Neural-Networks Control for Hover to High-Speed-Level-Flight Transition of Ducted Fan UAV With Provable Stability

In this paper, we focus on the transition control of a ducted fan vertical take-off and landing (VTOL) unmanned aerial vehicle (UAV). To achieve a steady transition from hover to high-speed flight, a neural-networks-based controller is proposed to learn the system dynamics and compensate for the tracking error between the aircraft dynamics and the desired dynamic performance. In prior, we derive the nonlinear system model of the aircraft full-envelope dynamics. Then, we propose a novel neural-networks-based control scheme and apply it on the underactuated aircraft system. Key features of the proposed controller consist of projection operator, state predictor and dynamic-formed adaptive input. It is proved and guaranteed that the tracking errors of both state predictor and neural-networks weights are upper bounded during the whole neural-networks learning procedure. The very adaptive input is formed into a dynamic structure that helps achieve a reliable fast convergence performance of the proposed controller, especially in high-frequency disturbance conditions. Consequently, the closed-loop system of the aircraft is able to track a certain trajectory with desired dynamic performance. Satisfactory results are obtained from both simulations and practical flight test in accomplishing the designed flight course.


I. INTRODUCTION
In recent years, vertical take-off and landing (VTOL) unmanned aerial vehicles (UAVs) have attracted more and more attentions in transportation, surveillance, detection and many other areas. Such a UAV is required to capable of VTOL, steadily hover and long-endurance high-speed flight. To meet this need, one preference is the ducted fan UAV. Ducted fan is well known for its compact layout which enables the vehicle working safely in crowed urban environments such as warehouses and alleys. During the past decades, many classical prototypes of ducted fan, like iSTAR [1], HoverEye [2] and GTSpy [3], have been invented and applied successfully. The typical configuration of a ducted fan contains one duct-fan and four control surfaces.
The associate editor coordinating the review of this manuscript and approving it for publication was Jiankang Zhang . To this layout, the fan is shrouded by a solid wall that 'ducted fan' is hence named. Comparing to a single rotor, a ducted fan features higher aerodynamic efficiency as the fan tip vortex is strongly blocked and weakened by the duct wall. Basically, fixed aerodynamic flaps and deflectable control surfaces are placed below the fan. Being exposed in the flow filed of the air exhausted from the fan, the fixed flaps collectively generate an anti-torque moment counteracting the fan torque, while the control surfaces create attitude control moments. Similar to a helicopter or a quadrotor, this typical configuration results in an underactuated property of the ducted fan dynamic system.
One important characteristic that distinguishes ducted fan from rotorcraft is the high-speed level flight capacity. Notice that, for rotorcrafts like helicopters or quadrotors, flying at high-speed (>10m/s) requires a large pitching angle of the rotor disc, which results in more power consumption than hover. Thus, this high-speed flying status is generally VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ considered as transient maneuvering for these rotorcrafts, not an economical working point. On the contrary, when flying horizontally at high-speed, a ducted fan is subject to an aerodynamic lift which is considerably larger than the associated drag. It is hence indicated that the power consumption in this high-speed-level-flight condition is less than that in hover. This special aerodynamic property can remain effective on a single duct-and-fan structure, even without a wing. Therefore, similar to a fixed-wing airplane, a ducted fan is capable of conducting long-distance missions with high-speed economical cruising. For instance, to the ducted fan studied in this paper, the power rate of steady flight at 20m/s is less than 1/3 of that in hover, and the aircraft is functioned of 15min hovering or 27km flight radius (54km maximum range). It is of great significance to study the transition control for hover to high-speed flight of ducted fan UAV. Till now, most previous works on this aircraft are concentrated on nearhover and low-forward-speed control. At near-hover point, the aircraft dynamics can be simplified into linear system with underactuated property [4]. In [5], a cascade PID controller is designed to achieve good performance in simulation. More complex control structures are adopted to handle the nonlinearity with good performance in practice [6], [7]. Taking consideration of uncertainty and disturbance, adaptive control schemes [8]- [10] are proposed for better control accuracy and disturbance rejection performance. However, it is challenging to design a proper controller for the hover to high-speed transition process. On one hand, as the relevant system values suffer great changes from hover to level flight, the equilibrium point of the aircraft dynamic system encounters a considerable deviation with highly nonlinearity. On the other hand, as the forward speed increases, the aerodynamic effects on the control surfaces are enhanced and the aircraft becomes more and more sensitive to input noises as well as relevant disturbances.
Till now, this transition control is mostly studied for tailsitter VTOL UAV. In [11], [12], a ducted-fan-pushed fixedwing tail-sitter UAV is studied. An L1 adaptive controller is applied on its transition control based on linearized model. In [13], a vectored-thrust tail-sitter UAV is studied, the system model of which is to an extent similar to the ducted fan. In this work, a robust controller is proposed to tackle the complexity of aerodynamic effects and conduct the transition process. In contrast to a typical tail-sitter configuration, a ducted fan without a wing features highly aerodynamic coupling and unstable properties. Thus, for the transition control of a ducted fan UAV, it is essential to deal with the nonlinearity and uncertainty of the aircraft system. Neural networks (NNs) have been widely employed to aircraft control for its excellent performance in nonlinearity matching and uncertainty compensation [14]- [18]. In [19], NNs are introduced to learn the uncertainties online and control a small quadrotor. In [20], a NNs based robust adaptive controller is proposed to identify the inertia matrix of a small helicopter. In [21], NNs are introduced to enhance the performance of a dynamic inversion control scheme.
