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Adaptive observer-based fault detection and active tolerant control for unmanned aerial vehicles attitude system
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ScienceDirect IFAC PapersOnLine 52-24 (2019) 47–52
Adaptive observer-based fault detection and active tolerant control for unmanned Adaptive detection and active tolerant Adaptive observer-based observer-based fault fault active tolerant control control for for unmanned unmanned aerialdetection vehicles and attitude system Adaptive observer-based fault detection and active tolerant control for unmanned aerial vehicles attitude system aerial vehicles attitude system aerial vehicles attitude system CAO Lijia*, TANG Yu*, ZHANG Guo*
CAO Lijia*, TANG Yu*, ZHANG Guo* CAO CAO Lijia*, Lijia*, TANG TANG Yu*, Yu*, ZHANG ZHANG Guo* Guo* * Artificial Intelligence Key Laboratory ofCAO Sichuan Province ; Sichuan Provincial Lijia*, TANG Yu*, Key ZHANG Guo*Research Base of Intelligent Tourism; ** Artificial Intelligence Key Laboratory of Sichuan Province ; Sichuan Key Provincial Research Base of Intelligent Tourism; Sichuan University of Science & Engineering , Yibin, 644000,China Artificial Intelligence Key Laboratory of Sichuan Province ; Sichuan Key Provincial Research * Artificial Intelligence Key Laboratory of SichuanofProvince ;&Sichuan Key,Provincial Research Base Base of of Intelligent Intelligent Tourism; Tourism; Sichuan University Science Engineering Yibin, 644000,China * Artificial Intelligence Key Laboratory of Sichuan Province ; Sichuan Key Provincial Research Base of Intelligent Tourism; (e-mail: [email protected]). Sichuan University of Science & Engineering , Yibin, 644000,China Sichuan University of Science & Engineering, Yibin, 644000,China Sichuan University (e-mail: of [email protected]). & Engineering, Yibin, 644000,China (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). Abstract: A robust adaptive observer combined with radial basis function neural network (RBFNN) is Abstract: A robust adaptive observer combined with radial basis function network (RBFNN) is designed the unmanned vehicles (UAVs) systemneural is proposed this paper. Abstract: A adaptive observer combined with radial neural network is Abstract:for A robust robust adaptive aerial observer combined with fault-detection radial basis basis function function neural networkin(RBFNN) (RBFNN) is designed for the unmanned aerial vehicles (UAVs) fault-detection system is proposed in this paper. Abstract: A robust adaptive observer combined with radial basis function neural network (RBFNN) is Firstly, the fault dynamics model with unknown disturbance of the unmanned aerial vehicle’s attitude designed for the unmanned aerial vehicles (UAVs) fault-detection system is proposed in this paper. designed for the unmanned aerial vehicles (UAVs) fault-detection system is proposed in this paper. Firstly, the fault dynamics model with unknown disturbance of the unmanned aerial vehicle’s attitude designed for the unmanned aerial vehicles (UAVs) fault-detection system is proposed in this paper. system is established, and a robust adaptive observer combined with radial basis function neural network Firstly, the fault model with unknown disturbance of the unmanned aerial attitude Firstly, fault dynamics dynamics with unknown disturbance ofwith theradial unmanned aerial vehicle’s vehicle’s attitude system isthe established, and aa model robust adaptive observer combined basis function neural network Firstly, theestablished, faultthedynamics model with unknown disturbance thecombined unmanned aerial vehicle’s attitude is designed for vehicle’s fault-detection, then, the detected of fault with a robust controller is system is and robust adaptive observer combined with radial basis function neural network system is established, and a robust adaptive observer combined with radial basis function neural network is designed for the vehicle’s fault-detection, then, the detected fault combined with a robust controller is system is established, and a robust adaptive observer combined with radial basis function neural network applied to design the fault-tolerant controller. In the end, the stability and effective of the fault detection is is designed designed for for the the vehicle’s vehicle’s fault-detection, fault-detection, then, then, the the detected detected fault fault combined combined with with aa robust robust controller controller is is applied to design controller. In the the stability and effective the fault detection is for thethe vehicle’s then, theend, detected fault combined with of a robust controller is anddesigned tolerant system is fault-tolerant provedfault-detection, by Lyapunov theory and simulation. applied to the fault-tolerant controller. In end, the and of the detection applied to design design the fault-tolerant controller. In the the end, the stability stability and effective effective of the fault fault detection and tolerant system is proved by Lyapunov theory and simulation. applied to design the fault-tolerant controller. In the end, the stability and effective of the fault detection and tolerant system is proved by Lyapunov theory and simulation. and tolerant system is proved by Lyapunov theoryobserver; and simulation. © 2019, IFAC (International ofadaptive Automatic Control) Hostingbasis by Elsevier Ltd.neural All rights reserved. Keywords: unmanned aerialFederation vehicle; radial function network; faultand tolerant system is proved by Lyapunov theory and simulation. Keywords: unmanned aerial vehicle; adaptive observer; radial basis function neural network; faultdetection; fault-tolerant control. Keywords: unmanned aerial vehicle; adaptive observer; radial basis function neural network; faultKeywords: unmanned aerial vehicle; adaptive observer; radial basis function neural network; faultdetection; fault-tolerant control. Keywords: unmanned aerial vehicle; adaptive observer; radial basis function neural network; faultdetection; control. detection; fault-tolerant fault-tolerant control. detection;1.fault-tolerant control. INTRODUCTION 2. MATHEMATICAL MODEL OF THE UNMANNED 1. INTRODUCTION 2. MODEL OF UNMANNED AERIAL VEHICLE’S ATTITUDE 1. INTRODUCTION INTRODUCTION 2. MATHEMATICAL MATHEMATICAL MODEL OF THE THESYSTEM UNMANNED 1. 2. MATHEMATICAL MODEL OF THE UNMANNED In the recent decades, the use of unmanned vehicles for AERIAL VEHICLE’S ATTITUDE SYSTEM 1. INTRODUCTION 2. MATHEMATICAL MODEL OF THESYSTEM UNMANNED AERIAL VEHICLE’S ATTITUDE AERIAL VEHICLE’S ATTITUDE SYSTEM In the recent decades, the use of unmanned vehicles for military and civilian applications has increased rapidly, since The vehicle’s attitude motion ATTITUDE has three degrees of freedom In the recent decades, the use of unmanned vehicles for AERIAL VEHICLE’S SYSTEM In the recent decades, the use of unmanned vehicles for military and civilian applications has increased rapidly, since The vehicle’s attitude motion has three degrees of freedom In the recent decades, the use of unmanned vehicles for which can be applied in the situation where manned missions about its centroid, including pitch, roll and yaw motion; the military and civilian applications has increased rapidly, since The vehicle’s attitude motion has three degrees of military and civilian applications has where increased rapidly, since The vehicle’s attitude motionpitch, has three degrees of freedom freedom which can be applied in the situation manned missions about its centroid, including roll and yaw motion; the military and civilian applications has increased rapidly, since The vehicle’s attitude motion has three degrees of freedom could introduce challenges or dangerous. During the vehicle’s configuration shown in Fig. 1 permits the pitch which can be applied in the situation where manned missions about its centroid, including pitch, roll and yaw motion; the which can be applied in the situation where manned missions about its centroid, including pitch, roll and yaw motion; the could introduce challenges or dangerous. During the vehicle’s configuration