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Unformatted text preview: Chapter 6 Dynamic Response Characteristics
of More Complicated Systems G (s) = Inverse Responses
Differences caused
by different zero in
G(s) = Fig. 6.3
G (s) K ( a s 1)
( 1s 1)( 2 s 1) slope If a 0 response faster than if = 0 a 0 .... " inverse " response K ( a s 1)
( 1s 1)( 2 s 1) K ( a s 1)
(4 s 1)(1s 1) (t =0) 0
(see Fig. 6.3) a : the “zero” of the transfer function is at –1/a
Use nonlinear regression for fitting step response data
(graphical method not available) – “step2g” program at OSU Fig. 6.3 Step response of an overdamped secondorder system
for different values of a G(s) has 4 poles “in the complex s plane”
(denominator is 4th order polynomial) Fig. 6.4 Two firstorder process elements acting in parallel.
(can lead to a 1st/2nd order transfer function.
Y ( s)
K1
K2
K ( s 1) K 2 ( 1s 1) G( s) 12 1s 1 2 s 1
( 1s 1)( 2 s 1)
U ( s) K K s K1 K 2 = 12 221 1 2 s 1 2 s 1 For inverse response K1 2 K 2 1
0
K1 K 2 (6 21) G ( s) K
s ( 1s 1)( s 2 s 1)
22 More General Transfer Function Models In the second, we easily see the values of the “zeros” and • Poles and Zeros: the “poles”. • 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: G s bm s z1 s z2 s zm an s p1 s p2 s pn (67) where {zi} are the “zeros” and {pi} are the “poles”. There are two common equivalent representations.
In the first, the coefficient (an a or a b) on the lowest order of s =
1 in numerator and denominator. • We will assume that there are no “polezero” cancellations. That
is, that no pole has the same numerical value as a zero. If there
is such an equivalence, (67) shows they “cancel” algebraically. m G s K bi s i
i 0
n (440) ai si
i 0 • Review: in order to have a physically realizable system.
n m Ususally there are more poles than zeros, but never fewer ! What are the “poles and zeroes” of a process?
The poles and zeros of the transfer
function model = One output / one input
What physically is a “zero” ?
A modification of simple dynamics =
a change in the “shape”, but not
the “speed” of response.
Where do zeroes “come from” ?
Process interactions.
Why or How ? Let’s look at an example. Fig. 6.13 Two tanks in series whose liquid levels interact.
H1 / Qi has “numerator” dynamics (a zero) (see 672)
See CHE 361 Interacting Tanks Handout for more details
on derivation of transfer functions with and without a zero. Time Delays
Time delays occur due to:
1. Fluid flow in a pipe
2. Transport of solid material (e.g., conveyor belt) Fig. 6.6 The effect of
a time delay is
a translation of the
function in time. 3. Chemical analysis
 Sampling line delay  Time required to do the analysis (e.g., online 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 (637) pg.98 G( s) e s Y (s) X ( s) (627) Approximation of HigherOrder Transfer Functions
In this section, we present a general approach for
approximating highorder transfer function models with
lowerorder models that have similar dynamic and steadystate
characteristics. Why? Because methods have been developed
to “design” industrial PID controllers for lowerorder models
with time delay.
In Eq. 64 we showed that the transfer function for a time
delay can be expressed as a Taylor series expansion. For small
values of s,
eθs 1 θs a single RHP zero (657) 1 s 2s2 2
12 = 2nd order Pade' approx. s 2s2
1 2
12 • An alternative firstorder approximation consists of the transfer
function,
e θs 1
e θs 1 a single LHP pole
1 θs (658) where the time constant has a value of
θ.
• These simple expressions can be used to approximate time
delay as a single pole or a single zero (and visa versa) in a
transfer function. Pade’ (rational) approximations use equal
number of poles and zeros, e.g. two poles and two zeros for a
secondorder Pade’ approximation, which can be much more
accurate. Example 6.4 Skogestad’s “half rule” Consider a transfer function, 1 RHP zero, 3 LHP poles: • Skogestad (2002) has proposed an approximation method for
higherorder 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 (658),
just add them to any pure time delay to get , the overall time
delay of the approximate model. Solution (a) Taylor series: The dominant (largest) time constant (5) is
retained. Applying the approximations in (657) and (658)
gives:
Approximate the RHP zero as a time delay 0.1s 1 e 0.1s
Approximate the smaller time constants (3 and 0.5) as
time delays.
1
1
(662) e 3s e 0.5 s
3s 1
0.5s 1
Substitution into(659) gives the Taylor Series (TS)
G s:
approximation, TS Ke0.1s e 3s e0.5 s Ke3.6 s GTS s 5s 1
5s 1 (663) G s K 0.1s 1 5s 1 3s 1 0.5s 1 (659) Derive an approximate firstorderplustimedelay model,
Keθs G s τs 1 (660) using two methods:
(a) The Taylor series expansions of Eqs. 657 and 658.
(b) Skogestad’s half rule
Compare the normalized responses of G(s) and the approximate
models for a unit step input. (b) To use Skogestad’s method, we note that the largest neglected
time constant in (659) has a value of three.
• According to his “half rule”, half of this neglected value is
added to the next largest time constant to generate a new
dominant 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 (661) provides an
additional time delay of 0.1.
• Approximating the smallest time constant of 0.5 in (659) by
(658) produces an additional time delay of 0.5.
• Thus the total time delay in (660) is,
θ 1.5 0.1 0.5 2.1 Multivariable Processes = MIMO and G(s) can be approximated as:
Ke2.1s GSk s 6.5s 1 (664) 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. 2 Outputs = T(t) and h(t)
4 Inputs = wh(t), wc(t), Th(t) and Tc(t)
For constant density, G23 and G24 = 0
G12
G
G ( s ) is a 2 x 4 matrix 11
G21 G22 12.8e X D ( s ) 16.7 s 1 X ( s) 7 s B 6.6e
10.9 s 1 s 18.9e 21s 1 R( s ) 3 s
19.4e S ( s ) 14.4 s 1 3 s G13 G14 G23 G24 SISO = single input / single output is the “common”
method for process control, but interactions
may present “tuning” problems (trial and error
iterations for controller settings may or may not work).
In multivariable control design, interactions are taken
into consideration, but controller design equations
are more complicated, from CHE 461 to CHE 550. ...
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 Winter '08
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