Another important issue taken into account is the feasibility for implementing the designed controller. In terms of controller design of a rotorcraft, a cascade PID controller is widely used for its convenient in practical parameter tuning at hover. However, controller design for a fixed-wing at high-speed flight takes much more efforts. Generally, taking safety into consideration, the parameters of this controller are directly tuned on the real aircraft in a wind tunnel, or pretuned on a mathematical model which is acquired through wind tunnel tests or computational fluid dynamics (CFD). Unfortunately, all of these methods are complicated and costly on both money and time. Therefore, the basic idea for the proposed controller in this paper is to build a reference system, which is easy to tuned such as cascade PD controller, and use NNs to learn and compensate for all the uncertainties between the real system and the reference system. This control structure is of great advantage in practical implementation.
One core objective for NNs design is fast convergence of the weights and reconstruction errors. In [22], an experience replay based learning algorithm is proposed to obtain fast learning performance with estimation error uniformly ultimately bounded (UUB). However, the UUB property cannot guarantee the transient performance in the early stage of parameter adaptation. In [23], a neural learning controller is proposed to improve this transient performance on controlling marine surface vessels. In order to achieve high tracking accuracy with fast uncertainty matching, a state predictor is introduced into the control scheme in [24], in which the NNs update from both tracking error and prediction error. Generally, enlarging the learning rate of the weights contributes to the increasing of estimation convergence rate of NNs. But this may also lead to great system oscillation or even an overshoot problem in the early adaptation process. To deal with this issue, projection operator is invoked into the weights-tuning law that constrains the NNs weights into certain compact sets. As a result, the NNs with projection operator show excellent performance in the reconstruction of time-varying disturbance and uncertainties [25]- [28].
Based on the concerns above, we propose a novel neuralnetworks-based control scheme. The controller is constructed to track a reference system with tracking error learned and compensated by NNs. Projection operators and state predictors are introduced into the NNs learning algorithm to achieve fast convergence performance. Additionally, we design an adaptive control law with a dynamic form to enhance the robustness of the closed-loop system. To show the advantage of the proposed method, a comparison between the proposed controller and standard approaches is given in Table 1.
In this paper, we study the transition control of a small ducted fan UAV with the typical one duct-fan and four control surfaces layout (See Fig.1 and Fig.2.). Additionally, the aircraft is installed with an extra front wing for balancing the  pitching moment. We first analyze the aerodynamic characteristics of the ducted fan model during the hover to highspeed transition process. Then a novel neural-networks-based control scheme for nonlinear system is proposed. Based on these, a cascade flight control strategy is designed for the transition control of the ducted fan UAV. The paper is organized as follows: the aerodynamic analysis and system modeling of the ducted fan studied in this paper are presented in Section II. In Section III, a neural-networks-based controller for nonlinear system is proposed with a nominal design and its proof of stability. In Section IV, based on the proposed NN controller, a cascade flight strategy is designed for the aircraft to track certain reference inputs with desired dynamic performance. Then simulations and flight test results are respectively shown in Section V and Section VI. Finally, Section VII draws the main conclusion.
The main contributions of this paper are summarized as follows: 1) A novel neural-networks-based control scheme for nonlinear system is proposed. To track a reference system, NNs and adaptive control law are designed to eliminate the tracking error between the original system and the reference system. With the introduction of projection operators, state predictors and high-order adaptive control law, the closed-loop system achieves a good robustness with fast convergence of the NN weights. The stability of the proposed controller is theoretically proved. 2) The proposed NNs control scheme is successfully applied on the transition control of a ducted fan UAV. Both simulation and practical flight test results verify the effectiveness and reliability of the proposed method. Additionally, we give a comparison result between the proposed method and an adaptive method mentioned in [29] in order to reveal the advantage of the proposed method.

II. AIRCRAFT SYSTEM MODELING
In this section, the 6 degree-of-freedom (DOF) motion dynamics of the ducted fan is derived from Newton and Euler Theorems. Then corresponding aerodynamic characteristics are presented and analyzed particularly for the transition process. Although neural-networks control is usually model free, a precise and reliable mathematical model is necessary and contributes to the follows: • To make the simulations more convincible and help choose the proper basis functions of the proposed NNs.
• To deeply indicate the advantage of using neuralnetworks by showing the complexity of the system model in transition.

