EE 4343/ Control System Design Project
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1 Copyright F.L. Lewi 2004 All right reerved EE 4343/ Control Sytem Deign Project Updated: Sunday, February 08, 2004 Background: Analyi of Linear ytem, MATLAB Review of Baic Concept LTI Sytem LT I Sytem Repreentation Tranfer function. a. Laplace Tranform and notation. b. Sytem zero, ytem pole, pole zero map. c. Invere Laplace tranform. d. Partial fraction expanion. e. Initial and final value theorem Differential equation. a. Solution of Ordinary Differential Equation with zero Initial Condition. b. ODE Tranfer function. Impule Repone. a. Time domain and frequency domain meaning. Block Diagram. a. Serie Interconnection. b. Parallel Interconnection. c. Parallel Interconnection. d. Superpoition theorem example. Flow Graph. MATLAB Introduction MATLAB REVIEW FOR BEGINNERS: % 3d PLOT GENERATION FOR SECOND ORDER SYSTEMS clear all; cloe all; n=;y=zero(200,);i= for zeta=.:.: d=[ 2*zeta ]; t=[0:.:9.9]';
2 y(:,i)=tep(n,d,t); i=i+; end meh(y,[7 8]); ; % SECOND ORDER APPROXIMATION BY STEP RESPONSE cloe all; clear all; %PLANT a=2; b=5; J=0.009;J2=0.002; c=0.007;c2=0.002; k=.32; khw=7.2; N=[c2 k]/khw; D=[J*J2 (c*j2+c2) (k*(j+j2)+c*c2) 5*(c+c2) 0]; G=tf(conv(N,[ a]),conv(d,[ b])); %open loop circuit W=feedback(G,); %cloed loop % % Step Repone
3 tep(w,0) title('repone without Approximation') %Step Repone Analyi %Period T= ; %natural Frequency wn=2*pi/t; %Rie time tr= ; %Percentage Overhoot POV=32.4; % Damping ratio z=((log(pov/00))^2/((log(pov/00))^2+pi^2))^.5 %Second Order Approximation W_ec=tf([wn^2],[ 2*z*wn wn^2]) %Step Repone tep(w_ec) title('second Order') hold tep(w,0); tep(w_ec,0); % SECOND ORDER SYSTEMS APPROXIMATION BY NEGLECTING %FAST POLES clear all; cloe all; p=poly([-+2*i --2*i -+0*i --0*i -0]); h=tf(5050,p) tep(h); approx_p=poly([-+2*i --2*i]); approx_h=tf(5,approx_p) tep(approx_h); hold tep(h,0) tep(approx_h,0) n=;y=zero(200,);i=
4 How to Ue lim LSIM Simulate time repone of LTI model to arbitrary input %SIMULATION USING LSIM clear all cloe all % imulate the repone of a ingle-input model SYS to the input % u(t)=in(t) during 5 econd. t = 0:0.0:5; u = in(t); y=tf([],[5 ]);% Defining the tranfer dunction y=lim(y,u,t); ize(y);% To know about the dimenion of y ize(t); %To know about the dimenion of y t=t'; plot(t,y); ; How to make a uer defined oure in SIMULINK To make a ource which i not available SIMULINK ource library, one can write the function or the eaier way to do it i by uing one of the exiting ource, for example we want to plot output of the ytem H() if input U ( ) =, Let u make a ytem C() which i having input a one of the available ource (tep U i ( ) = ) and output Y ( ) = i 2 + 2, later we can connect output of C() to input of H() to get the required output. Here we can write Y ( C( ) = Ui ( 2 ) + 2 ( ) i ) () %CONTENTS tutorial for EE434 Steven Scully cully@airmail.net %File tep_imp.m Plotting tep and impule repone of a ytem (phae plane alo) %File 2 root_ft.m Multiplying and factoring the root of a polynomial. %Partial fraction expanion. plotting a function of time
5 %File 3 Limit.m Converting a tranfer fn to tate pace % and back to a tranfer fn. plotting %File 4 call_.m and tatep.m ODE23 and phae plane %File 5 bodeit.m Bode plot %File 6 rootloc.m Root locu (auto range, manual, and finding root on the locu) %for help on any function type "help and the function name" %E.g. "help bode" will decribe how to generate Bode plot. %Save each file a a *.m file (go to menu and elect NEW m file) invoke %the file by typing the file name in the MatLab command window. %Note: you will need to change the directory to the location of your file; %the default directory of MatLab i C:\MatLab\Bin. MatLab directorie are %changed the ame way that they are in DOS; e.g. type CD\ CD your_dir at the %MatLab command window. %tep_imp.m %plot the tep repone for ( + 3)/(^2 + 5* +4) num=[ 3]; den=[ 5 4]; %define the ytem t=0:0.0:0; %et the time pan y=tep(num,den,t); %compute the tep repone plot(t,y); xlabel('time (ec)'); ylabel('sytem tep Repone'); title('sytem repone with a unit tep input'); grid; %plot the impule repone for ( + 3)/(^2 + 5* +4) y2=impule(num,den,t); ; %ue thi to create a new graph (without thi we draw over the old one) plot(t,y2); xlabel('time (ec)'); ylabel('sytem impule Repone'); title('sytem repone with a impule input'); grid; % plot tep, impule and phae plane on ame graph ; ubplot(2,2,) %tep rep plot(t,y); xlabel('time (ec)'); ylabel('sytem tep Repone'); title('sytem tep Repone'); grid; ubplot(2,2,2) %impule rep plot(t,y2); xlabel('time (ec)'); ylabel('sytem impule Repone'); title('sytem impule Repone'); grid;
6 ubplot(2,2,3) %phae plane (ydot v y) plot(y,y2); xlabel('y'); ylabel('ydot'); title('phae plane ydot v y'); grid; %root_ft.m %multiply 0 + % * % + 9 ^2 + 5* +6 clear %get rid of old value num = [0,] d=[,9] d2=[,5,6] den=conv(d,d2); %Leaving the ";" off will print the anwer %conv multiplie the polynomial together root_of_den = root(den) %find the root % ue reidue to find the partial fraction expanion of ()/(^2 + 3* +2) %the anwer i -/(+2) + /(+) clear [pfract_coef,pole,direct]= reidue(,[,3,2]) %partial fraction expanion %note that we are given the POLES not the expanded denominator %plot the function of time t=0:0.05:0; %note that a period i needed in (t.^2). and in exp(0.5*t). % y= (t.^2).*exp(-9.*t) + exp(0.5*t).*( 2*co(3*t)) ; plot(t,y); xlabel('time(ec)'); ylabel('y'); title('function of time'); grid; Problem:
7 A ytem ha tranfer function H ( ) = a. Find pole and zero of the ytem. b. Plot pole and zero of the ytem uing MATLAB. c. Plot Impule repone uing MATLAB. d. Plot tep repone uing MATLAB. t e. Find output y (t) if input iu( t) = e. 2. For the ytem in problem a. Make a SIMULINK Model file. b. Plot output y (t) if input i U ( ) = uing SIMULINK. + c. Plot output (e) and 2(b) on a ame plot uing MATLAB. d. Find the DC gain of the ytem.
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