DYCOPRO- Process Dynamics and Control
PRACTICE Quiz 2
Name:
ID #
Date: April 21, 2021
Professor: Dr. Joseph Auresenia
Part I. Multiple Choice. 10 points
Encircle the correct the letter of the correct answer. If the correct answer is not one of the given
choices, write the correct answer beside the number.
1.
For a system with open loop transfer function
G ( s) =
10
, find y(t) for a unit step change in the
s+2
input.
a.
5[ 1 – e3/2 t (cosh 1/2t – 3sinh 1/2t)]
d. 5(1 − e
b.
5 – e3/2 t
e. 10(1 − e )
c.
5[ 1 – e3/2 t (cos 1/2t – 3sin 1/2t)]
−2t
)
−t
(cosh 1/2t – 3sinh 1/2t)
A ________________ is an arrangement, of physical components connected or related in such as
manner as to command, direct or regulate itself or another system.
a) Regulator system b. Servomechanism system c. control system d. open loop system.
In control industry, the proportional band is a parameter that is used to relate the _______________ of
2.
3.
4.
5.
6.
7.
8.
the control system.
a) Integral time constant
b) reset rate
c) process gain
d) controller gain
This industry term is used to represent the effect of the integral control mode in the control system.
a) rate
b) reset rate
c) proportional band
d) integral time constant
In order to minimize the oscillation, derivative time constant of the derivative controller parameter is
usually increased, what is the industry term for this parameter?
a) rate
b) reset rate
c) proportional band
d) integral time constant
A control system mechanism wherein the aim is to have the value of the output follows the changes of
the value of the set point at all times.
a) closed loop mechanism
b) open loop mechanism
c) servo mechanism
d)
regulator mechanism
A control system mechanism wherein the aim is suppress the effect of any disturbances, the value of
output should always follow the fixed value of the set point.
a) closed loop mechanism
b) open loop mechanism
c) servo mechanism
d)
regulator mechanism
The transfer function of the derivative controller is
a)
𝐺𝑐 = 𝜏𝐷 𝑠
b. 𝐺𝑐 = 𝐾𝑐
d. 𝐺𝑐 =
1
1
𝜏𝑖 𝑆
d. 𝐺𝑐 =
1
𝜏𝑑 𝑆
9. The transfer function of the proportional controller is
a)
𝐺𝑐 = 𝜏𝐷 𝑠
b. 𝐺𝑐 = 𝐾𝑐
d. 𝐺𝑐 =
1
𝜏𝑖 𝑆
d. 𝐺𝑐 =
1
𝜏𝑑 𝑆
10. The transfer function of an integral controller is
a) 𝐺𝑐 = 𝜏𝐷 𝑠
b. 𝐺𝑐 = 𝐾𝑐
d. 𝐺𝑐 =
1
𝜏𝑖 𝑆
d. 𝐺𝑐 =
1
𝜏𝑑 𝑆
b)
Part II. Fill in the blanks. 10 pts.
Fill in the blank with the correct answer.
1.
2.
3.
4.
The transfer function of a P controller is ______________________________________.
The transfer function of a PI controller is ______________________________________.
The transfer function of a PD controller is ______________________________________.
The transfer function of a PID controller is ______________________________________.
5. The change in control action as a function of time for a proportional controller is ∆𝑐(𝑡) =
______________________________________________________
6. The change in control action as a function of time for an integral controller is
∆𝑐(𝑡) = ______________________________________________________
7. The change in control action as a function of time for a derivative controller is ∆𝑐(𝑡) =
______________________________________________________
8. The change in control action as a function of time for a proportional-integral controller is ∆𝑐(𝑡) =
______________________________________________________
9. The change in control action as a function of time for a proportional-derivative controller is ∆𝑐(𝑡) =
______________________________________________________
10. The change in control action as a function of time for a proportional-integral-derivative controller is
∆𝑐(𝑡) = ______________________________________________________
Part II. Problem Solving.
1.
Solve the following problems
Open Loop Response 40 pts
a. For a system with an open loop response to a unit step change shown in Figure 1,
i. What order can this system be?
ii.
