Use Of Mat lab Commands

Coursework Specification
Use Matlab commands and Simulink to perform the following tasks. You must submit
the Matlab commands you used as well as resulting outputs/plots in your report. For
Q10, you should also include a discussion of the results.
Submit the report in PDF format on Canvas by the deadline.
Block Diagram Models
Q1 Use Matlab to find the roots of the following polynomials:
a) p = s6+3s5+2s4+3s3+2s2+3s+5
b) p = 3s2+2s+3
[5 marks]
Q2 Use Matlab to find the pole(s) and zero(s) of the following transfer functions:
a)
10 3
5
( )
1 2
 


s s
s
G s
b)
2 5 2
5 7
( )
3 2
2
1
  
 

s s s
s s
G s
[5 marks]
Q3 Use Matlab to determine the open loop transfer function of the following system:
[5 marks]
Q4 Use Matlab to determine the closed loop transfer function of the following unity feedback
system:
[5 marks]
–
+
s s 5
2s 1
2
 

s s 3
1
2
 
Q5 Use Matlab to determine the closed loop transfer function of the following feedback system:
[5 marks]
Q6 In matlab, use block reduction techniques to determine the closed loop transfer function of
the following system:
[5 marks]
–
+
–
+ +
–
System Stability-Routh Hurwitz Criterion
Q7
a) Find the closed loop transfer function of the following unity feedback system:
The closed loop transfer function can be found using
1 ( ) ( )
( )
G s H s
G s

where, for this system,
H(s) = 1.
[3 marks]
b) Using the characteristic equation from above and the Routh Hurwitz Stability Criterion,
find the range of gain k for system stability.
[4 marks]
c) What is the value of k that makes the system oscillate and the frequency of oscillation?
Illustrate your results using Simulink and/or Matlab simulations.
[3 marks]
Q8 For the unity feedback system shown below, use the Routh Hurwitz Stability Criterion to find
the range of K for stability. Use Simulink to verify the stability range,
where G(s) =
( 2)( 3)
( 20)
 

s s s
K s
and T(s) =
1 ( ) ( )
( )
G s H s
G s

T(s) = Closed Loop Transfer Function and for unity feedback H(s) = 1
[10 marks]
U(s) Y(s)
(s 1) (s 4)
k
3
 
–
+
G(s)
U(s) Y(s)
Tracking Accuracy
Q9 A second order position control system is shown in the following diagram:
The output Y(s) gives the position in mm and is subject to the effect of disturbances. A proportional
plus integral controller is used.
a) With K = 0.5, plot the output y(t) for a disturbance input of magnitude 2. What does the plot
show?
[5 marks]
b) Let K take the following values: K = 0.5, 1.0, 1.5 and 2.0. Draw a table showing the effect of
the disturbance on the output response.
[5 marks]
D(s)
R(s) Y(s)
+
–
+
+
Controller Design
Q10 Introduction
The process to be controlled is a liquid level process consisting of 2 storage tanks.
The process dynamics are described by the following transfer function:
?(?) =
?2(?)
?(?)
=
?
(?1? + 1)
2
where H2 is the level in tank 2 and M is the input to the inflow valve of tank 1.
Choose values for your process parameters using your date of birth:
K = day of birth (in range 1 to 31)
T1 = month of birth (in range 1 to 12)
Open Loop Step Response
a) Simulate the process transfer function with a unit step change in input.
[5 marks]
b) Take required measurements from the response in a) for use in the Ziegler-Nichols open
loop PID tuning method.
[5 marks]
P Control
c) Design a P controller gain (Kc) for the process using your results in b) and Ziegler-Nichols
open loop method. Simulate the closed loop P control response to a unit step input.
[5 marks]
A PID block can be used to simulate the controller, shown in the following example diagram.
Transfer function is the process. PID, Multiplexer (MUX) and Sum simulation blocks are used.
Note The PID block implements a parallel form of the controller: ??
(?) = ?? +
??
?
+ ???.
This should be compared to the standard form: ??
(?) = ??(1 +
1
???
+ ???), to enter the
correct values for the block.
c) Adjust Kc and simulate the response to achieve a percent overshoot (%OS) of between 10 to
20%
[5 marks]
General Note: To get a more accurate simulation (more data points, smoother plots), set
the Max. Step Size (Simulation → Simulation Parameters menu on SIMULINK model) from
‘auto’ to a smaller value, e.g. 0.1.
PI Control
d) Design a PI controller (Kc and Ti) for the process using your results in part b) and ZieglerNichols
open loop method. Simulate the closed loop PI control response to a unit step input.
Compare the response with the previously designed P controller in terms of %OS, settling
time and steady state error.
[10 marks]
e) Adjust Kc if required to achieve a percent overshoot (%OS) of between 10 to 20%. With this
value of Kc, choose two other values for Ti to show the effect of changing the integral action
time (Ti) on the controlled response. Include:
i. plots of the step responses (max. 3),
ii. a table comparing the percent overshoot (%OS), settling time and steady state error
for the three values of Ti used.
[10 marks]
Routh Hurwitz Stability Criterion