1. Consider the feedback system below: \frac{100(s+4)}{(s+20)(s+10)(s^2+2s+2)} K (a) Sketch the root loci for $K \ge 0$. (b) If possible, find the range(s) of $K > 0$ such that the closed-loop system is stable. 2. Consider the system $G(s) = \frac{25}{s^2 + 5s + 10}$ (a) Find the damping ratio $\xi$. (b) Find the natural frequency $\omega_n$. (c) Find the exponential decay $\sigma = \xi\omega_n$. (d) Find the damped frequency oscillation $\omega_d = \omega_n\sqrt{1-\xi^2}$. (e) Find the peak value of the unit-step response. (f) Find the steady-state value of the unit-step response. (g) Find the 2% settling time. (h) Find the DC gain. 3. Consider the unity feedback system around the plant $\frac{K}{s+1}$ (a) Choose the gain $K$ such that the closed-loop system is stable and the step response reaches the steady state within 0.1 second. Assume that the time taking to reach the steady state is 5 times the time constant. (b) What value of $K$ the steady state error to the step input is maximum and what is the error in percentage do you expect?
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Sketching the root locus for K=0 means finding the locations of the poles of the closed-loop transfer function as K approaches 0. When K=0, the closed-loop transfer function becomes G(s)/(1+G(s)) = 0/(1+0) = 0. This means that the poles of the closed-loop system Show more…
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