## 2 Grinding Protocol and Problem Formulation
### Problem Formulation  
Consider the batch grinding system as a SISO nonlinear discrete-time system, which can be shown in Equation (9):  
\[\mathrm{D_{k}(t+1)}=\mathrm{f}(\mathrm{D_{k}(t)},\mathrm{u_{k}}) \tag{9}\]  
where \(\mathrm{t}=0,1,2,\ldots,\mathrm{N}\) is the sampling time index and \(\mathrm{N}\) is the finite time interval of batch grinding system; k indicates the system repetition number and \(\mathrm{D_{k}(t)}\in\mathrm{R}\) is the system output; \(\mathrm{u_{k}}\in\mathrm{R}\) denotes the system input, which is constant at all sampling time in the same iteration; and \(\mathrm{f}(\cdot)\) is an unknown function and continuously differentiable.  
In the batch grinding system, only the terminal dimension \(\mathrm{D_{k}(N)}\) is measurable at the end of each run. Over iterations, the Equation (9) can be rewritten as a continuously differential function \(\mathrm{g}(\cdot)\)[25]. It is shown in Equation (10):  
\[\begin{split}\mathrm{D_{k}(N)}=\mathrm{f}(\mathrm{D_{k}(N-1)}, \mathrm{u_{k}})=\mathrm{f}(\mathrm{f}(\mathrm{D_{k}(N-2)},\mathrm{u_{k}}), \mathrm{u_{k}})=\mathrm{f}(\ldots\mathrm{f}(\mathrm{f}(\mathrm{D_{k}(0)}, \mathrm{u_{k}}),\mathrm{u_{k}}),\ldots,\mathrm{u_{k}})\\ =\mathrm{g}(\mathrm{D_{k}(0)},\mathrm{u_{k}})\end{split} \tag{10}\]  
Here, \(\mathrm{D_{k}(0)}\) is the initial value of Equation (9). The system is controllable and the initial value \(\mathrm{D_{k}(0)}\) is identical for every iteration, \(\mathrm{D_{k}(0)}=\mathrm{D_{k-1}(0)}=\mathrm{c}\). c is a constant for the batch grinding system, which is the dimension of workpiece before grinding discussed in Section 4.  
**Assumption 1**.: _The partial derivative of \(f(\ldots)\) with respect to control input \(u_{k}\) is continuous._  
**Assumption 2**.: _The Equation (9) satisfies the generalized Lipschitz condition. That is, \(|\Delta D_{k}(N)|\leq b|\Delta u_{k-1}|\), here \(\Delta D_{k}(N)=D_{k}(N)-D_{k-1}(N)\), \(\Delta u_{k-1}=u_{k-1}-u_{k-2}\) and \(b\) is positive._  
Applying the mean value theorem into Equation (10), Equation (10) can be transformed to Equation (11),  
\[\Delta\mathrm{D_{k}(N)}=\frac{\partial\mathrm{g}}{\partial\mathrm{D}}\Delta \mathrm{D_{k}(0)}+\frac{\partial\mathrm{g}}{\partial\mathrm{u}}\Delta\mathrm{ u_{k}} \tag{11}\]  
Since \(\mathrm{D_{k}(0)}=\mathrm{D_{k-1}(0)}\), then \(\Delta\mathrm{D_{k}(0)}\) is 0. Therefore, Equation (11) can be simplified as Equation (12),  
\[\Delta\mathrm{D_{k}(N)}=\frac{\partial\mathrm{g}}{\partial\mathrm{u}}\Delta \mathrm{u_{k}}=\varphi_{k}\Delta\mathrm{u_{k}} \tag{12}\]  
Here, \(\varphi_{k}=\frac{\partial\mathrm{g}}{\partial\mathrm{u}}\). According to Assumption 2, \(|\varphi_{k}|\leq b\). It is called pseudo-partial-derivative (PPD), which is a DL parameter. For more details, please see [23].  
Then Equation (10) can be transformed to Equation (13),  
\[\mathrm{D_{k}(N)}=\mathrm{D_{k-1}(N)}+\varphi_{k}\Delta\mathrm{u_{k}} \tag{13}\]  
**Remark 1**.: _From a practical point, these assumptions imposed on the industrial system are reasonable and acceptable. Assumption 1 is a typical condition of control system design for general nonlinear systems. Assumption 2 imposes an upper bound limitation on the change rate of the system output driven by the changes in the control inputs. From the energy conservation point, the energy change rate in an industrial system is not infinite [24]._  
**Remark 2**.: \(\varphi_{k}\) _is a differential signal and bounded to any k, which is unknown and iteratively changes._An index function \(\mathrm{J}=||\mathrm{D_{k}(N)-D_{k-1}(N)}||^{2}+\upmu||\mathrm{D_{k}(N)}||^{2}\)[23] is introduced to estimate the value of \(\varphi_{\mathrm{k}}\). By optimal condition, an estimation algorithm for \(\varphi_{\mathrm{k}}\) is derived as Equation (14): \[\hat{\varphi}_{\mathrm{k}}=\hat{\varphi}_{\mathrm{k-1}}+\frac{\rho(\Delta \mathrm{D_{k-1}}-\hat{\varphi}_{\mathrm{k-1}}\Delta\mathrm{u_{k-1}})\Delta \mathrm{u_{k-1}}}{\upmu+||\Delta\mathrm{u_{k-1}}||^{2}}\] (14) Here it is bounded. The weighting factor \(\upmu>0,\ 0<\rho<2\).