## 4 Optimization of Adaptive Control Parameters for Permanent Magnet Synchronous Motors
### Parameter Optimization System
In this study, once the algorithm's initial parameters are established, they remain constant throughout the optimization process. This consistency guarantees stable performance during the search for the requisite particles, ultimately yielding optimal solutions for the adaptive control rate parameters (K1, K2, Kp, and Ki). The specific initial parameter settings are thoroughly delineated in Table 1.  
Figure 5: Block diagram of the two optimization algorithms NSGA-II and MOPSO. The Multi-objective Particle Swarm Optimization (MOPSO) algorithm is a multi-objective optimization algorithm that is an extension of the Particle Swarm Optimization algorithm and is specifically designed for solving multi-objective optimization problems. MOPSO aims to find the set of non-dominated solutions to a problem, i.e., a set of solutions in which none is dominated by any other solution across all objective functions. MOPSO uses a non-dominated sorting method to efficiently partition the solution into different levels on the Pareto frontier. This helps to provide multiple feasible solutions to the problem, forming a balanced set of Pareto-optimal solutions. MOPSO introduces the concept of congestion distance into the algorithm by adjusting the positions of the particles in order to maintain the diversity of solutions. This helps to maintain an even distribution of solutions on the Pareto front, thus providing a more comprehensive solution set. MOPSO can also improve search efficiency near the local Pareto front by introducing a local search mechanism. Due to the advantages of MOPSO in multi-objective optimization, it is used as the second optimization algorithm in this paper, and its principle is shown in Figure 5.  
The initial parameter settings for MOPSO are analogous to those employed in comparable optimization algorithms. Appropriately chosen fixed parameter settings within the algorithm ensure consistent operational behavior, thus averting instability in results induced by parameter fluctuations, ultimately resulting in optimal solutions for the adaptive control rate parameters (K1, K2, Kp, and Ki) through the optimization process. To attain favorable optimization outcomes, the initial parameter settings employed in this study are delineated in \[\mathbf{T}_{e}=\text{max}\bigg{\{}\frac{3}{2}P\Big{(}\psi_{f}I_{q}+\big{(}\mathbf{L}_{ d}-\mathbf{L}_{q}\big{)}\mathbf{I}_{d}I_{q}\Big{)}\bigg{\}} \tag{38}\]0  
Figure 6 depicts the evolution of the optimized control rates in this study. As observed, the motor control parameters K1, K2, Kp, and Ki exhibit gradual convergence under both optimization algorithms. Specifically, the NSGA-II algorithm yields optimal parameters of 48.53, 10.38, 0.041, and 4.07, respectively, whereas the MOPSO algorithm results in optimal parameters of 46.92, 11.71, 0.042, and 4.72, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline Population size & 10 \\ General scale & 20 \\ Crossover probability & 0.9 \\ Cross-distribution index & 10 \\ Variation distribution index & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameterization. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline Maximum Iterations & 20 \\ Number of Particles & 10 \\ Inertia & 0.9 \\ Global Increment & 0.9 \\ Particle Increment & 0.9 \\ Maximum Velocity & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameterization.