Dynamic behaviour of process

1
Dynamic Behavior
Chapter
5
In analyzing process dynamic and process control systems, it is
important to know how the process responds to changes in the
process inputs.
A number of standard types of input changes are widely used for
two reasons:
1. They are representative of the types of changes that occur
in plants.
2. They are easy to analyze mathematically.

2
1. Step Input
A sudden change in a process variable can be approximated by
a step change of magnitude, M:
Chapter
5
0 0
(5-4)
0
s
t
U
M t





• Special Case: If M = 1, we have a “unit step change”. We
give it the symbol, S(t).
• Example of a step change: A reactor feedstock is suddenly
switched from one supply to another, causing sudden
changes in feed concentration, flow, etc.
The step change occurs at an arbitrary time denoted as t = 0.

3
Chapter
5 Example:
The heat input to the stirred-tank heating system in Chapter 2 is
suddenly changed from 8000 to 10,000 kcal/hr by changing the
electrical signal to the heater. Thus,
     
   
8000 2000 , unit step
2000 , 8000 kcal/hr
Q t S t S t
Q t Q Q S t Q
 
    
and
2. Ramp Input
• Industrial processes often experience “drifting
disturbances”, that is, relatively slow changes up or down
for some period of time.
• The rate of change is approximately constant.

4
Chapter
5 We can approximate a drifting disturbance by a ramp input:
 
0 0
(5-7)
at 0
R
t
U t
t





Examples of ramp changes:
1. Ramp a setpoint to a new value. (Why not make a step
change?)
2. Feed composition, heat exchanger fouling, catalyst
activity, ambient temperature.
3. Rectangular Pulse
It represents a brief, sudden change in a process variable:

5
Chapter
5
 
0 for 0
for 0 (5-9)
0 for
RP w
w
t
U t h t t
t t



 

 

Examples:
1. Reactor feed is shut off for one hour.
2. The fuel gas supply to a furnace is briefly interrupted.
0
h
XRP
Tw Time, t

7
Chapter
5
Examples:
1. 24 hour variations in cooling water temperature.
2. 60-Hz electrical noise (in the USA)
4. Sinusoidal Input
Processes are also subject to periodic, or cyclic, disturbances.
They can be approximated by a sinusoidal disturbance:
 
 
sin
0 for 0
(5-14)
sin for 0
t
U t
A t t






where: A = amplitude,  = angular frequency

8
Chapter
5
Examples:
1. Electrical noise spike in a thermo-couple reading.
2. Injection of a tracer dye.
5. Impulse Input
• Here,
• It represents a short, transient disturbance.
• Useful for analysis since the response to an impulse input
is the inverse of the TF. Thus,
   .
I
U t t


 
 
 
 
 
u t y t
G s
U s Y s
Here,
      (1)
Y s G s U s


9
Chapter
5
The corresponding time domain express is:
     
0
τ τ τ (2)
t
y t g t u d
 

where:
   
1
(3)
g t G s

 
 
L
Suppose . Then it can be shown that:
   
u t t


    (4)
y t g t

Consequently, g(t) is called the “impulse response function”.

10
Chapter
5
The standard form for a first-order TF is:
where:
Consider the response of this system to a step of magnitude, M:
Substitute into (5-16) and rearrange,
First-Order System
 
 
(5-16)
τ 1
Y s K
U s s


steady-state gain
τ time constant
K
   
for 0
M
U t M t U s
s
   
 
 
(5-17)
τ 1
KM
Y s
s s



11
Chapter
5 Take L-1 (cf. Table 3.1),
   
/ τ
1 (5-18)
t
y t KM e
 
Let steady-state value of y(t). From (5-18),
y .
y KM
 
t ___
0 0
0.632
0.865
0.950
0.982
0.993
y
y
τ
2τ
3τ
4τ
5τ
Note: Large means a slow response.
τ
y
y
τ
t

12
Chapter
5
Consider a step change of magnitude M. Then U(s) = M/s and,
Integrating Process
Not all processes have a steady-state gain. For example, an
“integrating process” or “integrator” has the transfer function:
 
 
 
constant
Y s K
K
U s s
 
   
2
KM
Y s y t KMt
s
  
Thus, y(t) is unbounded and a new steady-state value does not
exist.
L-1

13
Chapter
5
Consider a liquid storage tank with a pump on the exit line:
Common Physical Example:
- Assume:
1. Constant cross-sectional area, A.
2.
- Mass balance:
- Eq. (1) – Eq. (2), take L, assume steady state initially,
- For (constant q),
 
q f h

(1) 0 (2)
i i
dh
A q q q q
dt
    
     
1
i
H s Q s Q s
As
  
 
 
 
  0
Q s
 
 
 
1
i
H s
Q s As




14
Chapter
5
• Standard form:
Second-Order Systems
 
  2 2
(5-40)
τ 2ζτ 1
Y s K
U s s s

 
which has three model parameters:
steady-state gain
τ "time constant" [=] time
ζ damping coefficient (dimensionless)
K
• Equivalent form:
1
natural frequency
τ
n

 

 
 
 
 
2
2 2
2ζ
n
n n
Y s K
U s s s

 

 

15
Chapter
5 • The type of behavior that occurs depends on the numerical
value of damping coefficient, :
ζ
It is convenient to consider three types of behavior:
Damping
Coefficient
Type of Response Roots of Charact.
Polynomial
Overdamped Real and ≠
Critically damped Real and =
Underdamped Complex conjugates
ζ 1

ζ 1

0 ζ 1
 
• Note: The characteristic polynomial is the denominator of the
transfer function:
2 2
τ 2ζτ 1
s s
 
• What about ? It results in an unstable system
ζ 0


18
Chapter
5
1. Responses exhibiting oscillation and overshoot (y/KM > 1) are
obtained only for values of less than one.
2. Large values of yield a sluggish (slow) response.
3. The fastest response without overshoot is obtained for the
critically damped case
Several general remarks can be made concerning the
responses show in Figs. 5.8 and 5.9:
ζ
ζ
 
ζ 1 .


20
Chapter
5 1. Rise Time: is the time the process output takes to first reach
the new steady-state value.
2. Time to First Peak: is the time required for the output to
reach its first maximum value.
3. Settling Time: is defined as the time required for the
process output to reach and remain inside a band whose width
is equal to ±5% of the total change in y. The term 95%
response time sometimes is used to refer to this case. Also,
values of ±1% sometimes are used.
4. Overshoot: OS = a/b (% overshoot is 100a/b).
5. Decay Ratio: DR = c/a (where c is the height of the second
peak).
6. Period of Oscillation: P is the time between two successive
peaks or two successive valleys of the response.
r
t
p
t
s
t

Dynamic behaviour of process

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