# A Polishing Processes Optimization Method for Ring-Pendulum Double-Sided Polisher
## 3 RDP Radial-Feed System Control Structure  
### Single-Tube RMPC  
Figure 3 shows the control structure of the single-tube RMPC. \(x_{i}\) is the actual input. \(x_{o}\) is the actual output. The difference between the input and output is defined as actual state \(x(k)\). The structure consists of three parts: nominal model predictive control, state estimation, and state feedback. The identification system model is utilized for the nominal MPC optimization. \(\hat{x}(k)\) represents the estimated state of the system, which is obtained through the utilization of the estimation matrix \(L\). The total control volume of the single-tube robust predictive controller consists of the nominal model predictive control (MPC) and the feedback control. Moreover, the system state gradually approaches the origin under the influence of the control quantity. \[x\in\mathbb{X},u\in\mathbb{U},w\in\mathbb{W},v\in\mathbb{V} \tag{4}\]  
Figure 3: Single-tube RMPC structure. \[\overline{x}(k+1)=A\ \ \overline{x}(k)+B\overline{u}(k) \tag{5}\] \[\overline{y}(k)=C\ \ \overline{x}(k)\]  
\(x(k)\), \(u(k)\), \(w(k)\), and \(v(k)\) in Equation (3) are constrained by Equation (4). \(\mathbb{X}\), \(\mathbb{U}\), \(\mathbb{W}\), and \(\mathbb{V}\) represent the constraint sets containing the origin. Equation (5) describes the nominal MPC system, which excludes the disturbances \(w(k)\) and \(v(k)\). \(\overline{x}(k)\) is the nominal prediction state \(\overline{x}(k)\). \(\overline{u}(k)\) is the nominal model prediction control quantity. \[ \begin{array}{l}\min\limits_{\overline{x}(k),\overline{u}(k)^{\Omega N-1}}V _{N}\Big{(}\overline{x}(k),\overline{u}(k|k)^{\Omega N-1}\Big{)}=F(\overline{ x}(N|k))+\sum\limits_{i=0}^{N-1}L(\overline{x}(i|k),\overline{u}(i|k))\\ F(\overline{x}(N|k))=\overline{x}\left(N|k\right)^{T}\!\!P\overline{x}(N\! \left|k\right)\\ L(\overline{x}(i|k),\overline{u}(i|k))=\overline{x}\left(i|0\right)^{T}\!Q \overline{x}(i|k)+\overline{u}\left(i|k\right)^{T}\!R\overline{u}(i\!\left|k\right) \\ \overline{x}(k)\in\mathbb{X}\ \ \ \overline{u}(k)^{\Omega N-1}\in\mathbb{U}\ \ \ \overline{x}(N)\in\mathbb{X}_{f}\\ i=k,k+1,\cdots,k+N-1\end{array}  \tag{6}\]  
\(x(k)=[\overline{x}(k|k),\overline{x}(k+1|k),\cdots\overline{x}(k+N|k)]\) is a sequence of the nominal system state and \(u(k)=[\overline{u}(k|k),\overline{u}(k+1|k),\cdots\overline{u}(k+N-1|k)]\) is a sequence of the optimal control quantity at k = [0,... \(k\)]. The integer \(N\) represents the prediction horizon length. At current time \(k\), \(\overline{x}(k)\) is the nominal MPC system state and \(\overline{u}(k)=\overline{u}^{*}(k|k)\) is the nominal MPC control quantity. \(\mathbb{X}_{f}\) represents the nominal MPC reachable set, while \(R\) and \(Q\) are two weight matrices in nominal MPC cost function \(V_{N}\). \[u(k)=\overline{u}(k)+K(\hat{x}(k)-\overline{x}(k)) \tag{7}\]  
\[A_{K}=A+BK \tag{8}\]  
The feedback matrix \(K\) can eliminate the distance between the actual state and the estimated state. To ensure system stability, the feedback matrix \(K\) must satisfy the condition that the spectral radius of matrix \(A_{K}\) is below a value of 1. \[\hat{x}(k+1)=A\hat{x}(k)+Bu(k)+L(C\hat{x}(k)-y(k)) \tag{9}\]  
\[A_{L}=A+LC \tag{10}\]  
\(\hat{x}(k)\) in Equation (9) is the estimated state. The parameters in estimated matrix \(L\) must guarantee that the spectral radius of matrix \(A_{L}\) is below a value of 1, ensuring system stability. The estimated error \(e(k)\) is defined as the difference between \(x(k)\) and \(\hat{x}(k)\). The prediction error \(\xi(k)\) is defined as the difference between \(\overline{x}(k)\) and \(\hat{x}(k)\).