What is the approximate transfer function?
b. For the system with an open loop response to a step change of magnitude 10 shown
i. What are the overshoot ratio and the decay ratio?
ii. What is the damping factor?
iii. What is the period of oscillation and the time constant?
iv. What is the process gain?
v. What is the approximate transfer function of the system?
2
in Figure 2,
c. A step change of magnitude 3 is introduced into the system with transfer function,
Gc =
Y ( s)
10
= 2
X ( s) 2s + .3s + .5 , what is the response of the system Y(t)? What is the overshoot ratio,
decay ratio and the period?
d.
A system with transfer function
Gc =
10
2s + 4 is undergoing a sinusoidal change in the input with
amplitude of 5 and radian frequency of 2,
2.
x(t ) = 5 sin 2t
what is the response of the system?
Block Diagram Representation 15 pts.
For a non isothermal system the final equations of concentration and temperature after linearization and
Laplace transform operation are
C A ( s) =
1
1
C Ai ( s) +
Ti ( s)
s+2
s+3
a) Draw the open loop transfer function showing all the blocks and transfer functions inside the blocks.
b) If the C Ai (s ) is the manipulated variable, Ti (s ) the disturbance variable and C A (s ) the controlled
output variable, what is Gd and Gp .
c) If G f = K c , and G f = Gm =
3.
1
s +1
, draw the closed loop block diagram.
15 pts. Closed Loop Transfer Function
For the closed loop system with PI controller shown in Figure 3 below
d (s )
Ysp (s )
-
+ X

1 

Gc = K c 1 +
 Is 
Gp =
s
Gf =
s +1
Gd =
+
1
X
s + 2s + 2 +
1
s +1
Y (s )
2
Gm = 1
Figure 3
If Kc =1 and τI= 1;
a. Find the servo control closed loop transfer function.
b. Using final value theorem find the final value of the response to unit step change in the setpoint.
c. Find the regulator transfer function.
3
d. Using final value theorem find the final value of the response to unit step change in the disturbance
Simplify closed loop transfer functions.
4.
(20 pts). Modeling and Block Diagram Representation and Open loop Response.
Consider the stirred-tank reactor. The reaction occurring is 𝐴 → 𝐵 and it proceed at a rate 𝑟 = 𝑘𝐶𝐴 ,
where
r = moles A reacting /(volume)(time)
k = reaction velocity constant = 2/min.
𝐶𝐴 = concentration of A in reactor at any time in moles/liter.
𝑉 = volume of mixture inside the reactor = 100 liters (held constant)
F = constant feed rate = 10 li/min, the steady state inlet flow rate
𝐶𝐴𝑖 = inlet concentration of A flowing into the reactor at any time in moles/liter, allowed to vary.
𝐶𝐴𝑖𝑠 = the steady state inlet concentration of A flowing into the reactor = 20 moles/liter;
Assuming constant density and constant V,
a. Formulate the differential equation relating the output concentrations of A and B to the varying
input.
b. What are the steady state concentrations of A and B?
c. Derive the transfer function relating the concentration of A and concentration of B inside the
reactor to the inlet feed stream concentration.
d. Draw the block diagram of the system.
e. If the inlet concentration undergo a step change of magnitude 5, what is the response; 𝛥𝐶𝐴 = 𝑓(𝑡)
and 𝛥𝐶𝐵 = 𝑓(𝑡)?
5.
If the system in 4 is controlled by a PID controller with; 𝛥𝐶𝐴 the controlled output and the flow rate as
the controlled variable and 𝐺𝑓 = 𝐺𝑚 = 1, assuming that all the PID parameters are unity, what is the
overall closed loop transfer function of the system? If the setpoint undergoes unit step change , find the
response of 𝛥𝐶𝐴 = 𝑓(𝑡).
4
Step Response
Process Reaction Curve for Unit Step Change in the Input
1
0.9
0.8
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
Time (sec.)
Figure 1
80
70
60
50
Y(t)
y(t)
Amplitude
0.7
40
30
20
5
15