shown in Fig. 1 permits the pitch which can be applied in the situation where manned missions about its centroid, including pitch, roll and yaw motion; the unmanned vehicles running, the attitude dynamic system has torque and the roll torque to be produced by ailerons and could introduce challenges or dangerous. During the vehicle’s configuration shown in Fig. 1 permits the pitch could introduce challenges or dangerous. During the vehicle’s configuration shown in Fig. 1 permits the pitch unmanned vehicles running, the attitude system torque and the roll torque to be produced by ailerons and could introduce orcope dangerous. During the configuration shown in Fig. 1 permits the pitch high possibility ofchallenges fault. To thedynamic system fault, has an vehicle’s elevators, the yaw torque to be produced by the rudder. unmanned vehicles running, the attitude dynamic system has torque and the roll torque to be produced by ailerons and unmanned vehicles running, the attitude dynamic system has torque andthethe rolltorque torque to produced be produced byrudder. ailerons and high possibility fault. To system fault, an elevators, yaw to be by the unmanned vehiclesof the cope attitudethe dynamic andthe rolltorque torque to produced be produced byrudder. ailerons and effective fault-detect and fault-tolerant method issystem needed to torque high possibility of fault. To cope the system fault, an elevators, yaw to by high possibility ofrunning, fault. To cope the system fault, has an elevators, thethe yaw torque to be be produced by the the rudder. effective fault-detect and fault-tolerant method is needed to high possibility of fault. To cope the system fault, an elevators, the yaw torque to be produced by the rudder. be designed. effective fault-detect effective fault-detect and and fault-tolerant fault-tolerant method method is is needed needed to to be designed. effective fault-detect and fault-tolerant method is needed to be designed. be designed. Currently, be designed.the fault diagnosis or detection of unmanned Currently, fault diagnosis or detection unmanned vehicles hasthe many scholar’s attentionof et al., Currently, the fault or of unmanned Currently, theattracted fault diagnosis diagnosis or detection detection of(Mao unmanned vehicles has attracted many scholar’s attention (Mao et al., Currently, the fault diagnosis or detection of unmanned 2017; Hui and Wang, 2017). (Wang et al., 2015; Abbaspour vehicles has attracted many scholar’s attention (Mao et vehicles has attracted many scholar’s attention (Mao et al., al., 2017; Hui and Wang, 2017). (Wang et al., 2015; Abbaspour vehicles has attracted many scholar’s attention (Mao et al., et al., Hui 2017) proposed a neural network adaptive structure 2017; and Wang, 2017). (Wang et al., 2015; Abbaspour 2017; Hui and Wang, 2017). (Wang et al., 2015; Abbaspour et al., 2017) proposed aa neural network adaptive structure 2017; Wang, 2017). (Wang et al.,(FDI). 2015; (Wu Abbaspour (NNAS) forand fault detection and isolation et al., et al., 2017) proposed network adaptive structure et al., Hui 2017) proposed a neural neural network adaptive structure (NNAS) for fault detection and isolation (FDI). (Wu et et al., 2017) proposed a neural network adaptive structure 2015) using an adaptive fault observer to detect the faults of (NNAS) for for fault fault detection detection and and isolation isolation (FDI). (FDI). (Wu (Wu et al., al., (NNAS) et al., 2015) using an adaptive fault observer to detect the faults of (NNAS) for fault detection and isolation (FDI). (Wu et al., the system online. In (Mostafa et al., 2018), Expert system 2015) using an fault to the of 2015) usingonline. an adaptive adaptive fault observer observer to detect detect the faults faults of the system In (Mostafa et al., 2018), Expert system 2015) using an adaptive fault observer to detect the faults of for the fault diagnosis of the vehicle system is proposed. In the system online. In (Mostafa et al., 2018), Expert system the system online. In (Mostafa et al., 2018), Expert system for the fault diagnosis of the vehicle system is proposed. In the system online. In (Mostafa et al., 2018), Expert system (Hasan and Johansen, 2018), the extend Kalman filter (EKF) for the diagnosis of vehicle system is In for the fault fault diagnosis 2018), of the the the vehicle system is proposed. proposed. In (Hasan and Johansen, extend Kalman filter (EKF) for the in fault diagnosis ofdetection. the the vehicle system is proposed. In is used vehicle’s fault In (Ghadhab etfilter al., 2019), (Hasan and Johansen, 2018), extend Kalman (EKF) (Hasan and Johansen, 2018), the extend Kalman filter (EKF) used in vehicle’s fault detection. In (Ghadhab et al., 2019), (Hasan and Johansen, 2018), the extend Kalman filter (EKF) ais dynamic fault tree is constructed to detection the vehicle is used in vehicle’s fault detection. In (Ghadhab et al., 2019), is dynamic used in vehicle’s fault detection. Into(Ghadhab etthe al.,vehicle 2019), aasystem fault is constructed detection is dynamic used in vehicle’s fault detection.have Into (Ghadhab al.,vehicle 2019), fault. Thetree above methods their ownetadvantages, fault tree is constructed detection the asystem dynamic fault tree is constructed to detection the vehicle fault. The above methods have their own advantages, asystem dynamic fault tree is constructed to detection the vehicle however, the influence of model error and external random fault. The above methods have their own advantages, system fault. The above of methods have their own advantages, Fig.1 The vehicle’s configuration and coordinate system however, the influence model error and external random system fault. aboveconsidered methods have their own advantages, disturbance isThe not been and qualified, and the fault Fig.1 The vehicle’s configuration and coordinate system however, the influence of model error and external random however, the influence of model error and external random Fig.1 The The vehicle’s vehicle’s configuration configuration and and coordinate coordinate system system disturbance not been considered and qualified, and the fault however, theis of be model error and external random detection algorithm should robust enough to cope this Fig.1 disturbance is not considered and qualified, and the fault disturbance is influence not been been considered and qualified, and with the fault Fig.1 The vehicle’s configuration and coordinate system The navigation frame, the body-fixed frame and the wind detection algorithm should be robust enough to cope with this disturbance is not been considered and qualified, and the fault disturbance. detection algorithm algorithm should should be be robust robust enough enough to to cope cope with with this this The navigation frame, the body-fixed frame and theand wind detection frame are shown in Figure 1 by the letters n, b, w disturbance. The navigation frame, the body-fixed frame and navigation frame, the body-fixed frame and the the wind wind detection algorithm should be robust enough to cope with this The disturbance. disturbance. frame are shown in Figure 1 by the letters n, b, and w The navigation frame, the body-fixed frame and the wind respectively. And the differential equations concern angle In this paper, according to Lyapunov stability theory, a robust frame are shown in Figure 1 by the letters n, b, and frame are shown in Figure 1 by the letters n, b, and w w disturbance. respectively. And the differential equations concern the angle In this paper, according to Lyapunov stability theory, a robust frame areα and shown in differential Figure 1 by theas letters n, b,the w of attack thethe sideslip angle βequations are follows: adaptive faultaccording observer to combined with radialtheory, basis function respectively. And concern angle In this paper, Lyapunov stability aa robust respectively. And the differential equations concern theand angle In this paper, according to Lyapunov stability theory, robust of attack α and the sideslip angle ββequations are as follows: adaptive fault observer combined with radial basis