**4. Parameter Optimization for Three Controllers with HPSO Algorithm**

In this section, the HPSO algorithm is developed and applied to optimize the controller parameters. Except for the SIFOFLC, the T-S FLC and FOPID control systems are optimized for further comparison in Section 5, as the SIFOFLC algorithm essentially originates from the T-S FLC, and because the FOPID controller is the most extensively used fractional order control algorithm.

It is necessary to mention that our optimization and research scenario is the spiral dive motion control of REMUS-100 AUV. It is performed using Marine Systems Simulator (MSS), which is a MATLAB toolbox and offers a set of tools for marine engineering researchers [23]. Some research details are as follows: The target depth of AUV motion linearly increases from 0 m to 30 m and the target yaw angle from 0◦ to 720◦. The simulation time is 300 s and the input constraint of fins is set to 13.8◦ in accordance with [24]. Furthermore, the external disturbances are ignored.

Figure 8 illustrates the block diagram of an AUV motion control system. The propeller speed is fixed at 1500 r/m so that the cruise speed maintains a constant value. In this case, the target trajectory is obtained with heading controller and depth controller.
