## 2 Preliminaries for Designing a Sliding Mode Controller
### SMC Schemes
#### 2.2.3 Fractional-Order SMC  
Fractional-order calculus extends the derivative and integration operations to fractional exponents [103; 104]. It has been widely used for the design of control systems and process modeling [105; 106; 107; 103]. Fractional-order controllers provide designers with flexibility in choosing a fractional power order and additional parameters for the tuning process that can enhance performance and robustness concerning integer-order controllers [105; 107]. Regarding fractional-order models, they can obtain better approximations of processes that exhibit inherent fractional-order behavior, for instance, phenomena of viscoelasticity, chaos systems, robotics, cell diffusion process, voltage-current relation of semi-infinitetransmission lines, fractals, resonant systems, HIV infection resistance, chemical processes, etc. [104; 105; 106; 108; 109].  
Thanks to the advances in computational tools, the number of fractional-order controllers has increased in automatic control systems with a high degree of complexity [106; 107]. The most common structures of fractional-order controllers include combinations such as integer order plant and controller, integer order plant and fractional-order controller, fractional-order plant and IO controller, and fractional-order plant and controller [105]. Moreover, fractional-order controller design can be approached from both the time and frequency domains, where the time domain uses optimization techniques more frequently. In contrast, the frequency domain methods are based on \(H\infty\) norm, isodamping, and loop-shaping [105].  
Various fractional-order control strategies have been documented, such as Tilted Integral Derivative (TID) controllers [110; 111; 112; 113], CRONE (Controle Robust d'Orde Non-Entier translating to non-integer robust control in French) controllers [114; 115; 116; 117], fractional-order PID controller [118; 119; 120; 121], fractional-order lead-leag compensators [122; 123; 124]. Additionally, there are proposals for fractional-order adaptive controllers based on model reference, fuzzy systems, and neural networks, among others [125; 126; 127; 128; 129].  
Within the field fractional-order SMC (FOSMC), the fractional order can be approached using the sliding surface and switching functions of the fractional order or the derivatives and integrals of the fractional order in the control law [130; 131; 132]. Some proposals combine fractional-order HOSMC or TSMC with adaptive techniques, fuzzy and neural network systems, observers, etc. [133; 134; 135; 136; 137; 138].  
Despite the increasing number of publications on fractional-order controllers [105; 139; 140; 141; 107], some significant aspects must be considered. The implementation of fractional-order function models exhibits a higher degree of complexity than IO controllers [105; 107; 109] due to the memory requirement for fractional-order functions, which need past values of signals, that is, a long memory effect [109; 139]. In addition, it is also common to use approximate models of integer order to implement fractional-order transfer function models [105; 139]. Furthermore, the mathematical operations and calculations of fractional-order control systems are inherently more complex than those of integer order [97; 105; 139].