Dynamic Response Characteristics
of More Complicated Systems
Chapter 6
Inverse Response
K(1 +  3 s )
G(s) =
(1 +  1s )(1 +  2 s)
slope (t = 0)  0
If 3  0 ….fast response
 3  0 ....invers e response
(see Fig. 6.3)
 3 : zero of transfer function
Use nonlinear regression for fitting data
(graphical method not available)
Chapter 6
Chapter 6
K1 2  K 2 1
For inverse response
0
K1  K 2
(6  21)
Chapter 6
K
G( s) 
s ( 1s  1)( 2 s 2  2 s  1)
4 poles (denominator is 4th order polynomial)
More General Transfer Function Models
• Poles and Zeros:
Chapter 6
• The dynamic behavior of a transfer function model can be
characterized by the numerical value of its poles and zeros.
• General Representation of a TF:
There are two equivalent representations:
m
G s 
i
b
s
i
i 0
n
i
a
s
i
i 0
(4-40)
s  zm 

G s 
an  s  p1  s  p2   s  pn 
Chapter 6
bm  s  z1  s  z2 
(6-7)
where {zi} are the “zeros” and {pi} are the “poles”.
• We will assume that there are no “pole-zero” cancellations. That
is, that no pole has the same numerical value as a zero.
• Review: n  m in order to have a physically realizable system.
Time Delays
Time delays occur due to:
1. Fluid flow in a pipe
Chapter 6
2. Transport of solid material (e.g., conveyor belt)
3. Chemical analysis
-
Sampling line delay
-
Time required to do the analysis (e.g., on-line gas
chromatograph)
Mathematical description:
A time delay, θ, between an input u and an output y results in the
following expression:
 0
y t   
u  t  θ 
for t  θ
for t  θ
(6-27)
Chapter 6
G( s)  e
 s
Y ( s)

U ( s)
Chapter 6
H1/Qi has numerator dynamics (see 6-72)
Chapter 6
Approximation of Higher-Order Transfer
Functions
In this section, we present a general approach for
approximating high-order transfer function models with
lower-order models that have similar dynamic and steady-state
characteristics.
In Eq. 6-4 we showed that the transfer function for a time
delay can be expressed as a Taylor series expansion. For small
values of s,
e θ 0 s  1  θ 0 s
(6-57)
Chapter 6
• An alternative first-order approximation consists of the transfer
function,
e
θ 0 s
1

e
θ0 s
1

1  θ0 s
(6-58)
where the time constant has a value of θ0 .
• These expressions can be used to approximate the pole or zero
term in a transfer function.
Skogestad’s “half rule”
Chapter 6
• Skogestad (2002) has proposed an approximation method for
higher-order models that contain multiple time constants.
• He approximates the largest neglected time constant in the
following manner.
One half of its value is added to the existing time delay (if
any) and the other half is added to the smallest retained time
constant.
Time constants that are smaller than the “largest neglected
time constant” are approximated as time delays using (6-58).
Example 6.4
Consider a transfer function:
Chapter 6
G s 
K  0.1s  1
 5s  1 3s  1 0.5s  1
(6-59)
Derive an approximate first-order-plus-time-delay model,
Keθs
G s 
τs  1
using two methods:
(6-60)
(a) The Taylor series expansions of Eqs. 6-57 and 6-58.
(b) Skogestad’s half rule
Compare the normalized responses of G(s) and the approximate
models for a unit step input.
Solution
(a) The dominant time constant (5) is retained. Applying
the approximations in (6-57) and (6-58) gives:
Chapter 6
0.1s  1  e0.1s
(6-61)
and
1
 e3s
3s  1
1
 e0.5 s
0.5s  1
(6-62)
Substitution into (6-59) gives the Taylor series
approximation, GTS  s  :
Ke0.1s e3s e0.5 s Ke3.6 s
GTS  s  

5s  1
5s  1
(6-63)
Chapter 6
(b) To use Skogestad’s method, we note that the largest neglected
time constant in (6-59) has a value of three.
• According to his “half rule”, half of this value is added to the
next largest time constant to generate a new time constant
τ  5  0.5(3)  6.5.
• The other half provides a new time delay of 0.5(3) = 1.5.
• The approximation of the RHP zero in (6-61) provides an
additional time delay of 0.1.
• Approximating the smallest time constant of 0.5 in (6-59) by
(6-58) produces an additional time delay of 0.5.
• Thus the total time delay in (6-60) is,
θ  1.5  0.1  0.5  2.1
and G(s) can be approximated as:
Chapter 6
Ke2.1s
GSk  s  
6.5s  1
(6-64)
The normalized step responses for G(s) and the two approximate
models are shown in Fig. 6.10. Skogestad’s method provides
better agreement with the actual response.
Figure 6.10
Comparison of the
actual and
approximate models
for Example 6.4.
Multivariable Processes
many examples:
distillation columns,
FCC,
boilers,
etc.
Chapter 6
Consider stirred tank with level controller
2 disturbances (Ti, wi)
2 control valves (A, B) manipulate ws, wo
2 measurements (T0, h)
controlled variables (T0, h)
change in w0 affects T0 and h
change in ws only affects T0
Three non-zero transfer functions
G11 
T0 (s)
Ws (s)
H(s)
Ws (s)
Chapter 6
G 21 
G12 
T0 (s)
W0 (s)
G 22 
H(s)
W0 (s)
Transfer Function Matrix
T0 (s) G11 G12   Ws (s) 



 H(s)   G
W
(
s
)
G

  21
0
22  

From material and energy balances,
G11 
K11
 1s  1
G22 
K 22
 2s 1
G12 
K12
( 1s  1)( 2 s  1)
Chapter 6
Chapter 6
Chapter 6
Normal method, but interactions may present
tuning problems.
In multivariable control, interactions are treated,
but controller design is more complicated.
Chapter 6
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