# Online Adaptive Parameter Estimation for Quadrotors
## 1 Introduction  
Adaptive control and parameter estimation have been hot topics in the control fields [1; 2], since, in control engineering practice, many unknown or immeasurable parameters and dynamics exist in a wide range of control plants. In order to address this problem, adaptive control was proposed as an effective and powerful method [3; 4; 5]. In this methodology, the unknown or immeasurable parameters and dynamics can be updated online or estimated by using control errors to ensure a closed-loop system's stability [5]. Nevertheless, in the past few years, standard adaptive control with a gradient-based algorithm has focused mainly on tracking a given reference trajectory rather than guaranteeing that the estimated parameters converge to their true values [6]. However, according to the equivalence theorem [7], accurate parameter estimation can be essential to enhancing the stability and robustness of adaptive systems for the purpose of tracking control. Hence, developing adaptive laws with guaranteed convergence is always a critical issue in adaptive control designs. Recently, in order to solve the problems of parameter estimation, some scholars have made great efforts, and many well-recognized results have been reported; e.g., the gradient descent, least squares (LS) [3; 4; 5], neural network [8; 9], and projection algorithms [10]. Moreover, a new adaptive parameter estimation framework has recently been proposed and applied in many applications [11; 12; 13; 14], where the adaptive laws are driven by the parameter estimation error such that exponential or even finite-time convergence can be achieved. In [6], a new adaptive law was also studied to reconstruct unknown system parameters when an intermediate matrix is positive. However, most of the existing results are suitable for linearly parameterized systems only, and cannot be directly used for nonlinearly parameterized systems. In parameter estimation, the parameter error convergence rate is also an important issue. To address this issue, a finite-time sliding neural network observer was studied in [8], where theobserved state derivatives are used to estimate parameters, and some known nonlinear functions in the system are needed. Moreover, the inverse of these matrices should exist and be available. In a recent research work [6], under the condition that the persistent excitation (PE) is satisfied, a novel parameter estimation was given, which allows exact reconstruction of the unknown parameters. However, this method needs to verify online the inevitability for a matrix and compute its inverse when it is appropriate. Besides this, an observer is needed to be designed for this parameter estimation method, such that the corresponding computational cost is also increased. Nowadays, quadrotors are playing a more and more important role in daily life, since they can perform versatile maneuvers and are able to complete various missions [15]. Therefore, a lot of attention has been paid to the modeling and control of quadrotors in the past few years [16; 17; 18; 19]. However, some parameters in quadrotors are unknown for the controller's design. In particular, the stability and robustness of control systems for quadrotors are always influenced by the unknown or immeasurable parameters. Hence, inspired by the above facts, in this paper, we will propose an adaptive parameter estimation technology for quadrotors with unknown parameters to enhance the stability and robustness of quadrotors. Firstly, a typical quadrotor mathematical model is given, and then an expression of the parameter estimation error is derived by introducing auxiliary filtered variables. Moreover, an augmented matrix is constructed based on the auxiliary filtered variables, which is further used to design the adaptive law to achieve convergence under the standard persistent excitation (PE) condition. Finally, simulation and experimental verification results for quadrotors are shown to illustrate the effectiveness of the proposed method.