Robust proportional–integral control of submersible autonomous robotized vehicles by backstepping-averaged sub-gradient sliding mode control

This study presents a novel robust controller strategy for submersible autonomous robotized vehicles (SARV). This controller applies the averaged sub-gradient (ASG) descendant method to optimize the tracking of well-posed reference trajectories. The motion control form of the SARV is done in three stages (cascade-like control design). The phases consist of using the translation velocities as pseudo controllers to adjust the SARV position, regulating the angular velocities to attain the requested translation velocities, and using the proposed ASG to control the thrusters to get the needed angular velocities. ASG implementation optimizes a convex cost function depending on the integral of the tracking error using the ASG method. The application of Barbalat's lemma justifies the tracking error is converging the origin asymptotically. A class of integral sliding mode controllers with a variable state-dependent gain solves the tracking of the designed reference trajectories in each of the three stages. The sliding surface depends on the tracking error for each pseudo-controller, its integral, and the cost function average. The optimization of the cost function can be done without complete knowledge of the SARV dynamics. A numerical example is presented in this study to confirm the suggested control design's effectiveness based on the cost function's time evolution analysis. The forced motion by the proposed controller is compared with the movement obtained by a proportional–integral–derivative (PID) controller. The proposed controller exhibits a better tracking of the reference trajectory than the PID version.