Robust fault diagnosis and fault-tolerant control for nonlinear quadrotor unmanned aerial vehicle system with unknown actuator faults

Abstract

This article addresses the problem that quadrotor unmanned aerial vehicle (UAV) actuator faults, including small-amplitude bias faults and gain degradation, cannot be detected in time. A hybrid observer, which combines the fast convergence from adaptive observer and the strong robustness from sliding mode observer, is proposed to detect and estimate UAV actuator faults accurately with model uncertainties and disturbances. A nonlinear quadrotor UAV model with model uncertainties and disturbances is considered and a more precise unified expression for actuator faults that do not require knowing where the upper or lower bound is provided. The original system is decomposed into two subsystems by coordinate transformation to improve detection accuracy for small amplitude bias faults and avoid external influences. The hybrid observer is then designed to estimate subsystem states and faults with good stability by selecting a Lyapunov function. A fault-tolerant controller is obtained depending on fault estimation by compensating the normal controller (proportion integral differential [PID] controller). Several numerical simulations confirmed that unknown actuator faults can be accurately detected, estimated, and compensated for even under disturbance conditions.

Keywords

Quadrotor UAV actuator faults hybrid observer fault diagnosis fault-tolerant control

Introduction

With the development of autonomous control technology, various unmanned aerial vehicles (UAVs) have been extensively used in logistics transportation, agricultural plant protection, and so on.^1

–4 Quadrotor UAVs have several advantages compared with UAV types, such as flight flexibility and fewer site requirements, and have attracted considerable research attention.^5,6 However, quadrotor UAV is a nonlinear control system with multiple and significantly coupled inputs and outputs, and control system complexity increases as performance improves.⁷ They inevitably encounter multiple uncertain factors, such as wind disturbances, rotor vibration, and accidental faults. Faults in any quadrotor UAV component, including actuators, sensors, and so on, can be catastrophic. Therefore, fault detection and diagnosis (FDD) has attracted widespread industrial and scientific attention^8

–11 to identify and compensate for faults so that the control system can remain stable and safe.

Fault diagnosis encompasses three main domains: fault detection, fault isolation, and fault identification. Fault diagnosis has been widely studied over recent years with many feasible methods proposed that can generally be classified into knowledge-based fault diagnosis,^12
–14 model-based fault diagnosis,^15

–18 and signal processing-based approaches.^19

–22 Various effective model-based methods have been proposed for the linear model or linearized the nonlinear model,^15,18,23,24 and many studies have proposed improved observers for FDD. For example, Han et al.²⁵ proposed a reduced-order fault estimation observer for a switched system to solve the problem of actuator and sensor faults. Actuator and sensor bias faults were accurately estimated under the assumption that these faults are completely unknown. Mostafa et al.²⁶ proposed a novel sliding mode observer to estimate the actuator bias faults for the linear system of a wind turbine. Osorio-Gordillo et al.²⁷ proposed a new generalized observer for general linear systems. Shen et al.²⁸ proposed an adaptive method to solve actuator gain and bias faults for the linearized system of a front-wheeled steered vehicle.

The above works solved the actuator fault in the linear system, but the fault types or external disturbances were not fully considered. Linearized systems are relatively simple to build and analyze, but they cannot reflect the system’s nonlinear nature. Therefore, Cen et al.²⁹ proposed an adaptive Thau observer to detect and estimate actuator bias faults, and Hongjun et al.³⁰ proposed a high-gain observer to estimate the bias faults of actuator and sensor for the nonlinear quadrotor UAV. Zhang et al.³¹ proposed a switching sliding mode observer for actuator fault detection and isolation for time-invariant systems with nonlinear perturbation that considered bias and loss of effectiveness faults, where fault upper and lower bounds were known constants. In the above work, actuator fault types considered for these nonlinear systems were insufficient and actual bias fault amplitudes are significantly larger than the disturbances considered.

According to the above works, it is very important to study the actuator faults for nonlinear quadrotor UAV. In fact, fault amplitudes are commonly very small initially, and uncertain disturbances and noise are unavoidable, making it difficult to detect small or potential faults in their initial stage. Faults tend to accumulate over time and eventually seriously impact system control. Therefore, reliable fault detection can significantly help avoid unnecessary losses caused by false alarms.

Motivated by the above analyses, the coordinate transformation was adopted by the literature³² to enhance the performance of fault estimation. The article proposes a hybrid observer that can deal with quadrotor UAV actuator faults leveraging advantages from adaptive and sliding mode observers and hence also providing better balance between speed and robustness. In addition, actuator faults and disturbances are separated by the coordinate transformation, with subsystem $\sum_{1}$ contains only actuator faults while subsystem $\sum_{2}$ contains both actuator faults and disturbances. An adaptive observer can be used to estimate actuator faults for subsystem $\sum_{1}$ , because this method can quickly and accurately estimate the states on the premise of ensuring the stability of the system. A sliding mode observer can estimate system states for subsystem $\sum_{2}$ by feeding output estimation error back to the observer using a nonlinear term and hence addresses nonzero output estimation error for unknown signals.^23,33,34 Thus, quadrotor UAV can operate in suboptimal states by estimating fault and compensating for it. Stability for the proposed adaptive and sliding mode observers is proven under actuator faults and external disturbance conditions using the Lyapunov functional approach. A fault-tolerant controller can be obtained depending on fault estimation by compensating the normal controller. The main contributions of this article are summarized as follows: (1) The considered quadrotor model is nonlinear and the exterior uncertainties and disturbances are fully considered; (2) a more precise unified expression for actuator bias fault and gain fault is provided, considering small or potential faults where knowing upper and lower bounds of faults are not required; and (3) the external disturbance and uncertainties are compensated by designing the discontinuity term of sliding mode observer to achieve the robustness of fault diagnosis, and adaptive laws of adaptive observer are designed to quickly estimate the actuator bias and gain faults.

Notations. Some notations will be utilized throughout the article. The transpose and the inverse of matrix A are defined by $A^{T}$ and $A^{- 1}$ , respectively. A > 0 represents a positive defined matrix. I is an identity matrix of appropriate dimensions. $λ (A)$ is the eigenvalue of A. Define R ⁿ as the n-dimensional Euclidean space. The Euclidean norm of the matrix or the vector x is expressed as $‖x‖$ . The diagonal matrix is constructed by $diag (\dots)$ .

