You are provided with the following magnitude Bode diagram o

You are provided with the following magnitude Bode diagram of a 2^nd order system Can you estimate the system\'s relative overshoot to a unit step input?

Solution

solution:

1) when for sinusoidal input of system will produce response then it is shown on with respect to frequency of input signal

2) here resonant peak occure of magnitude of 3.33 from graph,

hence Mr=3.33

where damping ratio is obtain as follows

Mr=(1/2*zeta*(1-zeta^2)^.5)

hence here putting value we egt

zeta^4-zeta^2+.02250=0

hence zeta1^2=.9769 and zeta2^2=.02303

hence for positive damping ratio we get

zeta2=.9884 and zeta1=.1517

3) here percentage overshoot mean maximum deviation from average as

Mp%=exp(-pi*zeta/(1-zeta^2)^.5)*100

for zeta2=.1517 we get

Mp1=exp(-pi*.1517/(1-.1517^2)^.5)*100=61.73%

where overshoot means error in input and output

output=input-Mp%

for input=100%

output=38.27%

4)here relative overshoot mean with respect input as

relative overshoot=(percentage overshoot/percentage output)=61.73/38.27=1.6130

hence it is giving response at overshoot 1.61 times average output for zeta1=.1517

5) fr zeta2=.9884

we get Mp%=1.3199*10^-7

output=input-Mp%=100-1.31*10^-7=99.999%

hence relative overshoot=1.3199*10^-7/99.999=1.31*10^-9 it is too small value hence

6) overshoot occure at zeta1=.1517 and relative overshoot is1.61 for step input of 100%