## 4 Non-Dominated Sorting Genetic Algorithm-II  
In the traditional approach, multiple-objective optimization problems can be converted to single-objective optimization problems by predefining the weighting for each objective function. Thus, the optimal solution depends crucially on the selected weights. In addition, certain profitable trade-off relationships may be ignored, as a slight sacrifice of one objective function may lead to a significant improvement in the others. Using a different approach, the NSGA-II algorithm selects the optimal solutions on a Pareto front, which includes multiple non-dominated solutions. The trade-off relationship between each objective can be clearly presented, allowing the decision-maker to pick the optimal solutions based on specific criteria. Note that there is no unique way to select the optimal individual solution among the nondominated solution set from NSGA-II, as the performance evaluation criteria vary with different requirements. In this work, reference point-based multiple-criteria decision-making (MCDM) [36] is applied by considering the origin point as the reference  
Figure 6: Structure of gain-scheduled PID controller.  
point, with equal weights for objectives \(J_{1}\) and \(J_{2}\). Thus, the point on the Pareto front with minimum distance to the origin (\(J_{1}=J_{2}=0\)) is selected as the ultimate solution.  
The detailed NSGA-II optimization method implemented in this paper is described below, along with the parameters predefined for calibrating the PID scheduled gains; see [28; 33] for details.  
1. **Initialization:** Letting \(n=0\), the initial parent population \(P_{n}\) is generated with size \(N_{p}\). For each individual, all decision variables \(x^{i}\) are randomly generated among the related search space \([x^{i}_{min},x^{i}_{max}]\) with \(i=1,2,3,\ldots,N_{p}\).
2. **Nondomination sorting:** The resulting individual is evaluated based on the nondomination level with the nondomination rank (rank 1 is the best level, rank 2 is the next-best level, etc.) assigned.
3. **Offspring generation:** The usual binary tournament selection is applied, with binary crossover at \(P_{C}\) probability and \(D_{C}\) distribution index, polynomial mutation at \(P_{M}\) probability, and \(D_{M}\) distribution index for generating an offspring population \(Q_{n}\) with size equal to \(N_{p}\).
4. **Combination and sorting:** An extended set is generated by combining \(P_{n}\) and \(Q_{n}\) with a size of \(2N_{p}\). After nondominated sorting (as stated in Step 2), selection starts from low ranked individuals and continues to form a new population \(P_{n+1}\) of size \(N_{p}\) until it is filled. If the rank of multiple individuals is the same and the remaining space in \(P_{n+1}\) is limited, individuals with larger crowding distance have priority in selection; for further details on crowding distance, see [28]. Lastly, the new population set \(P_{n+1}\) is filled with the selected individuals.
5. **Iteration:** After updating the generation \(n=n+1\), Steps 2 to 4 are repeated until the stopping criterion is satisfied.  
To demonstrate the advantages of the gain-scheduled PID controller, a fixed-gain PID controller was optimized for the high static friction model using NSGA-II. To ensure a fair comparison, an identical search space, population, and generation size to the gain-scheduled case were selected.