Problem formulation

Dynamic model of a quadrotor unmanned aerial vehicle

As the name implies, quadrotor UAVs have four cross-shaped rotors that are noncoaxially symmetrical. Figure 1 shows a simple diagram of the quadrotor UAV, where $B = \{O_{B}, X_{B}, Y_{B}, Z_{B}\}$ is the body coordinate system and $E = \{O_{E}, X_{E}, Y_{E}, Z_{E}\}$ is the inertial coordinate system. Euler angles ϕ, θ, and ψ, defined in the inertial coordinate system, represent the roll angle, pitch angle, and yaw angle of the quadrotor UAV, respectively. Rotors r ₂ and r ₄ rotate clockwise, whereas rotors r ₁ and r ₃ rotate counterclockwise to counteract torque and maintain UAV attitude stability. The position and attitude of the quadrotor UAV in three-dimensional space are controlled by rotor thrust, and various attitudes and positions can be obtained by adjusting rotor speeds. The quadrotor UAV is a nonlinear, underactuated, and coupled system with six degrees of freedom, which are all driven by four actuators. The modeling process is simplified by assuming that ϕ, θ, and ψ are small, and hence, $ω = {[p q r]}^{T} \approx {[\dot{ϕ} \dot{θ} \dot{ψ}]}^{T}$ . The moment of inertia along the directions x, y, and z can be expressed as $I_{inertia} = diag (I_{inertia}^{x}, I_{inertia}^{y}, I_{inertia}^{z})$ . The moment of each channel M_i can be expressed by

\sum_{i} M_{i} = I_{inertia} \dot{ω} = M_{g} + M_{r} + M_{d}

where M_g is the gyroscopic moment of the quadrotor UAV, M_r is the torque provided by the rotors, and M_d is the aerodynamic drag. Thus, the dynamics model of the quadrotor UAV can be obtained as follows

[\begin{matrix} I_{inertia}^{x} \ddot{ϕ} \\ I_{i nertia}^{y} \ddot{θ} \\ I_{inertia}^{z} \ddot{ψ} \end{matrix}] = [\begin{matrix} (I_{inertia}^{y} - I_{inertia}^{z}) \dot{θ} \dot{ψ} - K_{d x} {\dot{ϕ}}^{2} + l K_{l} (u_{4} - u_{2}) \\ (I_{inertia}^{z} - I_{inertia}^{x}) \dot{ψ} \dot{ϕ} - K_{d y} {\dot{θ}}^{2} + l K_{l} (u_{3} - u_{1}) \\ (I_{inertia}^{x} - I_{inertia}^{y}) \dot{ϕ} \dot{θ} - K_{d z} {\dot{ψ}}^{2} + K_{t c} (u_{1} + u_{3}) + K_{t n} (u_{2} + u_{4}) \end{matrix}]

where l is the distance from the rotor to UAV center of mass, $K_{t n}$ and $K_{t c}$ are rotor torques in clockwise and counterclockwise rotations, respectively, K_l represents lift coefficient of rotor measured by experiment; $u_{1}, u_{2}, u_{3}$ , and u ₄ are the output voltage of each motor; and $K_{d x}, K_{d y}$ , and $K_{d z}$ are the aerodynamic drag coefficients along the directions of x, y, and z axes.

Figure 1.

Coordinate system of a quadrotor UAV. UAV: unmanned aerial vehicle.

Let $x (t) = {[ϕ, \dot{ϕ}, θ, \dot{θ}, ψ, \dot{ψ}]}^{T}$ be the state vector, $y (t) = {[ϕ, \dot{ϕ}, θ, \dot{θ}, ψ, \dot{ψ}]}^{T}$ be the system output, and $u (t) = {[u_{1}, u_{2}, u_{3}, u_{4}]}^{T}$ be the system input. To fully consider the actual working conditions, disturbances and uncertainties are also considered, equation (2) is rewritten as in state-space form as

\{\begin{array}{l} \dot{x} (t) = A x (t) + B u (t) + F (x, u, t) + D Δ ξ (t) \\ y (t) = H x (t) \end{array}

According to the parameters of quadrotor UAV, we can conclude that $A, B, D$ , and H are known matrices, and D is the disturbance matrix with full column rank. The system matrices are given as $A = [a_{i j}]$ , $B = [b_{i j}]$ with $a_{12} = a_{34} = a_{56} = 1$ , $b_{24} = - b_{22} = l K_{l} / I_{inertia}^{x}$ , $b_{43} = - b_{41} = l K_{l} / I_{inertia}^{y}$ , $b_{61} = b_{63} = K_{t c} / I_{inertia}^{z}$ , $b_{62} = b_{64} = K_{t n} / I_{inertia}^{z}$ , and everything else is 0 and $F (x, u, t) = [h_{i j}]$ with $h_{21} = [(I_{inertia}^{y} - I_{inertia}^{z}) \dot{θ} \dot{ψ} - K_{d x} {\dot{ϕ}}^{2}] / I_{inertia}^{x}$ , $h_{41} = [(I_{inertia}^{z} - I_{inertia}^{x}) \dot{ψ} \dot{ϕ} - K_{d y} {\dot{θ}}^{2}] / I_{inertia}^{y}$ , $h_{61} = [(I_{inertia}^{x} - I_{inertia}^{y}) \dot{ϕ} \dot{θ} - K_{d z} {\dot{ψ}}^{2}] / I_{inertia}^{z}$ , and $H = I_{6}$ . The function $Δ ξ (t)$ represents the uncertainties and disturbances. Several assumptions must be established before commencing solving the problems.

Assumption 1. The function $F (x, u, t)$ is assumed to satisfy the Lipschitz condition with a constant $γ > 0$ , $\forall x_{1}, x_{2} \in X, ‖F (x_{1}, u, t) - F (x_{2}, u, t)‖ \leq γ ‖x_{1} - x_{2}‖$ .

Assumption 2. Without loss of generality, the uncertainty disturbance function $Δ ξ (t)$ satisfies the following constraint, $‖Δ ξ (t)‖ < ζ$ , where $ζ$ is a positive constant. However, the upper bound of the actuator fault is not a priori.

Theorem 1. There are matrices J ₁, J ₂ and a positive scalar $μ$ , the following inequality holds $2 J_{1}^{T} J_{2} \leq \frac{1}{μ} J_{1}^{T} J_{1} + μ J_{2}^{T} J_{2}$ .

Proof. There exist matrices J ₁ and J ₂ that satisfy ${(\frac{1}{\sqrt{μ}} J_{1} + \sqrt{μ} J_{2})}^{T} (\frac{1}{\sqrt{μ}} J_{1} + \sqrt{μ} J_{2}) \geq 0$ , so we can obtain that $\frac{1}{μ} J_{1}^{T} J_{1} - 2 J_{1}^{T} J_{2} + μ J_{2}^{T} J_{2} \geq 0$ , therefore, $2 J_{1}^{T} J_{2} \leq \frac{1}{μ} J_{1}^{T} J_{1} + μ J_{2}^{T} J_{2}$ . This completes the proof.

Actuator fault model

The actuator of UAV is mainly composed of motors and blades. In the long-term flight, the actuator is inevitably subject to wear, aging, and external impact, resulting in abnormal motor power, blade friction change, and even propeller damage. Actuator faults generally include loss of motor speed, reduction of lift force due to blade damage, stuck motor, and so on. In addition, the wear and aging of the actuators of UAV also cause small-amplitude faults. According to the degree of fault, actuator fault can be divided into three categories: out of control, complete failure, and partial failure. Complete failure means that the output voltage of the motor is always zero or the propeller in common axis is stuck and unable to rotate. Complete failure and out-of-control states can only be solved by redundant hardware. Partial failure means that the motor is in a controllable state, but the output voltage is lost. This article mainly considers a partial failure.

