Constraint-Aware Discrete-Time PID Gain Optimization for Robotic Joint Control Under Actuator Saturation

The precise regulation of rotary actuation is fundamental in autonomous robotics, yet practical PID loops deviate from continuous-time theory due to discrete-time execution, actuator saturation, and small delays and measurement imperfections. We present an implementation-aware analysis and tuning workflow for saturated discrete-time joint control. We (i) derive PI stability regions under Euler and exact zero-order-hold (ZOH) discretizations using the Jury criterion, (ii) evaluate a discrete back-calculation anti-windup realization under saturation-dominant regimes, and (iii) propose a hybrid-certified Bayesian optimization workflow that screens analytically unstable candidates and behaviorally unsafe transients while optimizing a robust IAE objective with soft penalties on overshoot and saturation duty. Baseline sweeps ($τ=1.0$~s, $Δt=0.01$~s, $u\in[-10,10]$) quantify rise/settle trends for P/PI/PID. Under a randomized model family emulating uncertainty, delay, noise, quantization, and tighter saturation, robustness-oriented tuning improves median IAE from $0.843$ to $0.430$ while keeping median overshoot below $2%$. In simulation-only tuning, the certification screen rejects $11.6%$ of randomly sampled gains within bounds before full robust evaluation, improving sample efficiency.

💡 Research Summary

This paper addresses the gap between continuous‑time PID theory and its practical implementation for robotic joint actuation, where discrete‑time execution, actuator saturation, small delays, sensor noise, and quantization significantly affect performance. The authors first derive explicit stability regions for PI controllers under both forward‑Euler and exact zero‑order‑hold (ZOH) discretizations using the Jury stability test. These analytic guardrails relate the proportional and integral gains to the sampling period, providing a quick pre‑screen for unsafe gain choices.

Next, the study evaluates a discrete back‑calculation anti‑windup scheme in saturation‑dominant scenarios. By augmenting the integral update with a correction term proportional to the difference between the saturated and unsaturated control signals, the anti‑windup reduces saturation dwell time and overshoot, especially when the actuator frequently hits its voltage/torque limits.

The core contribution is a hybrid‑certified Bayesian optimization (BO) workflow for robust gain selection. Candidate gain triples (Kp, Ki, Kd) are first filtered through the analytical PI stability test, discarding about 11.6 % of random samples before any simulation. The remaining candidates are evaluated across a Monte‑Carlo ensemble of plant models that vary time constant, static gain, input delay, measurement noise, quantization, and saturation bounds. The objective combines Integral of Absolute Error (IAE) with soft penalties on percent overshoot and saturation duty (λ_os·%OS + λ_sat·Duty). This formulation encourages low tracking error while keeping transient excursions and actuator clipping within acceptable limits.

Simulation results demonstrate the effectiveness of the approach. Baseline sweeps (τ = 1 s, Δt = 0.01 s, u∈