## 2 Study of a Robust Nonlinear Control Strategy
### Analytical Design for the SMC and I/O FBL Control Technique  
The SMC technique is an application of the relatively general variable structure control (VSC) technique that was first researched by [33]. This technique is interesting, especially because it can handle time-varying nonlinear systems and can deal with disturbances and uncertainties directly. Furthermore, it may facilitate the production of flexible control systems.  
VSC systems comprise controls whose structure can be changed, i.e., movement among the elements of a group comprising continuous feasible functions corresponding to the state of the system. The design problem concerns a specific choice of the factors corresponding to every structure and specifying the travel logic. The ability to integrate useful characteristics pertaining to the structure is a suitable tradeoff for complexity.  
Furthermore, a dynamic structure system can have characteristics, unlike individual structures. Hence, the sliding mode control approach is defined as such. Such control systems have been cemented by the efforts presented in [32].  
The SMC process requires the definition of a sliding structure \(P(t)\), which is typically locally asymptotically stable and linear time-varying; it is required that the sliding surface should be associated with the problem. The objective is to get to this surface and be present (slide) on the same. In the context of nonlinear systems having the Byrnes-Isidori standard form, the surface \(P(t)\) is selected as a function that acts on a specific error variable:  
\[P(t)=\left(d/dt+\theta\right)^{q-1}E(t) \tag{37}\]  
Here, \(\theta\) denotes the bandwidth corresponding to error dynamics, representing system performance on \(P(t)\). In the case where the process is not evolving on the sliding area \(P(t)\), there must be a constraint that makes the system dynamics move to reach \(P(t)\).  
Furthermore, referred to as the reaching condition or the attractiveness equation, it may be selected to set the dynamics corresponding to the switching function directly. The goal is met under the conditions specified by the reaching law technique [8; 9]. Consider the joining constraint seen as constant rate law and given by:  
\[\dot{P}\leq\theta|P|^{\rho}\text{sgn}P \tag{38}\]  
It can then be asserted that the system is expected to take a finite time to reach the surface \(P(t)\) and move along it. This indicates that at a bandwidth of the variable, \(\theta\) the error function \(E(t)\) is close to zero.  
If the objective is to integrate I/O FBL with SMC, the surface definition comprises a stable linear operator having an order \(q-1\):  
\[P=\sum\limits_{k=0}^{q-1}\sum\limits_{i=0}^{k}\Gamma_{k}E^{(i+1)} \tag{39}\]  
\[\Gamma_{q-1}=1\]  
From Equations (37) and (38), the altered control input is acquired as:  
\[U=-\frac{1}{\sum\limits_{k=0}^{q-1}\Gamma_{k}^{i}X^{[i]}}\left[\begin{array}[ ]{c}\sum\limits_{i=0}^{q}\Lambda_{i}^{j}X^{[i]}-\frac{\alpha y_{d}}{dq_{t}} \Gamma_{k}\sum\limits_{i=0}^{q}\Lambda_{j}^{i}X^{[i]}\\ -\sum\limits_{k=0}^{q}\Gamma_{k}\frac{\alpha y_{d}}{dq_{k}}-\theta|P|^{\theta }\text{sgn}P\end{array}\right] \tag{40}\]  
The association between the SMC technique and I/O FBL is specified in Equation (40) under the condition where the error dynamics corresponding to the sliding surface are linear and time-invariant.  
Sliding mode, in its original form, demonstrates chattering corresponding to the control signal, which is inadmissible for a majority of industrial applications. Some techniques help reduce chattering [8; 9]. For example, the control signal can be smoothened by regulating the parameters \(\vartheta\) and \(\rho\) corresponding to Equation (38).