A second-order system is excited by a unit step and the response is as illustrated in Figure E.37. Write an expression for the transfer function of the system.
From the graph it is apparent that system is highly underdamped and that the final value of the step response is 0.1. So this second-order system has no zeros at zero. Therefore the general form of the transfer function of this second-order system is
and . From the graph there are 10 ringing cycles of response between 0 and 10 seconds. Therefore the resonant frequency is approximately 1Hz or radians per second. The impulse response is of the form,
So the characteristic exponential decay has a time constant of 
From the graph, the time constant is approximately the time at which the ringing is at 36.8% of its maximum value. That is at about 10 seconds. Therefore
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