function respectively. And the differential concern the angle In this paper, according to Lyapunov stability theory, a robust neural network (RBFNN) is designed for the unmanned of attack α and the sideslip angle are as follows: adaptive fault observer combined with radial basis function angle β are as follows: adaptive fault observer combined with radial basisunmanned function of attack α and the sideslip ρVT SCZα ρVT SCY1 neural network (RBFNN) is designed for sideslip angle adaptive fault observer combined withberadial basis function of attack α and vehicle’s fault-detection, which can usedthe to unmanned accurately α the α，ββ =are =q+ −ras+ follows: neural network (RBFNN) is designed for the ρ V SC ρVT SCY1 β (1) neural network (RBFNN) is designed for the unmanned T Zα vehicle’s fault-detection, which can be used to accurately = q + ρVTT2SC α α r ， β = − + m m YY11 β ρV2TT SC neural network (RBFNN) is designed for the unmanned Zα estimate the interference of UAV attitude system in a very (1) vehicle’s fault-detection, which can be used to accurately Zα α q α r = + ， β = − + β vehicle’s fault-detection, which can be used to accurately (1) ρ V SC ρ V SC m m 2 2 estimate the interference of UAV attitude system in aa very vehicle’s fault-detection, which be used to accurately short time. Then, the detected faultcan and a robust controller are where m is the m Zα m Y1 β ρ is (1) α =quality q + T2of β = −r V+T is 2Tairspeed, estimate the interference of UAV attitude system in very air theα， vehicle, (1) estimate the interference of UAV attitude system in a very short Then, the fault aa robust are m the vehicle, VT is 2airspeed, m 2of ρ estimate interference of controller UAV attitude system a very is air m is the quality applied inthe the fault-tolerant design. Incontroller theinend, the where short time. time. Then, the detected detected fault and and robust controller are short time. Then, the detected fault and a robust controller are ρ where m is quality the vehicle, V ρ is is airspeed, airspeed, is air air where mThe is the the quality of of the vehicle, VTT is C applied in the fault-tolerant controller design. In the end, the density. coefficient like see in Table 1. short time. Then, the detected fault and a robust controller are stability and convergence properties of this method can be ΔΔ applied in in the the fault-tolerant fault-tolerant controller controller design. design. In In the the end, end, the the where ρ is air mThe is the quality of the vehicle, VT is airspeed, applied C density. coefficient like see in Table 1. stability and convergence properties of this method can be ΔΔ applied inand theconvergence fault-tolerant controller design. In the end, demonstrated by Lyapunov properties theory andof simulations. stability this can be density. The The coefficient coefficient like like CΔΔ see see in in Table Table 1. 1. stability and convergence properties of this method method can the be density. demonstrated by Lyapunov theory and simulations. density. The coefficient like CΔΔ stability and convergence properties of this method can be ΔΔ see in Table 1. demonstrated by Lyapunov theory and simulations. demonstrated by Lyapunov theory and simulations. demonstrated by Lyapunov theory and simulations. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.379
Cao Lijia et al. / IFAC PapersOnLine 52-24 (2019) 47–52
48
By definition Ω b = [ p q r ]T , where p, q, r are the vehicle’s roll rate, pitch rate, and yaw rate respectively. The relationship between the rates p, q, r expressed in the body-fixed frame and the Euler attitude rates φ , θ , ψ expressed in the navigation frame is 0 p 1 q = 0 cos φ r 0 − sin φ
− sin θ φ sin φ cos θ θ cos φ cos θ ψ
(2)
(3)
where M b is the total torque in body-fixed frame, I b is the body-fixed frame inertia matrix; which is defined as Lb qSbCL I xx 0 b M = M b = qScCM , I b = 0 I yy N b qSbCN I zx 0
where
Lb , M
b
an d N b
I xz 0 I zz
(4)
are the roll torque, pitch torque and yaw
2 torque respectively, q = ρVT 2 is dynamic pressure, b is wingspan, c is mean aerodynamic chord, S is wing surface. And
CL = CLα1δa1 +CLα 2δa 2 +CLe1δe1 + CLe2δe 2 + CLβ β + CLp p + CLr r CM = CM1 + CMe1δe1 + CMe 2δe 2 +CMa1δa1 + CMa 2δa 2 + CMq q + CMα α C N = CNδ δr + C Nr r + C Nβ β r
(5)
where δa1 and δa2 are the left and right aileron’s deflection
respectively, δe1 and δe2 are the left and right elevator’s
deflection respectively, δr is the rudder’s deflection. And the
2VT , dimensionless angular rates is defined as: p=bp
q=cq 2VT , r =br 2VT . The rest coefficient like CΔΔ see in
Table 1. Let x = [ p q r ] , u = [δa1 δa 2 δe1 δe 2 δr ] , the state-space model of (3) can be derived as:
Sc 2CMq q 2VT I yy N 3 p + N1r D1
( I zz CLr − I xzC Nr ) sb 2 q N 2 q + D1 2 D1VT −( I xx − I zz ) p − 2 I xz r I yy Sb 2 ( − I xzCLr + I xxCNr ) q N1q + D1 2 D1VT
SbI zz CLa1 D1 ScC Ma1 q D1 − SbI xz CLa1 D1
SbI zz CLa 2 D1
SbI zz CLe1 D1
SbI zz CLe 2 D1
ScCMa 2 D1
ScCMe1 D1
ScCMe 2 I yy
− SbI xz CLa 2 D1
− SbI xz CLe1 D1
− SbI xzCLe 2 D1
− SbI xz CN δ r D1 0 − SbI xxC N δ r D1
(7) where N1 = Ixz (Ixx − I yy + Izz ) , N 2 = I yy I zz − I − I , 2 xz
2 zz
D1 = I xx I zz − Ixz2 , N3 = I xz2 − I xx I yy + I xx2 . 3. DESIGN OF OBSERVER AND RBFNN BASED FAULT DETECTION METHOD 3.1 Problem Formulation In the system’s operation process, there will be various uncertain factors such as disturbance and system fault, the fault model of the vehicle’s attitude system derived from (6) can be defined as: = x Ax + Bu + Ef + Q + r y = Cx
(8)
where E ∈ R 3 × 3 , C ∈ R 3 × 3 ; f ∈ R 3 is time-varying unknown bounded fault, r is the modelling error or random external disturbance, and r ≤ σ , σ is a bounded constant. y is the output of vehicle’s attitude system.
T
x = Ax + Bu + Q
where
− N1 p + N 2 r D1
3.2 Design of Adaptive Fault Observer T
∂Ω b A= ∂x
I zz Sb 2CLp N1 q − D1 1 T 2 DV ( I xx − I zz ) r − 2 I zx p I yy Sb 2CLp I xz q N 3q + − D1 2 D1VT B=
According to Newtonian mechanics, the dynamic equation of the vehicle’s attitude system can be derived as:
Ω b = ( I b )−1 M b − Ωb × ( I b ⋅ Ωb )
A=
b , B = ∂Ω ∂u
An adaptive fault observer of (8) can be designed as: (6)
,
Sbβ q ( I zzCLβ β − I xzCN β ) Scα qCM α Sbβ q ( I xxCN β β − I xz CLβ ) T Q =[ , , ] , I xx I zz − I xz2 I yy I xx I zz − I xz2
and the complete form of A and B is
= xˆ m Axˆ m + Bu + Efˆ + K ( yˆ m - y ) + Q yˆ m = Cxˆ m
(9)
where xˆ m is the state vector of observer, yˆ m is the output of
observer, fˆ is used to detect the unknown fault f, K ∈ R 3×3 is observer gain matrix needed to be designed. 3.3 Design of Observer Gain Matrix
Let e m = xˆ m − x , ε m = yˆ m − y , f = fˆ − f , combined with (8) and (9), the observer error and output error equation can be obtained as:
Cao Lijia et al. / IFAC PapersOnLine 52-24 (2019) 47–52
( A + KC )em + Ef − r em = ε m = Cem
(10)
Let P ∈ R 3× 3 and Q ∈ R 3× 3 be positive definite symmetric matrices; According to Lyapunov theory in (Khail, 2015), the error dynamic system (10) can be asymptotically stable, that is lim em (t ) = 0 , if there existed: t →∞
(11)
P ( A + K C ) + ( A + K C )T P = − Q
Based on (11), the observer gain matrix K can be designed as: 1 K = −( P −1Q + A)C −1 2
(12)
3.4 RBF Neural Network Approximation
RBFNN is mostly chosen as an approximator of the unknown dynamics, according to (Luan et al., 2019; Wu et al., 2019), which can be described in the following form: N
Snn (Z ) = wiηi (Z ) = wTη(Z )
(13)
i=1
where Z ∈ R m is the input vector, w = [ w1,..., wN ] ∈ R
N ×n
is the
weight vector, N > 1 is the node number of neural network, m and n are the dimension of input and output vector T respectively, and η ( Z ) = [η1 ( Z ),...,η N ( Z ) ] ∈ R N is the regressor vector, with η i ( Z ) being a radial basis function. And 2
η i ( Z ) = exp( − Z − ξ i / ζ i2 ) ,where ξ i and ζ i are the centre and
width of the receptive field respectively. RBFNN, with suitable widths and centres, can approximate the unknown fault f in (8) precisely:
f = Snn* (Z) = w*Tη(Z) where w * is the ideal weight vector, Z Similarly, fˆ and f can be express as:
= em
(14) is the input vector.
