# Root Cause Tracing Using Equipment Process Accuracy Evaluation for Looper in Hot Rolling
## 2 EPAE Model of Looper
### Control Accuracy of the Looper
#### 2.3.1 Entry Process  
The starting process mainly focuses on three indicators, the starting angle, the rising time and the time to enter the steady state. (1) Starting angle  
This is the average value of the looper within 3-8 m of the head of the strip. The maximum value of the difference between the measured angle of the loop and the set value within the range of 3-8 m from the head of the rolling plate strip on the lower frame of the loop is calculated. The calculation process is as follows. The defined length of the strip head is 3-8 m. The starting and ending points of the head are calculated according to the rolling speed and sampling period:  
\[ \begin{split}\sum_{t=0}^{N_{\text{B}}}[v_{i}[t]T=3]\\ \sum_{t=0}^{N_{\text{B}}}[v_{i}[t]T=8]\end{split}  \tag{1}\]  
where \(v_{i}\) is the rolling speed of the finishing rolling \(F_{i}\) stand, \(T\) is the data sampling interval, \(N_{3}\) is the head 3 m data point, \(N_{\text{B}}\) is the head 8 m data point. When the rolling section reaches the 3 m data point and the 8 m data point, the lifting angles at both positions as \(\alpha_{i}\) and \(\alpha_{i}^{{}^{\prime}}\), respectively, are recorded. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Object** & **Name** & **Symbol** \\ \hline \multirow{4}{*}{Entry process} & Starting angle & \(\Delta\alpha\) \\  & Rising time & \(t\) \\  & Steady-state time & \(t_{n}\) \\ \hline \multirow{4}{*}{Steady-state process} & Oscillation amplitude & \(a\) \\  & Number of oscillations & \(f_{s}\) \\  & Looper tension & \(T_{f}\) \\ \hline \multirow{4}{*}{Exit process} & Falling time & \(d_{t}\) \\  & Steel-throwing tension & \(T_{t}\) \\ \cline{1-1}  & Small set time & \(t_{s}\) \\ \cline{1-1}  & Small set angle & \(a_{s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Evaluation index of process accuracy of the looper. The starting angle is calculated based on the obtained \(h_{3}\) and \(h_{8}\):  
\[\Delta\alpha_{i}=\max(|\alpha_{i}-\alpha_{i}^{{}^{\prime}}|) \tag{2}\]  
where \(\alpha_{i}\) is a vector composed of measured angles of the looper \(LP_{i}\), \(\alpha_{i}^{{}^{\prime}}\) is a vector composed of set angles of the looper \(LP_{i}\), and \(\Delta\alpha\) is the starting angle of the looper. (2) Rising time  
The time it takes for the looper to rise during the set-up phase. The time it takes for the actual measured value of the looper angle to go from the set value 10% to the set value 90% is calculated. The calculation method is as follows:  
\[ \begin{array}{l}\alpha_{i-set}\times 10\%=\alpha_{i}[t_{1}]\\ \alpha_{i-set}\times 90\%=\alpha_{i}[t_{2}]\\ t_{i}=t_{2}-t_{1}\end{array}  \tag{3}\]  
where \(\alpha_{i-set}\) is the set value of the looper angle, \(t_{i}\) is the rise time of the looper \(LP_{i}\), \(t_{1}\) is the data point reaching the 10% set value, and \(t_{2}\) is the time to reach 90% of the data points for the set values. (3) Steady-state time  
The time it takes for the looper to reach steady state. The time it takes for the looper to bite the steel from the lower frame until the actual measured value of the looper enters the \(\pm 2^{\circ}\) error band of the looper set value is calculated. All points where the upper and lower error bands of the measured value of the looper angle intersect with the set value of the looper are calculated:  
\[\alpha_{i}[\Xi[i]]=(\alpha_{i}^{{}^{\prime}}+2)||(\alpha_{i}^{{}^{\prime}}-2) \tag{4}\]  
where \(\Xi\) is the set of all points where the upper and lower error bands of the measured value of the looper angle intersect with the set value of the looper, and \"\(||\)\" is the \"OR\" operation. The two points with the largest distance in the set, which are the steady-state start time and steady-state end time is calculated:  
\[ \begin{array}{l}\Xi[k+1]-\Xi[k]=\max(\Xi[i+1]-\Xi[i])\\ d_{u}=\Xi[k]\\ d_{e}=\Xi[k+1]\end{array}  \tag{5}\]  
where \(d_{u}\) is the start data point of steady state, and \(d_{e}\) is the end data point of steady state.