Approaches | Operation principles |

Boundary layer approach | To insert a boundary layer near the sliding surface so that a continuous control action replaces the discontinuous one when the system is inside the boundary layer [38]. For this purpose, the discontinuous component of the controller: |

${u}_{N}(t)=-{K}_{s}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s(x))$, | |

is often replaced by the saturation control: | |

${u}_{N}(t)\approx -{K}_{s}{\displaystyle \frac{s(x)}{\Vert s(x)\Vert +\delta}},$ | |

for some, preferably small, $\delta >0$. The boundary layer approach has been utilized extensively to practical applications. However, this method has some disadvantages such as: | |

• It may give a chattering-free system but a finite steady-state error must exist; | |

• The boundary layer thickness has the trade-off between control performance of SMC and chattering mitigation; | |

• Within the boundary layer, the characteristics of robustness and the accuracy of the system are no longer assured. | |

Reaching law approach | Since the amplitude of chattering depends on the magnitude of control, the intuitive way of chattering reduction is to decrease the amplitude of the discontinuous control [39]. This technique affects the robustness property of the controller and degrades the transient response of the system. Therefore, exists a trade-off between the chattering reduction and the system performance. A compromised approach is to decrease the amplitude of the discontinuous control, ${u}_{N}(t)$, when the system state trajectories are near to sliding surface (to reduce the chattering), and to increase the amplitude when the system states are far from the sliding surface. |

Dynamic SMC approach | The main idea of the dynamic SMC approach is to insert an integrator (or any other strictly proper low-pass filter) between the SMC and the controlled plant [40]. The concept is illustrated here: |

The time derivative of the control input, ω, is treated as the new control input for the augmented system. Since the low-pass integrator filters out the high frequency chattering in ω, the control input to the real plant, u, becomes continuous which offers a possibility to reduce chattering. Such a method can eliminate chattering and ensure zero steady-state error, however, it should be noted that the system order is increased by one and the transient responses may be degraded [41]. | |

SOSMC approach | Second order sliding mode control (SOSMC) approach is one of the most popular methods to alleviate the chattering problems, which not only effectively eliminates the disadvantages of the conventional SMC, but also maintains its all major attributes such as strong robustness and finite time convergence. The common SOSMC algorithms include twisting algorithm, super-twisting algorithm, drift algorithm and prescribed convergence law algorithm, etc. All these algorithms guarantee that system state can converge $s=\dot{s}$ in finite time [42]–[45]. Among these second order sliding mode control algorithms, super-twisting algorithm viewed as nonlinear proportional-integral control is very effective to deal with the system in presence of strong nonlinearity effects [46]. The super-twisting algorithm can be expressed as, |

$\begin{array}{c}{u}_{sta}(t)={u}_{sta1}(t)+{u}_{sta2}(t),\\ {u}_{sta1}(t)=-{\lambda}_{1}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s(x)),\phantom{\rule{1em}{0ex}}{u}_{sta2}(t)=-{\lambda}_{2}|s(x){|}^{0.5}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s(x)),\end{array}$ | |

where ${\lambda}_{1}$ and ${\lambda}_{2}$ are positive design parameters. | |

Intelligent SMC approach | The main idea of the intelligent SMC approach is integration of SMC with intelligent control technologies, such as neural networks (NNs), fuzzy logic technologies, etc., to make it smarter [47]–[49]. Depending on the technology different goals can be defined as follows. |

• NNs allows one to approximate smooth nonlinear functions to arbitrary accuracy. Therefore, they can be used to approximate and compensate external disturbances and model uncertainties to soften the chattering. On the other hand, NNs in SMC can be also aimed to estimate the switching control term in which the discontinuous control signal is converted to a continuous control signal providing an efficient method to reduce the chattering. | |

• The purpose of adding fuzzy logic systems to SMC is similar to NNs. It allows one to approximate smooth nonlinear functions to arbitrary accuracy. Then it is used to estimate the external disturbance and modeling uncertainties to alleviate chattering. | |

• Evolutionary computation is an alternative to seek optimal control parameters. Thus it can be used to alleviate chattering. |