fˆ = Sˆ nn (em ) = wˆ Tη(em ) , f = Snn (em ) = w Tη(em ) (15)
Choosing a Lyapunov function candidate
γ
1 1 T Pem − r T Pem + tr ( w T w ) = − emT Qem + η (em )T wE 2 γ
Noting that
if we choose the weight update law as:
eT PEwˆ Tη(em ) wˆ = w = −γ η(em )emT PE − c0γ m wˆ Mw2 T T 1, if wˆ > Mw and emPEwˆ η(em ) > 0 with, c0 = otherwise 0,
(19)
where M w ≥ tr ( wˆ (0)T wˆ (0))1/ 2 . It is obtained that: 1 1 V1 = − e mT Qe m + tr (γ E T Pe mη (e m )T w + w T w ) − r T Pe m γ 2 1 1 = − e mT Qe m + tr (γ η (e m )e mT PE + w )T w − r T Pe m γ 2
(20)
e T PEwˆ T η (e m ) T 1 = − e mT Qe m − r T Pe m − c0 tr m w wˆ M w2 2
It can be seen that the last term in the above equation, e T PEwˆ Tη (em ) T w wˆ c0tr m M w2 emT PEwˆ Tη (em ) = c0 tr ( wˆ T wˆ − w*T wˆ ) ≥ 0 M w2
(21)
It’s noted that when wˆ < M w or wˆ = M w and emT PEwˆ Tη (em ) ≤ 0 , then c0 = 0 by (19), the above term is true and ignorable; when wˆ > M w and emT PEwˆ Tη (em ) > 0 , then, due to the fact that w * ≤ M w , one can have tr ( wˆ T wˆ − w *T wˆ ) ≥ 0 , therefore, the above inequality is also true. In other words, the projection will not make the derivative of the Lyapunov function more positive. So, it is obtained that: 1 V1 ≤ − emT Qem − r T Pem 2 1 2 ≤ − λmin (Q ) em + σ λmax ( P ) em 2 1 = − em λmin (Q) em − 2 σ λmax ( P ) 2
(22)
Consequently, it can be concluded that the error em will (16)
Differentiating V1 and substituted (10) yields: 1 1 V1 = (emT Pem + emT Pem ) + tr ( w T w ) γ 2 1 T = em ( P ( A + KC ) + ( A + KC )T P )em 2 1 T Pem − r T Pem + tr ( w T w ) +η (em )T wE
(18)
where λmin (Q ) and λmax ( P ) denote the minimum and maximum eigen-value of matrices Q and P respectively.
where w = wˆ − w * .
1 1 V1 = emT Pem + tr (w T w ) 2 2γ
T Pem = tr(ET Pemη(em )T w ) η(em )T wE
49
converge to the radius r0 = κ σ with κ = 2λmax ( P ) / λmin (Q ) . 4. DESIGN OF FAULT-TOLERANT CONTROLLER
(17)
Based on the system fault being estimated by the observer, and the fault-tolerant controller of fault system needs to be designed to ensure the closed-loop stability. So, the faulttolerant controller can be designed as: u = B + ( x r − λ1e − λ2 edt − Ax − Q − Efˆ − Eurc )
(23)
where B+ is the Moore-Penrose inverse of B, x r is the reference attitude’s rate, and e = x − x r . urc is a robust controller needed to be designed. λ 1 and λ 2 are constants.
Cao Lijia et al. / IFAC PapersOnLine 52-24 (2019) 47–52
50
Substituting the controller (23) into fault model (8), the following closed-loop state equations is got: E −1 (e + λ1e + λ2 edt ) = Δ − urc (24) −1 where Δ= f +E r − fˆ .
Let s = E − 1 ( e + λ1e + λ 2 edt ) s = s dt
, urc =
k2 +1 s 2k 2
(25)
/
ρ
1.29
kg/m3
S
1.8
m
VT
10
m/s
CMα
-0.09
/
C Lr
0.036
/
C Lp
-0.19
/
C Nr
-0.21
/
CY1
-0.38
/
CLα1
-0.03
/
C Mq
-9.83
/
CLα2
0.03
/
CZα
-3.25
/
CLe1
-0.05
/
CMe1
0.272
/
CLe2
0.05
/
CMa1
0.038
/
2
CMe2
0.272
/
CMa2
0.038
/
1
0
0
1
0
1 2 s 2
(26)
Differentiating V2 yields: 2
k +1 V2 = ss = s( Δ − urc ) = s( Δ − s) 2k 2 1 1 1 1 s2 = sΔ − s 2 − 2 s 2 = − s 2 − ( 2 − 2 sΔ) 2 2k 2 2 k 1 2 1 s2 1 = − s − ( 2 − 2 sΔ + Δ2 k 2 ) + Δ2 k 2 2 2 k 2 1 2 1 s 1 = − s − ( − Δk )2 + Δ2 k 2 2 2 k 2 1 2 1 2 2 ≤− s + k Δ 2 2
49 7 0 300 20 10 P = 7 50 7 , Q = 20 60 0 0 7 50 10 0 300
The reference attitudes in the navigation frame and its’ rate in the body-fixed frame are set as:
(27)
sin(t ) 0 1 ϕ r θ = 1 − exp( −t ) / 2 , x = 0 cos ϕ ) r r ( 0 − sin ϕ cos(t ) ψ r
− sin θ ϕ r sin ϕ cos θ θr cos ϕ cos θ ψ r
The reference system fault is set as: f = [1 − sin(2t ) 1 + cos(2t ) 1 + sin(2t )]
T
Assume Δ∈ L2 [0,T ) , ∀T ∈[ 0, ∞) ; integrating (25) from t=0 to
t=T yields:
To verify the performance of fault-tolerant controller (23), A contrasted controller without fˆ and urc is set as: u = B + ( x r − λ1e − λ2 edt − Ax − Q )
T 1 T 2 1 s dt + k 2 Δ 2 dt 0 2 0 2
(28)
Then, the simulation results are shown as follows.
Since V 2 (T ) ≥ 0 , (26) implies the following: T 1 T 2 1 s dt ≤ V2 (0) + k 2 Δ2 dt 0 0 2 2
Since
V 2 (0 )
is
finite,
error Δ∈ L2 [0,T ) , that is
and
T 0
if
the
(29)
approximation
Δ 2 dt ≤ ∞ , using the Barbalat’s
lemma in (Hsu, 2007; Lin and Wang, 2011), it implies that lim s = 0 . t →∞
5. SIMULATION RESULTS A simulation is provided to demonstrate the effectiveness of the control law on the Links-Box real-time simulation platform. And the parameters are as follows: Table 1. The Parameters of the vehicle’s mathematical model (Ducard, 2009; Cao, et al. 2018). Term
Value
Unit
Term
Value
Unit
Ixx Iyy
2.56 10.9
kg·m2 kg·m2
CLβ m
0.087 28
/ kg
Izz
11.3
kg·m2
b
3.1
m
Ixz
0.5
kg·m
c
0.58
m
Izx
0.5
kg·m2
CNβ
0.087
/
2
0
λ2 = 5, σ = 0.5 , E = 0 1 0 , C = 0 1 0 , 0 0 1 0 0 1 γ = 0.01
Proof: Define a Lyapunov function as
V2 (T ) − V2 (0) ≤ −
0.053
k = 0.5, λ1 = 5
where k is constant.
V2 =
CNδ
Fig.2 Vehicle attitude system fault and it’s detection
(30)
Cao Lijia et al. / IFAC PapersOnLine 52-24 (2019) 47–52
51
errors converge quickly in the presence of different reference attitudes; Combined with (27-29) and the fault detecting performance in Fig.2, it is concluded that, with fˆ and urc to detecting and compensating unknown faults, the favourable fault-tolerant performance can be fulfilled by controller (23). From the simulation results in Fig. 4, it is observed that the smoothness and continuity of aileron, elevator and rudder deflection under external disturbance can be guaranteed by using the method proposed in this paper. 6. CONCLUSIONS In this study, first, an observer and RBFNN based fault detection method is designed to detecting the unknow fault of vehicle’s attitude system, then, a robust controller and the output of RBFNN is applied to design the fault-tolerant controller; And the effective of the proposed fault detection and tolerant method is proved by Lyapunov theorem and simulations. In the end, this work is one of the first attempts in implementing observer and RBFNN for the unmanned aerial vehicle’s fault detection and tolerant control process; this study provides a new idea that combined the intelligence methods with the traditional control methods to solve the practical engineering problems.
