Most existing works focus on stationary environments at low speeds without significant disturbances.

Recently, incremental nonlinear dynamic inversion (INDI) control algorithms have emerged as promising solutions for quadrotors operating at high speeds with aggressive maneuvers [3], [4], [5], [6]. This approach utilizes sensor measurements and their derivatives to enhance robustness against model uncertainty and external disturbances.

Active disturbance rejection control (ADRC) [9] is similar to INDI in that it relies less on accurate model information. An extended state observer (ESO), an essential component in ADRC, estimates both the states and lumped disturbances, including internal disturbances caused by model uncertainty and external disturbances from the environment. A subsequent disturbance rejection controller can be designed accordingly once the disturbance has been estimated by the ESO. This method, following the separation principle, offers a superior modular design compared to direct robust methods such as the adaptive fuzzy approach [10]. There are several works on ADRC-based FTC [11], [12], however their focus is on stationary environments at low speeds.

Moreover, existing controllers based on ADRC typically require the original dynamic model to be divided into multiple SISO subsystems, which must have the form of cascades of integrators and matched disturbances. In this letter, we use the necessary and sufficient conditions for the existence of a model-based extended state observer (MB-ESO), as proposed in our recent work [13], as a criterion to design MB-ESOs for more general control-affine nonlinear systems subject to mismatched disturbances. This letter employs a disturbance decoupling controller design for multi-input, multi-output (MIMO) affine nonlinear systems with mismatched disturbances [14]. The idea of reduced attitude control is used to properly select controlled outputs [4] to ensure the stability of the zero dynamics of the quadrotor when two opposing rotors are available. Note that we use two opposing rotors as a case study to facilitate comparison with the state of the art [4]. By using reduced attitude for control output selection along with our general observer and controller design, it is straightforward to generalize to other cases, such as different numbers of missing rotors or two adjacent missing rotors. The main contribution is the extension of our MB-ESO theory to a challenging quadrotor FTC within a new framework. (1) Unlike existing ESO-based controls that require stringent structural constraints, our new design has flexibility for a broader range of nonlinear systems. (2) It utilizes all available measured outputs and model information for observer design. (3) We integrate the MB-ESO with mismatched disturbance decoupling under the principle of reduced attitude, demonstrating superior robustness and tracking performance in strong winds compared to the INDI [4]. II. PRELIMINARY AND PROBLEM FORMULATION Notation: Boldface symbols like P denote vectors or matrices, while non-boldface symbols like p denote scalars. Superscripts [ • ] I and [ • ] B indicate coordinates expressed in the inertial and body frames, respectively. Subscripts specify relationships or attributes; e.g., x I represents the x-axis in the inertial frame. 0 m×n denotes a m by n zero matrix and I m×m denotes an m-dimension identity matrix.

Let F I ={O I , x I , y I , z I } denote a right-hand inertial frame fixed to the ground, where x I , y I , and z I point to the north, east, and points down, respectively. As shown in Fig. 1, F B ={O B , x B , y B , z B } is a right-hand body frame fixed to the quadrotor with O B located at the center of mass, and x B , y B and z B pointing forward, right and down, respectively.

The complete dynamical model is as follows [4]:

where the translation dynamics are described in (1a) and (1b), and the rotation dynamics are described in (1c) and (1d);

T , and g I = [0, 0, g] T are the position, velocity, and gravity vectors in F I ; B = [p, q, r] T is the angular velocity vector in F B including pitch rate, roll rate, and yaw rate; R IB represents the rotation matrix from F B to F I ; B × denotes the skew-symmetric matrix such that B × a = B × a for the vector cross product × and any vector a ∈ R 3 ; m v and I v represent the gross mass and inertial matrix of the quadrotor. The resultant force F B and moment M B applied to the center of mass of the quadrotor in F B are modeled as:

where κ 0 and τ 0 are coefficients of the thrust and the reaction torque; b and l are geometric parameters of the quadrotor shown in Fig. 1; ω i is the angular speed of the ith rotor rotating about the z axis in F B ; γ is the damping coefficient on yaw; I p represents the moment of inertial of each rotor inducing the gyroscopic moment; F a and M a are additional forces and torques caused by quadrotor model uncertainty, unmodeled aerodynamic effects and external wind.

