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### hw7

Course: ASE 330M 18510, Spring 2011
School: University of Texas
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Word Count: 581

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330M ASE Linear System Analysis Unique Number: 12495, Spring 2006 Homework #7 Date given: April 27, 2006 Date Due: May 4, 2006 1. An LTI system has a step response (et cos2 t)u(t). Determine an ordinary dierential equation representation for this system. 2. An LTI system at zero initial conditions is subject to an input signal x(t) = [sin(2t)]u(t). The resulting output is designated by y (t) = [4et + 2 sin(2t) 4...

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330M ASE Linear System Analysis Unique Number: 12495, Spring 2006 Homework #7 Date given: April 27, 2006 Date Due: May 4, 2006 1. An LTI system has a step response (et cos2 t)u(t). Determine an ordinary dierential equation representation for this system. 2. An LTI system at zero initial conditions is subject to an input signal x(t) = [sin(2t)]u(t). The resulting output is designated by y (t) = [4et + 2 sin(2t) 4 cos(2t)]u(t). (a) Determine an ordinary dierential equation representation for this system. (b) Determine whether the system is BIBO stable. (c) Depending on your answer in part (b), evaluate (if possible) the frequency response function H (j ) for this system. Further, obtain expressions for the magnitude and argument (phase angle) of the frequency response function. (d) Analytically determine the frequency = max at which the magnitude of the frequency response function has a maximum value. Also calculate the magnitude of the frequency response function at frequency max . (e) Use MATLAB commands impulse and step to plot the impulse and step responses for this system on the range 0 t 30. Submit both the plots and your source code. 3. An LTI system has transfer function H (s) = s2 + 2s (s2 + 3s + 2)(s2 + s 2) (a) Determine an ordinary dierential equation representation for this system. (b) Determine whether the system is BIBO stable. (c) Depending on your answer in part (b), evaluate (if possible) the frequency response function for this system. Further, obtain expressions for the magnitude and argument (phase angle) of the frequency response function. 4. The transfer function of an LTI system is described by H (s) = sa , s+a a>0 where is a a real constant. (a) Determine whether the system is BIBO stable. What happens when a 0? (b) Sketch the pole-zero diagram for this system. (c) Do you notice any symmetry in the pole-zero diagram about the imaginary axis? 1 (d) Evaluate the frequency response function for this system. Further, prove that the magnitude of the frequency response function is unity for all frequencies. (e) Comment on the magnitude of the frequency response function for the general case where any BIBO stable LTI system has n number of zeros and n number of poles having a symmetry about the imaginary axis. Any system with such properties is known as an allpass system. Remember, since the so called all-pass system is stable, all poles are located on the left half of the complex plane and accordingly, all the zeros are (symmetrically) placed on the right-half of the complex plane. 5. A second-order causal LTI system with non-zero initial conditions is subject to an input x1 (t) = u(t). The resulting output is determined by y1 (t) = 1 + e2t for all t 0 . From the same initial conditions, when an input x2 (t) = [sin(t)]u(t) is applied, the system produces an output y2 (t) = 2 cos(t) sin(t) for all t 0 . (a) Derive the transfer function of this system. (b) Calculate the non-zero initial conditions of this system. (c) Now, suppose the same system is set back to zero initial conditions and an input x(t) = [sin(t)]u(t) is applied [same as x2 (t)!]. Use MATLAB lsim command to generate the output of the system on 0 t 50. Graphically compare this result with y2 (t) that was produced as output of the same system. Explain why there is a dierence between these two quantities. 2
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University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12495, Spring 2006Homework #7Date given: April 27, 2006Date Due: May 4, 20061. An LTI system has a step response (et cos2 t)u(t). Determine an ordinary dierential equation representation for this system.
