**1. Introduction**

Autonomous underwater vehicles (AUVs) are a crucial technical platform for ocean information acquisition and autonomous operation. They have extensive application prospects, such as marine environment observation, marine resources exploration and security defense. Nevertheless, motion control systems for AUV have become very challenging due to their high nonlinearity, strong coupling, model parameter uncertainties and external disturbances. In addition, an AUV system is usually designed to be underactuated to save cost and improve propulsion efficiency.

With regard to the motion control of underactuated AUV, a variety of control algorithms are available, including proportional-integral-derivative (PID) control, backstepping control, fuzzy logic control, and sliding mode control [1–7]. In [4], a single-input fuzzy logic controller (SIFLC) was proposed for AUV depth control. Simulation results show that the SIFLC gives an identical response as Mamdani and T-S type FLC to the same input sets, while its execution time is more than two orders of magnitude faster than the conventional FLC. In [6], a switching control algorithm based on active disturbance rejection control (ADRC) and fuzzy logic control was applied to the depth control of a self-developed AUV. Numerical simulations showed that the proposed method is more efficient in suppressing external disturbances and inner signal transmission disturbance than PID controller.

Fractional calculus is an extension of traditional integral calculus, it describes the fractal dimension of space. In recent years, its applications in the control field, such as

**Citation:** Cui, Z.; Liu, L.; Zhu, B.; Zhang, L.; Yu, Y.; Zhao, Z.; Li, S.; Liu, M. Spiral Dive Control of Underactuated AUV Based on a Single-Input Fractional-Order Fuzzy Logic Controller. *Fractal Fract.* **2022**, *6*, 519. https://doi.org/10.3390/ fractalfract6090519

Academic Editors: Thach Ngoc Dinh, Shyam Kamal and Rajesh Kumar Pandey

Received: 24 July 2022 Accepted: 29 August 2022 Published: 14 September 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

fractional-order model [8,9], fractional-order control algorithm [10–12], fractional-order optimization algorithm [13,14], have attracted significant research attention. Furthermore, stability analysis for fractional-order control systems have been proposed in several studies [15–17]. A fractional-order, proportional–integral–derivative (FOPID) controller has been proposed by Podlubny. Their proposed controller has two additional parameters: integral order and differential order compared with a PID controller [18]. In [19], an optimized FOPID controller for improved transient control performance was applied to an AUV yaw control system. In addition, a fractional-order Mamdani fuzzy logic controller has been proposed for vehicle nonlinear active suspension, which effectively improves ride comfort and handling stability [20]. However, there has been no report on the application of fractional-order fuzzy logic control in AUV motion.

In this paper, a single-input fractional-order fuzzy logic controller (SIFOFLC) is proposed and applied to an AUV motion control system. Its control input was simplified to a single variable known as distance variable by applying the signed distance method [21], which aims to reduce the computation burden and complex parameter tuning process. Furthermore, a fractional calculus operator was applied to the enhanced FLC due to its recognized ability to increase the controller's flexibility and adaptability. With respect to the controller parameters, we developed and applied a hybrid particle swarm optimization (HPSO) algorithm to obtain optimal control performance. Unlike a conventional PSO algorithm, this includes the local optimal particle term to avoid falling into local optimal region, and the fitness value function includes both steady-state performance and transient performance of an AUV motion control system. To verify the effectiveness of SIFOFLC, we conducted comparative numerical simulations of spiral dive motion control. The object of study, REMUS-100 AUV, was developed by Woods Hole Oceanographic Institution [22], while the simulation was performed using the marine systems simulator (MSS) by Fossen and Perez [23]. Simulation results show that, compared with a FOPID controller and conventional T-S FLC, the SIFOFLC is more efficient in reducing angular velocity oscillations, shortening settling time and improving control accuracy.

The remainder of this paper is organized as follows. Section 2 discusses the six degrees of freedom nonlinear motion equations of AUV. The SIFOFLC design is introduced in Section 3, along with its advantages compared with traditional T-S FLC. In Section 4, the HPSO algorithm is described and is applied to various control systems to obtain optimal parameters. To verify the effectiveness of the proposed method, simulations and numerical comparisons are carried out in Section 5. Finally, some concluding remarks are presented in Section 6.