Due to the inability to control the yaw angle in rotor failure cases, reduced attitude control [4], [8] simplifies the original attitude control to thrust vector pointing, compromising yaw control. n B in Fig. 1 is a unit vector fixed to the body frame, as a rotation axis of the damaged quadrotor, and pointing in the same direction as the averaged thrust. To achieve position control, n B must align with the reference unit vector n B d , which is generated by the outer-loop controller for n I d in F I and subsequently converted to F B .Eq.( 1c) is replaced by the first two of the following relaxed attitude kinematic equations, as n B d is a unit vector with two independent components [4]:

where

T is a disturbance vector, and n I d is n B d expressed in F I . For the quadrotor with two opposing rotors failure, n B = [0, 0, -1] T is chosen for the maximized energy efficiency. h 1 and h 2 should be both stabilized to zero to align n B with n B d .

To illustrate the design of MB-ESO based control for the inner-loop controller more clearly, the dynamical model of the quadrotor after replacement can be written in the following affine nonlinear system with disturbance w: (5) where

] T represents a disturbance vector in the inner-loop of the quadrotor,

is a diagonal inertia matrix, R 33 is the element at the third row and the third column of R IB , and the rotor induced gyroscopic moment is ignored. Every state can be measured or estimated. Since there remain two functional rotors, a maximum of two outputs can be controlled. The same outputs in [4] are used in (9), where the second output y 2 is the projection of n B d onto the axis x of a new coordinate system obtained by rotating the body frame F B about z B by an angle χ . The zero dynamics of the quadrotor can be stabilized by properly selecting χ ,see [4] for more details. Without loss of generality, in this letter, we consider the case that rotors 2 and 4 remain functional, i.e., ω 2 1 and ω 2 3 are assumed to be zero. The research problem addressed in this letter is how to maximally exploit all available measured outputs and known model information to better estimate the disturbance w, thereby improving control performance and robustness for the twoinput, two-output affine nonlinear system (5).

Fig. 2 shows a schematic overview of our framework, which consists of an outer-loop horizontal position controller and an inner-loop attitude and altitude controller.

To make a fair comparison with the INDI [4], our framework shares the same outer-loop PID controller generating desired rotation axis n I d in F I , which can be obtained as The objective of designing the MB-ESO is to maximally exploit all available measured outputs and model information to estimate disturbances in the dynamic model. The sufficient and necessary conditions for the existence of an MB-ESO proposed in [13] are: 1) the system is observable, and 2) there is no invariant zero between the output and the disturbance. For system (5) with disturbance vector w, the output vector for observers can be chosen as y o = [Z, h 1 , h 2 , p, q, r] T . It can be easily verified that the system (5) with output y o can be transferred in a normal form, with a disturbance vector relative degree of {2, 1, 1, 1, 1, 1}.Thesum of these degrees is 7, which is equivalent to the number of states. Therefore, the system is observable and has no zero dynamics between w and y o [15].

Reference [13] addresses only single-disturbance singleoutput (SDSO) linear systems, while the system in ( 5) is a multi-disturbance multi-output (MDMO) affine nonlinear system. To apply the results from [13], ( 5) is decomposed into six SDSO systems based on the disturbance vector relative degree, each containing a nonlinear term in the same channel as the disturbance. The disturbance vector w is treated as an extended state vector, enabling the design of six MB-ESOs:

where Ẑ, Vz , ĥ1 , ĥ2 , p, q, r, Fa,z , λ1 , λ2 , Ma,x , Ma,y , and Ma,z are estimates; G i (x) is the ith row of G(x) for i = 1,...,7; f i (x) is the ith row of f (x) for i = 1,...,7; L i,j is the ith tunable observer gain for the jth MB-ESO for i = 1,...,3 and j = 1,...,6, which can be determined by the place of eigenvalues of MB-ESO. For simplicity, the eigenvalues of the ith MB-ESO are all placed at -ν oi , where ν oi is the observer bandwidth of ESO [16].

Remark 1: Although [13] mainly considers discrete-time systems, it also generalizes to the case of continuous-time systems proposed in [17].

Next, we will show that the error dynamics of the MB-ESOs proposed for system (5) are the same as those in [13], [17]. For simplicity, we will use the MB-ESO described in (11) as an example. The augmented component of system (5) corresponding to (11) is

Subtracting ( 11) from ( 17) yields the error dynamics:

where

, and E o = [0, 0, 1] T . Therefore, we verify that the condition in [13] holds for this system, and by applying pole placement to the observer, we can guarantee the convergence of disturbance estimation.

Reference [13] examines the connection between the MB-ESO and the unknown input observer (UIO), also known as the disturbance decoupled observer [18]. Established UIO theory supports MB-ESO development. Reference [19] proves geometrically that the number of decoupled disturbances in an observer cannot exceed the number of measured outputs. In this letter, the number of disturbances equals the number of measured outputs, ensuring maximal utilization of measurements for disturbance estimation in the MB-ESOs.