A. 6-DOF KINEMATICS
Basically, we describe the 6-DOF motion of a rigid body by adopting two reference frames in Cartesian coordinates.
{X E , Y E , Z E } denotes the inertial frame defined by northeast-down. {X b , Y b , Z b } denotes the body frame attached to the aircraft. Position in the inertial frame is denoted as p = [x y z ] T . Velocity in the inertial frame and the body frame are respectively denoted as In this paper, we adopt Euler angles ξ = [ϕ θ ψ ] T to describe the aircraft attitude with 'ZXY' rotation order (See Fig.3). ω = [p q r ] T denotes the angular velocity with respect to the body frame. Based on the 'ZXY' rotation order, rotation matrix R from body frame to inertial frame and the derivatives of Euler angles are given by: where C, S, and T respectively denote cosine, sine and tangent functions for short. Remark 1: In this paper, we adopt 'ZXY' rotation order to define Euler angles rather than 'ZYX' which is commonused in attitude description. During the transition, since the aircraft pitches and the roll angle ϕ is around zero, Q defined by (1) is always invertible and far away from singularity (i.e. ϕ = 90 • ). This helps prevent the gimbal lock problem of a tail sitter UAV.
Velocity of environmental wind disturbance is denoted as D w = [D x (t) D y (t) D z (t) ] T in the body frame. Subsequently, we define airspeed V w , angle of attack (AOA) α and sideslip angle β by: F and M respectively denote the resultant force and moment, exclusive of gravity, in the body frame: where all the components are discussed below.
The fan is generally regarded as a disc acting a steady thrust and torque on the aircraft. Hence, according to Newton and Euler's Law of Motion, the dynamics of the aircraft is derived as follows: where m denotes the mass and J = diag(J x , J y , J z ) denotes the inertia matrix of the aircraft. g denotes the gravity acceleration. 'sk' denotes the skew symmetric matrix to a vector. The relevant definition in R 3 is given by: Basically, as a special type of tail-sitter VTOL aircraft, a ducted fan features the same as a rotorcraft at hover or vertical flight, and acts more like a fixed-wing airplane when levelly flying with high-speed [30], [31]. In this part, we give the detailed expressions of all the forces and moments in (3).

1) DRIVING FORCE AND MOMENT
Thrust F T is regarded as the driving force of the aircraft that can be expressed as: where denotes the fan rotation speed. k T (V w , α) is the thrust coefficient.
When the fan is operating, remote airflow of speed V c is ingested into the fan and accelerated to V e when exhausted from the fan. This exhaust velocity V e can be derived from the momentum theory of the duct-fan system: where V i denotes the induced velocity of the local flow at the fan. The area of the fan disc is denoted as A. Moment M cs created by the control surfaces is regarded as the driving moment of the aircraft. Its expression is given by: where c i denotes the deflection angle of the i-th control surface. k cs is a constant coefficient. l 1 is the lever arm with respect to roll/pitch axis and l 2 is the lever arm with respect to yaw axis.

2) DUCT EFFECTS
Momentum drag F m , duct body force F d and duct aerodynamic pitching moment M d are important factors that differ the ducted fan from a single exposed rotor (e.g. a helicopter) or a pure solid shape (e.g. a wing) [32]. They are presented on 2-D plane in Fig.4. Momentum drag is a reaction to momentum increment of the air component orthogonal to the fan axis [33]. Additionally, with relative motion to a fluid, the duct shape is subject to a body force which is proportional to the square of the airspeed.
where coefficient γ m is introduced as damping rate. V w is the airspeed defined by (2). k Ld ( , α) and k Dd ( , α) are duct lift and duct drag coefficients. Duct aerodynamic pitching moment M d is a result from asymmetric pressure distribution on the duct body [33],  mainly on the duct lip. Basically, this moment can be expressed as [3]: where ε m and ε d denote the lever arms of momentum drag and duct body force, respectively. The mathematic expression of this moment consists of both proportional and quadratic terms with respect to airspeed V w , the coefficients of which are complicatedly coupled with AOA α and fan rotation speed [34]. To demonstrate the complexity of this moment, CFD results for the ducted fan studied in this paper are shown in Fig.6, in which positive moment and negative moment correspond respectively with nose up and nose down of the aircraft.

3) FRONT WING EFFECTS
Aerodynamic effects on the front wing consist of wing force F w and wing moment M w . First, we defined the local airspeeds and local AOAs in associated with the left wing and the right wing (See Fig.5).
where l w is the lever arm from the aerodynamic center of the left/right wing part to the Z b axis. Wing force F w is the resultant force of the lifts and drags: where k Lw (α) and k Dw (α) are lift and drag coefficients of the half wing part which are functions of AOA.
Wing moment M w is given as: where k Mw (α) is moment coefficient of the half wing part which is a function of AOA.

4) OTHER TERMS
As for the rest terms of (3), fan torque M fan , anti-rotation torque M f generated by the fixed flaps, and fan gyroscopic effect M g are given as: where k Q (V w , α) is the torque coefficient. k f is a constant coefficient and J fan denotes the inertia of the fan.

