Underwater-Mission-Oriented Objective Function Design for Metaheuristic Tuning of PID and Fractional-Order PID Controllers

Metaheuristic tuning of depth controllers for remotely operated vehicles is usually performed against a generic error integral, which does not encode mission priorities such as battery use, actuator stress from chattering, direction-dependent overshoot risk and post-event recovery. This paper treats the objective function itself as the design object. A mission-oriented cost augments a time-weighted error integral with baseline-normalized effort, smoothness, asymmetric-overshoot and recovery terms on a one-degree-of-freedom BlueROV2 heave model with saturation, actuator lag and sensor noise. Adding the terms cumulatively raises the advantage of the fractional controller over PID from 7.0 to 22.7 percent over ten runs per level, and encoding survey, near-bottom inspection and long-endurance missions as weight profiles gives it the lower cost in every run, largest at 35.4 percent on inspection. The conclusion is then tested against the designer's choices: 160 random weight vectors, four normalization schemes, Monte-Carlo sweeps over noise and plant parameters, and eleven held-out scenarios. A linear analysis exposes one negative result: the cost carries no frequency-domain specification, so the tuned loops have small phase margins that saturation hides, and an explicit margin requirement is needed to restore them. The study is simulation-based and one-dimensional; no hardware validation is claimed.