Control Systems
Control Systems tests modelling (transfer functions, block diagrams, signal flow graphs), time-response specifications, stability, and frequency-domain design. Most questions reduce to one standard tool applied carefully.
Modelling
- Transfer function = Laplace ratio with zero initial conditions.
- Mason's gain formula: T = (1/Δ)Σ P_k Δ_k, with Δ = 1 − (sum of loops) + (sum of pairs of non-touching loops) − …
Stability and steady-state error
- Routh–Hurwitz: necessary that all coefficients be positive; the count of sign changes gives right-half-plane roots.
- K_p = lim G(s), K_v = lim sG(s), K_a = lim s²G(s). Step error 1/(1+K_p), ramp error 1/K_v, parabolic error 1/K_a - valid only if the closed loop is stable.
Frequency response
- Gain margin = −20 log|G(jω_pc)|; phase margin = 180° + ∠G(jω_gc).
- Root locus leaves open-loop poles and terminates on zeros or infinity; a real-axis point is on the locus when the number of poles and zeros to its right is odd.
Worked example: G(s) = 10/(s(s+2)) with unity feedback. This is a type-1 system, so K_p = ∞ and the step error is zero. For a unit ramp, K_v = lim_{s→0} s·10/(s(s+2)) = 10/2 = 5, hence e_ss = 1/5 = 0.2.
Traps that cost marks
- Applying the final-value theorem to an unstable closed loop.
- Forgetting the extra −90° from the integrator when reading the Bode phase.
- Confusing gain margin as a magnitude ratio with its decibel value.