1 − ⎜ 1 z ⎟ e τ1 + ⎜ 2 z
KM
⎝ τ1 − τ 2 ⎠
⎝ τ1 − τ 2
−t
5
−t
⎞ τ2
⎟ e for 1st/2nd overdamped step response
⎠
−t
τ2
= 1 − 0.75e − 0.25e = 1 − 0.2759 − 0.0017 = 0.7224 at t = 5 which is = the 0.5419 + 0.1805 from above.
SpeedUpEffectOfZero.slx
Gc = Ga + Gb
2s+1
5s2 +6s+1
1/2_Gc(s)
1
5s2 +6s+1
Mux
PC_graphY
0/2_Ga(s)
2s
5s2 +6s+1
2xDerivOf_0/2_Gb(s)
Mux
Note: Initial slope of 1st/2nd model
step response is 0.4, not 0, since
it is “net” first-order.
Dash 0/2 order + Dotted deriv. = Solid 1/2 order
U_step
Speed-Up Effect of Zero on 2nd-order Overdamped Response
1
0.9
0.8
0.7
0.6
2s + 1
5s 2 + 6 s + 1
GG(s)
= (2s+1)/(5s2+6s+1)
c (s) =
c
0.5
0.4
Gc = 1st/2nd
Ga = 0th/2nd
Gb = 2 x time derivative
of Ga step response
0.3
0.2
0.1
0
0
5
10
Time
15
20