## 3 Combined Inertial Control Based on CAE and DNN
### Particle Swarm Optimization  
Particle swarm optimization (PSO) was presented in 1995 and originated from the study of bird predation. Its design ideas are mainly inspired by the regularity of bird clusters activities, and then a simplified model established by swarm intelligence. PSO is a problem-solving technique that draws inspiration from the collective behavior of animal clusters. It leverages the sharing of information between individuals in the group to progressively move towards an optimal solution by moving from a state of disorder to order in the problem-solving space. PSO has a distinct biosocial background: cognitive behavior and social behavior; that is, in the process of seeking consistent cognition, individuals tend to remember their beliefs, while considering the beliefs of other peers, and when individuals perceive that the beliefs of their peers are better, they will make adaptive adjustments [17]. The solution of the optimization problem is represented by the position of the particles in PSO, and the performance level of each particle is determined by the objective function's adaptation value. Each particle is determined by a velocity that determines its direction of flight and the magnitude of its rate. Suppose that in a D-dimensional target search space, there are \(m\) particles forming a population. The position of the particle \(i\) at the \(t\)th iteration is expressed as \(X_{i}(t)=(x_{i1}(t),x_{i2}(t),\cdots,x_{it}(t))\). The corresponding flight speed is expressed as \(V_{i}(t)=(v_{i1}(t),v_{i2}(t),\cdots,v_{it}(t))\). When initiating PSO, the position and velocity of \(m\) particles are initially randomized, and an optimal solution is then reached through subsequent iterations. During each iteration, the particle updates both its position and velocity by tracing two extremums: one of the extremums is the optimal solution that the particle itself has searched for so far, called the individual extremum, expressed as \(P_{i}(t)=(p_{i1}(t),p_{i2}(t),\cdots,p_{it}(t))\); another extremum is the optimal solution found so far for the entire swarm of particles, called the global extremum, expressed as \(P_{g}(t)=(p_{g1}(t),p_{g2}(t),\cdots,p_{gd}(t))\). Specifically, at the \((t+1)\) th iteration of the calculation, particle \(i\) updates its velocity and position according to the following rules:  
\[V_{ik}(t+1)=\omega v_{ik}(t)+c_{1}\times\text{rand}()\times(p_{ik}(t)-x_{ik}(t ))+c_{2}\times\text{rand}()\times(p_{gk}(t)-x_{ik}(t)) \tag{7}\]  
\[x_{ik}(t+1)=x_{ik}(t)+v_{ik}(t+1) \tag{8}\]  
where \(\omega\) is inertia weights; \(c_{1}\), \(c_{2}\) represent the individual learning factor and social learning factor, respectively; rand() is a random number between (0, 1); \(i=1,2,\cdots,m\); \(k=1,2,\cdots,d\). To further improve the efficiency of the PSO, this paper proposes an advanced version that dynamically adjusts the inertia weight. By utilizing an exponential function to nonlinearly adjust the inertia weight and incorporating a random adjustment numberthat follows a beta distribution, the algorithm achieves dynamic adjustment and exhibits excellent global convergence capability. The specific improvement can be described as follows: using an exponential function to control the change of inertia weight \(\omega\), where \(\mathrm{e}^{-\frac{t}{\mathrm{max}}}\) nonlinearly decreases with the increase in iteration times. Betarnd is a random number generator in MATLAB that can generate random numbers that follow the beta distribution. Using this generator in the later stages of the iterations can increase the global search ability of the algorithm and reduce the likelihood of the algorithm getting trapped in local optima [18]. The improved expression for the inertia weight is:  
\[\omega=\omega_{\mathrm{min}}+(\omega_{\mathrm{max}}-\omega_{\mathrm{min}}) \mathrm{e}^{-\frac{t}{\mathrm{max}}}+\sigma B(p,q) \tag{9}\]  
\[\mathrm{B}(p,q)=\int_{0}^{1}x^{p-1}(1-x)^{q-1}\mathrm{d}x \tag{10}\]  
where \(\sigma\) is the inertia adjustment factor, which is set to 0.1.