## 4 Objective Function and Optimization Design
### Optimization Procedure  
To improve the search capability of the particle swarm algorithm, we use the variable inertia weight particle swarm optimization algorithm [37]. An optimization design method is described in Figure 6. The steps of the optimization procedure are presented as follows:  
**Step 1:** Initialization parameters, including the time-delay \(\tau^{*}\in[0,\;7.0196]\), feedback gain \(g^{*}\in[-1.7647,\;1.7647]\), population size (pop_size = 50), number of iterations (\(N=200\)), maximum velocity (\(V_{max}\)), initial position(\(X_{id}\)), initial velocity (\(V_{id}\)), inertia weight (\(w^{k}\)), and learning factors (\(c_{1}=c_{2}=1.2\)). The position of the particles in each dimension is restricted to the selected area and the maximum velocity is a certain fraction of the search space range in each dimension.  
**Step 2:** Randomly generate the initial population, and temporarily record it as the population optimum, and then find the optimal particle in the population and record it as the global optimum.  
**Step 3:** Record the number of iterations of the system and compare the current number of iterations with the maximum number of iterations.  
**Step 4:** Based on the original population, the population position is updated by the PSO for the iteration, as shown in Equation (27).  
\[\left\{\begin{array}{c}{{V_{id}}^{k+1}={w^{k}{V_{id}}^{k}}+{c_{1}{r_{1}}({P_{id }}^{k}-{X_{id}}^{k})}+{c_{2}{r_{2}}({P_{gd}}^{k}-{X_{id}}^{k})}}\\ {{X_{id}}^{k+1}={X_{id}}^{k}+{V_{id}}^{k+1}}\end{array}\right. \tag{27}\]  
where \(k\) is the current number of iterations; \({r_{1}}\) and \({r_{2}}\) are random parameters between \([0,\ 1]\); \({P_{id}}\) is the individual optimal particle position; \({P_{gd}}\) is the global optimal particle position; \({V_{id}}\) is the particle velocity; and \({X_{id}}\) is the current position of the particle. The start inertia weight is \({w_{start}}=0.9\), end inertia weight is \({w_{end}}=0.4\), and the current inertia weight is  
\[{w^{k}}={w_{start}}-{k({w_{start}}-{w_{end}})}/N. \tag{28}\]  
**Step 5:** In the optimization process, the objective function is also called the fitness function. The performance of the new structure is determined by the time delay (\(\tau^{*}\)) and the feedback gain coefficient (\(g^{*}\)). The minimum value of the fitness function is obtained by optimizing these two parameters.  
**Step 6:** The particle positions updated in Step 4 are brought into the function in Step 5, and the new fitness function values are calculated. Each particle adaptation value is compared in an iterative search process, and the particle that produces a smaller value of fitness function is retained.  
**Step 7:** When the number of cycles reaches the set number of iterations \(N\), the system outputs the optimal parameters.  
By the above steps, the global optimal control parameters under harmonic excitation and random excitation are obtained, respectively: \({g_{h}}^{*}=-1.0316\) and \({\tau_{h}}^{*}=0.1079\) represent the time-delay feedback control parameters under harmonic excitation. \({g_{r}}^{*}=0.7962\) and \({\tau_{r}}^{*}=3.0446\times 10^{-4}\) represent the time-delay feedback control parameters under random excitation. Then, transforming the obtained dimensionless parameters into dimensional parameters, we can obtain \({g_{h}}=-17537\)\(\mathrm{N}\cdot\mathrm{s}/\mathrm{m}\), \({\tau_{h}}=1.5373\times 10^{-2}\)\(\mathrm{s};{g_{r}}=13537\)\(\mathrm{N}\cdot\mathrm{s}/\mathrm{m}\), \({\tau_{r}}=4.3373\times 10^{-5}\)  
Figure 6: Flowchart of the optimization procedure.