YU Ziquan, ZHANG Youmin, and JIANG Bin

1.College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China;

2.Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal H3G 1M8, Canada

Abstract: This paper introduces a fault-tolerant control (FTC)design for a faulty fixed-wing unmanned aerial vehicle (UAV).To constrain tracking errors against actuator faults, error constraint inequalities are first transformed to a new set of variables based on prescribed performance functions.Then, the commonly used and powerful proportional-integral-derivative (PID) control concept is employed to filter the transformed error variables.To handle the fault-induced nonlinear terms, a composite learning algorithm consisting of neural network and disturbance observer is incorporated for increasing flight safety.It is shown by Lyapunov stability analysis that the tracking errors are strictly constrained within the specified error bounds.Experimental results are presented to verify the feasibility of the developed FTC scheme.

Keywords: unmanned aerial vehicle (UAV), fault-tolerant control(FTC), prescribed performance control (PPC), proportional-integral-derivative (PID), composite learning, actuator faults.

1.Introduction

Recently, unmanned aerial vehicles (UAVs) have been widely used to perform dangerous and tedious tasks, due to their high flexibility and large flight radius [1−4].However, with unexpected flight scenarios, various faults encountered by a UAV may significantly degrade the maneuverability performance or even cause catastrophic accidents.To react to faults, the fault-tolerant control(FTC) concept is investigated for increasing flight safety[5−10].In general, FTC methods can be classified into passive and active methods.With respect to the passive FTC, the robust control strategy is usually used.Regarding the active FTC, fault detection and diagnosis (FDD)is artfully incorporated into the FTC architecture to compensate for the faults [11,12].By using such a strategy,the faulty system can be stabilized in a timely manner.

By using terminal sliding mode control (TSMC) architecture and timescale separation principle involved within the faulty attitude dynamics, a passive FTC scheme was proposed in [13] for a hypersonic vehicle.Similarly,by considering the loss-of-effectiveness actuator faults and uncertainties, and using sliding-mode control (SMC)method, an FTC scheme was designed in [14] for nonlinear systems.In [15], to accommodate the actuator faults,an active FTC scheme was artfully investigated for electric vehicles by integrating a baseline controller, a set of reconfigurable controllers, and a decision mechanism.By simultaneously considering the actuator faults and constraints, an active FTC scheme was developed in [16] to obtain reliable FTC performance.To achieve a safe flight against actuator faults, numerous FTC methods have been designed for UAVs.To utilize the robustness inherent in nonsingular TSMC (NTSMC), a finite-time FTC scheme was designed for a quadrotor UAV by constructing an NTSMC-based inner controller and an NTSMC-based outer controller [17].By using neural network to learn the fault-induced terms in the backstepping control architecture, a fast FTC method was developed in [18] for UAVs.To counteract the faults and wind disturbances, an FTC was developed in [19] by incorporating fractional-order concept and composite learning algorithm.In [20], highorder sliding-mode differentiator and intelligent learning mechanism were integrated to design the FTC scheme for UAVs.More recently, to increase flight safety, a neural network disturbance observer-based finite-time FTC scheme was developed in [21].

Prescribed performance control (PPC) was initially developed in [22] for constraining tracking errors with predefined error bounds and convergence rates.Recently,various PPC methods have been reported [23−27].In[24], a PPC scheme was constructed for pure feedback systems.By combing dynamic surface control and adaptive fuzzy logic, a PPC method was proposed in [26] for constraining the tracking errors.Regarding the PPC of UAV, the authors in [28] proposed a robust PPC scheme for UAVs against payloads.Recently, the PPC concept has been integrated into FTC architecture for further constraining tracking errors when UAV was encountered by actuator faults.In [29], an active FTC strategy was investigated with the integration of fault diagnosis component, such that the prescribed attitude tracking error requirements can be satisfied.Recently, the PPC was further incorporated into the finite-time FTC structure for fixed-wing UAVs to significantly enhance the FTC performance [30].Although massive FTC methods are effective in handling actuator faults, more effective FTC schemes should be developed towards reliable flights of fixed-wing UAVs.

