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

TL;DR

This work tackles the gap between continuous-time PID theory and its discrete-time, hardware-constrained application in robotic joints. It develops analytic PI stability regions under Euler and exact ZOH discretizations via the Jury criterion, studies saturation-dominant behavior with discrete anti-windup, and proposes a hybrid-certified Bayesian optimization workflow for robust, implementation-aware gain tuning. By evaluating across a randomized model family that includes delays, noise, and tighter saturation, the approach achieves substantial IAE improvements (e.g., from $0.843$ to $0.430$) while constraining overshoot, and demonstrates simulation-only screening that enhances sample efficiency. The combination of analytic guards, anti-windup-aware design, and Safe-BO on both first-order and second-order actuator benchmarks provides a practical, hardware-free pathway for deploying robust PID gains in robotic joints.

Abstract

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 without hardware experiments.

Figures (21)

• Figure 1: Sampling-period dependence of the integrator gain guardrail for a first-order plant with PI control. The forward-Euler condition in \ref{['eq:ki_upper']} is compared to a ZOH-derived sufficient bound obtained from the discrete-time determinant condition (Sec. III-B).
• Figure 2: Nominal PI Schur-stable set (ZOH, $\Delta t=0.01$) and robust point-cloud screen under plant uncertainty and sample delays; sampled gain outcomes are overlaid to validate the analytic boundary.
• Figure 3: BO workflow: propose gains, simulate, score $J$, update surrogate.
• Figure 4: P-only step response at $K_p=0.5$.
• Figure 5: P-only step response at $K_p=3.0$.