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12495, Spring 2006Revised Homework #1Math ReviewComplex Numbers and Ordinary Dierential EquationsDate given: January 24, 2006Date Due: February 2, 20061. Convert the following complex numbers to polar c
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12495, Spring 2006Revised Homework #1Math ReviewComplex Numbers and Ordinary Dierential EquationsDate given: January 24, 2006Date Due: February 2, 20061. Convert the following complex numbers to polar c
University of Texas - ASE 330M - 18510
2/16/06 12:24 PM\ase-lrc-server\home\rdc55\Desktop\Linear\systems_2hw.mt=[-5:.01:5]d=[-2*pi:(4*pi/1000):2*pi]t1=t[y1]=fx(t1)figure (1)y1=sin(y1)plot (d,y1)t2=2*t[y2]=fx(t2)figure (2)plot (t,y2)t3=t[y3]=fx(t3)figure (3)plot (t,y3)t4=2*t[y
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE330MHomework#6DueNovember16th,2007.Problem#1Consideramassonaflatnonsmoothsurface,subjecttoaforce F ( t ) .Thesurfacefrictionisdirectlyproportionaltothevelocityoftheparticle, x ,throughafrictioncoefficient .Thus,theequationofmotionfortheparticlecan
University of Texas - ASE 330M - 18510
ASE330MHomework#6DueNovember16th,2007.Problem#1Consideramassonaflatnonsmoothsurface,subjecttoaforce F ( t ) .Thesurfacefrictionisdirectlyproportionaltothevelocityoftheparticle, x ,throughafrictioncoefficient .Thus,theequationofmotionfortheparticlecan
University of Texas - ASE 330M - 18510
ASE330MHomework#3Problem#1:Findtheequilibriumsolutions,iftheyexist,foreachofthefollowingsystems:(a)x1 = x2x2 = x1 + 1(b)(c)x1 = x2x2 = 1x1 = x2x2 = x1 x13(d)x1 = x2x2 = 0Problem#2:Obtainallequlilibriumsolutionsforthefollowingsystem:( cos
University of Texas - ASE 330M - 18510
ASE330MHomework#3Problem#1:Findtheequilibriumsolutions,iftheyexist,foreachofthefollowingsystems:(a)x1 = x2x2 = x1 + 1(b)(c)x1 = x2x2 = 1x1 = x2x2 = x1 x13(d)x1 = x2x2 = 0Problem#2:Obtainallequlilibriumsolutionsforthefollowingsystem:( cos
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330MProblem Set 71. Use Matlab to plot the following functions for 1 t 5 .a. x ( t ) = 3e2t 1( t )b. x ( t ) = e2t ( sin 3t )1( t )c.x ( t ) = e j 3t 1( t ) 1( t 3) 2. Sketch, by hand, the following signalsa. x ( t ) = 1( t + 1) 2 1( t 1) + 1(
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE330MHomework#7(ForExtraCreditOnly)Due:December7th,2007Problem#1:UsethemethodofpartialfractionexpansiontoidentifytheinverseLaplacetransformofthefollowingfunctions:(a) F ( s ) =1s ( s + 1)(b) F ( s ) =10e 3 s ( s + 2 )( s + 10 )(c) F ( s ) =(
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
% Homework 5, Problem 1clear allclose allclc% Problem 1a.) Plot x(t) = 3exp(-2*t)*1(t)t = (-1:0.01:5);jj_minus = find(t&lt;0);jj_plus = find(t&gt;=0);one = zeros(size(t);one(jj_plus) = ones(size(t(jj_plus);x = 3*exp(-2*t).*one;subplot(3,1,1)plot(t,x
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12880, Spring 2007Revised Homework #1Review on Complex Number VariablesDate given: January 25, 2007Date Due: February 1, 20071. Convert the following complex numbers to polar coordinate (exponential) for
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12880, Spring 2007Homework #2Math ReviewOrdinary Dierential Equations &amp; Matlab ExercisesDate given: February 6, 2007Date Due: February 15, 20071. State whether the following ordinary dierential equation
University of Texas - ASE 330M - 18510
% Problem B14w1=exp(j*pi/2)*ones(3,1)-[2*exp(j*pi/6);2*exp(3*j*pi/2);2*exp(17*j*pi/6)];% Problem B.22 (a)t1 = [0:0.01:10]';x1 = real(2*exp(-1+2*pi*j)*t1);% Problem B.22 (b)x2 = imag(3*ones(length(t1),1) - exp(1-2*pi*j)*t1);% Problem B.22 (c)x3 = 3
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12880, Spring 2007Homework #3Basics of Signals and SystemsDate given: February 22, 2007Date Due: March 6, 20071. The systems given below have input x(t) and output y (t) respectively. Determinewhether e
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12880, Spring 2007Homework #4Impulse and Step Response for LTI SystemsDate given: March 1, 2007Date Due: March 20, 20071. An LTI system has an impulse response h(t) = (et + sin t)u(t).(a) Evaluate the s
University of Texas - ASE 330M - 18510
