# Control Lab ECSE 403  Lab assignment1

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Control Lab
ECSE 403
Lab assignment1
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.
single PDF or MATLAB executable notebook file.
1
3 Model Description
The final goal of this lab is to model and control an inverted pendulum system. In that system
the input is applied to a DC motor connected to a cart-pole. The first step to model the whole
system is to model the DC motor. 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:
• 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 = 0.02Nm/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 0 ] , [ 1 4 1 ] )
and then use step command:
2
s t e p p l o t (G)
3. Using previous step response find rise time and settling time of the system.[10 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.[10 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 and
find the closed loop transfer function[10 marks].
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 the step response of the closed loop unity feedback
system choosing K = {0.1, 1, 10, 100}, in one figure. Describe the effect of proportional gain
on step response’s behavior [15 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(plot
closed loop unity feedback step response). Describe the effect of proportional gain on step
response’s behavior [15 marks].(In this case you should explain the effect of proportional gain
on the overshoot, rise-time, and settling time)
3

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