Solved–Lab Assignment 2 –Solution

Description

• Objective

The main goal of this assignment is to become familiar with Simulink environment and start working with the cart stations.

• Your responsibility

Your duty 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.

• Model Description

The equation of motion of a DC motor can be described by:

	KtKe	_	Kt
Jm + (b +	Ra	) =	Ra	va

where is the shaft angle (in radians) of the motor and v_a is the applied voltage.

The system parameters are as follows:

J_m = 0:01kgm² be the inertia of the rotor and the shaft.

b = 0:001N msec be the viscous friction coe cient

K_e = 0:02V sec be the back emf constant

(s)

K_t = 0:02N m=A be the motor torque constant. R_a = 10 be the armature resistance

Note that this document uses SI units for which K_e = K_t.

4 Questions

1. Implement the transfer function you derived in previous lab (_va(s) ) in simulink. You can use

blocks Transfer Fcn from continuous library, scope from Sinks library, and Signal Generator, and Step, from Sources library. [10 marks] (In order to include the diagrams you can use Print To File)

2. Using Step, nd the rise-time and steady-state response of the system to a unit step function. [5 marks]

2. Using Signal Generator or Sine Wave, measure response of the system to sine waves of amplitude 1 and frequencies w = [0:1; 1; 10; 100; 1000]. [5 marks] Using these data, plot Bode diagram(gain diagram) of the system. [10 marks] Using the transfer function and bode command, plot the theoretical Bode diagram and then compare it with the diagram you found by experiments. [10 marks]

Hint: Note that you have to use w = log10(input frequencies) as the x-axis and 20*log(gains) as y- axis to draw your data points Bode diagram.

2. Implement the unity Feedback loop in Simulink. Repeat the steps of Q2 to derive the Bode diagram of the feedback system. [10 marks] Using the closed-loop transfer function you derived in previous session, plot the Bode diagram of the closed-loop system then plot experimental bode diagram and theoretical one in one gure. [10 marks]

2. Implement a proportional controller in simulink. By changing the amount of proportional gain K, nd the corresponding K for which we get rise-time of 4 seconds. [10 marks]

2. Experimenting di erent proportional gains K in Simulink, nd the proportional gain K for which get, 20% overshoot. [10 marks]

2. Using a derivative block, and an integrator block, implement a PID controller for the system. Suppose we name the gain block before the derivative block K_d, and the gain block before the integrator block K_i, and proportional gain K_p, observe the output of the system to unit step function for di erent combinations of K_i; K_d; K_p 2 f0:1; 1; 10g(by di erent combinations we mean xing two of the gains and iterating over the last one). [15 marks]

Using the above experiment describe intuitively the impact of increment in K_d; K_i(individually), on the step response of the system. [15 marks]

8. Setting K_d = 0, Find the step response of the system. [5 marks]

9. Setting K_i = 0, Find the step response of the system. [5 marks]

10. Follow instructions of the Lab Manual, observe the output of the system to a sine wave[It would be checked by the TA]

11. Using a ruler nd the correct linear position gain.(Convert the sensor’s output to the position in mm)

12. Using previous linear position gain, plot the step-response of the system.[10 marks]

Hint: In order to bring the data out of Simulink environment you can use the To Workspace

block.

13. Alter the Simulink model such that you can observe velocity as output of the system.( ^{V (s)} ),

v_a(s)

and then plot the step-response. [10 marks]

14. Explain why the observed velocity on the scope is noisy.[5 marks]

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