Control Systems
Control Systems tests modelling and transfer functions, time-response specifications, steady-state errors, and stability by Routh–Hurwitz, root locus, Bode and Nyquist methods, plus compensator design and state-space basics.
Core rules
- Second order: ωn from the s² coefficient, and ζωn is half the s coefficient. Mp = e^(−ζπ/√(1−ζ²)); Ts (2% band) ≈ 4/(ζωn).
- Steady-state error: a type-0 system has finite position error 1/(1 + Kp); type 1 gives zero position error but finite velocity error 1/Kv; only type 2 and above track a parabola.
- Routh: the number of sign changes in the first column equals the number of right-half-plane poles; an all-zero row is handled with the auxiliary polynomial.
- Bode: gain margin is measured at the phase crossover (−180°), phase margin at the gain crossover (0 dB); both must be positive for closed-loop stability.
Worked example
G(s) = 25/(s² + 6s + 25). Comparing with s² + 2ζωns + ωn² gives ωn = 5 rad/s and ζ = 6/(2 × 5) = 0.6. Peak overshoot = e^(−0.6π/√(1 − 0.36)) = e^(−2.356) ≈ 9.5%. Settling time = 4/(0.6 × 5) = 1.33 s, and peak time = π/(ωn√(1 − ζ²)) = π/4 ≈ 0.785 s.
Traps
Reading ζ as the coefficient of s instead of half of it; quoting gain margin in degrees; forgetting that Bode stability is judged on the loop gain, not the plant alone; and using 4/(ζωn) when the 5% criterion (3/(ζωn)) was asked.