III. NOMINAL NEURAL NETWORKS DESIGN
In this section, a novel neural networks control strategy for nonlinear system is proposed. First, we divide the system dynamics into two parts. One is completely known and formulated into a reference system containing the reference inputs and the desired dynamics. The other part consists of all the uncertainties. Then neural networks are introduced to reconstruct the unknown system dynamics. In terms of NNs learning algorithm, projection operators and state predictors are invoked. To track the reference system, adaptive control law with a dynamic form is designed to cancel the tracking errors. Consider the following nonlinear system: where x = x(t) ∈ R n , u = u(t) ∈ R m (with n ≥ m), respectively denote the state vector, and the control input vector; Functions f : R n → R n , g : R n → R n×m , are continuously differentiable. Taking consideration of unknown system dynamics and parameter uncertainties, we divide system (14) into the following two parts: (15) wheref(x) andḡ(x) are completely known or measurable whilef(x) andg(x) contain all the uncertainties and measurement errors. Correspondingly, the control input is divided into a nominal termū and an adaptive termũ. Then, we can determine the desired dynamic performanceẋ m and nominal inputū to form a reference system such that: With the definitions (15)(16), we can rewrite the nonlinear system (14) into: In prior, we introduce an auxiliary system to handle the system uncertainties. First, we reconstruct the uncertain terms f(x) ,g(x) with parameterized neural networks: where W f ∈ R N f ×n and W g ∈ R N g ×n denote the unknown weight matrices. f : R n → R N f , g : R n → R N g ×m denote the proper basis functions with N f , N g denoting the numbers of neurons. Subsequently, the nonlinear system (14) can be reformed into the following: With this parameterization, the system uncertainty corresponds to the uncertainty of weight matrices. Based on the parameterized system, we design a state predictor to estimate the dynamics of the original nonlinear system: where K x = diag(K x1 , · · · , K xn ) is a positive definite gain matrix andŴ f ,Ŵ g are the estimations of W f , W g . The objective is that the predictor (20) tracks the nonlinear system dynamics (19). To achieve this, the following evolutions are designed:Ẇ where f = diag( f1 , · · · , fn ) and g = diag( g1 , · · · , gn ) are constant learning rate matrices which are positive definite. Proj denotes the projection operator of matrix. f ⊂ R n×N f , g ⊂ R n×N g are convex compact sets thatŴ f ∈ f ,Ŵ g ∈ g . See Appendix A for detailed definitions and properties of projection operator. Combining (20) with (21), an auxiliary system is established as [35]: Recalling (17), In order to track the reference systemẋ m , specific adaptive inputũ is to be adopted such that e = 0. On the first attempt, we can simply choose the adaptive term denoting the pseudo-inverse of non-square matrix. However, this control law is not expressed in affine and the inverse calculation could makeũ too sensitive to the noise ing(x) in some special cases, and probably results in unacceptable oscillation or divergence of the closed-loop system. Instead, we design the adaptive control inputũ by a dynamic form as: where u = diag( u1 , · · · , un ), ui > 0 is a constant gain matrix. Theorem 1: The states and dynamics of auxiliary system (22) converge to those of original nonlinear system. And the state prediction errorx =x − x is upper bounded by: (24) Proof: The proof of Theorem 1 is given in Appendix B. Theorem 2: The dynamics of the closed-loop system with auxiliary system (22) and control input (23) tracks the desired dynamic performanceẋ m with tracking error e → 0.
Proof: The proof of Theorem 2 is given in Appendix C. Remark 2: Known partsf(x) andḡ(x) can be arbitrarily chosen, usually constant to makeẋ m linear. Even if we remain them as nonlinear functions, there exists a certain inputū such thatẋ m can be linearized through feedback linearization technique. It is worth pointing out that more information about these two parts helps achieve more satisfactory control performance.
Due to the introduction of projection operator, it is guaranteed that the tracking errors of both state predictor and NNs weights are upper bounded during the NN learning procedure. According to Theorem 1, the upper bound of the state prediction error (24) is exclusive of time. This implies a stronger property than UUB that the transient performance of these tracking errors maintains upper bounded especially at the early stage of parameter adaption. Consequently, in order to achieve fast convergence of the tracking process, it is reliable to increase the learning rates f , g without the occurrence of parameter overshoot. In other perspective, large values of learning rates result in high-gain feedback control, which, in some case, could cause serious oscillations of the system states and reduce tolerance of time delay. This limits the highfrequency disturbance rejection ability of the closed-loop system. On the contrary, the designed adaptive control input (23) shares the same structure as a low-pass filter that prevents the actuators from vibrating with high-frequency. As a result, the deterioration in robustness resulting from large learning rates is neutralized. This also leads to the attenuation of noises from both states and input signals. In conclusion, the closedloop system with the auxiliary system (22) and adaptive control input (23) is capable of fast parameter convergence with good disturbance rejection performance, and works well under certain noises. In Section V, simulation results are shown in Fig.8, Fig.9 and Fig.10 to demonstrate this property.

