Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Homework 6 for Analytical Methods | EEL 3105, Study notes of Electrical and Electronics Engineering

University of Florida (UF)Electrical and Electronics Engineering
Prof. Pramod P. Khargonekar

HW 6 Material Type: Notes; Professor: Khargonekar; Class: ANALYTICAL METHODS; Subject: ENGINEERING: ELECTRICAL; University: University of Florida; Term: Fall 2011;

Typology: Study notes

2010/2011

Uploaded on 12/30/2011

stefan-castillo
stefan-castillo 🇺🇸

24 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
EEL3105Fall2011
Homework6
Note:Problem4willbegradedseparatelyforassessingthatoneofthecourseobjectivesis
met.PleasesubmitsolutiontoProblem4asaseparatedocument.
Considermatrices
12
123
13
4
01
010
001
Ak
Bbb
C
ccc








1. ForA,choosek=3.Findtheeigenvaluesoftheresultingmatrix.Aretheyreal?Why?
2. FindaformulaforeigenvaluesofAasafunctionofk.Supposekisadesignparameter
whichrangesfrom‐100to+100.UsetheformulatoplottheeigenvaluesofAasafunction
ofkinthecomplexplane.UseMATLABtoverifyyouranswer.Suchaplotiscalledroot
locus.Commentonyourresults.
3. FindcharacteristicpolynomialsofBandCintermsofb1,b2,c1,c2,andc2.Doyouseeany
patterninyouranswer?Canyouthinkofawaytogeneralizethistoannxnmatrix?
4. Fixb2=1.Supposeb1beadesignparameterthatrangesfrom0to100.Itmayrepresentthe
valueofaphysicalcomponent(e.g.,massorresistance).FindtheeigenvaluesofBasa
functionofb1andplottheminthecomplexplane.Commentonyouranswer.Supposefor
certaindesignconsiderations(havingtodowithtransientresponse),wewanttheangleof
theeigenvaluestobe3/4.
Choosethevalueofthedesignparameterb1tosatisfythis
constraint.
5. LetAbeannxnmatrix.ShowthateigenvaluesofAarethesameasthoseofAT.
6. Considermatrices
12
45
31
369
248
A
B

pf2

Partial preview of the text

Download Homework 6 for Analytical Methods | EEL 3105 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!

EEL 3105 Fall 2011

Homework 6

Note: Problem 4 will be graded separately for assessing that one of the course objectives is

met. Please submit solution to Problem 4 as a separate document.

Consider matrices

1 2 1 2 3

A

k

B

b b

C

c c c

  1. For A, choose k = 3. Find the eigenvalues of the resulting matrix. Are they real? Why?
  2. Find a formula for eigenvalues of A as a function of k. Suppose k is a design parameter which ranges from ‐ 100 to +100. Use the formula to plot the eigenvalues of A as a function of k in the complex plane. Use MATLAB to verify your answer. Such a plot is called root locus. Comment on your results.
  3. Find characteristic polynomials of B and C in terms of b 1 , b 2 , c 1 , c 2 , and c 2. Do you see any pattern in your answer? Can you think of a way to generalize this to an nxn matrix?
  4. Fix b 2 =1. Suppose b 1 be a design parameter that ranges from 0 to 100. It may represent the value of a physical component ( e.g., mass or resistance ). Find the eigenvalues of B as a function of b 1 and plot them in the complex plane. Comment on your answer. Suppose for certain design considerations (having to do with transient response), we want the angle of

the eigenvalues to be  3  / 4.Choose the value of the design parameter b 1 to satisfy this

constraint.

  1. Let A be an nxn matrix. Show that eigenvalues of A are the same as those of A T^.
  2. Consider matrices

A

B

Calculate eigenvalues of AB and those of BA. You can use MATLAB to compute these answers. Please comment on your answers.

  1. Now suppose A and B are two matrices (not necessarily square) such that AB and BA are both well defined. Suppose is a non‐zero eigenvalue of AB. Show that it is also an eigenvalue of BA. Not for grading [hard problem for exploration and challenge]:
  2. Let us return to Problem 3. Recall I told you in class that in order to ensure stability, all eigenvalues should have negative real parts. If we have numerical values for the parameters b 1 , b 2 , (or c 1 , c 2 , and c 2 ), then we can use MATLAB to compute the eigenvalues and see if the real parts of eigenvalues are negative. But if they are unknown parameters, then this task is harder. This is especially so as the matrix size gets larger. How could you check if all eigenvalues have negative real parts in terms of the parameters? Explore the web resources to find an answer to this question.