Motivated by the aforementioned analysis, a proportional-integral-derivative (PID)-type fault-tolerant PPC scheme is proposed for a UAV against actuator faults.First, the tracking errors are converted into a new set of errors using the prescribed performance functions.Then,a PID-type error filter is further adopted to transform the errors.To handle the unknown terms due to actuator faults, a composite learning algorithm with the integration of neural network and disturbance observer is utilized to facilitate the FTC design.Different from existing works, the distinct features of this paper are listed as follows:

(i) In contrast to the FTC methods for UAVs developed in [31−33], which designed the FTC scheme without the consideration of error constraints, PPC is introduced to strictly constrain the tracking errors against actuator faults.

(ii) Different from the composite learning algorithms presented in [20,34], this paper adopts the PID-type error dynamics to design the composite learning algorithm,such that the FTC performance can be significantly enhanced.

(iii) Regarding the massive FTC schemes, which mainly used simulation scenarios to verify the feasibilities of FTC methods, experiments are performed in this paper to show the effectiveness of the FTC scheme.

The remaining structure of this paper is organized as follows.In Section 2, UAV dynamics and fault models are presented.The main result including prescribed performance error transformation, development of composite learning algorithm, and the PID-type FTC design is given in Section 3.Section 4 presents the experimental results.Finally, Section 5 concludes the whole work.

2.Preliminaries and problem statement

2.1 Fixed-wing UAV dynamics

The dynamics of the fixed-wing UAV are as follows [35]:

where µ, α , and β are the bank angle, angle of attack, and sideslip angle, respectively.χ and γ represent the heading angle and flight path angle, respectively.p,q, andrdenote the angular rates.L, M , and N are the roll, pitch,and yaw moments.c1,c2, ···,c9represent the inertial terms [36].

wheremrepresents the mass.gis the gravity acceleration.LandYare the aerodynamic lift and side forces, respectively.Vis the velocity, which is updated by the following dynamics:

whereTandDdenote the thrust and aerodynamic drag forces, respectively.

The aerodynamic forcesL,D,Yand the moments L,M, N can be expressed as

where=0.5ρ0V2is the dynamic pressure with ρ0being the air density.s,b, andcare the wing area, wing span,and mean aerodynamic chord, respectively.δa, δe, and δrrepresent the aileron, elevator, and rudder deflection angles, respectively.The symbolsCL0,CLα,CD0,CDα,CDα2,CY0,CYβ,Cl0,Clβ,Clδa,Clδr,Clp,Clr,Cm0,Cmα,Cmδe,Cmq,Cn0,Cnβ,Cnδa,Cnδr,Cnp, andCnrrepresent the aerodynamic coefficients.

whereu=[δaδeδr]Trepresents the control input vector.,g1(see (10)), andare expressed as

withd1andd2being chosen as

2.2 Faulty UAV model against actuator faults

In this paper, the loss-of-effectiveness and bias faults are considered for the aileron, elevator, and rudder actuators of the fixed-wing UAV.The faulty actuator model is expressed by

where ρ=diag(ρ1,ρ2,ρ3) with 0<ρ1,ρ2,ρ3≤1 represents the remaining effectiveness matrix.uandu0=[δa0δe0δr0]Trepresent the applied and commanded input vectors, respectively.is the bounded bias fault vector.

By substituting (13) into (7), one can obtain the following faulty UAV model:

2.3 Control objective

The control objective is to design the control inputu0for the faulty UAV models (14) and (15), such that the attitude vectorx1can track the desired attitude vector against actuator faults, and the tracking errors are strictly constrained within the prescribed error bounds.

3.Main results

In this section, prescribed performance functions and a PID-type error filter are first used to transform the errors.Moreover, to counteract the faults, a composite learning algorithm with the neural network and the disturbance observer is artfully proposed for enhancing flight safety.

