## 4 Pareto Based Controller Parameter Optimization
### Further Optimization Based on Improved Particle Swarm Optimization Algorithm  
The PSO algorithm is used to search for the optimal parameters [21] in the POD controller and solve for the optimal parameters in the Pareto algorithm's optimal solution set. The main idea is to determine the adaptability of each particle in each generation by constructing several groups of particles and setting a fitness function. Then, compare the fitness values of each particle in each generation to obtain the global and local best particles. Finally, based on information sharing between particles, update the direction and velocity of each particle. The effectiveness and searchability of this method are related to population size, algebra, and fitness function design. This program can be summarized into the following four steps. Step 1: Initialize population attributes: Each particle's position contains n dimensions, corresponding to n control parameters in the POD controller. The initial velocity value of particle swarm optimization is formed by experience. The Particle number and iterations are both set to 50, the \(K_{\rm POD}\) gain search range is set to [\(-\)10,10], where the search range of the time constant \(T_{1}\) and \(T_{2}\) is set to [0.01,0.5], and \(T_{3}\) and \(T_{3}\) are set to [0.01,0.9]. The state attributes of particle i are as follows:  
Current position of particles: \(x_{\rm i}=(x_{11},x_  
Step 3: Update speed and position: The speed and position of each particle in its next state should be determined based on individual extremum, global extremum, and the speed and position of the previous state. \[v_{\rm i}^{\rm i+1}=\omega v_{\rm i}^{\rm t}+c_{1}r_{1}(PB_{\rm i}^{\rm t}-x_{ \rm i}^{\rm t})+c_{2}r_{2}(GB^{\rm t}-x_{\rm i}^{\rm t}) \tag{49}\]  
\[x_{\rm i}^{\rm i+1}=x_{\rm i}^{\rm t}+v_{\rm i}^{\rm t+1} \tag{50}\]  
In the equation, \(\omega\) is the inertia weight; \(v_{\rm i}^{\rm t}\) represents the current speed; \(c_{1}\) and \(c_{2}\) are acceleration constants (learning factors), usually taken as 2; \(r_{1}\) and \(r_{2}\) are uniformly distributed random numbers in [0,1]; \(r_{2}\) is the individual optimal position of the i-th particle; \(GB^{\rm t}\) is the current global optimal position. The equation consists of three parts. The first part, \(\omega v_{\rm i}^{\rm t}\), is the memory part of the particle, which explains the velocity of the particle's previous state; the second part, \(c_{1}r_{1}(PB_{\rm i}^{\rm t}-x_{\rm i}^{\rm t})\), is the cognitive part of particles, which determines their global search ability and prevents algorithms from falling into local optima; the third part \(c_{2}r_{2}(GB^{\rm t}-x_{\rm i}^{\rm t})\) is the social learning part of particles, reflecting the group learning process of information sharing between particles. Step 4: Determine whether to complete the program: If one of the following termination conditions is met, the optimization process will be completed. (1) The maximum number of iterations has been reached. (2) The fitness value of the global optimal solution is less than the set value, which is called iterative convergence. If neither of these situations is met, the program returns to the second step. Compared with another Evolutionary algorithm, PSO can quickly converge to the optimal solution and is immune to the local optimal solution. It has a simple principle, fewer parameters, and is easier to implement. To reduce the probability of the algorithm falling into the local optimum and improve the fitting accuracy, this paper proposes a new improved Particle swarm optimization (NIPSO) algorithm, which mainly improves the inertia weight \(\omega\) and the learning factor \(c_{1}\), \(c_{2}\). The inertia weight \(\omega\) expresses the degree to which the current particle search speed affects the subsequent particle speed, affecting the local and global search capabilities of the PSO algorithm.