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Bsb Assessment 2

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Submitted By mandeepboyal
Words 679
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Ian R. Manchester
April, 2012

AMME3500: Systems Dynamics & Control
Assignment 2
Note: This assignment contributes 10% towards your final mark. This assignment is due at 5pm on Tuesday, April 24th during Week 7. Submit your report to the assignment box on the 3rd floor outside of the drawing office in the Mechanical Engineering Building or via email by that time (i.manchester@acfr.usyd.edu.au). Late assignments will not be marked unless a doctor’s certificate or equivalent is provided. Plagiarism will be dealt with in accordance with the University of Sydney plagiarism policy. You must complete and submit the compliance statement available online. Mathematical derivations are expected to be done by hand except where the use of Matlab is indicated. The objective of this assignment is to familiarise yourself with the block diagram simplification, system specification, and feedback control techniques discussed in the lectures. It will also assess your understanding of the lab on system identification. This assignment should take an average student 15-20 hours to complete. Total Marks: 100 The front page of your report should include: • Your name and SID • Your tutorial group number 1. Consider the following block diagrams. Determine the Transfer Function T(s)=C(s)/R(s) relating the input and output of the systems. Show the steps taken in reducing the block diagram – i.e. do not simply write down the transfer function [10 marks] a. + 1
R(s)



K(s +10)

s+5

C(s)



b.

2. For the following second order systems, determine the damping ratio, natural frequency, rise time, settling time and percent overshoot. Show your derivation of these values. Sketch the location of the poles and zeros for the systems and use Matlab to generate the step response. Verify the response in terms of your rise time, settling time and percent overshoot calculations. [10 marks] a. c. b. d.

3. For the following systems, estimate the damping ratio, natural frequency, rise time, settling time and percent overshoot based on the standard second order assumptions. Show your derivation of these values. Sketch the location of the poles and zeros for the systems and use Matlab to generate the step response. Are the step responses consistent with the second order assumptions? Why or why not?[15 marks] 30 10 a. T(s) = 2 b. T(s) = 2 s + 4s +10 ( s + 3)(s + 4s +10) 100 4 ( s + 2) c. T(s) = d. T(s) = 2 2 ( s +10)(s + 4s +10) s + 4s +10 € € 4. You will now analyse a control system for a single link of a robot arm. To begin with, assume that the robot arm is completely rigid and has a moment of € € inertial of J=2kgm2. The motor, gearing, and joint mechanism has friction, which has been measured as c=0.25 Nms. Assume there is no gravity acting on the arm (e.g. the robot arm is in space, or operating horizontally). The ˙˙ ˙ equations of motion are: Jθ + cθ = T , where T is the total torque applied to the arm. Based on these system characteristics, answer the following [40 marks] a. Find the transfer function between the applied torque T and the € indicator angle θ.



b. Suppose the torque is computed so that θ tracks a reference command θr according to the proportional feedback law u = K (θ r − θ ) where K is the feedback gain. In interacting with the environment (picking up or pushing objects, etc) there is a reaction torque on the arm which we model as a disturbance torque w. Draw a block diagram of the resulting€ feedback system showing θ, the reference position θr as well as the disturbance torque w. So the total torque is T=u+w. Find the transfer function between θr and θ assuming zero disturbance torque. c. Find the transfer function between w and θ assuming θr=0. d. Assuming zero disturbance load (w=0) what is the percentage overshoot as a function of K? What is the maximum value of K that can be used if you wish to have an overshoot Mp

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