# Time-Delay Vibration Reduction Control of 3-DOF Vehicle Model with Vehicle Seat
## 4 Suspension Time-Delay Control Parameter Optimization
### Time-Delay Control Parameter Optimization Based on Adaptive Weighted Particle Swarm Algorithm
#### 4.2.2 The Steps of Adaptive Weighted Particle Swarm Optimization  
1. The position and velocity of each particle in the population are randomly initialized.
2. The fitness of each particle is evaluated. The current position and fitness value of each particle are stored in the Pbest of each particle. The location and fitness value of all the individuals are stored with the best fitness value in Pbest in Gbest.
3. The following formula is used to update the velocity and displacement of the particle: \[\begin{array}{c}x_{i,j}(t+1)=x_{i,j}(t)+v_{i,j}(t+1),j=1,2,\cdots,d\\ v_{i,j}(t+1)=wv_{i,j}(t)+l_{1}r_{1}[p_{i,j}-x_{i,j}(t)]+l_{2}r_{2}[p_{g,j}-x_{ i,j}(t)]\end{array}\] (21) where \(v_{i,j}\) is the update speed of the particle; \(x_{i,j}(t)\) is the updated displacement of the particle; r1 and r2 are random constants generated in [0,1]; and \(l_{1}l_{2}\) are learning factors.
4. The weight is updated: \(w=\left\{\begin{array}{cc}w_{\min}-\frac{(w_{\max}-w_{\min})(f-f_{\min})}{(f_ {avg}-f_{\min})},&f\leq f_{avg}\\ w_{\max},&f>f_{avg}\end{array}\right..\)
5. For each particle, its fitness is compared to the best position it passes through. If the result is good, it takes at the current best position. Comparing all current Pbest and Gbest values, Gbest is updated.
6. If the stop condition (generally the number of iterations or the preset accuracy of operation) is met, the search will stop and the results will be output. Otherwise, return to 3 to continue the search.  
Flow chart of adaptive weight particle swarm optimization algorithm (as shown in Figure 4).  
In the optimization process, the suspension parameters of a vehicle are referred to (as shown in Table 1[46]). To more accurately search for the optimal individual extreme value and global optimal solution, 40 particles were randomly selected for iterative optimization. The maximum inertial weight of the learning factor \(l_{1}=l_{2}=1.49445\) is 0.9, and the minimum inertial weight is 0.6. After 100 iterations, the variation diagram of the iteration number of the fitness function of the suspension performance index is shown in Figure 5. It can be seen that with the continuous evolution of the population, the fitness function value of the optimal individual decreases rapidly. The number of invalid iterations is lower, and the convergence is fast. Finally, the global optimal control parameters under harmonic excitation and random excitation are obtained: g = 18,815.1534673518, \(\tau\) = 0.868044394480619. g\({}_{1}\) = -33,108.79988701554, \(\tau_{2}\) = 0.992053286298810 It is carried to the I region of \(\zeta_{2}=\frac{\zeta_{2}}{\sqrt{m_{p}}\zeta_{2}}\), \(g_{1}=\frac{\zeta}{K_{1}}\), in which the feedback control parameters can be determined in the time-delay stability region. According to Theorem 1, the control system at this time is stable with full delay, and the system does not have stability switching; that is, the system is stable.  
Figure 4: Flow chart of adaptive weight particle swarm optimization algorithm.