Control Systems Theory Flashcards

The phase margin of a stable feedback system is measured at the frequency where the open-loop magnitude is:

Answer: 0 dB (unity gain)

Phase margin is the additional phase lag at the gain crossover frequency (where |G(jω)H(jω)| = 1, or 0 dB) before the system becomes unstable.

Which of the following statements about the root locus is correct?

Answer: Branches start at open-loop poles and end at open-loop zeros (or infinity)

Root locus branches begin at open-loop poles (K=0) and terminate at open-loop zeros or travel to infinity along asymptotes as K→∞.

A second-order system with ζ = 0.5 and ωn = 10 rad/s has a damped natural frequency (ωd) of:

Answer: 8.66 rad/s

The damped natural frequency is ωd = ωn√(1−ζ²) = 10√(1−0.25) = 10√0.75 ≈ 8.66 rad/s.

In the Nyquist stability criterion, encirclements of the −1+j0 point in the clockwise direction are counted to determine:

Answer: The number of unstable closed-loop poles

By the Nyquist criterion, N = Z − P, where N is clockwise encirclements of −1, P is open-loop RHP poles, and Z is closed-loop RHP poles (instability count).

A system is said to be observable if:

Answer: The initial state can be determined from the output over a finite time interval

Observability means that every initial state x(0) can be uniquely determined by observing the output y(t) over a finite time interval.

Which compensator is most effective at improving the steady-state accuracy of a control system without significantly affecting transient response?

Answer: Lag compensator

A lag compensator increases low-frequency gain to reduce steady-state error while placing its pole-zero pair far below the crossover frequency, minimally affecting transient response.

For a unity-feedback system with open-loop transfer function G(s) = K/[s(s+4)], what value of K places both closed-loop poles at s = −2?

Answer: K = 4

Closed-loop characteristic equation: s² + 4s + K = 0; for double root at s = −2, we need (s+2)² = s² + 4s + 4, so K = 4.