Control Systems: Transfer Functions, Stability & PID
State space, transfer functions, Bode/Nyquist stability and PID tuning — the shared language of EEE, ECE and Mech interviews.
Key formulas
T(s) = ωn² / (s² + 2ζωn·s + ωn²)
Standard second-order system: Every transient-spec question (Mp, ts, tr) is derived from this pair of parameters.
ts ≈ 4 / (ζωn); %Mp = e^(−πζ/√(1−ζ²)) × 100
Settling time & overshoot: Converts design specs (e.g. 5% overshoot) into ζ and ωn requirements.
ẋ = Ax + Bu; y = Cx + Du; TF = C(sI − A)⁻¹B + D
State-space model: State space handles MIMO and initial conditions; transfer function is its scalar projection.
Cheat sheet
- ▸Transfer function G(s) = output/input in the Laplace domain with zero initial conditions; poles are roots of the denominator, zeros of the numerator.
- ▸Stability: all closed-loop poles must lie in the open left-half s-plane. Routh-Hurwitz checks this from the characteristic polynomial without solving it.
- ▸Bode plot: gain margin = how much gain can increase before instability (measured at phase crossover −180°); phase margin = extra phase lag allowed at gain crossover 0 dB. PM 30–60° is a healthy design target.
- ▸Nyquist: count encirclements of −1+j0 by the open-loop Nyquist plot; Z = N + P determines closed-loop instability.
- ▸Second-order standard form: ωn (natural frequency) and ζ (damping). ζ < 1 overshoots; %overshoot = e^(−πζ/√(1−ζ²)) × 100.
- ▸PID: P speeds response but adds offset and overshoot; I removes steady-state error but worsens stability; D adds damping/anticipates but amplifies noise. Ziegler-Nichols tunes from ultimate gain/period or step response.
Effect of P, I and D terms
| Term | Rise time | Overshoot | Settling | Steady-state error | Stability |
|---|---|---|---|---|---|
| P | Decreases | Increases | Small change | Decreases | Degrades |
| I | Decreases | Increases | Increases | Eliminates | Degrades |
| D | Small change | Decreases | Decreases | No effect | Improves (if noise-free) |
Visual
Bode sketch of an open loop with one integrator:
|dB slope -20 dB/dec (integrator)
| ____/
| 0dB----/----\_______ slope -40 after 2nd pole
| wc (gain crossover)
+----|----------|---- log ω
phase → −90°, then −180° at phase crossoverTop 4 campus interview questions
MCQ mastery quiz
1. A closed-loop system is stable only if all its poles lie…
2. Adding a derivative term to a controller mainly…
3. For a second-order system with ζ = 0.7, the response is…
4. Phase margin is measured at the frequency where the open-loop gain is…