Control Systems Flashcards

For a system with open-loop transfer function G(s) = K/[s(s+1)(s+2)], the angle of departure from complex poles is calculated using the:

Answer: Angle condition: sum of angles to poles minus sum of angles to zeros = ±180°(2k+1)

Angle of departure at a complex pole is found by applying the root locus angle condition at a point infinitesimally close to that pole.

The Z-transform of a unit step sequence u[n] is:

Answer: z/(z-1) for |z| > 1

The Z-transform of u[n] = Σ z^(-n) for n≥0 sums to z/(z-1), valid for |z| > 1.

Kalman's rank condition for controllability of the pair (A, B) states that the system is controllable if and only if:

Answer: Rank[B AB A²B … A^(n-1)B] = n

The controllability matrix C = [B AB A²B … A^(n-1)B] must have full rank n for the n-dimensional system to be completely controllable.

In a Nichols chart, the M-circles (constant closed-loop magnitude) are plotted in terms of:

Answer: Open-loop phase vs. open-loop gain in dB

The Nichols chart plots open-loop phase (degrees) on the x-axis vs. open-loop gain (dB) on the y-axis, with M and N contours overlaid.

An all-pass system has a transfer function characterized by:

Answer: Zeros that are mirror images (RHP) of its LHP poles

An all-pass filter has RHP zeros that are conjugate reflections of its LHP poles, giving constant magnitude |H(jω)| = 1 at all frequencies.

The sensitivity function S(s) in a feedback control system is defined as:

Answer: S(s) = 1 / (1 + G(s)H(s))

The sensitivity function S(s) = 1/(1 + L(s)) quantifies how the closed-loop transfer function changes with variations in the plant, where L = GH.

For a PID controller, the derivative action primarily provides:

Answer: Anticipatory control that reduces overshoot and improves stability

The derivative term responds to the rate of change of error, providing a predictive (anticipatory) action that damps oscillations and reduces overshoot.