Fig.3 Vehicle’s attitudes Tacking
ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation of China (Nos.61703409, 11705122), Sichuan Science and Technology Program (Nos. 2018JY0197, 19ZDZX0037); Nature Science Foundation of Sichuan University of Science & Engineering (Nos.2017RCL12, 2018RCL18);Sichuan Key Provincial Research Base of Intelligent Tourism (No.ZHZJ18-01); Aeronautical Science Foundation of China (No.201605U8002); Xi'an Scientific and Technological Program (No:201805048YD26CG32(2)); Research Foundation of Department of Education of Sichuan Province (No.17ZA0271). REFERENCES
Fig.4 Vehicle’s deflections of aileron, elevator and rudder From the simulation results in Fig.2, it can be seen that favourable fault detecting performance can be achieved by the proposed observer based RBFNN fault detection method. −1 So the term Δ= f +E r − fˆ satisfy
T 0
Δ 2 dt ≤ ∞ , combined
with (27-29), it can be concluded that lim s = 0 . t →∞
From the simulation results in Fig.3, it is seen that, compared to the controller (30), the controller (23) make the tracking
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ScienceDirect IFAC PapersOnLine 52-24 (2019) 47–52
Adaptive observer-based fault detection and active tolerant control for unmanned Adaptive detection and active tolerant Adaptive observer-based observer-based fault fault active tolerant control control for for unmanned unmanned aerialdetection vehicles and attitude system Adaptive observer-based fault detection and active tolerant control for unmanned aerial vehicles attitude system aerial vehicles attitude system aerial vehicles attitude system CAO Lijia*, TANG Yu*, ZHANG Guo*
CAO Lijia*, TANG Yu*, ZHANG Guo* CAO CAO Lijia*, Lijia*, TANG TANG Yu*, Yu*, ZHANG ZHANG Guo* Guo* * Artificial Intelligence Key Laboratory ofCAO Sichuan Province ; Sichuan Provincial Lijia*, TANG Yu*, Key ZHANG Guo*Research Base of Intelligent Tourism; ** Artificial Intelligence Key Laboratory of Sichuan Province ; Sichuan Key Provincial Research Base of Intelligent Tourism; Sichuan University of Science & Engineering , Yibin, 644000,China Artificial Intelligence Key Laboratory of Sichuan Province ; Sichuan Key Provincial Research * Artificial Intelligence Key Laboratory of SichuanofProvince ;&Sichuan Key,Provincial Research Base Base of of Intelligent Intelligent Tourism; Tourism; Sichuan University Science Engineering Yibin, 644000,China * Artificial Intelligence Key Laboratory of Sichuan Province ; Sichuan Key Provincial Research Base of Intelligent Tourism; (e-mail: [email protected]). Sichuan University of Science & Engineering , Yibin, 644000,China Sichuan University of Science & Engineering, Yibin, 644000,China Sichuan University (e-mail: of [email protected]). & Engineering, Yibin, 644000,China (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). Abstract: A robust adaptive observer combined with radial basis function neural network (RBFNN) is Abstract: A robust adaptive observer combined with radial basis function network (RBFNN) is designed the unmanned vehicles (UAVs) systemneural is proposed this paper. Abstract: A adaptive observer combined with radial neural network is Abstract:for A robust robust adaptive aerial observer combined with fault-detection radial basis basis function function neural networkin(RBFNN) (RBFNN) is designed for the unmanned aerial vehicles (UAVs) fault-detection system is proposed in this paper. Abstract: A robust adaptive observer combined with radial basis function neural network (RBFNN) is Firstly, the fault dynamics model with unknown disturbance of the unmanned aerial vehicle’s attitude designed for the unmanned aerial vehicles (UAVs) fault-detection system is proposed in this paper. designed for the unmanned aerial vehicles (UAVs) fault-detection system is proposed in this paper. Firstly, the fault dynamics model with unknown disturbance of the unmanned aerial vehicle’s attitude designed for the unmanned aerial vehicles (UAVs) fault-detection system is proposed in this paper. system is established, and a robust adaptive observer combined with radial basis function neural network Firstly, the fault model with unknown disturbance of the unmanned aerial attitude Firstly, fault dynamics dynamics with unknown disturbance ofwith theradial unmanned aerial vehicle’s vehicle’s attitude system isthe established, and aa model robust adaptive observer combined basis function neural network Firstly, theestablished, faultthedynamics model with unknown disturbance thecombined unmanned aerial vehicle’s attitude is designed for vehicle’s fault-detection, then, the detected of fault with a robust controller is system is and robust adaptive observer combined with radial basis function neural network system is established, and a robust adaptive observer combined with radial basis function neural network is designed for the vehicle’s fault-detection, then, the detected fault combined with a robust controller is system is established, and a robust adaptive observer combined with radial basis function neural network applied to design the fault-tolerant controller. In the end, the stability and effective of the fault detection is is designed designed for for the the vehicle’s vehicle’s fault-detection, fault-detection, then, then, the the detected detected fault fault combined combined with with aa robust robust controller controller is is applied to design controller. In the the stability and effective the fault detection is for thethe vehicle’s then, theend, detected fault combined with of a robust controller is anddesigned tolerant system is fault-tolerant provedfault-detection, by Lyapunov theory and simulation. applied to the fault-tolerant controller. In end, the and of the detection applied to design design the fault-tolerant controller. In the the end, the stability stability and effective effective of the fault fault detection and tolerant system is proved by Lyapunov theory and simulation. applied to design the fault-tolerant controller. In the end, the stability and effective of the fault detection and tolerant system is proved by Lyapunov theory and simulation. and tolerant system is proved by Lyapunov theoryobserver; and simulation. © 2019, IFAC (International ofadaptive Automatic Control) Hostingbasis by Elsevier Ltd.neural All rights reserved. Keywords: unmanned aerialFederation vehicle; radial function network; faultand tolerant system is proved by Lyapunov theory and simulation. Keywords: unmanned aerial vehicle; adaptive observer; radial basis function neural network; faultdetection; fault-tolerant control. Keywords: unmanned aerial vehicle; adaptive observer; radial basis function neural network; faultKeywords: unmanned aerial vehicle; adaptive observer; radial basis function neural network; faultdetection; fault-tolerant control. Keywords: unmanned aerial vehicle; adaptive observer; radial basis function neural network; faultdetection; control. detection; fault-tolerant fault-tolerant control. detection;1.fault-tolerant control. INTRODUCTION 2. MATHEMATICAL MODEL OF THE UNMANNED 1. INTRODUCTION 2. MODEL OF UNMANNED AERIAL VEHICLE’S ATTITUDE 1. INTRODUCTION INTRODUCTION 2. MATHEMATICAL MATHEMATICAL MODEL OF THE THESYSTEM UNMANNED 1. 