# Nonsingular Terminal Sliding Mode Based Finite-Time Dynamic Surface Control for a Quadrotor UAV
## 1 Introduction
An adaptive PID-SMC method was proposed for quadrotor UAVs subject to external disturbances to achieve finite-time stability of the tracking control system in [29]. In applications, disturbances in the system will reduce the accuracy of the control scheme. One of the most effective ways to overcome this is to use an observer-based approach. To estimate the external disturbances, an AC-based terminal sliding mode observer for quadrotor UAVs was proposed to achieve appointed-fixed-time stability of the attitude system [30]. A disturbance observer (DO) based tracking control strategy was proposed to ensure the asymptotic stability of a wheeled mobile robot in [31]. For typical nonlinear systems, the authors of [32] presented a DO-based control framework to achieve asymptotic stability of the closed-loop tracking system. In [33], a DO was designed for mechanical systems to estimate the total uncertainty, and a controller was proposed to ensure the stability of the system. In [34], an extended state observer (ESO) for robot manipulators was proposed to estimate the velocity measurement uncertainty in finite time. In [35; 36], a finite-time ESO was designed based on the NTSMC technique to address the fault tolerant control problem, and this could be used to estimate the lumped uncertainties of the spacecraft. For a quadrotor with disturbances and model uncertainties, a filtered observer-based Interconnection and Damping Assignment-Passivity Based Control scheme was designed to address the tracking problem [37]. Utilizing the filter to attenuate noise, a tracking controller was proposed to achieve asymptotic stability of the closed-loop system for quadrotor UAVs [38]. This paper addresses the tracking control problem associated with quadrotor UAVs subject to external disturbances and parameter uncertainties. The main contributions of this paper are as follows: (1) For the position and attitude subsystems, two NTSMC-based finite-time ESOs are proposed to estimate external disturbances and/or parameter uncertainties. Compared with the asymptotically stable nonlinear ESO presented in [39], the proposed ESO is finite-time stable. The finite-time scheme is not only robust and highly precise, it can also estimate the upper bound of the settling time. (2) Inspired by [14], two finite-time controllers based on NTSMC and DSC are designed for the position and attitude subsystems. The estimation error of the filter converged to a neighborhood of the origin in [14]. The advantage of this paper is that the filter can precisely estimate the derivative of the virtual control law, and the estimation error can converge to zero in finite time. Hence, the convergence rate and the tracking error are improved in the proposed control scheme. The rest of this paper is organized as follows. In Section 2, the quadrotor model is established. In Section 3, the NTSMC technique-based finite-time ESOs and dynamic surface controllers are proposed, and the stabilities are analyzed in detail using the Lyapunov criteria. A numerical simulation of the proposed control algorithm is performed in Section 4. The conclusions are drawn in Section 5. **Notation. ** The superscript T represents the transpose of a matrix. Denote \(\text{sig}^{b}(x)=|x|^{b}\text{sign}(x)\), where \(\text{sign}(\cdot)\) is a standard symbolic function and \(|\cdot|\) is the absolute value. \(\|\cdot\|\) stands for the Euclidean norm of a vector. \(\mathbb{R}\) and \(\mathbb{R}^{+}\) respectively represent the sets of real numbers and positive real numbers. \(\mathbb{R}^{n}\) denotes an n-dimensional real vector. \(\mathbb{U}\backslash\{0\}\) represents the set, where \(\{0\}\) is removed from the set \(\mathbb{U}\).