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### lecture8

Course: EE 221A, Fall 2007
School: Berkeley
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Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 1Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2007Issued 9/4; Due 9/13Problem 1: Fields.(a) Dene addition and multiplication on cfw_0, 1 to form a eld. Show th
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 2Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2007Issued 9/18; Due 9/27Problem 1: Useful properties of eigenvalues. Let A Rnm , B Rmn and let n m. Observe that
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 3Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2007Issued 9/27; Due 10/4Problem 1: Dynamical systems, time invariance.Suppose that the output of a system is rep
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 4Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2006Issued 10/9; Due 10/18Dear 221A folks: Some problems like the ones below may be included on the midterm on Oct
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 5Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2007Issued 10/23; Due 11/1Problem 1.Suppose A Cnn is such that det(A) = 0. Is det(eA ) = 0? Explain why or why no
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 6Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2007Issued 11/6; Due 11/15Problem 1: Sti Dierential Equations.In the simulation of several engineering systems we
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 7Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2007Issued 11/15; Due 11/27Problem 1. Show thatA0c0,b0where A Rnn , b Rn , cT Rn is completely controllable
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 8Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2007Issued 11/27; Due 12/6Problem 1: State vs. Output Feedback.Consider the plant described by:XywhereA=07
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 9Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2007Issued 12/4; Due 12/15221A folks: Homework 9 is due at the nal exam. However, if youd like to hand in your hom
Berkeley - EE - 221A
1._ _ _ .-L.L-.-L-+-.) E GCS2-2-1Levl-vr'~ LA(G0A L~:0. nj Y1~cmU-fkhr 0 a&lt;vu e.,V-, mojmo~of fA'lJ il'\J.Lr i1-cr.O J -.~c: . :.'-'. . J.\ O~fa'-I1v- b-v- () fA.- tJLa yv~r Artd~/T1 w1J):-mo~Jzw- ()( Y1~ a nicfw_!;O'YZ
Berkeley - EE - 221A
EE221A Linear System TheoryFinal ExamProfessor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 201012/14/10, 3-6pmYour answers must be supported by analysis, proof, or counterexample.There are 9 questions: Pleas
Berkeley - EE - 221A
Berkeley - EE - 221A
EE221A Linear System TheoryMidterm TestProfessor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 200710/16/07, 9.30-11.00amYour answers must be supported by analysis, proof, or counterexample.There are 6 questio
Berkeley - EE - 221A
EE221A Linear System Theoryhttp:/inst.eecs.berkeley.edu/ee221a/Course OutlineProfessor C. TomlinDepartment of Electrical Engineering and Computer SciencesUniversity of California at BerkeleyFall 2011Lecture InformationLectures: TuTh 9.30-11, 521 C
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 1Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 9/1; Due 9/8Problem 1: Functions. Consider f : R3 R3 , dened1f (x) = Ax, A = 00as00110 , x R3
Berkeley - EE - 221A
EE221A Problem Set 1 Solutions - Fall 2011Note: these solutions are somewhat more terse than what we expect you to turn in, though the important thing isthat you communicate the main idea of the solution.Problem 1. Functions. It is a function; matrix m
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 2Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 9/8; Due 9/16All answers must be justied.Problem 1: Linearity. Are the following maps A linear?(a) A(
Berkeley - EE - 221A