If there is a partial failure of the actuator, there is a difference between the input command and the actual output of the fault actuator. The main feature is that the output voltage from the motor cannot reach the expected value, and hence, UAV attitude will be affected. The relationship between fault and actuator input signal can be divided into bias (additive) and gain (multiplicative) faults.

Let $P (t) = diag (ρ_{1} (t), \dots, ρ_{m} (t))$ , $N (t) = {[η_{1} (t), \dots, η_{m} (t)]}^{T}$ , then

u^{f} (t) = (I_{m} - P (t)) u (t) + N (t), t > t_{f}

where $u (t)$ is the actuator input, $u^{f} (t)$ is the fault actuator output, and $ρ_{i} (t)$ is the effective factor of actuators, which are assumed to be unknown, $η_{i} (t)$ is an unknown bias fault function and is bounded but the bounds are not a priori known, and t_f is the time of failure.

Thus, from equation (3), the system with actuator bias fault and actuator gain fault can be expressed as

\{\begin{array}{l} \dot{x} (t) = A x (t) + B [u (t) + N (t)] + F (x, u^{f}, t) + D Δ ξ (t) t > t_{f} \\ \dot{x} (t) = A x (t) + B [(I_{m} - P (t)) u (t)] + F (x, u^{f}, t) + D Δ ξ (t) t > t_{f} \end{array}

Nonlinear quadrotor model with actuator faults

Consider the following system with a general actuator fault

\{\begin{cases} \dot{x} = A x (t) + B u (t) + F (x, u, t) + D Δ ξ (t) t < t_{f} \\ \dot{x} = A x (t) + B u^{f} (t) + F (x, u^{f}, t) + D Δ ξ (t) t > t_{f} \\ y = H x (t) \end{cases}

There are transfer matrices T and S that satisfy $χ = T x = [\begin{matrix} χ_{1} \\ χ_{2} \end{matrix}]$ and $ο = S y = [\begin{matrix} ο_{1} \\ ο_{2} \end{matrix}]$ . Then, we have $T A T^{- 1} = [\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}]$ , $T B = [\begin{matrix} B_{1} \\ B_{2} \end{matrix}]$ , $T F = [\begin{matrix} {\bar{F}}_{1} \\ {\bar{F}}_{2} \end{matrix}]$ , $T D = [\begin{matrix} D_{1} \\ D_{2} \end{matrix}]$ , and $S H T^{- 1} = [\begin{matrix} H_{11} & 0 \\ 0 & H_{22} \end{matrix}]$ .

Consider the nonsingular transformation matrix T, $T = [\begin{matrix} I_{1} & - D_{1} D_{2}^{- 1} \\ 0 & I_{2} \end{matrix}]$ , with $I_{1} \in R^{(n - r) \times (n - r)}$ and $I_{2} \in R^{r \times r}$ . Then, equation (6) can be described as

Σ_{1} : \{\begin{array}{l} {\dot{χ}}_{1} (t) = A_{11} χ_{1} (t) + A_{12} χ_{2} (t) + B_{1} u^{f} (t) + {\bar{F}}_{1} (T^{- 1} χ, u^{f}, t) \\ ο_{1} (t) = H_{11} χ_{1} (t) \end{array}

Σ_{2} : \{\begin{array}{l} {\dot{χ}}_{2} (t) = A_{21} χ_{1} (t) + A_{22} χ_{2} (t) + B_{2} u^{f} (t) + {\bar{F}}_{2} (T^{- 1} χ, u^{f}, t) + D_{2} Δ ξ (t) \\ ο_{2} (t) = H_{22} χ_{2} (t) \end{array}

where $χ_{1} \in R^{n - r}$ , $χ_{2} \in R^{r}$ , ${\bar{F}}_{1} \in R^{n - r}$ , ${\bar{F}}_{2} \in R^{r}$ , $A_{11} \in R^{(n - r) \times (n - r)}$ , $A_{12} \in R^{(n - r) \times r}$ , $A_{21} \in R^{r \times (n - r)}$ , $A_{22} \in R^{r \times r}$ , $B_{1} \in R^{(n - r) \times m}$ , $B_{2} \in R^{r \times m}$ , $D_{1} \in R^{(n - r) \times r}$ , $D_{2} \in R^{r \times r}$ , and according to the model of the quadrotor UAV, we have n = 6, r = 2, and m = 4. Based on the above analyses, it is necessary to design corresponding observers for subsystems $\sum_{1}$ and $\sum_{2}$ to estimate the system states.

Design of fault detection based on hybrid observer

Based on equation (7), a Thau observer is chosen for fault detection in subsystem $\sum_{1}$

\{\begin{array}{l} {\dot{\hat{χ}}}_{1} (t) = A_{11} {\hat{χ}}_{1} (t) + A_{12} H_{22}^{- 1} ο_{2} (t) + B_{1} u (t) + {\bar{F}}_{1} (T^{- 1} \hat{χ}, u, t) + L_{1} (ο_{1} - {\hat{ο}}_{1}) \\ {\hat{ο}}_{1} (t) = H_{11} {\hat{χ}}_{1} (t) \end{array}

where $\hat{χ}$ is defined as $col ({\hat{χ}}_{1}, H_{22}^{- 1} ο_{2})$ . It is important to know that $\hat{χ}$ does not represent $col ({\hat{χ}}_{1}, {\hat{χ}}_{2})$ .

For subsystem $\sum_{2}$ , a sliding mode observer is designed to estimate the system states and the reasons are as follows. The sliding mode observer combines the theory of sliding mode control with the observer because the sliding mode algorithm has good robustness for the external uncertainty and disturbance. Output estimation error is fed back to the observer as a nonlinear term to eliminate disturbance influences, which can solve the problem that output estimation error is nonzero when an unknown disturbance signal exists. Thus, a sliding mode observer is designed based on equation (8) as

\{\begin{array}{l} {\dot{\hat{χ}}}_{2} (t) = A_{21} {\hat{χ}}_{1} (t) + A_{22} {\hat{χ}}_{2} (t) + B_{2} u (t) + {\bar{F}}_{2} (T^{- 1} \hat{χ}, u, t) + L_{2} (ο_{2} - {\hat{ο}}_{2}) + τ (t) \\ {\hat{ο}}_{2} (t) = H_{22} {\hat{χ}}_{2} (t) \end{array}

where the estimation of $χ_{1} (t)$ , $χ_{2} (t)$ , $ο_{1} (t)$ , and $ο_{2} (t)$ are ${\hat{χ}}_{1} (t)$ , ${\hat{χ}}_{2} (t)$ , ${\hat{ο}}_{1} (t)$ , and ${\hat{ο}}_{2} (t)$ , respectively, L ₁ and L ₂ are the observer gain. The discontinuous term $τ (t)$ is chosen as

τ (t) = \{\begin{array}{l} α \frac{P_{2} H_{22}^{- 1} (ο_{2} - {\hat{ο}}_{2})}{‖P_{2} H_{22}^{- 1} (ο_{2} - {\hat{ο}}_{2})‖ + δ}, if ‖ο_{2} - {\hat{ο}}_{2}‖ \neq 0 \\ 0 otherwise \end{array}

where $δ$ is a smaller positive constant to suppress chattering in sliding mode motion, $α = ‖D_{2}‖ ζ + μ$ , and $μ$ is a positive constant, which drives the error to a predetermined sliding surface.

