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1 Objective
The main goal of this assignment is to become familiar with Cart systems, and find system parameters.
2 Your responsibility
Your responsibility is to answer all questions which have been asked throughout this assignment
and submit all your answers in addition to Matlab codes and Simulink results.
Notice: You have to first finish Lab assignment 2.
3 Model Description
The equation described in lab 1 and lab 2, is the equation of the DC motor, on the cart systems.
In order to derive the equations for the mechanical system, we need following set of parameters in
addition to parameters given in the previous sessions:
• kg = 3.7 to 1 be the internal gear ratio
• r = 0.0064 m be the motor gear radius
• mc = 0.526 Kg be the cart mass.
• kth = 0.00153 rad/counts be the pendulum angle gain.
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• mp = 0.106 Kg be the pendulum mass.
• kx =(use the gain you found in previous assignment) be the linear position gain.
4 Questions
4.1 Linearity of the system
1. Consider two arbitrary input signals x1(t),x2(t). Find the response of the system to these two
signals H{x1(t)},H{x2(t)}.(voltage as input and velocity as the output). Plot an arbitrary
linear combination of the response signals aH{x1(t)} + bH{x2(t)}. Now use the expression
ax1(t)+bx2(t) as input signal, Plot the response of the system to this signal H{ax1(t)+bx2(t)}
and compare it with previous plot. [15 marks]
Hint:In order to export the data from Simulink environment you can use To Works space
block.
2. As you know, sine waves are eigenfunction of an LTI system. Using a sine wave with amplitude 2 and frequency of 1 Hz, check whether or not the system is LTI. [5 marks]
3. Suppose someone argue that the divergence from practical measurements and theoretical
ones is due to a hidden feedback loop. Do you agree with that person? Justify your answer.
[10 marks]
4. Explain possible sources of non-linearity of the system. [10 marks]
4.2 System Identification
1. Derive the open loop transfer function of the cart from mechanical and electrical first principles.(consider voltage as the input and velocity as the output). [ You have to submit this
part next week along with Lab assignment 2] [10 marks]
2. Calculate the coefficients of these transfer function using given constants in Model description
section. [10 marks]
3. Find the step response of the system, defining cart’s velocity as the output of the system.(voltage as input and velocity as the output). [10 marks]
Hint: Note that in order to achieve the velocity of the cart you need to differentiate cart’s
position in Simulink.
Hint: You can refer to lab manual to find a way to filter out the noise in the output signal.
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4. Estimate the coefficients of the transfer function by measuring the time constant and gain
of the step response. [10 marks]
Hint: The system with velocity as the output, is a first order system.
5. Using To workspace command, plot your estimated step response and actual step response
of the system on one figure. [10 marks]
6. Explain possible reasons for the differences between aforementioned plots. [10 marks]
7. Using frequencies w = [0.1, 1, 10, 100], plot the bode diagram of the system experimentally.
[10 marks]
8. Using your estimated parameters, define the transfer function in Matlab and plot bode
diagram of the system and compare it with previous question. [10 marks]
9. Find the step response of the system with defining position as the output.(voltage as the
input and position as the output). [10 marks]
10. Using frequencies w = [0.1, 1, 10, 100], plot the bode diagram of the system experimentally.(voltage as the input and position as the output) [10 marks]
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