## 3 Observer and Controller Design for Scratch Formation Reduction  
### Spo  
In hot rolling systems, the gap control and rotation velocity control of the rollers should be capable of overcoming intense disturbances. The SMCSPO is a robust controller suitable for controlling nonlinear systems with disturbances. The SPO estimates the load torque of the roller's rotating motor for velocity synchronization; it also renders the SMCSPO controller to operate more robustly by estimating the nonlinear terms of the system to compensate for the control input. Furthermore, the perturbation estimated by the SPO can provide a standard for evaluating the probability of scratch formation.  
In this section, the structure of an SPO is briefly discussed. The governing equation for the general second-order dynamics, including disturbance, is given as  
\[\ddot{\theta}=f(\theta)+\Delta f(\theta)+(b(\theta)+\Delta b(\theta))u_{c}+d( t), \tag{3}\]  
where \(\theta\) is the rotation angle of the roller; \(f(\theta)\) and \(\Delta f(\theta)\) are the system dynamics matrices and their nonlinear terms, respectively; \(b(\theta)\) and \(\Delta b(\theta)\) are the control input matrices and their nonlinear terms, respectively; and \(d(t)\) is the disturbance term caused by excessive friction between the roller and steel plate.  
Perturbation is defined as the combination of all nonlinear terms and disturbances in (3), as follows:  
\[\Psi=\Delta f(\theta)+\Delta b(\theta)u_{c}+d(t), \tag{4}\]  
where the perturbations are assumed as upper-bounded by a known continuous function of state [20].  
Based on the definition in (4), the nonlinear term is separated from the dynamics. For convenience, the linear term of the system is also separated. An arbitrary positive number (\(\alpha_{3}\)) and a new control variable (\(\overline{u}\)) are defined to decouple the control variable.  
\[f\big{(}\dot{\theta}\big{)}+b\big{(}\dot{\theta}\big{)}u_{c}=\alpha_{3} \overline{u}. \tag{5}\]  
The state space representation is given by the following:  
\[\begin{array}{c}\dot{x}_{1}=x_{2},\\ \dot{x}_{2}=\alpha_{3}\overline{u}+\Psi,\\ \mathrm{y}=x_{1}=\theta.\end{array} \tag{6}\]  
Using (4) and (5), (3) can be simplified as (6). A new state variable (\(x_{3}\)) for deriving the perturbation from the sliding observer is defined as  
\[x_{3}=\alpha_{3}x_{2}-\frac{\Psi}{\alpha_{3}}. \tag{7}\]  
The perturbation is calculated using (7). The perturbation of the entire sliding observer structure [17] for monitoring the system states is given by  
\[\begin{array}{c}\dot{\dot{x}}_{1}=\dot{x}_{2}-k_{1}sat(\widetilde{x}_{1})- \alpha_{1}\widetilde{x}_{1},\\ \dot{\dot{x}}_{2}=\alpha_{3}\overline{u}-k_{2}sat(\widetilde{x}_{1})-\alpha_ {2}\widetilde{x}_{1}+\dot{\Psi},\\ \dot{\dot{x}}_{3}=\alpha_{3}^{2}(-\dot{x}_{3}+\alpha_{3}\dot{x}_{2}+\overline {u}).\end{array} \tag{8}\]  
The perturbation (\(\dot{\Psi}\)) can be calculated as  
\[\dot{\Psi}=\alpha_{3}(-\dot{x}_{3}+\alpha_{3}\dot{x}_{2}), \tag{9}\]where \(k_{1},\ k_{2},\ \alpha_{1},\ \alpha_{2}\), and \(\alpha_{3}\) are positive values, and \(\widetilde{x}_{i}=\hat{x}_{i}-x_{i}\) is the estimated state error. Throughout the text, \"\(\sim\)\" refers to estimation errors. The saturation function is used to reduce chattering in the sliding surface and is given by  
\[sat(\widetilde{x}_{1})=\left\{\begin{array}{c}\widetilde{x}_{1}/|\widetilde {x}_{1}|,\ \ if\ |\widetilde{x}_{1}|\geq\varepsilon_{0}\\ \widetilde{x}_{1}/\varepsilon_{0},\ if\ |\widetilde{x}_{1}|<\varepsilon_{0} \end{array}\right., \tag{10}\]  
where \(\varepsilon_{0}\) is the SPO boundary layer.