IV. FLIGHT CONTROL DESIGN
In this section, aircraft dynamics (4) is reconstructed into a better arranged nonlinear cascade form. Then, we apply the proposed NNs-based control scheme on two cascade closedloop systems. With all the efforts, the aircraft is able to track a certain trajectory from hover to high-speed flight.

A. SYSTEM RECONSTRUCTION
Define the state variables and the regulated system inputs: whereas . T ∈ R marks the thrust input and u a ∈ R 3 marks the regulated moment inputs. The regulated moment inputs are obtained by: where mapping matrix H and deflection angles of control surfaces c are given in (7). Since there are no derivative terms in expression (7), we can conclude that virtual control input u a shares the same response dynamics with practical movements of the actuators. Subsequently, we separate the input terms apart from the dynamic system (4) and integrate the rest into standard nonlinear forms, let: Eventually, we complete the system descriptions by substituting (25)(26)(27) into (4):

B. FLIGHT CONTROL DESIGN
Essentially, the hover to high-speed transition is a process that the aircraft continuously pitches down and accelerates to a VOLUME 8, 2020 high cruising speed. To this end, the objective of controller design is to force the aircraft tracking a certain trajectory that formulates this process.
The first 2 rows of (28) are position and velocity dynamics of the aircraft, which is generally considered as system outer-loop. Meanwhile, the last 2 rows of (28) demonstrate the attitude dynamics of the rigid body, which is generally regarded as system inner-loop. Together, these two loops establish an underactuated cascade structure of the aircraft system. Furthermore, although equations (28) can describe the whole transition process without constructively change, component values of the system vary greatly from the original hover condition to the final high-speed flight condition as aforementioned. In terms of control strategy, we apply the nominal neural networks design in Section III on both innerloop and outer-loop.

1) INNER-LOOP DESIGN
The attitude dynamics is extracted from the last 2 rows of system (28): Note that the input signals are coupled with both thrust input T and moment inputs u a . In this paper, we only consider u a to be the control inputs of the inner-loop while a relevant neural network is to be designed to estimate and compensate for the effect of T . Meanwhile, Since Q is measurable and consist of no uncertainty, we only introduce neural networks to reconstruct the dynamics of angular velocity x 4 : In practice, neither f 2 , the aerodynamic moment, nor g 2 , the efficiency of control surfaces is measurable. In this case, let:f 2 = 0,ḡ 2 = g 2h = const whereas g 2h is the value of g 2 (x, T ) at hover working point which can be easily determined by experiments. Then, the reference system of x 4 is designed as linear feedback system: where K 4 is a positive feedback gain matrix, and x 4d is the reference input of x 4 . Hence, the nonlinear dynamics of angular velocity x 4 can be rewritten into: Subsequently, the auxiliary system is designed as: Note that g 2 represents the efficiency of the control surfaces which is positive definite. Thus, the application of (23) on adaptive control inputũ a can be simplified as: Eventually, we complete the inner-loop into PD controller by defining the reference system of x 3 : where K 3 is a positive feedback gain matrix. The notation ' (.)' denotes a special operator for a vector x ∈ R 3 that: Theorem 3: To the closed-loop system for (29) with control strategy of (31)(33)(34)(35), the state x 3 (Euler angles) tracks the reference input x 3d with desired dynamicsẍ 3m and tracking error goes to zero with time.
Proof: The proof of Theorem 3 is given in Appendix D.

2) OUTER-LOOP DESIGN
By the accomplishment of the inner-loop, the Euler angle x 3 always tracks the reference input x 3d with bounded tracking errorx 3 which converges to zero. In terms of outer-loop design, we consider x 3 , T as the inputs to the translation dynamics, and x 3d , u 1 as the inputs to the dynamic system of x 3 , T . The tracking errors between x 3d , u 1 and x 3 , T are respectively denoted asx 3 ,T , which are considered as input uncertainties of the translational dynamics.
where k T 0 is a constant thrust coefficient that could be arbitrarily chosen. In this paper, k T 0 is determined as the value of k T at hover condition which is easily acquired by experiments. Subsequently, the translation dynamics is extracted from the first 2 rows of system (28): Note that the gravity-excluded resultant force acting on the aircraft can be completely measured and calculated by an IMU sensor in practice. Based on this, different from the inner-loop design, the main issue for outer-loop design is to compensate for the input uncertainties defined by (37). Let: where ϒ = [ϒ 1 ϒ 2 ϒ 3 ] T denotes the acceleration due to the gravity-excluded resultant force which are completely measurable. In terms of the rotation matrix g 1 (x 3 ), we divide it into pure u 3 -related terms and uncertainty containing terms by using the first-order Taylor expansion. First, notice that there always exists a NN that: where the notation ' (.)' is pre-defined by (36). Then we reconstruct the translation dynamics (38) into: where the reference system with desired dynamicsẋ 2m for accelerationẋ 2 is given by: And the notation ' ' denotes a special operator for a vector x ∈ R 3 that: Subsequently, the auxiliary system is designed as: To complete the outer-loop controller, the nominal control inputs and adaptive control inputs are given as: where a 2d = [a 2d1 a 2d2 a 2d3 ] T is the desired acceleration vector in the inertial frame, which is obtained by a feedback form of velocity x 2 including reference input x 2d and diagonal positive definite gain matrix K 2 : Since there are no model-linked dynamics on the first row of (38), the controller design ends up with velocity tracking.
Theorem 4: To the closed-loop system for (38) with control strategy of (39)(44)(45)(46)(47), the state x 2 tracks the reference input x 2d with desired dynamicsẋ 2m = a 2d and tracking error goes to zero with time.
Proof: The proof of Theorem 4 is given in Appendix D. It is also implied in Appendix D that the inputs x 3 , T of the translational dynamics tracks the nominal inputsū 3 ,ū 1 defined by (45) with all the input uncertainties eliminated.
The complete control structure is shown in Fig.7. Combining the inner-loop and outer-loop designs, the closedloop system is able to track a certain trajectory which is describe by four independent state variables [x 2d x 3d3 ] = [V xE V yE V zE ψ ] (i.e., velocity in the inertial frame and yaw angle). It is also guaranteed that during the tracking process, the system performance follows a desired reference system. In terms of engineering practice for the aircraft, the reference system is usually determined as linear system formed with PD controllers that the corresponding parameters can be easily tuned at hover. During the hover to highspeed transition process, the system performance always tracks this reference system while encountering momentous changes in component values as well as time-varying disturbance.