The estimate errors are defined as $ε_{1} = χ_{1} (t) - {\hat{χ}}_{1} (t)$ , $ε_{2} = χ_{2} (t) - {\hat{χ}}_{2} (t)$

\begin{array}{l} {\dot{ε}}_{1} = A_{11} ε_{1} + {\bar{F}}_{1} (T^{- 1} χ, u, t) - {\bar{F}}_{1} (T^{- 1} \hat{χ}, u, t) + L_{1} H_{11} ε_{1} \\ = (A_{11} - L_{1} H_{11}) ε_{1} + {\bar{F}}_{1} (T^{- 1} χ, u, t) - {\bar{F}}_{1} (T^{- 1} \hat{χ}, u, t) \end{array}

\begin{array}{l} {\dot{ε}}_{2} = A_{21} ε_{1} + A_{22} ε_{2} + {\bar{F}}_{2} (T^{- 1} χ, u, t) - {\bar{F}}_{2} (T^{- 1} \hat{χ}, u, t) + L_{2} H_{22} ε_{2} + D_{2} Δ ξ - τ \\ = A_{21} ε_{1} + (A_{22} - L_{2} H_{22}) ε_{2} + {\bar{F}}_{2} (T^{- 1} χ, u, t) - {\bar{F}}_{2} (T^{- 1} \hat{χ}, u, t) + D_{2} Δ ξ - τ \end{array}

Let $Δ {\bar{F}}_{1} = {\bar{F}}_{1} (T^{- 1} χ, u, t) - {\bar{F}}_{1} (T^{- 1} \hat{χ}, u, t)$ , $Δ {\bar{F}}_{2} = {\bar{F}}_{2} (T^{- 1} χ, u, t) - {\bar{F}}_{2} (T^{- 1} \hat{χ}, u, t)$ , from Assumption 1 and $\hat{χ} : = c o l ({\hat{χ}}_{1}, H_{22}^{- 1} ο_{2})$ , then $‖Δ {\bar{F}}_{1}‖ \leq γ_{1} ‖T^{- 1} χ - T^{- 1} \hat{χ}‖ = γ_{1} ‖T^{- 1}‖ ‖ε_{1}‖$ and $‖Δ {\bar{F}}_{2}‖ \leq γ_{2} ‖T^{- 1} χ - T^{- 1} \hat{χ}‖ = γ_{2} ‖T^{- 1}‖ ‖ε_{1}‖$

\begin{array}{l} ‖Δ {\bar{F}}_{1}‖ \leq γ_{1} ‖T^{- 1} χ - T^{- 1} \hat{χ}‖ = γ_{1} ‖T^{- 1}‖ ‖ε_{1}‖ \\ ‖Δ {\bar{F}}_{2}‖ \leq γ_{2} ‖T^{- 1} χ - T^{- 1} \hat{χ}‖ = γ_{2} ‖T^{- 1}‖ ‖ε_{1}‖ \end{array}

where $γ_{1} = ‖T_{1}‖ γ$ , $γ_{2} = ‖T_{2}‖ γ$ .

Theorem 2. From Assumptions 1 and 2, the system is detectable, and there exist matrices $P_{1}^{T} = P_{1} \in R^{(n - r) \times (n - r)} > 0$ , $P_{2}^{T} = P_{2} \in R^{r \times r} > 0$ , $L_{1} \in R^{(n - r) \times (n - r)}$ , $L_{2} \in R^{r \times r}$ , and positive scalars $μ_{1}$ and $μ_{2}$ , such that

[\begin{matrix} A_{11}^{T} P_{1} + P_{1} A_{11} - H_{11}^{T} R_{1}^{T} - R_{1} H_{11} + σ I_{n - r} & P_{1} & A_{21}^{T} P_{2} & 0 \\ P_{1} & - μ_{1} I_{n - r} & 0 & 0 \\ P_{2} A_{21} & 0 & A_{22}^{T} P_{2} + P_{2} A_{22} - H_{22}^{T} R_{2}^{T} - R_{2} H_{22} & P_{2} \\ 0 & 0 & P_{2} & - μ_{2} I_{r} \end{matrix}] < 0

Then, the errors $ε_{1}$ and $ε_{2}$ converge asymptotically to zero, where $R_{1} = P_{1} L_{1}$ , $R_{2} = P_{2} L_{2}$ , $σ = μ_{1} γ_{1}^{2} {‖T^{- 1}‖}^{2} + μ_{2} γ_{2}^{2} {‖T^{- 1}‖}^{2}$ , $Λ_{1} = {(A_{11} - L_{1} H_{11})}^{T} P_{1} + P_{1} (A_{11} - L_{1} H_{11})$ , and $Λ_{2} = {(A_{22} - L_{2} H_{22})}^{T} P_{2} + P_{2} (A_{22} - L_{2} H_{22})$ .

Proof. The Lyapunov function is considered as

V_{1} (t) = ε_{1}^{T} (t) P_{1} ε_{1} (t)

From Theorem 1, let $J_{1} = P_{1}^{T} ε_{1}$ and $J_{2} = Δ {\bar{F}}_{1}$ , we have

2 ε_{1}^{T} P_{1} Δ {\bar{F}}_{1} \leq \frac{1}{μ_{1}} ε_{1}^{T} P_{1} P_{1}^{T} ε_{1} + μ_{1} Δ {\bar{F}}_{1}^{T} Δ {\bar{F}}_{1}

From equations (12) and (17), the time derivative of equation (16) is

\begin{array}{l} {\dot{V}}_{1} = ε_{1}^{T} [{(A_{11} - L_{1} H_{11})}^{T} P_{1} + P_{1} (A_{11} - L_{1} H_{11})] ε_{1} +2 ε_{1}^{T} P_{1} Δ {\bar{F}}_{1} \\ \leq ε_{1}^{T} ({(A_{11} - L_{1} H_{11})}^{T} P_{1} + P_{1} (A_{11} - L_{1} H_{11}) + \frac{1}{μ_{1}} P_{1} P_{1}) ε_{1} + μ_{1} γ_{1}^{2} {‖T^{- 1}‖}^{2} {‖ε_{1}‖}^{2} \end{array}

In the same way, consider the following Lyapunov function for equation (13)

V_{2} (t) = ε_{2}^{T} (t) P_{2} ε_{2} (t)

