## 6 Practicality Comparison
### Focus on Different Algorithm Choices  
In addition, the PID parameters have a certain physical meaning of engineering regulation; the realization of the PID control process can be realized through many ready-made commercial controller hardware devices in addition to the aforementioned PLC. This article can also provide a linear solution method for nonlinear systems. Since we were not concerned about how the third variable, \(x=\theta t\), changed over time, we only treated it as an independent variable. Therefore, Equation (12) can be regarded as a linear system, which can be solved by using certain multivariable linearization algorithms. The LMI control mentioned earlier can achieve the multivariable controlled tasks in this study, in theory, and even achieve better control effects, in theory. However, due to the large number of parameters within the control algorithms, and the fact that these parameters often have no engineering physical significance, it is difficult to realize them in physical control hardware, and it is difficult to realize the on-site connection and configuration with practical engineering objects. The LMI control has strong robustness and can realize control algorithm designs under control input constraints [36; 37]. To deal with the governing system in Equation (12), the LMI algorithm is briefly analyzed as follows:  
The control input signal is presented as \(u=KY\), with the auxiliary Lyapunov function presented as \(V=Y^{\mathrm{T}}QY\). By the proper design of \(Q\), the convergence effect of \(Y\) can be effectively adjusted. To achieve an exponential convergence, we assume that \(\dot{V}\leq-\alpha V\). Due to \(V(0)=Y_{0}^{\mathrm{T}}QY_{0}\); if \(\overline{\omega}>0\) exists, the inequality holds \(Y_{0}^{\mathrm{T}}PY_{0}\leq\overline{\omega}\), then \(V(0)\leq\overline{\omega}\) is guaranteed; hence, \(V(t)\leq\overline{\omega}\) can be guaranteed. The four LMIs for the calculation can be obtained as follows [36]:  
\[ \begin{bmatrix}\overline{\omega}&Y_{0}^{\mathrm{T}}\\ Y_{0}&N\end{bmatrix} \geq 0,\ \begin{bmatrix}k_{0}N&F^{\mathrm{T}}\\ F&1\end{bmatrix} \geq 0,\ 0<Q=Q^{\mathrm{T}},\alpha N+AN+BF+NA^{\mathrm{T}}+F^{ \mathrm{T}}B^{\mathrm{T}}<0 \tag{27}\]  
where \(N=Q^{-1}\), \(F=KQ^{-1}\), \(k_{0}=u_{\mathrm{max}}^{2}/\overline{\omega}\); herein, the theoretical process parameters are \(\alpha\), \(u_{\mathrm{max}}\), and \(\overline{\omega}\). In the present study, to limit the change rate of control input \(u\), we assumed that \(k_{1}=\dot{u}_{\mathrm{max}}^{2}/u_{\mathrm{max}}^{2}\); then, the fifth LMI could be designed as [37]:  
\[ \begin{bmatrix}k_{1}N&(AN+BF)^{\mathrm{T}}\\ AN+BF&N\end{bmatrix} \geq 0 \tag{28}\]  
solving the five LMIs in Equations (27) and (28) based on the selected theoretical process parameters, \(\alpha\), \(u_{\mathrm{max}}\), and \(\overline{\omega}\), the convergent state variable \(Y\) could be obtained. Figure 11 illustrates the results of the controlled displacements by solving the LMIs; the control input signals (\(u_{1}\) and \(u_{2}\)) and the changes in the control input signals can be obtained with the selected process parameters of \(\alpha=200\), \(u_{\mathrm{max}}=0.001\), and \(\overline{\omega}=0.001\). In terms of appearance, LMI control achieved a better result than the previous iterative algorithm control result. However, it should be noted that this is an ideal result after many manual selection and testing performances of the process parameters. In fact, such manual and artificial selections of three process parameters had no practical engineering significance. In terms of the implementation of the algorithm itself, it was also difficult to be successfully implemented in the PLC, which also increased the difficulty of engineering applications.