Question

Unit Step Response .A plant has the response, c(), to a unit step, as shown. 3.5 a. From the graph, estimate 3 3 the system's time constant, 5 % overshoot and DC gain. 2 1.5 c. Using the information, find o.5 b. What is the system's damped natural frequency and damping ratio? the second order transfer function C(s)/R(s). 0.2 0.4 0.6 0.8 1.2 Time (sec)

Answer 1

From the step response plot, the peak overshoot defined as

3-2 henk-ysteady-state My ะ ysteady-state

Also, the relationship between M_p and damping ratio $\zeta(0 \leq \zeta<1)$   is given by:

in terms of ζ:

$\zeta=\sqrt{\frac{\ln ^{2} M_{p}}{\ln ^{2} M_{p}+\pi^{2}}}$

0.215

Also, from the step response plot, the damped natural frequency is approx. 3.33 Hz or 20.92 rad/sec. The relationship with the undamped natural frequency is:

ะ 21 .42rad/s

the time constant of exponential decay is

= 0.217 sec 7t

the gain

GDcysteady-state

A standard second-order transfer function has the form:

$H(s)=G_{D C} \frac{\omega_{n}^{2}}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}}$.

so our transfer function is

917.97 H()9.116s s2 9.116s 458.95

VERIFYING

the plot is done in Matlab

File Edit View Insert Tools Desktop Window Help Step Response 3.5 1.5 0.5 0.5 Time (sec)

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