Then, its time derivative is

\begin{array}{l} {\dot{V}}_{2} = ε_{2}^{T} [{(A_{22} - L_{2} H_{22})}^{T} P_{2} + P_{2} (A_{22} - L_{2} H_{22})] ε_{2} +2 ε_{2}^{T} P_{2} A_{21} ε_{1} +2 ε_{2}^{T} P_{2} Δ {\bar{F}}_{2} +2 ε_{2}^{T} P_{2} D_{2} Δ ξ - 2 ε_{2}^{T} P_{2} τ \\ \leq ε_{2}^{T} ({(A_{22} - L_{2} H_{22})}^{T} P_{2} + P_{2} (A_{22} - L_{2} H_{22}) + \frac{1}{μ_{2}} P_{2} P_{2}) ε_{2} +2 ε_{2}^{T} P_{2} A_{21} ε_{1} + μ_{2} γ_{2}^{2} {‖T^{- 1}‖}^{2} {‖ε_{1}‖}^{2} +2 ε_{2}^{T} P_{2} (D_{2} Δ ξ - τ) \end{array}

From Assumption 2 and definition of $τ (t)$ , we obtain that $ε_{2}^{T} P_{2} τ (t) < ε_{2}^{T} P_{2} \frac{P_{2} H_{22}^{- 1} (ο_{2} - {\hat{ο}}_{2})}{‖P_{2} H_{22}^{- 1} (ο_{2} - {\hat{ο}}_{2})‖} = α ‖P_{2} ε_{2}‖$ , furthermore, we can determine that $2 ε_{2}^{T} P_{2} (D_{2} Δ ξ - τ) < 0$ . Thus, equation (20) can be simplified as

{\dot{V}}_{2} \leq ε_{2}^{T} (Λ_{2} + \frac{1}{μ_{2}} P_{2} P_{2}) ε_{2} +2 ε_{2}^{T} P_{2} A_{21} ε_{1} + μ_{2} γ_{2}^{2} {‖T^{- 1}‖}^{2} {‖ε_{1}‖}^{2}

According to equations (18) and (21), equation (22) can be obtained

\dot{V} = {\dot{V}}_{1} + {\dot{V}}_{2} \leq ε_{1}^{T} (Λ_{1} + \frac{1}{μ_{1}} P_{1} P_{1} + μ_{1} γ_{1}^{2} {‖T^{- 1}‖}^{2} + μ_{2} γ_{2}^{2} {‖T^{- 1}‖}^{2}) ε_{1} + ε_{2}^{T} (Λ_{2} + \frac{1}{μ_{2}} P_{2} P_{2}) ε_{2} +2 ε_{2}^{T} P_{2} A_{21} ε_{1}

Therefore, if Theorem 2 is satisfied, then $\dot{V} < 0$ , and equations (12) and (13) are asymptotically stable. This completes the proof.

Then, the detection residual can be defined as $J = ‖ο_{1} - {\hat{ο}}_{1}‖$ , and we can distinguish fault occurrences by comparing the detection residual with the safety threshold.

Fault estimation based on hybrid observer and fault-tolerant control

Fault estimation

Fault isolation is required after detection for multiple actuators and we can detect each actuator by designing a bank of observers. Depending on the relationship between residuals and threshold, we can determine which actuators have faults similar to fault detection. Details are not provided here due to length limitations. After judging the occurrence of the fault by the fault detection module in “Design of fault detection based on hybrid observer” section, the fault estimation observer is triggered to estimate the real-time value of the fault, which can be used for compensation to avoid the damage caused by the fault. Each actuator can be assessed for fault status using a bank of observers. Furthermore, the results of fault estimation can be prepared for fault-tolerant control (FTC), as shown in Figure 2.

Figure 2.

The diagram of fault diagnosis for actuators.

The design method of adaptive observer is combined with the adaptive control theory and state observer theory. Depending on state change, the system can be adjusted online by setting an appropriate gain matrix and adaptive regulation law. Assuming the s’th $(1 \leq s \leq m)$ actuator is fault, an adaptive observer for real-time estimating the magnitude of the fault based on equation (7) can be presented as follows

\{\begin{array}{l} {\dot{\hat{χ}}}_{1} (t) = A_{11} {\hat{χ}}_{1} (t) + A_{12} H_{22}^{- 1} ο_{2} (t) + B_{1} u (t) - b_{s} [{\hat{ρ}}_{s} (t) u_{s} (t) - \hat{η} (t)] + {\bar{F}}_{1} (T^{- 1} \hat{χ}, u, t) + L_{3} (ο_{1} - {\hat{ο}}_{1}) \\ {\hat{ο}}_{1} (t) = H_{11} {\hat{χ}}_{1} (t) \end{array}

where b_s represents the s’th column of B ₁, and u_s represents the s’th control input u.

Define $ε_{1} = χ_{1} (t) - {\hat{χ}}_{1} (t)$ , ${\tilde{η}}_{s} = η_{s} - {\hat{η}}_{s}$ , ${\tilde{ρ}}_{s} = ρ_{s} - {\hat{ρ}}_{s}$ , and $ε_{ο 1} = ο_{1} (t) - {\hat{ο}}_{1} (t)$ .

Theorem 3. If there exist matrices $P_{3} = P_{3}^{T} > 0$ , $Q > 0$ , $L_{3} \in R^{(n - r) \times (n - r)}$ , and positive scalars $ϑ_{1}$ and $γ_{3}$ , the following conditions are satisfied

{(A_{11} - L_{3} H_{11})}^{T} P_{3} + P_{3} (A_{11} - L_{3} H_{11}) + \frac{1}{ϑ_{1}} P_{3} P_{3} + ϑ_{1} γ_{3}^{2} I_{n - r} = - Q

P_{3} B_{1} = {(G H_{11})}^{T}

and the error system is asymptotically stable. The following adaptive fault estimation algorithm

\begin{array}{l} {\dot{\hat{ρ}}}_{s} = - 2 β_{1} g_{s} ε_{ο 1} u_{s} \\ {\dot{\hat{η}}}_{s} = 2 β_{2} g_{s} ε_{ο 1} \end{array}

can realize $lim_{t \to \infty} ε_{1} (t) =0$ , $lim_{t \to \infty} \tilde{ρ} =0$ , and $lim_{t \to \infty} \tilde{η} =0$ , where g_s is the s’th row of G, and $β_{1}$ and $β_{2}$ are the adaptive learning rates.

Equation ( 24) can be converted into an linear matrix inequality (LMI) problem using the Schur complement

[\begin{matrix} A_{11}^{T} P_{3} + P_{3} A_{11} - H_{11}^{T} Θ_{1}^{T} - Θ_{1} H_{11} & P_{3} & I_{n - r} \\ P_{3} & - ϑ_{1} I_{n - r} & 0 \\ I_{n - r} & 0 & - {(ϑ_{1} γ_{3}^{2})}^{- 1} I_{n - r} \end{matrix}] < 0

where $γ_{3}$ is the known Lipschitz constant, $Θ_{1} = P_{3} L_{3}$ , and $Λ_{3} = {(A_{11} - L_{3} H_{11})}^{T} P_{3} + P_{3} (A_{11} - L_{3} H_{11})$ .