2. MATHEMATICAL MODEL OF THE UNMANNED In the recent decades, the use of unmanned vehicles for AERIAL VEHICLE’S ATTITUDE SYSTEM 1. INTRODUCTION 2. MATHEMATICAL MODEL OF THESYSTEM UNMANNED AERIAL VEHICLE’S ATTITUDE AERIAL VEHICLE’S ATTITUDE SYSTEM In the recent decades, the use of unmanned vehicles for military and civilian applications has increased rapidly, since The vehicle’s attitude motion ATTITUDE has three degrees of freedom In the recent decades, the use of unmanned vehicles for AERIAL VEHICLE’S SYSTEM In the recent decades, the use of unmanned vehicles for military and civilian applications has increased rapidly, since The vehicle’s attitude motion has three degrees of freedom In the recent decades, the use of unmanned vehicles for which can be applied in the situation where manned missions about its centroid, including pitch, roll and yaw motion; the military and civilian applications has increased rapidly, since The vehicle’s attitude motion has three degrees of military and civilian applications has where increased rapidly, since The vehicle’s attitude motionpitch, has three degrees of freedom freedom which can be applied in the situation manned missions about its centroid, including roll and yaw motion; the military and civilian applications has increased rapidly, since The vehicle’s attitude motion has three degrees of freedom could introduce challenges or dangerous. During the vehicle’s configuration shown in Fig. 1 permits the pitch which can be applied in the situation where manned missions about its centroid, including pitch, roll and yaw motion; the which can be applied in the situation where manned missions about its centroid, including pitch, roll and yaw motion; the could introduce challenges or dangerous. During the vehicle’s configuration shown in Fig. 1 permits the pitch which can be applied in the situation where manned missions about its centroid, including pitch, roll and yaw motion; the unmanned vehicles running, the attitude dynamic system has torque and the roll torque to be produced by ailerons and could introduce challenges or dangerous. During the vehicle’s configuration shown in Fig. 1 permits the pitch could introduce challenges or dangerous. During the vehicle’s configuration shown in Fig. 1 permits the pitch unmanned vehicles running, the attitude system torque and the roll torque to be produced by ailerons and could introduce orcope dangerous. During the configuration shown in Fig. 1 permits the pitch high possibility ofchallenges fault. To thedynamic system fault, has an vehicle’s elevators, the yaw torque to be produced by the rudder. unmanned vehicles running, the attitude dynamic system has torque and the roll torque to be produced by ailerons and unmanned vehicles running, the attitude dynamic system has torque andthethe rolltorque torque to produced be produced byrudder. ailerons and high possibility fault. To system fault, an elevators, yaw to be by the unmanned vehiclesof the cope attitudethe dynamic andthe rolltorque torque to produced be produced byrudder. ailerons and effective fault-detect and fault-tolerant method issystem needed to torque high possibility of fault. To cope the system fault, an elevators, yaw to by high possibility ofrunning, fault. To cope the system fault, has an elevators, thethe yaw torque to be be produced by the the rudder. effective fault-detect and fault-tolerant method is needed to high possibility of fault. To cope the system fault, an elevators, the yaw torque to be produced by the rudder. be designed. effective fault-detect effective fault-detect and and fault-tolerant fault-tolerant method method is is needed needed to to be designed. effective fault-detect and fault-tolerant method is needed to be designed. be designed. Currently, be designed.the fault diagnosis or detection of unmanned Currently, fault diagnosis or detection unmanned vehicles hasthe many scholar’s attentionof et al., Currently, the fault or of unmanned Currently, theattracted fault diagnosis diagnosis or detection detection of(Mao unmanned vehicles has attracted many scholar’s attention (Mao et al., Currently, the fault diagnosis or detection of unmanned 2017; Hui and Wang, 2017). (Wang et al., 2015; Abbaspour vehicles has attracted many scholar’s attention (Mao et vehicles has attracted many scholar’s attention (Mao et al., al., 2017; Hui and Wang, 2017). (Wang et al., 2015; Abbaspour vehicles has attracted many scholar’s attention (Mao et al., et al., Hui 2017) proposed a neural network adaptive structure 2017; and Wang, 2017). (Wang et al., 2015; Abbaspour 2017; Hui and Wang, 2017). (Wang et al., 2015; Abbaspour et al., 2017) proposed aa neural network adaptive structure 2017; Wang, 2017). (Wang et al.,(FDI). 2015; (Wu Abbaspour (NNAS) forand fault detection and isolation et al., et al., 2017) proposed network adaptive structure et al., Hui 2017) proposed a neural neural network adaptive structure (NNAS) for fault detection and isolation (FDI). (Wu et et al., 2017) proposed a neural network adaptive structure 2015) using an adaptive fault observer to detect the faults of (NNAS) for for fault fault detection detection and and isolation isolation (FDI). (FDI). (Wu (Wu et al., al., (NNAS) et al., 2015) using an adaptive fault observer to detect the faults of (NNAS) for fault detection and isolation (FDI). (Wu et al., the system online. In (Mostafa et al., 2018), Expert system 2015) using an fault to the of 2015) usingonline. an adaptive adaptive fault observer observer to detect detect the faults faults of the system In (Mostafa et al., 2018), Expert system 2015) using an adaptive fault observer to detect the faults of for the fault diagnosis of the vehicle system is proposed. In the system online. In (Mostafa et al., 2018), Expert system the system online. In (Mostafa et al., 2018), Expert system for the fault diagnosis of the vehicle system is proposed. In the system online. In (Mostafa et al., 2018), Expert system (Hasan and Johansen, 2018), the extend Kalman filter (EKF) for the diagnosis of vehicle system is In for the fault fault diagnosis 2018), of the the the vehicle system is proposed. proposed. In (Hasan and Johansen, extend Kalman filter (EKF) for the in fault diagnosis ofdetection. the the vehicle system is proposed. In is used vehicle’s fault In (Ghadhab etfilter al., 2019), (Hasan and Johansen, 2018), extend Kalman (EKF) (Hasan and Johansen, 2018), the extend Kalman filter (EKF) used in vehicle’s fault detection. In (Ghadhab et al., 2019), (Hasan and Johansen, 2018), the extend Kalman filter (EKF) ais dynamic fault tree is constructed to detection the vehicle is used in vehicle’s fault detection. In (Ghadhab et al., 2019), is dynamic used in vehicle’s fault detection. Into(Ghadhab etthe al.,vehicle 2019), aasystem fault is constructed detection is dynamic used in vehicle’s fault detection.have Into (Ghadhab al.,vehicle 2019), fault. Thetree above methods their ownetadvantages, fault tree is constructed detection the asystem dynamic fault tree is constructed to detection the vehicle fault. The above methods have their own advantages, asystem dynamic fault tree is constructed to detection the vehicle however, the influence of model error and external random fault. The above methods have their own advantages, system fault. The above of methods have their own advantages, Fig.1 The vehicle’s configuration and coordinate system however, the influence model error and external random system fault. aboveconsidered methods have their own advantages, disturbance isThe not been and qualified, and the fault Fig.1 The vehicle’s configuration and coordinate system however, the influence of model error and external random however, the influence of model error and external random Fig.1 The The vehicle’s vehicle’s configuration configuration and and coordinate coordinate system system disturbance not been considered and qualified, and the fault however, theis of be model error and external random detection algorithm should robust enough to cope this Fig.1 disturbance is not considered and qualified, and the fault disturbance is influence not been been considered and qualified, and with the fault Fig.1 The vehicle’s configuration and coordinate system The navigation frame, the body-fixed frame and the wind detection algorithm should be robust enough to cope with this disturbance is not been considered and qualified, and the fault disturbance. detection algorithm algorithm should should be be robust robust enough enough to to cope cope with with this this The navigation frame, the body-fixed frame and theand wind detection frame are shown in Figure 1 by the letters n, b, w disturbance. The navigation frame, the body-fixed frame and navigation frame, the body-fixed frame and the the wind wind detection algorithm should be robust enough to cope with this The disturbance. disturbance. frame are shown in Figure 1 by the letters n, b, and w The navigation frame, the body-fixed frame and the wind respectively. And the differential equations concern angle In this paper, according to Lyapunov stability theory, a robust frame are shown in Figure 1 by the letters n, b, and frame are shown in Figure 1 by the letters n, b, and w w disturbance. respectively. And the differential equations concern the angle In this paper, according to Lyapunov stability theory, a robust frame areα and shown in differential Figure 1 by theas letters n, b,the w of attack thethe sideslip angle βequations are follows: adaptive faultaccording observer to combined with radialtheory, basis function respectively. And