3.1 Prescribed performance-based error transformation

Define the desired attitude vector asx1d=[µdαdβd] and the attitude tracking error as,then one has

One can further obtain

To constrain the attitude tracking error, the following error constraints are first introduced:

wherei=1,2,3,ki>0 andki>0 are design parameters.andare the prescribed lower and upper bounds, respectively.εi(t) is the prescribed performance function, given by

where εi0and εi∞represent the initial and final values of the prescribed performance function.ηidenotes the allowable convergence rate.

By using the prescribed performance function (19), the prescribed error bounds at the initial time and the steadystate phase can be described asandrespectively.The parametersandkiare set to satisfy, whereis the initial value,i=1,2,3.

To design the PPC scheme, the error constraint inequality (18) is changed to the following equality:

wherei=1,2,3 ,s1iis the transformed error.Λi(·) is the transformation function with the following properties:

In this paper, Λi(·) is chosen as

Then, one has

Taking the time derivative of (22) yields

By writing (23) into the compact form, one has

wheres1=[s11s12s13]T, ξ=diag(ξ1,ξ2,ξ3) , and ε=diag(ε1,ε2,ε3).

3.2 Composite learning-based PID fault-tolerant prescribed performance control (PIDFTPPC)design and stability analysis

In this subsection, the prescribed performance error (22)is first transformed by using the PID-type filter, and then a composite learning algorithm is developed to handle the fault-induced nonlinear terms within the PID-type error dynamics by integrating the neural network and the disturbance observer.

Taking the time derivative of (24) gives

Introduce the following PID-type error filter:

wherek1andk2are positive constants, such that the transfer functions2+2k1s+is Hurwitz.

By considering (24), (25), and (26), one has

By substituting (17) into (27), one has

where the strongly nonlinear functionFis expressed as

In this paper, the composite learning algorithm presented in [34] is used to approximate the strongly nonlinear functionFby integrating neural network and disturbance observer.From (29), it can be seen thatu0is involved inF, which causes the algebraic loops.To address this, the Butterworth low-pass filter technique is further used to break the algebraic loop by filtering the signalu0within (29) before sending it to the composite learning algorithm, such thatF=Fϱ+ef, whereFϱis the radial basis function neural network (RBFNN),efis the bounded filter error [37].

Introduce the error prediction Υ=e−, whereis updated by the following expression:

whereFϱ=W*Tφ+ϵ is used in (30),W*and φ represent the optimal weighting matrix and the Gaussian function vector, respectively.k3>0 is a positive parameter.ϵ represents the bounded approximation error vector.D=ef+ϵ is the lumped error.andrepresent the estimations ofW*andD, respectively.

Based on the prediction error Υ=e−, the following disturbance observer is developed to estimateD:

wherek4>0 is a positive parameter.

wherek5>0,k6>0 , andk7>0 are positive parameters.

Design the control signalu0as

wherek8is a positive diagonal matrix.

The time derivative of Υ =e−is

Similarly, one has

To this end, the proposed FTC scheme can be summarized as Fig.1.

Fig.1 Structure of the developed FTC scheme

Theorem 1Consider the UAV systems (1) and (2)against actuator faults, if the FTC scheme is developed by the prescribed performance error transformation (22), the PID-type error filter (26), the disturbance observer (31),the neural adaptive law (32), and the control signal (33),then the attitude vector of the fixed-wing UAV can reach to the desired attitudes.Moreover, the attitude tracking errorsand errorsare uniformly ultimately bounded (UUB).The attitude tracking errorsare strictly confined within the prescribed error bounds.

ProofChoose the following Lyapunov function candidate:

Differentiating (36) with (28) yields

By using the following inequalities:

one has

Equation (39) can be further transformed to the following expression:

where ζ1and ζ2are respectively expressed as

Remark 1In the previous work [34], a composite adaptive FTC scheme was proposed for UAVs with prescribed error requirements.The disturbance observer and neural network are designed to learn the fault-induced nonlinear terms within the prescribed performance error dynamics.In this paper, a PID-type error filter is further introduced for improving the FTC performance, and then the composite learning algorithm presented in [34] is modified to learn the fault-induced unknown terms within the PID-type error dynamics.