Proof. Consider the following Lyapunov function

V_{3} = ε_{1}^{T} P_{3} ε_{1} + \frac{1}{2 β_{1}} {\tilde{ρ}}_{s}^{2} + \frac{1}{2 β_{2}} {\tilde{η}}_{s}^{2}

Then, the time derivative of equation (28) is obtained

\begin{array}{l} {\dot{V}}_{3} = ε_{1}^{T} [{(A_{11} - L_{3} H_{11})}^{T} P_{3} + P_{3} (A_{11} - L_{3} H_{11})] ε_{1} - 2 ε_{1}^{T} P_{3} b_{s} {\tilde{ρ}}_{s} u_{s} + 2 ε_{1}^{T} P_{3} b_{s} {\tilde{η}}_{s} \\ + 2 ε_{1}^{T} P_{3} [{\bar{F}}_{1} (T^{- 1} χ, u, t) - {\bar{F}}_{1} (T^{- 1} \hat{χ}, u, t)] + \frac{1}{β_{1}} {\tilde{ρ}}_{s} {\dot{\tilde{ρ}}}_{s} + \frac{1}{β_{2}} {\tilde{η}}_{s} {\dot{\tilde{η}}}_{s} \end{array}

Let $J_{1} = P_{3}^{T} ε_{1}$ and $J_{2} = Δ {\bar{F}}_{1}$ , then from Theorem 1 and Assumption 1

2 ε_{1}^{T} P_{3} Δ {\bar{F}}_{1} \leq \frac{1}{ϑ_{1}} ε_{1}^{T} P_{3} P_{3} ε_{1} + ϑ_{1} Δ {\bar{F}}_{1}^{T} Δ {\bar{F}}_{1} \leq \frac{1}{ϑ_{1}} ε_{1}^{T} P_{3} P_{3} ε_{1} + ϑ_{1} γ_{3}^{2} ε_{1}^{T} I_{n - r} ε_{1}

Equation (29) can be simplified as

\begin{array}{l} {\dot{V}}_{3} = ε_{1}^{T} Λ_{3} ε_{1} - 2 ε_{1}^{T} P_{3} b_{s} {\tilde{ρ}}_{s} u_{s} + 2 ε_{1}^{T} P_{3} b_{s} {\tilde{η}}_{s} + \frac{1}{ϑ_{1}} ε_{1}^{T} P_{3} P_{3} ε_{1} + ϑ_{1} γ_{3}^{2} ε_{1}^{T} I_{n - r} ε_{1} + \frac{1}{β_{1}} {\tilde{ρ}}_{s} ({\dot{ρ}}_{s} - {\dot{\hat{ρ}}}_{s}) + \frac{1}{β_{2}} {\tilde{η}}_{s} ({\dot{η}}_{s} - {\dot{\hat{η}}}_{s}) \\ = ε_{1}^{T} Λ_{3} ε_{1} + \frac{1}{ϑ_{1}} ε_{1}^{T} P_{3} P_{3} ε_{1} + ϑ_{1} γ_{3}^{2} ε_{1}^{T} I_{n - r} ε_{1} - {\tilde{ρ}}_{s} (2 ε_{1}^{T} P_{3} b_{s} u_{s} + \frac{1}{β_{1}} {\dot{\hat{ρ}}}_{s}) + {\tilde{η}}_{s} (2 ε_{1}^{T} P_{3} b_{s} - \frac{1}{β_{2}} {\dot{\hat{η}}}_{s}) + \frac{1}{β_{1}} {\tilde{ρ}}_{s} {\dot{ρ}}_{s} + \frac{1}{β_{2}} {\tilde{η}}_{s} {\dot{η}}_{s} \end{array}

With the help of simple mathematical derivation, we can get

\begin{array}{l} \frac{1}{β_{1}} {\tilde{ρ}}_{s} \dot{ρ} = - \frac{{\tilde{ρ}}_{s}^{2}}{β_{1}} + \frac{{\tilde{ρ}}_{s} ({\tilde{ρ}}_{s} + {\dot{ρ}}_{s})}{β_{1}} = - \frac{{\tilde{ρ}}_{s}^{2}}{β_{1}} + \frac{(ρ_{s} - {\hat{ρ}}_{s}) (ρ_{s} - {\hat{ρ}}_{s} + {\dot{ρ}}_{s})}{β_{1}} \leq - \frac{{\tilde{ρ}}_{s}^{2}}{β_{1}} + \frac{(|ρ_{s}| + |{\hat{ρ}}_{s}|) (|ρ_{s}| + |{\hat{ρ}}_{s}| + |{\dot{ρ}}_{s}|)}{β_{1}} \end{array}

and similarly,

\frac{1}{β_{2}} {\tilde{η}}_{s} {\dot{η}}_{s} = - \frac{{\tilde{η}}_{s}^{2}}{β_{2}} + \frac{{\tilde{η}}_{s} ({\tilde{η}}_{s} + {\dot{η}}_{s})}{β_{2}} \leq - \frac{{\tilde{η}}_{s}^{2}}{β_{2}} + \frac{(|η_{s}| + |{\hat{η}}_{s}|) (|η_{s}| + |{\hat{η}}_{s}| + {\dot{η}}_{s})}{β_{2}}

From Assumption 2, there exist some unknown positive constants $\bar{ρ}$ , $\bar{\bar{ρ}}$ , $\bar{η}$ , and $\bar{\bar{η}}$ , such that the following conditions hold, $|ρ_{i} (t)| \leq \bar{ρ}$ , $|{\dot{ρ}}_{i} (t)| \leq \bar{\bar{ρ}}$ , $|η_{i} (t)| \leq \bar{η}$ , and $|{\dot{η}}_{i} (t)| \leq \bar{\bar{η}}$ . Thus, from equations (32) and (33), one can obtain

\begin{array}{l} \frac{1}{β_{1}} {\tilde{ρ}}_{s} \dot{ρ} \leq - \frac{{\tilde{ρ}}_{s}^{2}}{β_{1}} + \frac{2 \bar{ρ} (2 \bar{ρ} + \bar{\bar{ρ}})}{β_{1}} \\ \frac{1}{β_{2}} {\tilde{η}}_{s} {\dot{η}}_{s} \leq - \frac{{\tilde{η}}_{s}^{2}}{β_{2}} + \frac{2 \bar{η} (2 \bar{η} + \bar{\bar{η}})}{β_{2}} \end{array}

Therefore,

{\dot{V}}_{3} \leq ε_{1}^{T} Λ_{3} ε_{1} + \frac{1}{ϑ_{1}} ε_{1}^{T} P_{3} P_{3} ε_{1} + ϑ_{1} γ_{3}^{2} ε_{1}^{T} I_{n - r} ε_{1} - {\tilde{ρ}}_{s} (2 ε_{1}^{T} P_{3} b_{s} u_{s} + \frac{1}{β_{1}} {\dot{\hat{ρ}}}_{s}) + {\tilde{η}}_{s} (2 ε_{1}^{T} P_{3} b_{s} - \frac{1}{β_{2}} {\dot{\hat{η}}}_{s}) + \frac{1}{β_{1}} {\tilde{ρ}}_{s} {\dot{ρ}}_{s} + \frac{1}{β_{2}} {\tilde{η}}_{s} {\dot{η}}_{s}

