## 4 Multi-Objective Optimization Based on NSGA-II  
Based on above control design, a PFA-RRR is developed with obvious engineering significance. However, the practical application of PFA-RRR is still trapped in the restrictions caused by the model perturbation, external disturbance and the performance trade-off (see Figure 5). To achieve optimal performances and solve the trade-offs among them, multi-objective optimization is recommended to get the control parameter strategy library by off-line optimization in this note.  
Multi-objective optimization methods have been applied to solve many engineering problems. One of the most prominent Pareto evolutionary algorithms is NSGA-II. To achieve the optimal performances, multi-objective optimization based on NSGA-II [28; 29] is firstly used to solve the set of Pareto-optimal solutions of the aforementioned design parameters, and each solution represents an optimal feasible control strategy that cannot be outrun by any other considering all criteria.  
Three criteria are considered for the optimization. The minimum track error and roll amplitude are the most important control objectives. However, there is an obvious trade-off between them. On the other hand, the minimum movement of steering gear is considered as another criterion. The criteria are defined as follows  
\[\begin{array}{l}\min(f_{d},f_{\phi},f_{\delta})\\ f_{d}=\sqrt{\frac{1}{N_{\mathrm{s}}}\sum\limits_{i=1}^{N}{(d_{i})^{2}}}\\ f_{\phi}=\sqrt{\frac{1}{N_{\mathrm{s}}}\sum\limits_{i=1}^{N}{(\phi_{i})^{2}}} \\ f_{\delta}=\sqrt{\frac{1}{N_{\mathrm{s}}}\sum\limits_{i=1}^{N}{(\delta_{i})^{2}} }\end{array} \tag{42}\]  
where \(f_{d},f_{\phi},f_{\delta}\) stands for the root mean square (RMS) values of \(d_{i},\phi_{i},\delta_{i}\). \(N_{s}\) is the sample number. Constraints are as follows.  
\[\begin{array}{l}0\leq d_{i}<Er\\ 0\leq\phi_{i}<\phi_{\mathrm{max}}\\ |\delta|\leq 30^{\circ}\\ \left|\dot{\delta}\right|\leq 10^{\circ}/s\end{array} \tag{43}\]  
The problem is considered for searching the optimal control strategies of PFA-RRR as a Pareto-front. NSGA-II is used to find the optimal solutions set.  
Figure 6 illustrates the Pareto optimal set distribution, which appears two typical bands. Firstly, the path following performance varies from 2 to 4 m with a 0.5\({}^{\circ}\) perturbation of roll angle and rudder movement. Secondly, when the roll reduction rate doubles, the track error only increases 0.2 m with a triple rudder movement.  
Figure 5: The restrictions of model perturbation, external disturbance and the performance trade-off.  
**Remark 4**.: _The performance perturbation between \(f_{d}\) and \(f_{\phi}\) is limited to a small positive amount, which indicates the trade-off between the two is little and tolerable. Note that, the trade-off between RRR and course is obvious [14; 15; 16] but does not pass on to the indirect path following control with RRR. On the other hand, the performance of RRR is directly related to the rudder servo movements, and higher RRR rate results in greater weight in rudder movement. In other words, using the proposed control strategies, the rudder can achieve both good tracking accuracy and high RRR rate with a substantial increase in rudder movements._