Control Systems Basics — Cheat Sheet
Transfer functions, PID, first/second-order response, stability and steady-state error.
Blocks & transfer functions
| Transfer function | G(s) = Y(s) / U(s) |
| Series (cascade) | G = G₁·G₂ |
| Parallel | G = G₁ + G₂ |
| Closed loop (neg. fb) | T = G / (1 + G·H) |
| Characteristic eqn | 1 + G·H = 0 |
PID controller
| Ideal PID | u = Kp·e + Ki∫e dt + Kd·(de/dt) |
| Parallel (s-form) | C(s) = Kp + Ki/s + Kd·s |
| Integral time | Ti = Kp / Ki |
| Derivative time | Td = Kd / Kp |
| Effect | P: speed · I: kills offset · D: damps |
First-order response
| Standard form | G = K / (τs + 1) |
| Time constant τ | 63% of final in 1τ, ~settled in 4τ |
| DC gain | K = Δoutput / Δinput (steady) |
Second-order response
| Standard form | ωₙ² / (s² + 2ζωₙs + ωₙ²) |
| Damping ζ | <1 under · =1 critical · >1 over |
| Damped frequency | ω_d = ωₙ√(1−ζ²) |
| % overshoot | PO = e^(−πζ/√(1−ζ²)) × 100 |
| Settling (2%) | t_s ≈ 4 / (ζωₙ) |
| Peak time | t_p = π / ω_d |
Stability & error
| Stable if | all poles have Re(s) < 0 (left-half plane) |
| Routh–Hurwitz | no sign change in 1st column |
| Margins | gain & phase margin from Bode/Nyquist |
| Final value thm | y(∞) = lim(s→0) s·Y(s) |
| SS error (step, type 0) | e_ss = 1 / (1 + Kp) |
Formula reference — verify against the governing standard. Not a substitute for engineering judgment.