Control Systems
What the exam tests
Control Systems tests transfer-function derivation, stability margins, time-domain specifications, root locus, and compensator action. Numerical work usually involves Routh–Hurwitz, second-order response, Bode plots, and steady-state error.
Core rules
- Closed-loop poles determine stability; for a unity-feedback system, 1 + G(s)H(s) = 0.
- Routh array gives the number of right-half-plane roots. A row of zeros indicates symmetric roots; use the auxiliary equation.
- For a standard second-order denominator s² + 2ζωn s + ωn², percent overshoot depends only on ζ; settling time ≈ 4/(ζωn) for 2% band.
- Type number determines steady-state error: type 0 gives finite position error, type 1 zero position error, type 2 zero velocity error.
- Gain margin is the negative of gain at phase crossover; positive GM and PM indicate stability for minimum-phase systems.
Worked example
For G(s) = K/[s(s+2)] with unity feedback, the characteristic equation is s² + 2s + K = 0. Comparing with s² + 2ζωn s + ωn² gives 2ζωn = 2 and ωn² = K. For ζ = 0.5, ωn = 2, hence K = 4. The closed-loop poles are at −1 ± j√3, giving about 16.3% overshoot.
Traps
Do not equate system order with type number. Do not read Bode margins without checking the −180° crossing and the 0 dB crossing. Non-minimum-phase zeros can make positive PM misleading. Routh special cases require the auxiliary polynomial, not a guess.