EE221A Problem Set 2 Solutions - Fall 2011Problem 1. Linearity.a) Linear: A(u(t) + v (t) = u(t) + v (t) = A(u(t) + A(v (t)b) Linear:te (u(t ) + v (t )d =A(u(t) + v (t) =0te u(t )d +0te u(t )d0= A(u(t) + A(v (t)c) Linear:2s2A(a1 s + b1 s
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 3Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2010Issued 9/22; Due 9/30Problem 1. Let A : R3 R3 be a linear map. Consider two bases for R3 : E = cfw_e1 , e2 , e
Berkeley - EE - 221A
EE221A Problem Set 3 Solutions - Fall 2011Problem 1.a) A w.r.t. the standard basis is, by inspection,000 4 .022AE = 10b) Now consider the diagram from LN3, p.8. We are dealing with exactly this situation; we have one matrixrepresentation, and tw
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 4Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 9/30; Due 10/7Problem 1: Existence and uniqueness of solutions to dierential equations.Consider the fo
Berkeley - EE - 221A
EE221A Problem Set 4 Solutions - Fall 2011Problem 1. Existence and uniqueness of solutions to dierential equations.TTCall the rst system f (x, t) = x1 x2and the second one g (x) = x1 x2.a) Construct the Jacobians:D1 f (x, t) =1 et sin (x1 x2 )et
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 5Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 10/18; Due 10/27Problem 1: Dynamical systems, time invariance.Suppose that the output of a system is r
Berkeley - EE - 221A
EE221A Problem Set 5 Solutions - Fall 2011Problem 1. Dynamical systems, time invariance.i) To show that this is a dynamical system we have to identify all the ingredients:First we need a dierential equation of the form x = f (x, u, t): Let x(t) = y (t)
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 6Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 10/27; Due 11/4Problem 1: Linear systems. Using the denitions of linear and time-invariance discussed i
Berkeley - EE - 221A
EE221A Problem Set 6 Solutions - Fall 2011Problem 1. Linear systems.a) Call this dynamical system L = (U , , Y , s, r), where U = Rni , = Rn , Y = Rno . So clearly U , , Y are alllinear spaces over the same eld (R). We also have the response map(t, t0
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 7Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 11/3; Due 11/10Problem 1.A has characteristic polynomial (s 1 )5 (s 2 )3 , it has four linearly indepe
Berkeley - EE - 221A
EE221A Problem Set 7 Solutions - Fall 2011Problem 1.With the given information, we can determine the Jordan form J = T AT 1 of A to be,10J =1110111200120.012Thus,cos e10=cos eJe1 sin e1cos e1and cos eA = T 1 cos eJcos e01
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 8Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 11/10; Due 11/18Problem 1: BIBO Stability.fH , THiTHTCfC , TCVHVCiFigure 1: A simple heat ex
Berkeley - EE - 221A
EE221A Problem Set 8 Solutions - Fall 2011Problem 1. BIBO Stability.a) First write this LTI system in state space form,x = Ax + Bu=( +fC )VCVH=0.3 0.20.2 0.3y = Cx =10VC( +fH )VH01fCVCx+0.10x+0fHVH000.1u,uxwhere x := (TC
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 9Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 11/21; Due 12/1Problem 1: Lyapunov Equation.(a) Consider the linear map L : Rnn Rnn dened by L(P ) = A
Berkeley - EE - 221A
EE221A Linear System TheoryProblem Set 10Professor C. TomlinDepartment of Electrical Engineering and Computer Sciences, UC BerkeleyFall 2011Issued 12/2; Due 12/9Problem 1: Feedback control design by eigenvalue placement. Consider the dynamic system:
Berkeley - EE - 221A
8/26/11EE221A Section 11Administrivia1.1Section and Oce Hours SchedulingIf you have a conict, please contact Prof. Tomlin and/or Pat by email and let us know about it.Well try to accomodate everyone but it may not be possible.1.2Homework collabor
Berkeley - EE - 221A
9/2/11EE221A Section 21Fields1. Show that the set cfw_0, 1, with multiplication dened as binary AND and additiondened as binary XOR, is a eld. (AND)01+ (XOR) 0 10011100100012. Show that F, 0 = 0 = 0.2Vector Spaces1. Does C form a vecto
