Description
1 Objective
The main goal of this assignment is to review some concepts from the linear control course(ECSE
307), and to become familiar with some useful tools in matlab which helps us in designing and
implementing controllers.
2 Your duty
Your duty is to answer all questions which have been asked throughout this assignment and submit
all your answers in addition to matlab codes in mycourses website.
3 Model Description
The equation of motion of a DC motor can be described by:
Jm
¨θ + (b +
KtKe
Ra
)
˙θ =
Kt
Ra
va
where θ is the shaft angle (in radians) of the motor and va is the applied voltage.
System’s parameters are as following:
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• Jm = 0.01kgm2 be the inertia of the rotor and the shaft.
• b = 0.001Nmsec be the viscous friction coefficient
• Ke = 0.02V sec be the back emf constant
• Kt = 0.002Nm/A be the motor torque constant.
• Ra = 10Ω be the armature resistance
Note that using SI units Ke = Kt
.
4 Questions
1. Find the transfer function between the input voltage and the speed of the motor shaft( w(s)
va(s)
,
where w(s) = ˙θ(s)).(Plug in the coefficients above).[10 marks]
2. Using transfer function derived above and matlab, find the steady-state and time constant
of the response of the motor to a step function.[10 marks]
Hint: You can use following matlab commands for defining a transfer function: (random
coefficient)
s = t f ( ’ s ’ )
G = s / ( s ˆ2 + 4∗ s +1)
Or instead, you can just use polynomial coefficients.
G = t f ( [ 1 ] , [ 1 4 1 ] )
and then use step command:
s t e p p l o t (G)
3. Using previous step response find rise time and settling time of the system.[5 marks]
4. Using Final Value Theorem, calculate the steady state speed of the motor to step response
theoretically. Compare theoretical value and values you found in matlab.[5 marks]
5. Find the transfer function between the shaft’s angel and input voltage( θ(s)
va(s)
).(Identify the
order of the system with respect to new definition of input-output signals)[10 marks]
6. Consider the transfer function in question 1. Apply a unity feedback loop to the system.
Draw block diagram of the system[5 marks], find the closed loop transfer function[5 marks].
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7. Suppose a proportional controller is added to the system such that open-loop transfer
function has changed from G(s) to K.G(s). Plot step response of the system choosing
K = {0.1, 1, 10, 100}, in one figure [5 marks]. Describe the effect of proportional gain on step
response’s behaviour [5 marks].
Hint: One way to plot different graphs on one figure is as following:
f i g u r e ( 1 ) ;
s t e p p l o t ( h1 ) ;
hold on ;
s t e p p l o t ( h2 ) ;
.
.
hold o f f ;
8. Consider the transfer function in question 5, repeat steps of question 7 for this system[5
marks.] Describe the effect of proportional gain on step response’s behaviour [5 marks].(In
this case you should explain the effect of proportional gain on the overshoot, rise-time, and
settling time)
9. Analytically find the maximum value of K that can be used if an overshoot of at most 20%
is desired?[10 marks] Hint: You can refer to formulas in chapter 3 of [1]
10. Ignoring previous constraint, find the value for K, which provides a rise time of 4 seconds.[10
marks] Hint: You can refer to formulas in chapter 3 of [1]
11. For the transfer function which you derived in question 5, plot the Bode diagram of the
system[5 marks]. From the Bode plot, find Gain-Margin and Phase-Margin of the system[5
marks].
Write the definition of Gain-Margin and Phase-Margin[5 marks]. Describe the importance
of these two numbers[5 marks].
Hint : In order to plot Bode diagram you can use following command
bo d e plo t ( h1 )
12. For the transfer function which you derived in question 5, plot the root locus of the system[5
marks]. Explain application of root locus diagrams[5 marks]. Using root locus diagram, find
the gain in which system becomes unstable[5 marks].
13. Consider the transfer function you derived in question 5. We add a controller block with
following transfer function instead of proportional controller.
s + 0.35
s + 0.75
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Plot root lucos of the system considering this additional block[5 marks]. Using the root locus
diagram, Is there a gain for which system becomes unstable?[5 marks]
References
[1] G. F. Franklin, D. J. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems,
4th ed. Upper Saddle River, NJ, USA: Prentice Hall PTR, 2001.
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