hw8_sol - MEEN 364 Fall 2004 Homework Set 8 Due October 28,...

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MEEN 364 Homework Set 8 Fall 2004 October 21, 2004 1 Homework Set 8 Due October 28, 2004 @ 5:00 PM 1. Consider the following system: 1 11 2 22 111 222 1 1 6. 5 0 10 1 0 00 0 1 x xu x y xu --    =+       (a) Plot all the unit-step responses. You may use Matlab for plotting purpose, but must derive the transfer-function matrix with only pencil and paper (Of course Matlab has this capability, but we still urge you to do it by hand.). Attach the MATLAB code along with the plots. Make sure to label the plots and add comments in the code. Note that the system has two inputs and two outputs, so you will find a 2 by 2 transfer-function matrix and need to plot four unit-step responses. Although it is not necessary to obtain the transfer matrix expression for the system to obtain the unit-step response curves with MATLAB, we shall derive such an expression for reference. For the system defined by x A x Bu y C x Du (1) The transfer matrix G(s) is a matrix that relates Y(s) and U(s) as follows: ( ) () () Y s GsUs = (2) Taking Laplace transforms of the state-space equations, we obtain ( ) (0 ) ( ) () ( ) ( ) sX s x AX s BUs Y s CX s DUs - (3) In deriving the transfer matrix, we assume that x(0)=0. Then, from equation (3), we get 1 1 ( ) ( ) ( )( ) X s s IA Y s C s IAB DUs - - =- =-+ (4) Thus the transfer matrix G(s) is given by 1 ( ) G s IABD - (5) The transfer matrix G(s) for the given system becomes
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MEEN 364 Homework Set 8 Fall 2004 October 21, 2004 2 1 1 2 2 ( ) () 1 0 1 1 1 1 1 11 1 0 1 6. 5 1 0 6. 5 110 6.5 1 1 7. 5 6.5 6.5 G s C s IAB ss s - - =- +-  ==  -+ ++  - = + Hence 22 11 1 ( ) 6. 5 6.5 ( ) 7. 5 6.5 6. 5 6.5 Y s Us s s Y s s s s - + + = + + + Since this system involves two inputs and two outputs, four transfer functions may be defined depending on which signals are considered as input and output. Note that, when considering the signal u1 as the input, we assume that signal u2 is zero, and vice versa. The four transfer functions are 12 ( ) 1 , ( ) 5 ( ) 6.5 ( ) 7. 5 6.5 , ( ) 5 ( ) 6.5 Y s Ys U s U sss Y s s U s U - + + + + + The four individual step-response curves can be plotted by the use of command Step(A,B,C,D) 0 2 4 6 8 10 12 -0.4 -0.2 0 0.2 0.4 G11 from u1 (u2=0) time (s) y1 0 2 4 6 8 10 12 -0.2 -0.1 0 0.1 0.2 0.3 G12 from u2 (u1=0) time (s) y2 0 2 4 6 8 10 12 0 0.5 1 1.5 2 G21 from u1 (u2=0) time (s) 0 2 4 6 8 10 12 0 0.5 1 1.5 2 G22 from u2 (u1=0) time (s) Fig. 1. Step response of four transfer functions
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MEEN 364 Homework Set 8 Fall 2004 October 21, 2004 3 Matlab Script 1 % Figure 1 % Step-response curves for system defined in state space. % In this problem we plot step-response curves of a system % having two inputs (u1 and u2) and two outputs (y1 and y2) clear all;close all;clc; % Define the state space form A=[-1 -1;6.5 0]; B=[1 1;1 0]; C=[1 0;0 1]; D=[0 0;0 0]; %sys=ss(A,B,C,D) % Define transfer functions [num1,den1]=ss2tf(A,B,C,D,1); % For the input u1 G11=tf(num1(1,:),den1) % u1 to y1 G21=tf(num1(2,:),den1) % u1 to y2 [num2,den2]=ss2tf(A,B,C,D,2); % For the input u2 G12=tf(num2(1,:),den2) % u2 to y1 G22=tf(num2(2,:),den2) % u2 to y2 % Step response curves when the input is u1.
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hw8_sol - MEEN 364 Fall 2004 Homework Set 8 Due October 28,...

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