V. SIMULATIONS
In this section, simulation results are presented in order to show the performance of the proposed NNs-based controller. The ducted fan UAV studied in its paper is numerically modeled and equipped with the proposed control strategy in Section IV. Dynamic system and its simulation are built and programed in MATLAB TM . The relevant aerodynamic coefficients of the aircraft model are identified by CFD and functional-established by cubic spline interpolation. Then, three different simulations are conduct as follows:

A. SIMULATION 1: HOVER UNDER DISTURBANCE
In the first simulation, we intend to study the effect of parameter uncertainties on the dynamic response of proposed control scheme by verifying the fast convergence capacity of the ducted fan UAV in high-frequency disturbance condition. The aircraft is commanded to steadily hover at 10 meters' height and head to the north (i.e. yaw angle ψ = 0). Time-varying environmental wind disturbance, which is set as cosine function along Y E axis with amplitude of 5m/s, is added to the dynamic model of the aircraft. Simulation results on roll channel under different control parameters are shown in Fig.8, including performance of angular velocity p and NN matching result for exogenous moment effect f 21 .
In this simulation, taking consideration of simplicity, the basis function is chosen as 1 = [1], and the associated learning rate is then 1 = 1 I 3×3 . The associated NN weight W 11 is constrained into the compact set 11 = {−3 ≤Ŵ 11 ≤ 3} by projection operator. Effect of exogenous moment f 21 is the first component of f 2 defined by (27). ua1 is the first component of gain matrix ua of the adaptive control input u a defined by (34). Fig.8 (a) shows the environmental disturbance act on the closed-loop system of the aircraft. Fig.8 (b) shows the performance of angular velocity p in counteracting this disturbance. In Fig.8 (b), the red dash curve marks the reference input x 4d1 , which is the first component of x 4d defined by (25). This reference input is generated from the outer-loop of the controller. The black curve marks the system output merely acquired from the reference system without any influence of the NN controller. The green curve, brown curve, and blue curve mark the corresponding system output under different learning rates 1 and adaptive gains uc1 . Fig.8 (c) shows the NN matching results for exogenous moment effect f 21 . In Fig.8 (c), the red dash curve marks the real value of f 21 which is calculated from the aircraft model. The green curve, brown curve, and blue curve mark the corresponding NN matching results under different parameters. Apparently, increasing the learning rate helps improve the performance for uncertainty identification of the proposed NNs, while the adaptive control input guarantees a competent robustness. It is seen from the blue curve in Fig.8 (c) that the NN weightŴ 11 is limited to 11 as the learning rate 1 increasing. As a result, we achieve an excellent and reliable NN matching ability for unknown terms. Consequently, shown in Fig.8 (b), as the disturbance identification and rejection ability being enhanced, performance for tracking certain reference input of the closed-loop system improves. This reveal an excellent performance and robustness of the proposed control scheme under parameter uncertainties.