4.HIL experimental results

In this section, hardware-in-the-loop (HIL) experiments are performed for showing the effectiveness of the proposed PIDFTPPC scheme.The structural parameters and aerodynamic coefficients of the fixed-wing UAV can be referred to [35].The developed HIL testbed is shown in Fig.2, which has been used to verify the feasibility of the control scheme presented in [21] and consists of an opensource Pixhawk 4 autopilot hardware with the STM32F765 processor and a mobile workstation Thinkpad P52.In the HIL experiment, the Pixhawk 4 autopilot hardware and the P52 workstation are used to run the FTC algorithm and the fixed-wing UAV dynamics, respectively.

Fig.2 HIL testbed

The design parameters are chosen ask1=20 ,k2=11.6 ,k3=2 ,k4=1.5 ,k5=3.2 ,k6=10 ,k7=0.4 ,k8=diag(20,20,20) , η1=0.2 , η2=0.2 , η3=0.2,=0.6,=0.6 , ε10=1.72,ε1∞=0.57 , ε20=4.58 , ε2∞=2.29 , ε30=1.72 , and ε3∞=0.29.To show the superiority of the proposed PIDFTPPC scheme, the comparative backstepping control (BSC)scheme is adopted by removing the prescribed performance functions, PID-type error filter, and composite learning algorithm.

Fig.3 illustrates the response curves of the bank angle,angle of attack, and sideslip angle of the fixed-wing UAV under the PIDFTPPC and BSC schemes.By using the PIDFTPPC strategy, the states can reach to the desired attitude references even under the initial tracking errors and the fixed-wing UAV is encountered by the aileron, elevator, and rudder faults att= 15 s, 30 s, 45 s sequentially.From Fig.3(b), it is observed that large angle of attack deviations are caused by employing the BSC scheme.

Fig.3 Reseponse curves

Fig.4 presents the attitude tracking errors=µ−µd,=α−αd, and=β−βdunder the PIDFTPPC and BSC schemes.At the beginning of the HIL experiment,the initial bank angle, angle of attack, and sideslip angle tracking errors are 0.57°, −1.43°, and 0.57°, respectively.Then, with the developed PIDFTPPC scheme, these initial tracking errors are pulled into the very small region containing zero.When the aileron, elevator, and rudder actuators of the fixed-wing UAV are abruptly injected by the faults att= 15 s, 30 s, 45 s, respectively, slightly degraded performance is induced.Then, the composite learning unit is activated to compensate for the faults,such that the errors are reduced and flight safety can be enhanced.Moreover, with the help of the incorporated prescribed performance functions, the attitude errorsandare strictly confined within the prescribed boundsrespectively.However, large tracking errors occur when the comparative BSC scheme is used to steer the fixed-wing UAV to track its desired references.It can be seen from Fig.4(b) that the angle of attack tracking errors are outside of the prescribed lower bound.

Fig.4 Attitude tracking erros

Fig.5 presents the aileron, elevator, and rudder deflection angles under the PIDFTPPC and BSC schemes.By using the PIDFTPPC scheme, the control inputs, i.e., the aileron, elevator, and rudder deflection angles, adjust the signals to attenuate the adverse effects caused by the faults att= 15 s, 30 s, 45 s.It can be observed that the comparative BSC scheme has a weak adjustment capability to react to the actuator faults, leading to weaken FTC performance, which can be seen from the Fig.3 and Fig.4.

Fig.5 Aileron, elevator and rudder deflection angles

5.Conclusions

In this paper, a PIDFTPPC scheme has been developed for a fixed-wing UAV.The prescribed performance functions and PID-type filter are integrated to convert the attitude tracking errors and then a composite learning algorithm with neural network and disturbance observer has been developed to compensate for the fault-induced nonlinear terms.Lyapunov stability analysis has shown that the tracking errors are uniformly ultimately bounded and thoroughly confined within the specified ranges.Comparative HIL experiments have been conducted to show the superiority of the proposed control scheme.