Question

Case the confidence in the second-order approxima- tion during design, but then simulate the completed design. Let us look at an example that compares the responses of two different three- pole systems with that of a second-order system. Example 4.8 Comparing Responses of Three-Pole Systems PROBLEM: Find the step response of each of the transfer functions shown in Eqs. (4.62) through (4.64) and compare them. 24.542 T1 (s) 4s+24.542 (4.62) 245.42 T2(s) = (4.63) (s+10)(s24s+ 24.542) 73.626 +3) (s2 + 4s+ 24.542) (4.64) T3(s) SOLUTION: The step response, C;(s), for the transfer function, Ti(s), can be found by multiplying the transfer function by 1/s, a step input, and using partial-fraction expansion followed by the inverse Laplace transform to find the response, c(t). With the details left as an exercise for the student, the results are (4.65) C1(t) 1-1.09e2cos(4.532t -23.8°) c2(t) 1-0.29e10-1.189e 2cos(4.532t-53.34) C3(t)= 1-1.14e 0.707ecos(4.532t+78.63°) (4.66) (4.67) The three responses are plotted in Figure 4.24. Notice that c2(t), with its third pole at -10 and farthest from the dominant poles, is the better approximation of c1 (t), 1.4 T C1(t) 1.2 /c2(t) 1.0 e 0,8 C3(t) 0.6 Z 0.4 0.2 FIGURE 4.24 Step of system Ti(s), sys and system T3(s) 3.0 0.5 1.5 2.0 2 5 0 1.0 Time (seconds) Normalized response

Can you solve this ?

Answer 1

Solution:

The tranfr fanchin can be nmilten in pdnami fom 24.5h 2 s+S t 21.542 T(S) 245.42 S 3 65423t 245.42 73.626 52t 7536.5425t 73.626

The problem is completed by MATLAB tool

%-------- MATLAB CODE --------%

%--------- code starts here ----------%

%----- solution for T1-------%
num1=[24.542];
den1=[1 4 24.542];
T1=tf(num1,den1)
step(T1)
%-----------END ------------%
hold on
%----- solution for T2-------%
num2=[245.42];
den2=[1 14 64.542 245.42];
T2=tf(num2,den2)
step(T2)
%-----------END ------------%
hold on
%----- solution for T3-------%
num3=[73.626];
den3=[1 7 36.542 73.626];
T3=tf(num3,den3)
step(T3)
%-----------END ------------%
grid on;

%--------------- END -------------------%

PLOT:

Step Response 1.4 1,2 0.8 0.6 0,4 0.2 0.5 1.5 2 2.5 3 3.5 Time (seconds) Amplitude LO