B. SIMULATION 2: HOVER TO HIGH-SPEED TRANSITION
In the second simulation, we intend to give an overview of the hover to high-speed transition of the ducted fan UAV. Driven by a certain trajectory, the aircraft is commanded to accelerate at 2m/s 2 along X E axis with altitude stabilized at 10m. This course is set to start at hover and end with horizontal velocity of 20m/s. Since the flight control strategy in Section IV ends up with velocity control, we introduce a simple feedback controller to generate the desired velocity commands along Y E , Z E axis. Thus, the complete driven trajectory is described by: where K p2 and K p3 are constant positive feedback gains.
x 2d1f is the final horizontal velocity and x 1d3 is the , cos α 0 = −w/ |V E |,and the associated learning rate is determined as 1 = diag(2,2,0.1,0.1). It is worth pointing out that these basis functions are particularly determined based on the prior knowledge of f 2 in order to reduce the neurons and show the effectiveness of the proposed method. For more accurate and reliable approximation results, more higher-order terms should be involved into 1 .
The corresponding simulation results are shown in Fig.9. Fig.9 (a)-(c) present the relevant system states during the transition. Throughout the whole process, the aircraft gradually heads down and continuously accelerates to the final velocity and adjusts itself to cruising flight at 20m/s. A more visualized result of the flight trajectory is presented in Fig.9 (d). Additionally, typical trajectories of NN weights are shown in Fig.9 (e)-(h). It is seen that NN weights adapt fast from 0s to 10s and quickly converge after 10s. This due to the fact that the flight data for NN training is not complete until 10s. After this time node, the transition is finished and the aircraft states are stabilized. Hence, as the NN is sufficiently trained, fast convergence for corresponding NN weights are then observed.

C. SIMULATION 3: TRANSITION UNDER DISTURBANCE AND NOISES
In the third simulation, we intend to verity the disturbance rejection ability of the proposed control scheme in the transition control. During this process, sine environmental wind disturbance with amplitude of 5m/s is invoked into the system. To simulate a real IMU sensing result, Gaussian noises are added into the measurements of angular velocity ω and resultant force ϒ in the body frame. The associated transition course is set to start at hover and end with horizontal VOLUME 8, 2020 velocity of 15m/s. The driven trajectory is defined by (48) with final speed x 2d1f = 15m/s and altitude x 1d3 = −10.
Additionally, to show the advantage of the proposed NNs control scheme, we conduct a comparison between the proposed method and the adaptive method for tail-sitter UAV proposed in [29]. The main differences between these two methods are concluded as follows: • We use NNs to estimate the uncertainties in this paper.
While in [29], parameters for estimate the relevant uncertainties are designed based on the prior knowledge of the system model.
• The adaptive input in this paper is designed into a dynamic form to avoid adaptational actions in the denominator. While in [29], the adaptive control law is designed by a standard form with adaptive parameters occurred in the denominator.
The corresponding simulation results including more inner details about the NNs-based controller are shown in Fig.10. Fig.10 (a) shows the environmental disturbance act on the closed-loop system of the aircraft. Fig.10 (b)-(f) shows the transition control results of the proposed method, while the comparison results are given in Fig.10 (g)-(k). Fig.10 (b)(g) show the performance of horizontal velocity V xE . In Fig.10 (b)(g), the red dash curve marks the reference input x 2d1 and the black curve marks the system state V xE = x 21 . Fig.10 (c)(h) show the performance of altitude z. In Fig.10 (c)(h), the red dash curve marks the reference input −10 and black curve marks the system state z = x 13 . The corresponding position feedback controller is defined by (48). Fig.10 (d)(i) show the performance of pitch Euler angle θ. In Fig.10 (d), the red curve marks the reference input which is the nominal attitude inputū 32 defined by (45), and the black curve marks the system state θ = x 32 . The blue curve marks the attitude input u 32 of the outer-loop system defined by (37). Fig.10 (e)(j) show the performance of angular velocity q. In Fig.10 (e)(j), the red curve marks the reference input x 4d2 and the black curve marks the system state q = x 42 . The blue curve marks the corresponding measurement of the state with measurement noise. Fig.10 (f)(k) show the action of thrust. In Fig.10 (f), the red curve marks the reference input which is the nominal thrust inputū 1 defined by (45) and the black curve marks the actual thrust T . The blue curve marks the thrust input u 1 of the outer-loop system defined by (37).
According to the simulation results shown in Fig.10, the closed-loop system of the ducted fan reveals a good disturbance rejection performance. The aircraft is able to follow the desired trajectory to accomplish the hover to high-speed transition. Although suffering great changes on system values, the closed-loop system on attitude control (inner-loop) still exhibits a fast and accurate performance for tracking the reference signals, which is shown in Fig.10 (e). In the meanwhile, it is seen from Fig.10 (d)(f) that the adaptive inputsũ 3 ,ũ 1 of the outer-loop effectively compensate for the input uncertainties and make the actual input track the corresponding reference signals. Consequently, in strongly gusty environment, the aircraft's attitude and thrust is adjusted automatically to counteract the effect of wind disturbance. A good robustness of the closed-loop system is hence revealed.
Comparing the results of the proposed method ( Fig.10 (b)-(f)) with the results of the adaptive method in [29]  ( Fig.10 (g)-(k)), the advantage of the proposed NNs control scheme is obvious. Sever tracking errors of the attitude are observed at high-speed flight in Fig.10 (i)(j). This due to fact that the approximations of the uncertain terms f 2 and g 2 by the adaptive method are not accurate enough comparing with the NNs method in this paper. In the meanwhile, since there are no uncertainty compensations to the outer-loop in the adaptive method, horizontal velocity at high-speed in Fig.10 (g) suffers greater oscillations than that in Fig.10 (b). And the altitude loss of the adaptive method ( Fig.10 (h)) is nearly double of that in the proposed method (Fig.10 (c)). Based on these results, we can conclude that the proposed NNs control scheme guarantees a more steady flight under disturbances and noises.

