## 3 MPC Strategy
### Proof of MPC System Stability  
The stability of the MPC system has a significant influence on the control effect. Therefore, it is necessary to prove the stability of the MPC system. [PERSON]'s theory is used to prove the stability of the system [37], and the [PERSON] function \(\mathbf{V}(\mathbf{x}(t\,|\,t))\) is:  
\[\mathbf{V}(\mathbf{x}(t\,|\,t))=\sum\limits_{i=1}^{Np}\parallel\mathbf{\eta}(t+\mathrm{i }|\,t)-\mathbf{\eta}_{ref}(t+\mathrm{i}|\,t)\parallel^{2}\,_{Q}+\sum\limits_{i=0}^ {\text{Nc}-1}\parallel\Delta\mathbf{u}(t+\mathrm{i}|\,t)\parallel^{2}\,_{RR}+ \rho\varepsilon^{2} \tag{20}\]  
Combining the \(\mathbf{Q}\) matrix and the reference output \(\mathbf{\eta}_{ref}(t+\mathrm{i}|\,t)\)(1:4,1) \(\equiv\mathbf{zeros}\)(4,1), it can be known that \(\mathbf{V}(\mathbf{x}(t\,|\,t))\) is positive definite. The objective function (Equation (15)) always has an optimal solution that satisfies the constraints, and the control increment obtained by its optimization at time \(\mathrm{k}\) is \(\Delta\mathbf{U}_{t}=\Delta\left[\mathbf{u}(t\,|\,t);\,\Delta\mathbf{u}(t+1\,|\,t);\,\ldots,\,\Delta\mathbf{u}(t+Nc-1\,|\,t);\,\ldots\ ;\,\Delta\mathbf{u}(t+Np-1\,|\,t)\right]\).  
At sampling time \(t+1\), the Lyapunov function \(\mathbf{V}(\mathbf{x}(t+1\,|\,t+1))\) becomes:  
\[\begin{array}{l}\mathbf{V}(\mathbf{x}(t+1|t+1))=\sum\limits_{i=1}^{Np}\parallel\mathbf{ \eta}(t+1+\mathrm{i}|\,t+1)-\mathbf{\eta}_{ref}(t+1+\mathrm{i}|\,t+1)\parallel^{2 }\,_{Q}+\\ \sum\limits_{i=0}^{\text{Nc}-1}\parallel\Delta\mathbf{u}(t+1+\mathrm{i}|\,t+1) \parallel^{2}\,_{RR}+\rho\varepsilon^{2}\end{array} \tag{21}\]  
The control increment obtained by optimization at \(k+1\) is \(\Delta\mathbf{U}_{t+1}=[\Delta\mathbf{u}(t+1\,|\,t+1);\,\Delta\mathbf{u}(t+1\,|\,t+1);\, \ldots\ ;\,\Delta\mathbf{u}(t+Nc\,|\,t+1);\,\ldots\ ;\,\Delta\mathbf{u}(t+Np\,|\,t+1)]\).  
The connection between the Lyapunov functions \(\mathbf{V}(\mathbf{x}(t\,|\,t))\) and \(\mathbf{V}(\mathbf{x}(t+1\,|\,t+1))\) must be established and the intermediate function \(\mathbf{M}\mathbf{V}(\mathbf{x}(t+1\,|\,t+1))\) constructed. The control increment in the intermediate function is \(\mathbf{M}\Delta\mathbf{U}_{t}=\Delta\left[\mathbf{u}(t+1\,|\,t);\,\Delta\mathbf{u}(t+2\,|\,t); \,\ldots\,\,\Delta\mathbf{u}(t+Nc-1\,|\,t);\,\ldots\ ;\,\Delta\mathbf{u}(t+Np-1\,|\,t);\mathbf{0}_{3 \times 1}\right]\). This control increment is directly provided and not obtained through optimization. In addition, let \(\textbf{MV}(\textbf{x}(t+1\,|\,t+1))\) and the right-hand side of Equation (21) be equal, so:  
\[\textbf{V}(\textbf{x}(t+1|t+1))\leq\textbf{MV}(\textbf{x}(t+1|t+1)) \tag{22}\]  
Therefore,  
\[\textbf{V}(\textbf{x}(t+1|t+1))-\textbf{V}(\textbf{x}(t|t))\leq\textbf{MV}( \textbf{x}(t+1|t+1))-\textbf{V}(\textbf{x}(t|t)). \tag{23}\]  
Combining Equations (13) and (23), it can be obtained that:  
\[\textbf{V}(\textbf{x}(t+1|t+1))-\textbf{V}(\textbf{x}(t|t))\leq\parallel\textbf {\eta}(t+Np|t)\parallel^{2}\text{q}^{-}\parallel\textbf{\eta}(t+1|t)\parallel^ {2}\text{q}^{-}\parallel\Delta\textbf{u}(t|t)\parallel^{2}\textbf{{}_{RR}} \tag{24}\]  
Since Equation (19) is a constraint on the \(Np\)-th prediction vector of the quadratic programming (Equation (17)), it can be obtained that \(\textbf{\eta}(t+Np|\,t)\)(1:4) = _zeros_ (4,1). Combining the **Q** matrix, it can be obtained that the \(\parallel\textbf{\eta}(t+Np|t)\parallel^{2}\text{q}\) is equal to zero, that is:  
\[\textbf{V}(\textbf{x}(t+1|t+1)-\textbf{V}(\textbf{x}(t|t))\leq-\parallel \textbf{\eta}(t+1|t)\parallel^{2}\text{q}^{-}\parallel\Delta\textbf{u}(t|t) \parallel^{2}\textbf{{}_{RR}}\leq 0 \tag{25}\]  
Therefore, the MPC system is stable.