*4.2. Optimization Experiments and Result Analysis*

Here we conduct optimization of three control systems by using the HPSO algorithm mentioned above. To the SIFOFLC system, the particle dimension is four with the fractional order of the error signal and the scale factor of the depth controller and heading controller. We adopted the standard Oustaloup filter module deriving from FOTF toolbox [29] to perform fractional derivative operator, the frequency band was set to [0.001,1000] and the filter order was 4. As for the FOPID control system, the dimension of particle is 10, so that each FOPID controller has five unknown parameters. For the T-S FLC control system, only two scale factors are to be optimized. The membership functions and fuzzy rules of T-S FLC were determined through empirical approach, and its control surface is shown in Figure 6.

The coefficients of the HPSO algorithm are set as follows: the particle size is 10 and the maximum number of evolution is 20. The limit of the inertia weight, *ω*, is set to between 1 and 0.7. The learning factors *c*<sup>1</sup> and *c*<sup>2</sup> are set as value 2, the social factor *q* is set as value 0.7 and the time interval coefficient *dt* is set as value 0.5. The searching range for parameters are presented in Appendix A. Furthermore, we implement multiple optimizations and adopt the optimal result to solve randomness.

Figures 10–12 respectively show the convergence curve of optimal fitness of three control systems and the obtained target controller parameters are illustrated in Table 3. It can be observed that all the curves tend to decline through evolution, which illustrates the effectiveness of optimization. Actually, the optimal fitness of three control systems respectively decreases by 19.8%, 37.9% and 9.6%. The FOPID controller clearly outperforms the other two controllers as it has the highest degrees of freedom. The optimization effect of the SIFOFLC system is twice as good as the T-S FLC system and its ultimate fitness is significantly less than the others. These results demonstrate that the introduction of a fractional differential operator not only increases degrees of freedom and the flexibility of the controller but also improves the control performance, as is particularly demonstrated in the next section.



**Figure 10.** Convergence curve of SIFOFLC system.

**Figure 11.** Convergence curve of FOPID control system.

**Figure 12.** Convergence curve of T-S FLC system.
