Controls & Feedback Calculators
PID tuning, Bode margins, damping ratios, and state-space fundamentals.
- Ziegler-Nichols PID TuningTurn an ultimate gain and oscillation period into starting P, PI, or PID gains.P: Kp=0.5·Ku , PI: Kp=0.45·Ku Ti=Tu/1.2 , PID: Kp=0.6·Ku Ti=Tu/2 Td=Tu/8 , Ki=Kp/Ti Kd=Kp·Td
- IMC-Based PID Tuning (FOPDT)Turn a first-order-plus-dead-time model into PID gains using the Rivera-Morari IMC rules — one tuning knob, λ.Kc = (2τ+θ) / [K(2λ+θ)] , τI = τ+θ/2 , τD = τθ/(2τ+θ)
- Damping Ratio → % OvershootTranslate a second-order damping ratio into peak overshoot, peak time, and settling time.%OS = 100 · exp( −π·ζ / √(1−ζ²) ) , tp = π / (ωₙ√(1−ζ²)) , ts ≈ 4 / (ζ·ωₙ)
- Phase & Gain MarginTurn a Bode reading — gain at the phase-crossover, phase at the gain-crossover — into gain and phase margins with a stability verdict.GM(dB) = −20·log₁₀|L(jω₁₈₀)| PM = 180° + ∠L(jω_gc)
- 2×2 State-Space EigenvaluesEigenvalues of a 2×2 system matrix A — the poles that set stability, natural frequency, and damping.det(A − λI) = 0 ⇒ λ² − (tr A)λ + det A = 0 , λ = [tr A ± √((tr A)² − 4·det A)] / 2
- Final-Value Theorem — Steady-State ErrorSteady-state error of a unity-feedback loop from system type and static error constant — step, ramp, or parabolic input.e(∞) = lim s→0 s·E(s) e_step = A/(1+Kp), e_ramp = A/Kv, e_parab = A/Ka Kp = lim G(s), Kv = lim s·G(s), Ka = lim s²·G(s)
- Controllability / Observability Rank TestBuild the Kalman controllability and observability matrices for a 2nd- or 3rd-order LTI system and check their rank.Controllability: 𝒞 = [B AB A²B … Aⁿ⁻¹B] , observable iff rank 𝒞 = n , Observability: 𝒪 = [C ; CA ; CA² … CAⁿ⁻¹] , rank 𝒪 = n
- Lead / Lag Compensator DesignPlace the zero and pole of a lead or lag compensator to hit a target phase or gain at your crossover frequency.Lead: α = (1−sinφₘ)/(1+sinφₘ), ωz = ωc√α, ωp = ωc/√α , Lag: ωz = ωc/10, ωp = ωz/α