## 2 Preliminaries for Designing a Sliding Mode Controller
### Methodology Principles
#### 2.1.2 Control Law Determination  
The structure of a sliding mode controller consists of two parts: one called the continuous part (\(U_{eq}(t)\)) and the second one named the discontinuous or nonlinear part \(U_{n}(t)\); hence,  
\[U(t)=U_{eq}(t)+U_{n}(t), \tag{2}\]  
where \(U_{eq}(t)\) is obtained from the equivalent control method [6; 77] and is used to control the variable on the sliding surface \(\sigma(t)=0\). Moreover, \(U_{eq}(t)\) is obtained by combining the previous equation and the process model, which could be expressed by a differential equation such as the following:  
\[\frac{dx(t)}{dt}=f(t,x(t),u(t,x(t))),\ \ \ t\in\mathbb{R}, \tag{3}\]  
where \(x(t)\in\mathbb{R}^{n}\) denotes the vector of system states; \(u(t)\in\mathbb{R}^{m}\) represents the vector of control inputs, which depends on time and state, and it can be discontinuous in its arguments but locally bounded; and \(f\) is a continue function for simplicity.  
Thus, the equivalent control is deduced, considering that the derivative of the surface is zero. Therefore, the sliding condition is given as follows:  
\[\frac{d\sigma(t)}{dt}=0. \tag{4}\]  
Moreover, \(U_{eq}(t)\) is derived by combining the previous equation and the process system model. The equivalent control method involves determining a control law that makes the system's dynamics on the sliding surface equivalent to a more straightforward system. It transforms the original system into an equivalent system with simplified dynamics, making it easier to design and analyze the controller. This equivalent system typically exhibits simpler behavior on the sliding surface, facilitating the application of control strategies. In summary, the equivalent control method in sliding mode control aims to simplify the analysis and design of the control law by transforming the system dynamics on the sliding surface into an equivalent and more manageable form.  
On the other hand, \(U_{n}(t)\) is the discontinuous control part; it allows the system to reach the sliding surface. \(U_{n}(t)\) incorporates a nonlinear element that includes a switching element of the control law and is given as follows:  
\[U_{n}(t)=K_{D}\operatorname{sgn}(\sigma(t)). \tag{5}\]  
Finally, \(K_{D}\) is a tuning parameter responsible for the reaching mode.  
The chattering phenomenon is the main concern in SMC due to the jumps of the sliding surface [1]. In practice, chattering produces inadmissible effects, such as neglected excitation of system dynamics in modeling, wear-induced vibration of actuators, and performance degradation. Several approaches have been proposed to reduce such effects. One of these possibilities lies in smoothing the nonlinear switching function using soft logistic activation functions, as the sigmoid function preferentially [78]. Therefore, the discontinuous part of the proposed controller uses the sigmoid function as follows:  
\[U_{n}(t)=K_{D}\frac{\sigma(t)}{|\sigma(t)|+\delta}, \tag{6}\]  
where the parameter \(\delta\) is associated with the chattering effect; because of this, it is necessary to assign an appropriate value.