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題組內容

3. Consider the following feedback system with plant and controller

(e) As in part (d), design the closed-loop poles at -1± j, design the controller

相關申論題

(a) Choose x1 as the output of the system. Write down the state space representation using as the state vector, where denotes the time derivative of for i = 1,2.

(b) Find the transfer function from f to x1.

(c) Suppose that Find the resonant frequencies of the systen in terms of r.

(d) Let f(t) =sin(ω0t) and ·. Find ω0, x2(0), and such that x1(t) = 0 for all t≥0.

(a) Find the conditions on k and a such that the closed-loop system is stable.

(b) Find k and a to satisfy the following requirements simultaneously:i. The closed-loop transfer function from r to y is a stable, order rational function.ji. The steady-state error with respect to the unit-ramp input r(t) = t, t ≥0, is less than 0.2.iii. The percent maximum overshoot with respect to the unit-step input r(t) = 1, t≥0, isless than 10%.

(c) Let Draw the root locus when k increases from 0 to ㆀo.If k = 1, what is the gain margin of the system?

(a) (6%) Plot the root locus for K＞0 and K＜0, respectively.

(b) Sketch the Nyquist plot for K＞0.

(c) Analyze the stability from Nyquist plot for all K.

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