VI. FLIGHT TEST
In this section, experimental results are presented from practical flight tests (Fig.12). The ducted fan UAV studied in this paper is mounted with inertial measurement unit (IMU) and GPS to measure the relevant system states. Modules including data fusion, state estimation, communication and control scheme are coded into the airborne processor. The controller update and data download frequency is set as 100Hz. Taking environmental disturbance and flight safety into consideration, the transition course is set to start at hover and end with horizontal velocity of 15m/s, similar to that in simulation 3 in Section V. To implement the hover to high-speed transition, we first launch the aircraft and make it hover at an acceptable altitude. Then the driven trajectory defined by (48) with final speed of x 2d1f is sent to the airborne processor to control the aircraft accomplishing a transition process. In terms of implementation, we first choose a small final speed such as x 2d1f = 5m/s to pre-train the neural networks. Convergent weights are set as initial values of the weights in the next move. Continue this step until the final speed gradually increases to the desired one x 2d1f = 15m/s. After that, the NNs are completely trained and a steady flight is then guaranteed.
The corresponding flight test results are shown in Fig.11, in which relevant definitions are similar to that in simulation 3. In Fig.11 (a), the red dash curve marks the reference input x 2d1 and the black curve marks the system state V xE = x 21 . In Fig.11(b), the red dash curve marks the reference input −5.3 and black curve marks the system state z = x 13 . In Fig.11 (c), the red dash curve marks the reference inputū 32 and the black curve marks the system state θ = x 32 . The blue dash curve marks the attitude input u 32 of the outer-loop system. In Fig.11 (d), the red dash curve marks the reference input x 4d2 and the black curve marks the system state q = x 42 . It is seen in Fig.11 (b) that the actual altitude loss is less than 0.4m. Consistent with the simulation results in Section V, these flight test results demonstrate that the aircraft is capable to closely follow the mission trajectory.

VII. CONCLUSION
This paper concentrates on the transition control for hover to high-speed-level-flight of a ducted fan UAV. During this special process, nonlinearity and uncertainty of system values are emphasized as main issues. To tackle the problem, we propose a novel neural-networks-based controller that performs well in identifying uncertainties and tracking reference signals. In prior, key aerodynamic characteristics of the transition process are presented and analyzed. Then we model the dynamic system of the aircraft and reform it into a cascade underactuated nonlinear structure. In the meanwhile, the nominal controller design for nonlinear system based on neural networks is proposed and proved for stability. VOLUME 8, 2020 Subsequently, the proposed control strategy is applied on the ducted fan model. As a result, the closed-loop system of the aircraft is capable of tracking a certain trajectory with desired dynamic performance. Three simulations are conduct in this paper. Simulation 1 shows a fast convergence capacity of the proposed NNs controller by hover experiments under highfrequency disturbance. In simulation 2 and 3 we exhibit the hover to high-speed transition, the results of which show a good robustness of the closed-loop system. Furthermore, this transition process is successfully performed by practice flight test. In all, we conclude that the proposed control scheme is feasible and effective.

APPENDIX A PROJECTION OPERATOR
Definition A.1: Consider a convex compact set with a smooth boundary given by: where f : R n → R is a smooth convex function given by: with θ max being the norm bound imposed on the vector θ , and ε 0 > 0 is the projection tolerance bound of our choice. The projection operator is defined as: if f (θ ) ≥ 0 and ∇f T y > 0 where = S 1 = {θ ∈ R n |f (θ) ≤ 1}. Property A.1: The projection operator does not alter y if θ belongs to S 0 . In the set {θ ∈ R n |0 ≤ f (θ ) ≤ 1}, if ∇f T y > 0, the projection operator subtracts a vector normal to the boundary¯ f (θ) = {θ ∈ R n |f (θ ) = f (θ )}, so that we get a smooth transformation from the original vector filed y to an inward or tangent vector field for [36].
Property A.2: Given the vectors y ∈ R n , θ ∈ and θ * ∈ S 0 ⊂ ⊂ R n we have [36]: In this paper, we extend these definition and property to a matrix form.

APPENDIX B PROOF OF THEOREM 1
For the nonlinear system (14), recall the parameterized system (19): x = f (x) + W T f f (x) + ḡ(x) + W T g g (x) ū +ũ and the auxiliary system (22): Define the tracking errorx =x − x,W f =Ŵ f − W f and W g =Ŵ g − W g . Then, we can derive the error dynamics as: Consider the following Lyapunov function candidate: Then, we calculate the time derivative of the Lyapunov function along the trajectories of the system as: + tr W T g Proj g (Ŵ g , − g ux T ) = tr( fx TWT f ) + tr( g ux TWT g ) −x T K xx Then the equilibrium point of x 3 is resolved fromẍ 3m = 0, which is consequently x 3 = x 3d . This complete the proof of Theorem 3.