concern angle In this paper, Lyapunov stability aa robust respectively. And the differential equations concern theand angle In this paper, according to Lyapunov stability theory, robust of attack α and the sideslip angle ββequations are as follows: adaptive fault observer combined with radial basis function respectively. And the differential concern the angle In this paper, according to Lyapunov stability theory, a robust neural network (RBFNN) is designed for the unmanned of attack α and the sideslip angle are as follows: adaptive fault observer combined with radial basis function angle β are as follows: adaptive fault observer combined with radial basisunmanned function of attack α and the sideslip ρVT SCZα ρVT SCY1 neural network (RBFNN) is designed for sideslip angle adaptive fault observer combined withberadial basis function of attack α and vehicle’s fault-detection, which can usedthe to unmanned accurately α the α，ββ =are =q+ −ras+ follows: neural network (RBFNN) is designed for the ρ V SC ρVT SCY1 β (1) neural network (RBFNN) is designed for the unmanned T Zα vehicle’s fault-detection, which can be used to accurately = q + ρVTT2SC α α r ， β = − + m m YY11 β ρV2TT SC neural network (RBFNN) is designed for the unmanned Zα estimate the interference of UAV attitude system in a very (1) vehicle’s fault-detection, which can be used to accurately Zα α q α r = + ， β = − + β vehicle’s fault-detection, which can be used to accurately (1) ρ V SC ρ V SC m m 2 2 estimate the interference of UAV attitude system in aa very vehicle’s fault-detection, which be used to accurately short time. Then, the detected faultcan and a robust controller are where m is the m Zα m Y1 β ρ is (1) α =quality q + T2of β = −r V+T is 2Tairspeed, estimate the interference of UAV attitude system in very air theα， vehicle, (1) estimate the interference of UAV attitude system in a very short Then, the fault aa robust are m the vehicle, VT is 2airspeed, m 2of ρ estimate interference of controller UAV attitude system a very is air m is the quality applied inthe the fault-tolerant design. Incontroller theinend, the where short time. time. Then, the detected detected fault and and robust controller are short time. Then, the detected fault and a robust controller are ρ where m is quality the vehicle, V ρ is is airspeed, airspeed, is air air where mThe is the the quality of of the vehicle, VTT is C applied in the fault-tolerant controller design. In the end, the density. coefficient like see in Table 1. short time. Then, the detected fault and a robust controller are stability and convergence properties of this method can be ΔΔ applied in in the the fault-tolerant fault-tolerant controller controller design. design. In In the the end, end, the the where ρ is air mThe is the quality of the vehicle, VT is airspeed, applied C density. coefficient like see in Table 1. stability and convergence properties of this method can be ΔΔ applied inand theconvergence fault-tolerant controller design. In the end, demonstrated by Lyapunov properties theory andof simulations. stability this can be density. The The coefficient coefficient like like CΔΔ see see in in Table Table 1. 1. stability and convergence properties of this method method can the be density. demonstrated by Lyapunov theory and simulations. density. The coefficient like CΔΔ stability and convergence properties of this method can be ΔΔ see in Table 1. demonstrated by Lyapunov theory and simulations. demonstrated by Lyapunov theory and simulations. demonstrated by Lyapunov theory and simulations. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.379
Cao Lijia et al. / IFAC PapersOnLine 52-24 (2019) 47–52
48
By definition Ω b = [ p q r ]T , where p, q, r are the vehicle’s roll rate, pitch rate, and yaw rate respectively. The relationship between the rates p, q, r expressed in the body-fixed frame and the Euler attitude rates φ , θ , ψ expressed in the navigation frame is 0 p 1 q = 0 cos φ r 0 − sin φ
− sin θ φ sin φ cos θ θ cos φ cos θ ψ
(2)
(3)
where M b is the total torque in body-fixed frame, I b is the body-fixed frame inertia matrix; which is defined as Lb qSbCL I xx 0 b M = M b = qScCM , I b = 0 I yy N b qSbCN I zx 0
where
Lb , M
b
an d N b
I xz 0 I zz
(4)
are the roll torque, pitch torque and yaw
2 torque respectively, q = ρVT 2 is dynamic pressure, b is wingspan, c is mean aerodynamic chord, S is wing surface. And
CL = CLα1δa1 +CLα 2δa 2 +CLe1δe1 + CLe2δe 2 + CLβ β + CLp p + CLr r CM = CM1 + CMe1δe1 + CMe 2δe 2 +CMa1δa1 + CMa 2δa 2 + CMq q + CMα α C N = CNδ δr + C Nr r + C Nβ β r
(5)
where δa1 and δa2 are the left and right aileron’s deflection
respectively, δe1 and δe2 are the left and right elevator’s
deflection respectively, δr is the rudder’s deflection. And the
2VT , dimensionless angular rates is defined as: p=bp
q=cq 2VT , r =br 2VT . The rest coefficient like CΔΔ see in
Table 1. Let x = [ p q r ] , u = [δa1 δa 2 δe1 δe 2 δr ] , the state-space model of (3) can be derived as:
Sc 2CMq q 2VT I yy N 3 p + N1r D1
( I zz CLr − I xzC Nr ) sb 2 q N 2 q + D1 2 D1VT −( I xx − I zz ) p − 2 I xz r I yy Sb 2 ( − I xzCLr + I xxCNr ) q N1q + D1 2 D1VT
SbI zz CLa1 D1 ScC Ma1 q D1 − SbI xz CLa1 D1
SbI zz CLa 2 D1
SbI zz CLe1 D1
SbI zz CLe 2 D1
ScCMa 2 D1
ScCMe1 D1
ScCMe 2 I yy
− SbI xz CLa 2 D1
− SbI xz CLe1 D1
− SbI xzCLe 2 D1
− SbI xz CN δ r D1 0 − SbI xxC N δ r D1
(7) where N1 = Ixz (Ixx − I yy + Izz ) , N 2 = I yy I zz − I − I , 2 xz
2 zz
D1 = I xx I zz − Ixz2 , N3 = I xz2 − I xx I yy + I xx2 . 3. DESIGN OF OBSERVER AND RBFNN BASED FAULT DETECTION METHOD 3.1 Problem Formulation In the system’s operation process, there will be various uncertain factors such as disturbance and system fault, the fault model of the vehicle’s attitude system derived from (6) can be defined as: = x Ax + Bu + Ef + Q + r y = Cx
(8)
where E ∈ R 3 × 3 , C ∈ R 3 × 3 ; f ∈ R 3 is time-varying unknown bounded fault, r is the modelling error or random external disturbance, and r ≤ σ , σ is a bounded constant. y is the output of vehicle’s attitude system.
T
x = Ax + Bu + Q
where
− N1 p + N 2 r D1
3.2 Design of Adaptive Fault Observer T
∂Ω b A= ∂x
I zz Sb 2CLp N1 q − D1 1 T 2 DV ( I xx − I zz ) r − 2 I zx p I yy Sb 2CLp I xz q N 3q + − D1 2 D1VT B=
According to Newtonian mechanics, the dynamic equation of the vehicle’s attitude system can be derived as:
Ω b = ( I b )−1 M b − Ωb × ( I b ⋅ Ωb )
A=
b , B = ∂Ω ∂u
An adaptive fault observer of (8) can be designed as: (6)
,
Sbβ q ( I zzCLβ β − I xzCN β ) Scα qCM α Sbβ q ( I xxCN β β − I xz CLβ ) T Q =[ , , ] , I xx I zz − I xz2 I yy I xx I zz − I xz2
and the complete form of A and B is
= xˆ m Axˆ m + Bu + Efˆ + K ( yˆ m - y ) + Q yˆ m = Cxˆ m
(9)
where xˆ m is the state vector of observer, yˆ m is the output of
observer, fˆ is used to detect the unknown fault f, K ∈ R 3×3 is observer gain matrix needed to be designed. 3.3 Design of Observer Gain Matrix
Let e m = xˆ m − x , ε m = yˆ m − y , f = fˆ − f , combined with (8) and (9), the observer error and output error equation can be obtained as:
Cao Lijia et al. / IFAC PapersOnLine 52-24 (2019) 47–52
( A + KC )em + Ef − r em = ε m = Cem
(10)
Let P ∈ R 3× 3 and Q ∈ R 3× 3 be positive definite symmetric matrices; According to Lyapunov theory in (Khail, 2015), the error dynamic system (10) can be asymptotically stable, that is lim em (t ) = 0 , if there existed: t →∞
(11)
P ( A + K C ) + ( A + K C )T P = − Q
Based on (11), the observer gain matrix K can be designed as: 1 K = −( P −1Q + A)C −1 2
(12)
3.4 RBF Neural Network Approximation
RBFNN is mostly chosen as an approximator of the unknown dynamics, according to (Luan et al., 2019; Wu et al., 2019), which can be described in the following form: N
Snn (Z ) = wiηi (Z ) = wTη(Z )
(13)
i=1
where Z ∈ R m is the input vector, w = [ w1,..., wN ] ∈ R
N ×n
is the
weight vector, N > 1 is the node number of neural network, m and n are the dimension of input and output vector T respectively, and η ( Z ) = [η1 ( Z ),...,η N ( Z ) ] ∈ R N is the regressor vector, with η i ( Z ) being a radial basis function. And 2
η i ( Z ) = exp( − Z − ξ i / ζ i2 ) ,where ξ i and ζ i are the centre and
width of the receptive field respectively. RBFNN, with suitable widths and centres, can approximate the unknown fault f in (8) precisely:
f = Snn* (Z) = w*Tη(Z) where w * is the ideal weight vector, Z Similarly, fˆ and f can be express as:
= em
(14) is the input vector.
