## 4 The Proposed Method  
### Hybrid Optimization Algorithm  
While standard PSO has a fast rate of convergence, it is easy to fall into the local optimum. In order to ensure the diversity of particles, this paper introduces selection and crossover operations into PSO by quoting the principles of GA. After calculating the fitness values of individuals, the values are ranked and divided into part \(A\) and part \(B\). At each iteration, part \(A\) with low fitness is selected for random crossover to produce new generations. The amount of the offspring is the same as the parents. Part \(B\) is undated by the PSO algorithm. The amount of hybrid individuals is determined by the selection probability \(\delta\).  
In order to avoid destroying the individual with a high fitness value as well as accelerating the convergence speed, the paper adopts the adaptive crossover probability. The crossover probability depends on the compared value between fitness values of selected individuals and the average value. The formula is presented as:  
\[P=\left\{\begin{array}{c}P_{1},f<f_{avg}\\ P_{1}-\frac{\left(P_{1}-P_{2}\right)\left(f-f_{avg}\right)}{f_{max}-f_{avg}},f \geq f_{avg}\end{array}\right. \tag{10}\]  
where \(f_{max}\) is the biggest fitness value among the particles; \(f_{avg}\) is the average fitness value; \(f\) is the bigger fitness value among the two hybrid individuals; \(P_{1},P_{2}\) are the range of cross probability.  
The step of PSO is modified as follows:  
Step 1.1: Randomly initialize the particles including the search position and velocity. Define all PSO algorithm parameters such as inertia weight \(\omega_{max},\omega_{min},\ \omega\), learning factors \(c_{1},c_{2}\), swarm size \(N\), maximum iterations \(t_{max}\), etc.  
Step 1.2: Calculate the fitness of each particle using the evaluation function. All the particles are ranked according to the fitness values. In this paper, ITAE is adopted for the performance criteria.  
Step 1.3: Update the value of inertia weight using Equation (7).  
Step 1.4: Update the _pbest_ particles by comparing each individual fitness value with its experience. The best value among all _pbest_ is the _gbest_.  
Step 1.5: Divide the particles into part \(A\) and part \(B\) according to the selection probability \(\delta\). The particles with low fitness presented as part \(A\) are randomly conducted in the crossover operation. The crossover probability \(P\) is calculated with Equation (10). If the crossover probability of two selected particles is above the random value in the range of \((0,1)\), then, the crossover is performed using Equation (9). Otherwise, the selected particles are copied by the next generation. Part \(B\) is undated by the conventional PSO algorithm. The velocity and position of particles are modified according to Equations (5) and (6). The inertia weight is determined by step 1.3.  
Step 1.6: After updating all the particles, the iteration number reaching the maximum is judged. If the termination condition of the algorithm is achieved, the latest _gbest_ is the optimal controller parameter. Otherwise go back to step 1.2.  
The flowchart of the hybrid algorithm is presented in Figure 1.  
Figure 1: Flowchart of the hybrid algorithm.