Control Systems
Control systems questions test transfer functions, block-diagram reduction, stability, transient response, steady-state error, PID tuning, root locus and frequency response. You should be able to move between time-domain specifications and s-domain parameters.
Core rules
- For unity negative feedback with forward path G(s), the closed-loop transfer function is G/(1+G); the characteristic equation is 1+G=0.
- Second-order standard form: denominator s²+2ζωn s+ωn². Percent overshoot depends mainly on ζ; settling time (2%) ≈4/(ζωn).
- A type-0 system has finite position error 1/(1+Kp); type-1 has zero step error and finite velocity error 1/Kv.
- Routh-Hurwitz gives stability without solving roots; any sign change in the first column means right-half-plane poles.
- PID: proportional acts on present error, integral removes offset but adds phase lag, derivative anticipates but amplifies noise.
Worked example
G(s)=K/[s(s+4)] in unity feedback. Characteristic equation s²+4s+K=0, so ωn=√K and 2ζωn=4, giving ζ=2/√K. For ζ=0.7, K=(2/0.7)²=8.16. The closed-loop poles are then at −2±j2.04, a stable underdamped pair.
Traps
- Confusing open-loop gain with closed-loop gain.
- Using ω in Hz in formulas that require rad/s.
- Assuming a negative sign in feedback is always present.
- Forgetting integrator windup and derivative kick in practical PID.
- Predicting steady-state error from a non-unity feedback configuration without recalculating.