To design a control law u that makes the system output y follow the reference trajectory [Z ref , h 1,ref cos χ +h 2,ref sin χ ] T , the effects of disturbance w on system (5) should be eliminated from y. However, w cannot be canceled out directly in the input channel since w is not in the same channel as the control input u, making w mismatched disturbances. The disturbance decoupling method proposed in [14] is adopted in this letter to reject mismatched disturbances.

The system ( 5) is converted into the normal form via a coordinate transformation (x) = η, ξ

T [15], where η represents the states of the zero dynamics, see [4],

H i (x) denotes the ith real-valued function in the vector field H(x), and L f H i (x) is the first order Lie derivative of H i (x) along the vector field f (x). Since the zero dynamics of system ( 5) can be stabilized by properly selecting χ and control input u cannot change the place of zeros, the following subsystem under the new coordinates (x) without zero dynamics is used to design the control law:

where

T are the inputs of functional rotors 2 and 4,

is the first order Lie derivative of H i (x) first along the vector field f (x) and then along the vector filed Q j (x), L 2 f H i is the second order Lie derivative of H i (x) along the vector field f (x).

Since there exists a disturbance term D c (x)w in the system (20), a disturbance decoupling control law to compensate for the disturbances in the outputs y in steady state is given in the following [14]:

where

determined such that the effects of w are eliminated in the output channels in steady state, and c i 1 and c i 2 can be calculated by placing all poles of the ith feedback loop at -ν ci , where ν ci is called the controller bandwidth [16].

For simplicity, the first subsystem is used to illustrate how to derive the first row of (x). Substituting (21) into (20),t h e first subsystem can be written as

TABLE I PARAMETERS OF THE QUADROTOR TABLE II CONTROL PARAMETERS OF THE MB-ESO BASED CONTROL

The last two terms in (22) should have zero impact on the output y 1 in steady state, which can be formulated as

Solving ( 23) for γ 1i yields

To make a fair comparison between our approach and the notable INDI fault-tolerant control [4], we use the same opensource simulator 1 developed by the authors. The parameters of the quadrotor in Table I are used in the designs of the INDI and ours. The maximum tilt angle of the quadrotor is set to 30 degrees. The angular speed range of each rotor lies in 0 rad/s to 1200 rad/s. We use the same noise settings as [4], with power values of 1 × 10 -8 ,1× 10 -8 ,2× 10 -4 , and 1 × 10 -5 applied to position, attitude, angular velocity, and rotor speed measurements, respectively, using Band-Limited White Noise in Simulink R . The observer bandwidths and control bandwidths in the inner-loop and the PID gains in the outerloop for the proposed control method are given in Table II. The control gains in the inner-loop for the INDI method are given in Table III with the same outer-loop parameters.

Fig. 3(a) shows that the tracking performances on the X axis for ours and INDI are similar without any over-shooting. However, due to aggressive maneuvering and significant unmodeled aerodynamics, substantial fluctuations occur on the Y and Z axes and y 2 when the quadrotor tries to follow the step reference on the X axis, as illustrated in Fig. 3 TABLE III CONTROL PARAMETERS OF THE INDI Fig. 3. Trajectory tracking comparison under windless condition.

and Fig. 3(d). Fig. 3 also verifies that the tracking performance of ours is much better than that of INDI. Moreover, Fig. 4 shows that even with better control performance the control signals of ours are less noisy than that of INDI.

To test the robustness of ours and INDI against model uncertainty and external disturbances with aggressive maneuvering, the maximum tilt angle of the quadrotor is changed to 60 degrees and a wind gust (-15, 0, 0) m/s filtered by a low-pass filter with a transfer function 1/(4s+1), the same setting in the environment 1 , is simulated continuously. The same reference trajectory as in Fig. 3 is used in the simulation. Fig. 5 shows the trajectory tracking results of ours and INDI simulating for 10 seconds. Both methods perform similarly during the first 7 seconds, when the wind strength remains relatively low. The position deviation from the reference is primarily caused by the outer-loop controller, which lacks the disturbance rejection capacity. After that, the quadrotor controlled by the INDI approach crashes. The robustness of INDI against the strong wind can be improved by using higher attitude control gains (i.e., k 2,p and k 2,d ). However, this improvement is achieved by sacrificing the tracking performance, as illustrated in Fig. 6 with different k 2,p = 200 and k 2,d = 30.

This letter proposes a novel FTC design based on the necessary and sufficient conditions for the existence of MB-ESO and the mismatched disturbance decoupling for a quadrotor experiencing two opposing rotor complete failures. Simulation results demonstrate that the proposed controller outperforms the well-known INDI against disturbances caused by significant aerodynamic effects in extreme conditions.