###### Abstract  
In this study, a fractional-order sliding mode backstepping control method was proposed, which involved the use of a fractional-order command filter, an interval type-2 fuzzy logic system approximation method, and a grey wolf and weighted whale optimization algorithm for multi-input multi-output nonlinear dynamic systems. For designing the stabilizing controls of the backstepping control, a novel fractional-order sliding mode surface was suggested. Further, the transformed errors that occurred during the recursive design steps were easily compensated by the controllers constructed using a new fractional-order command filter. Thus, the differentiation issue of the virtual control in the conventional backstepping control design could be bypassed with a simpler controller structure. Subsequently, the unknown plant dynamics were approximated by an interval type-2 fuzzy logic system. The uncertainties, such as the approximation error and the external disturbance, were compensated by the fractional-order sliding mode control that was added in the backstepping controller. Furthermore, the controller parameters and the fuzzy logic system were optimized  
controller design, and the stability proof appears due to the compensation of the filtering errors. Thus, bypassing the repeated differentiation of a virtual control can alter the BSC system into a better controller as compared to the DSC system. Based on this concept, in recent years, the finite-time command filter method, was studied to partially overcome these issues [11; 12]. This method ensures the finite-time control convergence and prevents the repeated differentiation of the intermediate virtual control. However, the complexity issues associated with the design and robustness of the controller have not been addressed. In this study, a fractional-order finite-time command filter was considered to achieve an enhanced performance and a simpler controller structure as compared to those of the previous command filter based BSC control [11; 12]. In recent decades, fractional-order operators [13; 14] have garnered increasing significance in designing proportional-integral-derivative (PID) controllers [15], sliding mode controllers (SMC) [16; 17; 18], and adaptive controllers [19], owing to their better flexibility, robustness, and performance than those exhibited by their integer-order counterparts. We know that the sliding mode control (SMC) [20; 21; 22] exhibits a higher robustness to uncertainty if the states are reached within the sliding surface. In the conventional BSC, SMC was not directly combined to construct the virtual tracking error because the structure of the sliding mode surface in SMC is difficult to match the tracking error structure in the BSC system. In this paper, the fractional-order based sliding mode surface is used to construct the virtual tracking error of the BSC scheme, which imparts the benefits of SMC to the BSC system. Next, a novel FLS, the interval type-2 FLS (IT2-FLS) [23], has been studied and extensively applied to control systems [24; 25; 26; 27]. IT2-FLS is an extension of the type-1 FLS [28; 29], which comprises the interval membership fuzzy functions. IT2-FLS has benefits such as more smoothness and robustness against uncertainty and more flexibility owing to its numerous tunable parameters, rendering it dealing with control applications. In this study, unknown dynamics of the nonlinear system are approximated via IT2-FLS. Controller parameters are generally selected by the trial and error tuning method, depending on the experience of the designer. For numerous gain numbers to be adjusted, tuning processes become more time-consuming and require hard works. Thus, bio-inspired artificial optimal techniques, such as ant colony optimization (ACO) [30] and particle swarm optimization (PSO) [31], were developed to alleviate and optimize unknown system parameters under the condition of minimizing or maximizing the objective function.