ASE330MHomework4Lasttime(springmassdamper):4,5eigenvaluesatcosForsin0015Therefore,,5where, 15,,152ForHomework:(1) ForthesameproblemdiscussedaboveFindfor2;a)b) sinc);givenby01;003;01;1;00(2) Verifythesolutionsint
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12880, Spring 2007Homework #5Frequency Response FunctionBIBO StabilityDate given: March 29, 2007Date Due: April 10, 20071. An LTI system has a step response (et sin t)u(t).(a) Determine whether the sys
University of Texas - ASE 330M - 18510
% Homework 5, Problem 1clear allclose allclc% Problem 1a.) Plot x(t) = 3exp(-2*t)*1(t)t = (-1:0.01:5);jj_minus = find(t&lt;0);jj_plus = find(t&gt;=0);one = zeros(size(t);one(jj_plus) = ones(size(t(jj_plus);x = 3*exp(-2*t).*one;subplot(3,1,1)plot(t,x
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12880, Spring 2007Homework #6Date Given: Wednesday, April 11, 2007Due Date: Monday, April 23, 2007For this homework, when necessary, you are permitted use of MATLAB and/or calculator inevaluating roots o
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12880, Spring 2007Homework #7Date Given: April 24, 2007Due Date: May 1, 2007For this homework, unless otherwise indicated, you are permitted use of MATLAB and/or calculatorin evaluating roots of polynomi
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330M Linear System AnalysisUnique Number: 12880, Spring 2007Homework #8Date Given: May 1, 2007Due Date: May 9, 2007 at 10.00am1. An LTI system at zero initial conditions is subject to an input signal x(t) = te3t u(t). Theresulting output is desi
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE 330M Linear Systems: Fall 2008Homework #3: Due 10/7/2008 by 5PMConsider the linear systems described by the differential equations below:a)D2 5 D 6 y t D 1 u t u t 6y 0 2, y 0 1b)D2 4 D 4 y t Du t u t sin 3ty 0 3, y 0 4c)D D 1 y t D 2
University of Texas - ASE 330M - 18510
ASE330MHomework#4Problem#1:1.22(Textbook)Problem#2:Consideradifferentialequationin y t characterizedbydifferentialoperatorsQ D D 2 D 3 ,P D 1.Supposethesystemisinitiallyatrest( y 0 0 , y 0 0 ).Findthecompletesolutioniftheforcinginputsignalisgiven x
University of Texas - ASE 330M - 18510
ASE330MHomework#6DueDecember5th,2008ReadingAssignment:Sections4.5,4.7,and4.8Problem#1,Fortheunityfeedbacksystembelow,identifythefollowingquantities:(a)Theplanttransferfunction, Y s / U s (b)Thecontrollertransferfunction, U s / E s (c)Theclosedloopt
University of Texas - ASE 330M - 18510
ASE 330M Homework #1 (Review of Vectors &amp; Dynamics)Due: Friday, September 14th.Reading Assignment: Read Chapters 1-3 of Torby (available as PDF from blackboard)Problem #1:During the encounter with the asteroid Gaspra, the Galileo spacecrafts position
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
ASE330MFall2008:Homework#2DueMonday,September29thThe diagram below illustrates a particle P of mass m that is constrained to move radially along arecessed channel on a spinning turntable. The motion of the mass is subject to some stiffness andfriction
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
University of Texas - ASE 330M - 18510
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University of Texas - ASE 330M - 18510
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University of Texas - ASE 330M - 18510
~\i\ ~ ca M~\s ~ 0 oh. A.r, e=:5(tF BD: l+- e~&quot;bl' - v'-~e -op'PA13r~&quot; 'e3&quot;'-tl-0H--0M =~'D-0M. a&quot;'AA:-=-)v\b~-~-~-\JX:it 13-t A\'L b&quot;z +.&quot;' +~f\b~ + ~1-b~--~-~-At=JJ\'J,00twJ,z'e :~-.t\-lfoM.\1&amp;-2~-f/
University of Texas - ASE 330M - 18510
-LA01cfw_\ ~Cc:~l)c/c.l.A1- 00D0:=.bii&quot;,1tz,US! 1tu&quot;A.lIJA-xISlkoWv--=To0,;:c:= C./ 0Thl'2--.B,,, C&gt; IIl:)tJ~0olC&gt;D\&quot;3~'2M0=--_0+tIe.~cw-C/oetC;e~cw00eo- 0_0t\-2.1 n ,3t\~ee-1)H-,
University of Texas - ASE 330M - 18510
l et-iu~ e- =tt= lR.9/ tol~oJLoAQ.(2i(~ [N+eJ'(a-h'~ = 7 A-PPR-at(.of- ~ '1Jw-llv\&amp;rl&quot; oldN/vv\l(! bdw~. t -.- -l ~ tLll\.tho.R &quot;I\cfw_)J(\J~ ~.Ie i. ~ t\M-tt f ex):=; ; = .;.~te I/t'Ll-ov- ~ ilt;)f~L- ~.of- 1:Xlt~ : I lUIW cY cDNd. i~' ~-=
University of Texas - ASE 330M - 18510
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