# A Polishing Processes Optimization Method for Ring-Pendulum Double-Sided Polisher
## 3 RDP Radial-Feed System Control Structure
### Stability Analysis  
As part of our analysis, we prove that the nominal MPC shows recursive feasibility. Moreover, we prove that \(z(k)\) will be contained in the terminal constraint set \(\mathbb{Z}_{\infty}\) and eventually approach zero.  
Using Equation (6), we obtain the difference between the nominal predictive optimization cost functions \(V_{N}(\overline{x}(k))\) and \(V_{N-1}(\overline{x}(k+1))\) as follows:  
\[\begin{split}& V_{N-1}(\overline{x}(k+1))-V_{N}(\overline{x}(k)) \\ &=L(\overline{x}(0\mid k+1),\overline{u}(0\mid k+1))+\cdots+L \Big{(}\overline{x}(N-2\mid k+1),\overline{u}(N-2\mid k+1)_{N-2\mid k+1}\Big{)} \\ &+F(\overline{x}(N-1\mid k+1))-(L(\overline{x}(0\mid k),\overline {u}(0\mid k))+\cdots+L(\overline{x}(N-1\mid k),\overline{u}(N-1\mid k))+F( \overline{x}(N\mid k)))\end{split} \tag{27}\]  
Then, by combining Equation (27) and the condition \(\overline{x}(i+1|k)=\overline{x}(i|k+1)\), \(i\in[1,N-1]\), we obtain:  
\[\begin{split}& V_{N-1}(\overline{x}(k+1))-V_{N}(\overline{x}(k))=L( \overline{x}(1\mid k),\overline{u}(1\mid k))+\cdots\\ &\quad+L(\overline{x}(N-1\mid k),\overline{u}(N-1\mid k))+F( \overline{x}(N\mid k))-(L(\overline{x}(0\mid k),\overline{u}(0\mid k))+\cdots \\ &\quad+L(\overline{x}(N-1\mid k),\overline{u}(N-1\mid k))+F( \overline{x}(N\mid k)))\\ &=-L(\overline{x}(0\mid k),\overline{u}(0\mid k))=-\Big{(} \overline{x}(0\mid k)^{T}\overline{Q}\overline{x}(0\mid k)\Big{)}-\Big{(} \overline{u}(0\mid k)^{T}R\overline{u}(0\mid k)\Big{)}\leq 0\end{split} \tag{28}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Steps** & **Single-Tube RMPC Method** \\ \hline
1 & Define the predictive control step size \(N\) and the constraints \(\mathbb{X}\),\(\mathbb{U}\),\(\overline{W}\), and \(\mathbb{V}\). Set the system state \(x(0)\) and matrixes \(L\), \(K\). \\
2 & Compute \(\mathbb{Z}(\alpha,s)\) as the single-tube constraint set by utilizing \(\mathbb{Z}_{\infty}\). \\
3 & Drive \(\overline{x}(k)\) to the nominal MPC reachable set \(\mathbb{X}_{f}\) by the nominal model prediction control quantity \(\overline{u}(k)\). \\
4 & Combine the \(\overline{u}(k)\) and the feedback to obtain the general control law \(u(k)\) and drive \(x(k)\) into the single-tube constraint set. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Steps of RDP radial-feed system control method.  
Figure 4: Flowchart of RDP radial-feed system control method.  
The inequality of nominal model prediction optimization cost functions is extended as:  
\[\begin{array}{l}V_{N-2}(\overline{x}(k+2))-V_{N-1}(\overline{x}(k+1))=-L( \overline{x}(1\mid k+1),\overline{u}(1\mid k+1))\\ =-\left(\overline{x}(1\mid k)^{T}Q\overline{x}(1\mid k+1)\right)-\left( \overline{u}(1\mid k+1)^{T}R\overline{u}(1\mid k+1)\right)\leq 0\end{array} \tag{29}\]  
Due to the nominal MPC ignoring the uncertain disturbance, we demonstrate that \(V_{N}(\overline{x}(k))\) is a bounded and non-increasing sequence; therefore, the nominal MPC shows recursive feasibility.  
As the nominal MPC shows recursive feasibility, when \(k\geq 0\), \(\overline{z}(k)\in\overline{\mathbb{Z}}\), and \(\overline{u}(k)\in\overline{U}\) are satisfied, \(\overline{z}(k)\in\overline{\mathbb{Z}}\Rightarrow z(k)\in\mathbb{Z}\) and \(\overline{u}(k)\in\overline{U}\Rightarrow u(k)\in\overline{U}\). Then, using Equation (23), we can obtain \(z(0)\in\mathbb{Z}\), \(z(k)\in\mathbb{Z}\). For \(k\geq 0\), we can derive \(z(k)\in\mathbb{Z}\) from \(z(0)\in\mathbb{Z}\); for \(k\geq 0\), \(x(k)=\begin{bmatrix}I&I\end{bmatrix}z(k)+\overline{x}(k)\) is satisfied. When \(\overline{x}(k)\to 0\), we can obtain \(e(k)+\xi(k)\rightarrow\mathbb{Z}_{\infty}\); then, \(x(k)\rightarrow\mathbb{Z}_{\infty}\). As condition \(\lim\limits_{N\rightarrow\infty}x(k+N)=0\), the system state is asymptotically stable. The system states will gradually approach zero and stabilize in the \(\mathbb{Z}_{\infty}\).