Figure 2 displays the proposed entire system that implemented in the presented study. As observed, all components of the system details are tabulated in Table 1. For precise speed and position control applications, BLDC motor drives' speed regulation is a crucial component. In this study, the BLDC motor was operated under three speed controller techniques to compare the performance under different conditions. The classical PID method, SMC method and the proposed improved fast terminal slide mode control (IFTSMC). The three phase voltage source inverter (VSI) is applied for driving the BLDC motor via Hall sensors and Control commutation logic scheme. The system is simulated under MATLAB/Simulink software.

**Figure 2.** The proposed system diagram.

However, a discontinuous controlling technique that takes the time-varying manifold into account is the SMC, which has its roots in the variable structure control theory. Because of its key characteristics, which include stability-based disturbance rejection, robustness, and agility, the SMC is preferred [37]. The SMC architecture features two phases: the reaching phase and the sliding phase. The system's states push in the direction of a predefined sliding surface (s) that corresponds to the starting point on the phase plane during the reaching phase. The simple or the classical slide surface can be written as follows [38]:

$$\:\dot{S}={x}_{1}+{k}_{1}\:{x}_{2}$$

17

where \(\:{k}_{1}>0\) is the constant system. In the proposed BLDC motor system, the state variables of \(\:{x}_{1}\:\text{a}\text{n}\text{d}\:{x}_{2}\) can be expressed as:

$$\:\left.\begin{array}{c}{x}_{1}={\:\omega\:}_{m,ref}-{\:\omega\:}_{m}\\\:{x}_{2}=\dot{{x}_{1}}=\dot{{\:\omega\:}_{m,ref}}-\dot{{\:\omega\:}_{m}}\end{array}\right\}$$

18

where, \(\:{\:\omega\:}_{m,ref}\)is the reference BLDC motor speed.

It is imperative to enhance the reaching law technique in SMC for multiple reasons such as improving system performance, strengthening resilience, decreasing chattering, and guaranteeing a quicker and more seamless convergence to the sliding surface are the main goals. For this reason, the modified SMC called terminal SMC (TSMC) method was developed. This method can be expressed as follows [39]:

(19)

where \(\:{k}_{1}>0\:\)is a positive constant and \(\:0<\alpha\:<1\). As presented in Eq. (19) the relation between the variables is nonlinear.

However, modifying the TSMC can help reduce or eliminate chattering by providing a smoother control action, thereby preserving system components and improving performance. Accordingly, the fast TSMC (FTSMC) control method is presented in different studies such [40–42]. The slide surface of this method can be written as:

$$\:\dot{S}={x}_{2}+{{k}_{1}\:x}_{1}+{k}_{2}\:{{x}_{1}}^{\alpha\:}\:$$

20

where \(\:{k}_{2}>0\:\)is a positive constant. As mentioned in Eq. (20), the FTSMC sliding surface combines the advantages of traditional and TSMC by providing a trade-off between classical SMC and TSMC. Moreover, the exponential equation will become non-real if \(\:{x}_{1}\) is negative, hence the issue with negative values is still present.

The stable conditions that achieved by designing the optimal surface is the main goal in the SMC design. Moreover, an optimal control rule should be computed to drive the sliding states towards sliding surface that achieved under searching process with finite time. In the proposed of improved FTSMC (IFTSMC), the surface is designed by presenting a new reaching law as follows:

$$\:\dot{S}={x}_{1}+{{k}_{1}\:x}_{2}+{k}_{2}\:{\left|{x}_{1}\right|}^{\alpha\:}\times\:sign\left(s\right)$$

21

The time derivate of the surface can expressed as follows:

$$\:\dot{S}={x}_{1}+{{k}_{1}\:x}_{2}+\:f\left(x\right)+g\left(x\right)u+d(x,t)$$

22

The proposed suggested sliding surface for the BLDC motor case can be written as:

$$\:\dot{S}=\left({\:\omega\:}_{m,ref}-{\:\omega\:}_{m}\right)+{k}_{1}\left(\dot{{\:\omega\:}_{m,ref}}-\dot{{\:\omega\:}_{m}}\right)+{k}_{2}\:{\left|\left({\:\omega\:}_{m,ref}-{\:\omega\:}_{m}\right)\right|}^{\alpha\:}\times\:sign\left(s\right)$$

23

From equations (21, 22 and 23), the control unit of the system can achieved as:

$$\:u=\frac{1}{g\left(x\right)}\left[-{x}_{1}-{k}_{1}{x}_{2}-\left(f\left(x\right)+D\right)-\:{k}_{2}\:{\left|{x}_{1}\right|}^{\alpha\:}\times\:sign\left(s\right)\:\right]$$

24

Or the control rule can be written as:

$$\:u=\frac{1}{g\left(x\right)}\left[-\left({\:\omega\:}_{m,ref}-{\:\omega\:}_{m}\right)-{k}_{1}\left(\dot{{\:\omega\:}_{m,ref}}-\dot{{\:\omega\:}_{m}}\right)-\left(f\left(x\right)+D\right)-\:{k}_{2}\:{\left|\left({\:\omega\:}_{m,ref}-{\:\omega\:}_{m}\right)\right|}^{\alpha\:}\times\:sign\left(s\right)\:\right]$$

25

Moreover, to design the stability analysis of the used sliding surface law, Lyaponov function is provided as:

The conditions of the system stability is proven as follows:

$$\:\underset{s\to\:0}{\text{lim}}\dot{V}=\underset{s\to\:0}{\text{lim}}\dot{SS}\:\le\:0$$

27

As mentioned above in Eq. (20), the following formula was obtained:

$$\:S\dot{S}=S\left[{x}_{1}+{{k}_{1}\:x}_{2}+\:f\left(x\right)+g\left(x\right)u+d(x,t)\:\right]$$

28

By substitute the control unit into Eq. (28):

$$\:S\dot{S}=\left[\:\left|S\right|.D-\:{k}_{2}\:{\left|S\right|}^{\alpha\:}\times\:sign\left(s\right).\left|S\right|+d\left(x,t\right).\:S\right]$$

29

where \(\:{\left|S\right|}^{\alpha\:}\ge\:0,\:\:D\ge\:\left|d\right|,\:\text{a}\text{n}\text{d}\:sign\left(s\right)\cong\:\left|S\right|\ge\:0\), all gains in this equation greater than zero. As a result, \(\:\dot{V}<0\) and only when \(\:S=0\). For this reason, the suggested sliding surface reached the stability conditions.

Table 1

BLDC motor and controllers parameters.

Parameter | Value |

Input DC voltage | \(\:100\:V\) |

Reference speed | \(\:100-500\:RPM\) |

Stator resistance | \(\:1.43\:{\Omega\:}\) |

Stator inductance | \(\:9.4\:mH\) |

Rotor flux | \(\:0.215\:wb\) |

Friction coefficient | \(\:2{e}^{-3}\) |

Motor’s inertia | \(\:5.5{e}^{-3}\:kg.m²\) |

Number of poles | \(\:4\) |

\(\:{\varvec{k}}_{1}\) | \(\:0.5\) |

\(\:{\varvec{k}}_{2}\) | \(\:2000\) |

\(\:\varvec{\alpha\:}\) | \(\:0.9\) |