Berkeley - EE - 221A
9/9/11EE221A Section 311.1Functions, linear mapsSolutions to linear equationsTheorem. (range and nullspace of linear operators) [LN3 p. 4]Consider A : U V with (U, F ), (V, F ) linear spaces. Let b V . Then:a) A(u) = b has at least one solution b
Berkeley - EE - 221A
9/16/11EE221A Section 41Change of basisExercise 1. [LN3, p. 10]Let A : R3 R3 be a linear map. Consider 1B = cfw_b1 , b2 , b3 = 0 , 0 1C = cfw_c1 , c2 , c3 = 1 , 001 ,001 ,100 ,110 .1Clearly B and C are bases for R3 . Suppose A
Berkeley - EE - 221A
9/23/11EE221A Section 51NormsExercise 1. Prove that x Rn , xx1n xExercise 2. In R2 , sketch the unit sphere B =p = . What about 0 &lt; p &lt; 1?2x: xp=1for p = 1, p = 2,Complete (Banach) SpacesExercise 3. Let X be the space of real-valued contin
Berkeley - EE - 221A
9/30/11EE221A Section 61Singular Value DecompositionExercise 1. Show that the eigenvalues of a Hermitian matrix are all real.Exercise 2. Show that AA , for A Cmn , is positive semidenite.Exercise 3. Consider a real unitary matrix U R33 . Give a geom
Berkeley - EE - 221A
10/7/11EE221A Section 71Practice midtermProblem 1. Injectivity and surjectivitya) Suppose that T : V W is an injective, linear map, and that cfw_v1 , . . . , vn is a linearly independent set in V . Prove that cfw_T (v1 ), T (v2 ), . . . , T (vn ) is
Berkeley - EE - 221A
10/21/11EE221A Section 81Administrivia Midterms still being graded. HW5 is out, due next Thurs (Oct 27) GSI oce hours Mon Oct 24th time change to 1 PM (still in 504 Cory)2Dynamical systemsExercise 1. Show that the following system is time invaria
Berkeley - EE - 221A
10/28/11EE221A Section 91Administrivia Midterm avg 28, median 29, std dev 4.9 (out of 34) HW6 is out, due next Fri (Nov 4)2Cayley-Hamilton TheoremRecall:Characteristic polynomial of A: A (s) := det (sI A) = sn + d1 sn1 + + dnCharacteristic equat
Berkeley - EE - 221A
11/04/11EE221A Section 101Direct sum of subspacesExercise 1. Show that if V = V1 V2 Vn , then Vi Vj = cfw_ for i = j .Exercise 2. Let M and N be two subspaces of V . Let cfw_m1 , . . . , mp be a basisfor M , and cfw_n1 , . . . , nk be a basis for
UMBC - ENME - 110
Part II Answer the following questions in the space provided and then tear off this pageand turn it in with your scantron answer sheet for Part I.Name:(Please print first and last name neatly)1. (14 points) Consider the following reaction.HH 3COCH
UMBC - ENME - 110
Part II Answer the following questions in the space provided and then tear off this pageand turn it in with your scantron answer sheet for Part I.Name:KEY(Please print first and last name neatly)1. (14 points) Consider the following reaction.HH3CO
UMBC - ENME - 110
Part I, K=01. What statement about the bonding in aspirin (shown below) is true?OCOOHOA)B)C)D)it has 11 bonds and 4 bondsit has 14 bonds and 5 bondsit has 18 bonds and 4 bondsit has 21 bonds and 5 bonds2. Consider the following acid-base equi
UMBC - ENME - 110
Part II Answer the following questions in the space provided and then tear off this pageand turn it in with your scantron answer sheet for Part I.Name:(Please print first and last name neatly)1. (18 points) Monochlorination of 2-methylbutane produces
UMBC - ENME - 110
Part II Answer the following questions in the space provided and then tear off this pageand turn it in with your scantron answer sheet for Part I.Name:(Please print first and last name neatly)1. (20 points) Monochlorination of 2-methylbutane produces
UMBC - ENME - 110
Part I, K=01. What is the relationship between the following structures?Cl BrCl BrCH 3H 3CCH 3H 3CHHA)B)C)D)HHidenticalenantiomersdiastereomersnone of the above2. Consider the following energy versus reaction coordinate diagram:EnergyR
UMBC - ENME - 110
Part II Answer the following questions in the space provided and then tear off this pageand turn it in with your scantron answer sheet for Part I.Name:(Please print first and last name neatly)1. (30 points) Draw the MAJOR product(s) for each of the fo