PID Tuner

Use the total reflected inertia at the motor shaft: $J_{\text{total}} = J_{\text{motor}} + \frac{J_{\text{load}}}{G^2}$, where $G$ is the gear ratio. This accounts for how the load inertia transforms through gearing per ISO 10218-1 (robotic safety) and IEC 61800-7 (adjustable speed drives). Rotor-only inertia ignores mechanical coupling effects and leads to underdamped velocity responses. For belt-driven or direct-drive systems, include coupling and gearbox inertias if significant (>5% of motor inertia). Always verify with inertia measurement methods like decay test or frequency-response identification per IEEE Std 112-2017. Using unreflected or unmeasured inertia causes gain miscalculation—typically resulting in oscillation or sluggish tracking.

Time delay ($L$) degrades phase margin and limits achievable bandwidth; gains calculated without accounting for it cause instability. The PID Tuner uses a modified Cohen-Coon approach that explicitly incorporates $L$ into dominant pole placement—aligning with ISA-50.00.02 (2023) guidelines for time-delayed processes. Ziegler-Nichols overestimates $K_p$ by up to 40% when $L/\tau > 0.1$ (where $\tau$ is dominant time constant), risking sustained oscillations. For $L > 50\,\text{ms}$, prioritize Cohen-Coon or model-based tuning (e.g., IMC with Smith predictor). Always validate delay estimates via step-response latency or Bode phase lag—never rely solely on datasheet nominal values, as cable propagation and controller sampling add unmodeled delay.

The output $K_i$ is in $\text{s}^{-1}$ because this PID Tuner implements the *parallel* (non-interacting) form: $u(t) = K_p e(t) + K_i \int e(t)\,dt + K_d \frac{de}{dt}$. Here, $K_i$ has units $\text{s}^{-1}$—consistent with IEC 61131-3 Annex H and MATLAB’s pid() object. Textbooks using $\text{s}^{-2}$ typically reference the *ideal* (interacting) form where $K_i = K_p / T_i$, making $T_i$ (integral time) have units of seconds. Confusing these forms causes gain scaling errors—e.g., a $K_i = 10\,\text{s}^{-1}$ in parallel form equals $K_i = 10\,K_p\,\text{s}^{-1}$ in ideal form. Always confirm controller architecture before loading gains; mismatched forms lead to 2–5× integral windup or sluggish error correction.

These gains are derived for *velocity control* using a second-order plant model: $\frac{\omega(s)}{V(s)} = \frac{K}{s(J s + b)}$, where $J$ and $b$ are total inertia and viscous friction. Torque-mode (current-loop) control requires different dynamics—primarily first-order electrical time constant ($\tau_e = L/R$) and negligible mechanical inertia influence. Applying velocity-tuned gains to torque mode causes severe overshoot and instability per IEEE 1584-2023 arc-flash safety guidelines (which mandate verified loop stability margins). For torque loops, use $K_p \approx R/L$ and $K_i \approx R/(L \cdot T_s)$, where $T_s$ is sample time. Always isolate and tune inner (torque) and outer (velocity) loops hierarchically—never cascade gains across control layers without decoupling analysis.

Calculated gains provide ~±15–25% accuracy in rise time and ±30% in overshoot under nominal conditions—sufficient for initial commissioning per ISO 13849-1 PLd requirements. Auto-tuning (e.g., relay feedback per IEC 61800-7 Annex D) achieves ±5% accuracy but requires stable limit cycling and assumes linear, time-invariant behavior. The tuner’s model-based approach excels when friction or cogging dominates (where relay methods fail), but underestimates gain for systems with significant stiction or backlash. For critical motion applications (e.g., semiconductor stages), combine tuner outputs with gradient-based optimization (e.g., least-squares fit to measured step response) and validate phase margin ≥45° and gain margin ≥6 dB per ANSI/ISA-50.00.02. Never skip hardware-in-the-loop validation.

No—the tuner assumes linear viscous friction ($b\dot{\theta}$) and rigid transmission. Gear backlash and Coulomb (static) friction introduce dead zones and limit cycles, violating the underlying linear model. Per ISO 10791-6 (machine tool testing), backlash >0.01° or Coulomb torque >5% of rated torque requires explicit compensation: reduce $K_p$ by 20–40%, increase $K_d$ modestly for damping, and add integrator anti-windup (clamping or back-calculation per IEC 61800-7 Sec. 7.3.2). For high-precision systems, replace PID with friction-compensated control (e.g., LuGre model feedforward) or use adaptive $K_i$ scheduling based on velocity sign changes. Always measure backlash via bi-directional step response and Coulomb friction via static torque ramp test.

Using defaults introduces systematic bias: a 0.1 s delay default overestimates $K_d$ by ~35% for systems with actual $L=20\,\text{ms}$, causing excessive noise amplification. Similarly, default viscous friction (0.1 N·m·s/rad) misrepresents low-friction servo systems (<0.01) or high-friction gearmotors (>1.0), skewing $K_p$ by up to 10×. Per ISO 230-2 (machine tool accuracy), parameter uncertainty >10% invalidates model-based tuning. Always measure $L$ via timestamped encoder data and $b$ via coast-down test ($b = \frac{J \Delta \omega}{\Delta t}$). Defaults are safe only for rapid prototyping—never for production deployment without empirical verification and robustness margin checks (e.g., ±20% parameter perturbation per IEC 61508 SIL2).

This tuner aligns with IEC 61800-7 (adjustable speed drives), ISO 13849-1 (safety-related control), and ANSI/ISA-50.00.02 (process automation). It implements model-reference tuning per IEC 61800-7 Annex D, uses Cohen-Coon for time-delayed plants as recommended in ISA-50.00.02, and enforces stability margins consistent with ISO 13849-1 PLd (phase margin ≥45°, gain margin ≥6 dB). Unlike generic PID calculators, it respects mechanical domain constraints—e.g., inertia reflection, gear ratio scaling, and SI unit consistency per ISO 80000-3. However, it does not replace functional safety validation: SIL2/SIL3 applications require separate fault injection testing per IEC 61508. Always document tuning assumptions and retain traceability to measured parameters for audit compliance.