## 1 Introduction
However, due to the existence of constraints, the upper bound of fault estimation error was related to the parameters. In References [20; 21], an active fault-tolerant control algorithm based on the interpolation gain scheduling PI was proposed for the actuator fault of the UAV. It was verified on the UAV experimental platform. The test results showed that the active fault-tolerant control algorithm based on the interpolation gain scheduling PI can effectively improve the fault-tolerant performance of UAV, compared with the simple PID controller. In References [22; 23], the robust linear variable parameter observer is applied to the rotating subsystem to detect actuator fault of the UAV. Numerical experiments show the effectiveness of the method. In Reference [24], the problem of distributed fault detection and isolation is studied, and a local fault diagnosis algorithm is designed for each unit of UAV through the linear model. This algorithm is effective in most of common fault situations. In past research, the linear algorithm of state estimation was widely used. Although this method has advantages in hardware implementation, there is a contradiction between the tracking time and the tracking effect in the performance. In this regard, a nonlinear state estimation algorithm can effectively improve the tracking effect and fundamentally solve the contradiction between the tracking time and the tracking effect. In this paper, fault diagnosis algorithms based on the nonlinear observer is used to improve the effect of the additive single sensor fault estimation. The adaptive parameter updating algorithm is given, which effectively improves the fault estimation effect. This paper is organized as follows. In the Section 2, the system model and the fault diagnosis algorithm based on the PI state observer are introduced. In the Section 3, the fault estimation of the nonlinear state observer is analyzed. In the Section 4, combined with the UAV system, the simulation verification is carried out, and the fault estimation effect is expected. **Theorem 1**. : _For the error system Equation (6), the positive definite symmetric matrix \(\mathbf{P}_{1}\) and the parameter \(\gamma>0\) can be chosen, so that the inequality_  
\[\left[ \begin{array}{cc}\mathbf{A}_{e}^{\mathbf{T}}\mathbf{P}_{1}+\mathbf{P}_{1} \mathbf{A}_{e}+\mathbf{I}&\mathbf{P}_{1}\mathbf{B}_{e}\\ \mathbf{B}_{e}^{\mathbf{T}}\mathbf{P}_{1}&-\gamma\mathbf{I}\end{array} \right]<0 \tag{7}\]  
_holds. Then, the error system equation Equation (6) is bounded, and the bound satisfies_  
\[\left\|\varepsilon\right\|^{2}<\gamma\left\|\mathbf{\phi}\right\|^{2}+V(0)\leq \gamma\left\|\mathbf{f}\right\|^{2}+\gamma\left\|\dot{\mathbf{f}}\right\|^{2}+V(0), \tag{8}\]  
_where \(\left\|\mathbf{g}\right\|^{2}=\int_{0}^{\tau}(\mathbf{g}^{T}\mathbf{g})dt\)._  
Proof. : Consider the Lyapunov function \(V=\mathbf{\varepsilon}^{\mathbf{T}}\mathbf{P}_{1}\mathbf{\varepsilon}\), then \(\dot{V}\) is as follows  
\[\dot{V}=\mathbf{\varepsilon}^{\mathbf{T}}\mathbf{P}_{1}\mathbf{\varepsilon}=\mathbf{ \varepsilon}^{\mathbf{T}}\Big{(}\mathbf{A}_{e}^{\mathbf{T}}\mathbf{P}_{1}+\mathbf{ P}_{1}\mathbf{A}_{e}\Big{)}\mathbf{\varepsilon}+2\mathbf{\varepsilon}^{\mathbf{T}} \mathbf{P}_{1}\mathbf{B}_{e}\mathbf{\phi}.\]  
Further, the performance tracking index \(J=\int_{0}^{\tau}\big{(}\mathbf{\varepsilon}^{\mathbf{T}}\mathbf{\varepsilon}-\gamma \mathbf{\phi}^{\mathbf{T}}\mathbf{\phi}\big{)}dt\) is examined. \[J-V(0)=\int_{0}^{\tau}\big{(}\mathbf{\varepsilon}^{\mathbf{T}}\mathbf{ \varepsilon}-\gamma\mathbf{\phi}^{\mathbf{T}}\mathbf{\phi}\big{)}dt-V(0)\] \[\leq\int_{0}^{\tau}\left(\left[ \begin{array}{c}\mathbf{\varepsilon }\\ \mathbf{\phi}\end{array} \right]^{\mathbf{T}}\left[ \begin{array}{c}\mathbf{I}&\mathbf{0}\\ \mathbf{0}&-\gamma\mathbf{I}\end{array} \right]\left[ \begin{array}{c}\mathbf{ \varepsilon}\\ \mathbf{\phi}\end{array} \right]\right)dt+V(\tau)-V(0)\] \[=\int_{0}^{\tau}\left(\left[ \begin{array}{c}\mathbf{\varepsilon} \\ \mathbf{\phi}\end{array} \right]^{\mathbf{T}}\left[