\begin{array}{l} {\dot{V}}_{3} \leq ε_{1}^{T} (Λ_{3} + \frac{1}{ϑ_{1}} P_{3} P_{3} + ϑ_{1} γ_{3}^{2} I_{n - r}) ε_{1} - \frac{{\tilde{ρ}}_{s}^{2}}{β_{1}} + \frac{2 \bar{ρ} (2 \bar{ρ} + \bar{\bar{ρ}})}{β_{1}} - \frac{{\tilde{η}}_{s}^{2}}{β_{2}} + \frac{2 \bar{η} (2 \bar{η} + \bar{\bar{η}})}{β_{2}} \\ \leq ε_{1}^{T} (Λ_{3} + \frac{1}{ϑ_{1}} P_{3} P_{3} + ϑ_{1} γ_{3}^{2} I_{n - r}) ε_{1} - \frac{{\tilde{ρ}}_{s}^{2}}{2 β_{1}} - \frac{{\tilde{η}}_{s}^{2}}{2 β_{2}} + σ \\ \leq - \frac{λ_{min} (Q)}{λ_{max} (P_{3})} ε_{1}^{T} P_{3} ε_{1} - \frac{{\tilde{ρ}}_{s}^{2}}{2 β_{1}} - \frac{{\tilde{η}}_{s}^{2}}{2 β_{2}} + σ \\ \leq - μ V_{3} + σ \end{array}

where $μ = min \{1, \frac{λ_{min} (Q)}{λ_{max} (P_{3})}\}$ and $σ = \frac{2 \bar{ρ} (2 \bar{ρ} + \bar{\bar{ρ}})}{β_{1}} + \frac{2 \bar{η} (2 \bar{η} + \bar{\bar{η}})}{β_{2}}$ . Therefore, the residual $ε_{1}$ is bounded and asymptotically converges within a small neighborhood of zero. The proof is completed.

Fault-tolerant control

After obtaining the fault magnitude, as discussed in “Fault estimation” section, it is necessary to reconstruct the original controller and design a fault-tolerant controller. This article adopted a simple and practical PID to control quadrotor UAVs under normal operation. First of all, we use the fault detection module to monitor the health of the quadrotor UAV actuators in real time. The fault detection and fault estimation module are triggered when a fault occurs to estimate bias fault and efficiency loss fault. Finally, the FTC module is triggered after fault estimation convergence to control the UAV and maintain stability.

The fault-tolerant controller based on PID controller is as follows

u_{i}^{T} = \frac{u_{i}^{P I D} - {\hat{η}}_{i} (t)}{[1 - {\hat{ρ}}_{i} (t)]}, i = 1, \dots, 4

where $u_{i}^{P I D}$ is the i’th PID control input, ${\hat{ρ}}_{i} (t)$ and ${\hat{η}}_{i} (t)$ are estimated values of $ρ_{i} (t)$ and $η_{i} (t)$ , which can be obtained in the previous section. When there is no fault in the quadrotor UAV, the designed normal controller needs to ensure the system state tracking error converges asymptotically to zero. The normal controller adopted in this article is a practical PID controller, which can ensure the stability of the quadrotor UAV system. The expression of quadrotor UAV with bias fault or gain fault is shown in the above equation (5), as long as the designed normal controller can stabilize the system, the actuator faults can be compensated in time following the FTC strategy (equation (37)) after accurate fault estimation is obtained, and the system state tracking error asymptotically converges to zero.

Simulation results

Parameters setting

Based on the description in “Actuator fault model” section, we know that UAV actuators mainly comprise blades and motors, with typical actuator faults including blade damage and motor speed loss. UAV actuator wear, aging, and external impact will result in abnormal motor power, characterized by that the output of the motor voltage cannot reach the expected value. Two actuator fault types are considered in this article, depending on the relationship between fault and input signals, as shown in equation (4).

Actuator mechanical fatigue and aging create small-amplitude faults. An appropriate algorithm can effectively deal with and isolate these small-amplitude faults if they can be detected in time, avoiding catastrophic results. We superimposed a small fault signal on the motor output voltage to simulate small-amplitude faults and observe whether the hybrid observer could estimate the faults in time. Motor and propeller faults have a similar influence on UAV performance, hence, we considered motor faults as illustrative examples. Actuator gain faults, such as reduced motor speed, attenuate the lift force provided, and gain faults are introduced by adding gain attenuation to the controller output. Sudden additive faults can be introduced by reducing the motor voltage duty cycle, producing constant, and time-varying bias fault.

Several simulations are performed to verify the proposed method reasonableness. The parameters of the quadrotor UAV³⁵ are $I_{inertia}^{z} = 0.11 kg \cdot m^{2}$ , $I_{inertia}^{x} = I_{inertia}^{y} = 0.0552 kg \cdot m^{2}$ , $l = 0.197 m$ , $K_{t n} = - K_{t c} = 0.0036 N \cdot m / V$ , $K_{l} = 0.1188 N / V$ , $K_{d x} = K_{d y} = 0.008$ , and $K_{d z} = 0.0091$ . The reference trajectory is set as follows: $x_{r} = 0.5 cos (0.5 t)$ , $y_{r} = 0.5 sin (0.5 t)$ , $z_{r} = 2 + 0.1 t$ , and $ψ_{r} = 1$ .

We assumed the following actuator faults could occur to match actual situations.

Case 1 (gain fault): The following scenario is designed to model motor gain attenuation due to reduced motor speed during flight. Partial loss of effectiveness (30%) is injected into the first motor at t = 7 s, with the other motors running normally. The actuator gain fault can be expressed as

u_{1}^{f} (t) = (1 - ρ_{1} (t)) u_{1} (t), ρ_{1} (t) = \{\begin{matrix} 0, t < 7 s \\ 0.3, t \geq 7 s \end{matrix}

Case 2 (bias fault): Motor output voltage cannot be accurately guaranteed relative to the reference voltage due to external disturbances causing weak jumps. We modeled this by considering a constant bias fault in the second motor, expressed as follows

u_{2}^{f} (t) = u_{2} (t) + η_{2} (t), η_{2} (t) = \{\begin{matrix} 0, t < 6 s \\ 0.04, t \geq 6 s \\ 0.07, t \geq 10 s \end{matrix}

Case 3 (bias fault): Long-term operation will result in mechanical fatigue in the motor, with the increasing error between motor output and expected voltage over time. Therefore, we considered a small and time-varying actuator fault in the third motor, expressed as follows

u_{3}^{f} (t) = u_{3} (t) + η_{3} (t), η_{3} (t) = \{\begin{matrix} 0, t < 7 s \\ 0. 1 (1 - e^{- (t - 7)}), t \geq 7 s \end{matrix}

The original system matrices A, B, H, and $F (x, u, t)$ can be obtained from equations (2) and (3). The perturbation term $Δ ξ (t)$ , the fault distribution matrix D, and the transfer matrix T are designed as follows: $Δ ξ (t) = {[0.4 sin (0.2 π t) 0 .3cos (0.2 π t)]}^{T}$ , $D = [\begin{array}{l} 0.3 & 1 \\ 0.2 & 0.5 \\ 0.2 & 0 \\ 0 & 0.7 \\ 0.5 & 1 \\ 1 & 1 \end{array}]$ , and $T = [\begin{array}{l} 1 & 0 & 0 & 0 & - 1.4 & 0.4 \\ 0 & 1 & 0 & 0 & - 0.6 & 0.1 \\ 0 & 0 & 1 & 0 & 0.4 & - 0.4 \\ 0 & 0 & 0 & 1 & - 1.4 & 0.7 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}]$ .

