Modern Control Systems Exams 1999-2003 Solutions - p99

Description

Modern Control Systems Exams 1999-2003 + Solutions - p99

-------------
Written Exam
Modern Control Systems
(Et 3-015)
OnWednesday the 3rd of November 1999 from 14:00 PM to 17:00 PM
Read the following VERY carefully.
?The exam is composed of 2 Parts: theory and exercises. The £rst part is WITHOUT any
books or notes and it will last 40 minutes. At the end of the 40 minutes, the exam will be
collected and after 10 minutes the second part of 130 minutes will start. You are allowed
to use anything you want for the second part:
?Part I, 40 minutes (end at 14:40) , no books, hand in at the end.
?Break, 10 minutes, (end at 14:50) collect books, sheets and the material you are
allowed to use for Part II.
?Part II, 130 minutes (end at 17:00), with books, sheets.....
If you are ready with Part I earlier than the time given, you can start working on Part II,
but WITHOUT BOOK and on a NEW SHEET OF PAPER.
?NEVER talk with your neighbor.
?If the answers are not easily readable, the corresponding answer will be given 0 points.
Therefore WRITE CLEARLY.
?Read every question well before answering.
?Write ALL your reasoning steps on paper.
?Write your name and student number clearly readable on EACH piece of paper.
?Good luck !
Technische Universiteit Delft
Faculteit der Informatie Technologie en Systemen
Vakgroep Regeltechniek
Mekelweg 4
2628 CD Delft
Part I: theory
(Closed Book)
Question 1 (weight:3)
Given: For a transfer function F(s), due to the Cauchy’s theorem of complex analysis, we have
that:
1
2πj
F
(s)
F(s)
ds = Z − P
where the integral is along a closed curve encircling Z zeros and P poles of F(s).
Asked: Prove the Nyquist criterium formula
Nenc = Z − P
where Nenc is the number of clockwise encirclements around the origin of F(s) for values of s
along a closed curve enclosing Z zeros of F(s) and P poles of F(s).
http://rapidshare.de/files/15090333/Modern_Control_Systems_Exams_1999-2003___Sol...9.rar.html
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