## 4 Closed-Loop Stability Analysis  
**Theorem 1**. : _There exist matrices \(X\) and \(X\in\mathbb{R}^{n\times n}\), any positive constant \(a\), symmetric positive definite matrix \(Y\), and \(x_{1},x_{2}\in\mathbb{R}^{n}\); so, we have:_  
\[2x_{1}^{T}Xx_{2}\leq ax_{1}^{T}XY^{-1}X^{T}x_{1}+\frac{1}{a}x_{2}^{T}Yx_{2}\]  
**Proof. ** For any constant \(a>0\) and any symmetric matrix \(Y>0\),  
\[0\leq a\Big{[}X^{T}x_{1}-\frac{1}{a}Yx_{2}\Big{]}^{T}Y^{-1}\Big{[}X^{T}x_{1}- \frac{1}{a}Yx_{2}\Big{]}=ax_{1}^{T}XY^{-1}X^{T}x_{1}-2x_{1}^{T}Xx_{2}+\frac{1 }{a}x_{2}^{T}Yx_{2}.\]  
\(\Box\)  
In combination with Equations (17), (22), and (23), the first-order derivative of the error is:  
\[ \begin{array}{ll}\dot{E}&=\dot{X}-\dot{X}_{d}=A_{1}X+A_{2}X(t-\tau)+B\Big{(} -\frac{\mu_{0}A_{m}N_{m}^{2}}{4}u+mg+n_{u}\Big{)}-\phi_{m}X_{d}-A_{2}X_{d}(t- \tau)-\varphi_{m}\sigma\\ &\qquad=A_{1}E+A_{2}E(t-\tau)-BK_{3}\sigma-BK_{1}X_{d}+BK_{2}X_{d}+B\Big{(}- \frac{\mu_{0}A_{m}N_{m}^{2}}{4}u+mg+n_{u}\Big{)}\end{array}  \tag{26}\]  
Equation (26) can be rewritten with controller (24) as follows:  
\[\dot{E}=(A_{1}-BK_{2})E+A_{2}E(t-\tau)-B\Big{[}\hat{\eta}\text{sgn}(B^{T}PE)- mg-n_{u}\Big{]} \tag{27}\]  
**Theorem 2**. : _The proposed adaptive levitation control law (24) with adaptive law (25) can guarantee that \(\hat{\eta}\) has an upper bound and can also drive error vector \(E(t)\) to be zero within a finite time._  
**Proof. ** The Lyapunov-Krasovskii function was selected as:  
\[V=\frac{1}{2}E^{T}PE+\frac{1}{2}\int_{t}^{t-\tau}E^{T}(\delta)JE(\delta)d \delta+\frac{1}{2\xi}\overline{\eta}^{2}, \tag{28}\]  
where \(\overline{\eta}=\eta-\hat{\eta}\) and \(\xi\) is a positive constant. The derivative of Equation (28) with respect to time yields can be obtained as follows:  
\[\dot{V}=\frac{1}{2}\dot{E}^{T}PE+\frac{1}{2}E^{T}P\dot{E}+\frac{1}{2}E^{T}JE- \frac{1}{2}E^{T}(t-\tau)JE(t-\tau)-\frac{1}{\xi}\overline{\dot{\eta}}\dot{\eta}. \tag{29}\]  
\(\Box\)  
Substitute error Equation (27) into Equation (29) and obtain the following:  
\[ \begin{array}{ll}\dot{V}=&\frac{1}{2}\dot{E}^{T}JE-\frac{1}{2}E^{T}(t-\tau) JE(t-\tau)-\frac{1}{\xi}\overline{\dot{\eta}}\dot{\eta}-\hat{\eta} \big{\|}E^{T}PB\big{\|}+E^{T}PB(mg+n_{u})\\ &\qquad+E^{T}PA_{2}E(t-\tau)+\frac{1}{2}E^{T}\Big{[}P(A_{1}-BK_{2})+(A_{1}- BK_{2})^{T}P\Big{]}\end{array}  \tag{30}\]  
controller were respectively applied to the maglev system in the simulation model. MATLAB/Simulink was utilized to provide the simulation environment. The values of the system parameters are outlined in Table 1.  
Without loss of generality, the values of \(n_{u}\) and \(A_{2}\) were selected as below:  
\[n_{u}=e^{-0.5t}\sin(t)\]  
\[A_{2}=diag(\left[ \begin{array}{cc}0.03\text{randi}(10)&0.001\text{randi}(20) \end{array} \right])\]  
For the proposed controller in this paper, gains were chosen as: \(K_{1}=0\), \(K_{2}=\left[ \begin{array}{cc}35.4&8.12\end{array} \right]\), \(K_{2}=\left[ \begin{array}{cc}0.065&0.0971\end{array} \right]\), and \(\xi=66.72\). The reference signal was selected as: \(\sigma=\left[ \begin{array}{cc}0.6&0\end{array} \right]^{T}\). The initial state of the system was \(X_{0}=\left[ \begin{array}{cc}0.0016&0\end{array} \right]^{T}\). The control target trajectory was \(X_{d}=\left[ \begin{array}{cc}0.0085&0\end{array} \right]^{T}\). The traditional PID controller is selected, and the control parameters are \(k_{p}=1800\), \(k_{i}=150\), and \(k_{d}=1150\) based on the Root Locus method. The parameters of the SMC controller are chosen as: \(c_{1}=20\), \(c_{2}=5\), \(k_{\text{smc}}=15\), \(\xi_{\text{smc}}=20\). The performance of the proposed controller, the SMC controller, and the PID controller were simulated under the two following cases. Case 1: Levitation without disturbance. Case 2: Levitation with disturbance. (1) Levitation without disturbance and with uncertain time delay. The Matlab/SimEvent(r) is utilized to build the CAN model to represent the signal transmission, which CAN reflect the bus load rate and the uncertain time delay.