Fault detection

Solving Theorem 2 with $γ_{1} = 3.8358$ and $γ_{2} = 0.2190$ , we can obtain $P_{2} = [\begin{matrix} 6.5416 & 0 \\ 0 & 6.5416 \end{matrix}]$ , $L_{2} = [\begin{matrix} 1.5 & 0.5 \\ 0.5 & 1.5 \end{matrix}]$ , $P_{1} = [\begin{array}{l} 6.5416 & 0 & 0 & 0 \\ 0 & 6.5416 & 0 & 0 \\ 0 & 0 & 6.5416 & 0 \\ 0 & 0 & 0 & 6.5416 \end{array}]$ , and $L_{1} = [\begin{array}{l} 156.7597 & 0.5000 & 0 & 0 \\ 0.5000 & 156.7597 & 0 & 0 \\ 0 & 0 & 156.7597 & 0.5000 \\ 0 & 0 & 0.5000 & 156.7597 \end{array}]$ .

Figure 3 shows the results of fault detection for the proposed methods, where the simulation threshold is chosen as 0.03, which is reasonable since equation (7) is not subject to disturbance terms. By comparing the residual mentioned above with the threshold, we can see that the residual is less than the threshold when no fault occurs. In Figure 3(a), when the gain fault is injected in the second motor at t =7 s, we can see that the residual is greater than the set threshold at t =7.012 s in a locally enlarged figure, which indicates that the actuator effectiveness loss can be detected in time.

Figure 3.

Results of fault detection by the proposed method: (a) Case 1: detection result of actuator efficiency factor loss; (b) case 2: detection result of constant bias fault of actuator; and (c) case 3: detection result of time-varying bias fault of actuator.

Figure 4 shows that the actual trajectory fails to track the reference trajectory since the first motor speed suddenly reduced. UAV lift forces become less than gravity, and hence, the UAV suddenly drops. At the same time, the pitch torque of the first and third motors is different, which leads to the increase of error in X direction. There will also be increasing Y direction error since counterclockwise torque from the first and third motors is no longer balanced by the second and fourth motors.

Figure 4.

Case 1: Result of quadrotor UAV trajectory tracking. UAV: unmanned aerial vehicle.

Figure 3(b) shows that a small amplitude fault is detected at t = 6.024 s, following the first motor being injected with bias fault at t = 6 s. In Figure 3(c), for case 3, as the amplitude of the bias fault is small and changes slowly, the bias fault of the actuator is detected at t =7.402 s (the bias fault occurs at 7 s).

Figure 5 shows the results of the actual and reference trajectory of case 3. Since the fault occurs with small amplitude, it is difficult to determine when the small amplitude fault occurred from the trajectory tracking. However, over time, we can find that UAV cannot track the reference trajectory properly.

Figure 5.

Case 3: Result of quadrotor UAV trajectory tracking and local enlarged results of track tracking in directions y and z: (a) Case 3: 3D trajectory tracking and (b) case 3: local enlarged results (y and z axes). UAV: unmanned aerial vehicle.

Fault estimation

To maintain the stability of UAV to a certain extent under actuator faults, fault estimation is required to compensate for the fault system. The following is the calculation of some parameters in the design process of fault estimation. By solving equation (27) with $γ_{3} = 3.8358$ and $ϑ_{1} = 1.5$ , we have

\begin{array}{l} P_{3} = [\begin{array}{l} 7.0499 & 0 & 0 & 0 \\ 0 & 7.0499 & 0 & 0 \\ 0 & 0 & 7.0499 & 0 \\ 0 & 0 & 0 & 7.0499 \end{array}] \\ L_{3} = [\begin{array}{l} 7.6331 & 0.5000 & 0 & 0 \\ 0.5000 & 7.6331 & 0 & 0 \\ 0 & 0 & 7.6331 & 0.5000 \\ 0 & 0 & 0.5000 & 7.6331 \end{array}] \\ G = [\begin{array}{l} - 0.0922 & - 0.0922 & 0.0922 & 0.0922 \\ - 0.0231 & - 0.0231 & 3.0115 & - 2.9654 \\ 0.0922 & 0.0922 & - 0.0922 & - 0.0922 \\ 2.8271 & - 3.1498 & 0.1614 & 0.1614 \end{array}] \end{array}

Good fault estimation can be achieved by selecting appropriate adaptive learning rates $β_{1}$ and $β_{2}$ . We set $β_{1} = 15$ and $β_{2} = 20$ for the simulations used here.

In Figure 6, the solid line is the actual injected actuator fault and the dotted line is the estimation of fault, we can see that the proposed method achieves fast convergence and can accurately estimate the injected fault. Figure 6(a) shows that the proposed method can estimate gain faults at approximately 8 s. Figure 6(b) shows that the method requires only approximately 0.8 s to converge to the constant bias fault. Figure 6(c) shows that the proposed method responds quickly to the time-varying bias fault and follows it well.

Figure 6.

Result of fault estimation by the proposed method: (a) Case 1: estimation of actuator efficiency factor loss; (b) case 2: estimation of constant bias fault of the actuator; and (c) case 3: estimation of time-varying bias fault of the actuator.

Fault-tolerant control

In Figure 7, the solid line is the reference trajectory and the dotted line is the trajectory adopted FTC; it can be seen that the fluctuation of trajectory has been compensated in time after adopting FTC strategy.

Figure 7.

Result of trajectory tracking of FTC and distance difference of y axis: (a) Case 1: trajectory tracking of FTC in case of efficiency loss; (b) case 2: trajectory tracking of FTC under constant bias; and (c) case 3: trajectory tracking of FTC under time-varying bias. FTC: fault-tolerant control.

Conclusion

This article proposed robust fault detection and estimation observers for quadrotor UAV actuator faults in the presence of model uncertainties and disturbances. Based on the actual situation, two types of actuator faults are considered, including small-amplitude bias faults and partial loss of effectiveness fault. The original system is converted into subsystems $\sum_{1}$ and $\sum_{2}$ by coordinate transformation to improve fault detection reliability and robustness to external disturbances. Corresponding fault detection and estimation observers are designed for subsystems $\sum_{1}$ and $\sum_{2}$ , observer stability proved using the Lyapunov function with unknown upper and lower bounds of actuator fault. The PID controller is compensated after estimating the faults to reduce impacts on the system. Simulation results verified that the proposed method achieved fast convergence and robustness. Future work will relax Assumption 2 and limitations on sliding mode observer design, and test the proposed algorithm through real-flight experiments.

Footnotes

Acknowledgement

The authors also would like to thank the National Natural Science Foundation of China.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [Grant No. 61873182] and the Science and Technology on Space Intelligent Control Laboratory [Grant no. HTKJ2019KL502009].

ORCID iD

Jinjin Guo

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