Control Theory: transfer functions, time response, stability margins and state space
Control Theory in GATE AE spans classical frequency-domain design and basic state-space analysis: modelling with transfer functions, time-response specifications, stability tests, root locus, Bode/Nyquist margins and controllability/observability.
Core rules
- Second-order standard form: denominator s² + 2ζωn s + ωn². Peak overshoot Mp = exp(−πζ/√(1−ζ²)); settling time (2%) ≈ 4/(ζωn); peak time = π/(ωd).
- Steady-state error depends on system type and input: step error = 1/(1+Kp), ramp error = 1/Kv, with Kv = lim s→0 sG(s) for unity feedback.
- Routh–Hurwitz gives the number of right-half-plane roots from sign changes in the first column; a zero row is handled with the auxiliary polynomial from the row above.
- Gain margin is measured at the phase-crossover frequency (−180°), phase margin at the gain-crossover frequency (0 dB).
- Controllability: rank[B AB A²B …] = n; observability: rank[Cᵀ AᵀCᵀ …] = n.
Worked example
G(s) = K/[s(s+2)] with unity feedback. Characteristic equation s² + 2s + K = 0, so ωn = √K and 2ζωn = 2. For ζ = 0.5, ωn = 2, hence K = 4. The resulting overshoot is exp(−π·0.5/√0.75) ≈ 16.3%.
Traps
- Reading the damping ratio from the wrong coefficient, or using degrees in the overshoot formula.
- Applying the steady-state error table to a non-unity-feedback loop without redrawing it.
- Forgetting that adding an open-loop pole pulls the root locus to the right (destabilising), while a zero pulls it left.
- Sign errors in phase margin when the plant has more than two poles, and confusing gain crossover with phase crossover.