fˆ = Sˆ nn (em ) = wˆ Tη(em ) , f = Snn (em ) = w Tη(em ) (15)
Choosing a Lyapunov function candidate
γ
1 1 T Pem − r T Pem + tr ( w T w ) = − emT Qem + η (em )T wE 2 γ
Noting that
if we choose the weight update law as:
eT PEwˆ Tη(em ) wˆ = w = −γ η(em )emT PE − c0γ m wˆ Mw2 T T 1, if wˆ > Mw and emPEwˆ η(em ) > 0 with, c0 = otherwise 0,
(19)
where M w ≥ tr ( wˆ (0)T wˆ (0))1/ 2 . It is obtained that: 1 1 V1 = − e mT Qe m + tr (γ E T Pe mη (e m )T w + w T w ) − r T Pe m γ 2 1 1 = − e mT Qe m + tr (γ η (e m )e mT PE + w )T w − r T Pe m γ 2
(20)
e T PEwˆ T η (e m ) T 1 = − e mT Qe m − r T Pe m − c0 tr m w wˆ M w2 2
It can be seen that the last term in the above equation, e T PEwˆ Tη (em ) T w wˆ c0tr m M w2 emT PEwˆ Tη (em ) = c0 tr ( wˆ T wˆ − w*T wˆ ) ≥ 0 M w2
(21)
It’s noted that when wˆ < M w or wˆ = M w and emT PEwˆ Tη (em ) ≤ 0 , then c0 = 0 by (19), the above term is true and ignorable; when wˆ > M w and emT PEwˆ Tη (em ) > 0 , then, due to the fact that w * ≤ M w , one can have tr ( wˆ T wˆ − w *T wˆ ) ≥ 0 , therefore, the above inequality is also true. In other words, the projection will not make the derivative of the Lyapunov function more positive. So, it is obtained that: 1 V1 ≤ − emT Qem − r T Pem 2 1 2 ≤ − λmin (Q ) em + σ λmax ( P ) em 2 1 = − em λmin (Q) em − 2 σ λmax ( P ) 2
(22)
Consequently, it can be concluded that the error em will (16)
Differentiating V1 and substituted (10) yields: 1 1 V1 = (emT Pem + emT Pem ) + tr ( w T w ) γ 2 1 T = em ( P ( A + KC ) + ( A + KC )T P )em 2 1 T Pem − r T Pem + tr ( w T w ) +η (em )T wE
(18)
where λmin (Q ) and λmax ( P ) denote the minimum and maximum eigen-value of matrices Q and P respectively.
where w = wˆ − w * .
1 1 V1 = emT Pem + tr (w T w ) 2 2γ
T Pem = tr(ET Pemη(em )T w ) η(em )T wE
49
converge to the radius r0 = κ σ with κ = 2λmax ( P ) / λmin (Q ) . 4. DESIGN OF FAULT-TOLERANT CONTROLLER
(17)
Based on the system fault being estimated by the observer, and the fault-tolerant controller of fault system needs to be designed to ensure the closed-loop stability. So, the faulttolerant controller can be designed as: u = B + ( x r − λ1e − λ2 edt − Ax − Q − Efˆ − Eurc )
(23)
where B+ is the Moore-Penrose inverse of B, x r is the reference attitude’s rate, and e = x − x r . urc is a robust controller needed to be designed. λ 1 and λ 2 are constants.
Cao Lijia et al. / IFAC PapersOnLine 52-24 (2019) 47–52
50
Substituting the controller (23) into fault model (8), the following closed-loop state equations is got: E −1 (e + λ1e + λ2 edt ) = Δ − urc (24) −1 where Δ= f +E r − fˆ .
Let s = E − 1 ( e + λ1e + λ 2 edt ) s = s dt
, urc =
k2 +1 s 2k 2
(25)
/
ρ
1.29
kg/m3
S
1.8
m
VT
10
m/s
CMα
-0.09
/
C Lr
0.036
/
C Lp
-0.19
/
C Nr
-0.21
/
CY1
-0.38
/
CLα1
-0.03
/
C Mq
-9.83
/
CLα2
0.03
/
CZα
-3.25
/
CLe1
-0.05
/
CMe1
0.272
/
CLe2
0.05
/
CMa1
0.038
/
2
CMe2
0.272
/
CMa2
0.038
/
1
0
0
1
0
1 2 s 2
(26)
Differentiating V2 yields: 2
k +1 V2 = ss = s( Δ − urc ) = s( Δ − s) 2k 2 1 1 1 1 s2 = sΔ − s 2 − 2 s 2 = − s 2 − ( 2 − 2 sΔ) 2 2k 2 2 k 1 2 1 s2 1 = − s − ( 2 − 2 sΔ + Δ2 k 2 ) + Δ2 k 2 2 2 k 2 1 2 1 s 1 = − s − ( − Δk )2 + Δ2 k 2 2 2 k 2 1 2 1 2 2 ≤− s + k Δ 2 2
49 7 0 300 20 10 P = 7 50 7 , Q = 20 60 0 0 7 50 10 0 300
The reference attitudes in the navigation frame and its’ rate in the body-fixed frame are set as:
(27)
sin(t ) 0 1 ϕ r θ = 1 − exp( −t ) / 2 , x = 0 cos ϕ ) r r ( 0 − sin ϕ cos(t ) ψ r
− sin θ ϕ r sin ϕ cos θ θr cos ϕ cos θ ψ r
The reference system fault is set as: f = [1 − sin(2t ) 1 + cos(2t ) 1 + sin(2t )]
T
Assume Δ∈ L2 [0,T ) , ∀T ∈[ 0, ∞) ; integrating (25) from t=0 to
t=T yields:
To verify the performance of fault-tolerant controller (23), A contrasted controller without fˆ and urc is set as: u = B + ( x r − λ1e − λ2 edt − Ax − Q )
T 1 T 2 1 s dt + k 2 Δ 2 dt 0 2 0 2
(28)
Then, the simulation results are shown as follows.
Since V 2 (T ) ≥ 0 , (26) implies the following: T 1 T 2 1 s dt ≤ V2 (0) + k 2 Δ2 dt 0 0 2 2
Since
V 2 (0 )
is
finite,
error Δ∈ L2 [0,T ) , that is
and
T 0
if
the
(29)
approximation
Δ 2 dt ≤ ∞ , using the Barbalat’s
lemma in (Hsu, 2007; Lin and Wang, 2011), it implies that lim s = 0 . t →∞
5. SIMULATION RESULTS A simulation is provided to demonstrate the effectiveness of the control law on the Links-Box real-time simulation platform. And the parameters are as follows: Table 1. The Parameters of the vehicle’s mathematical model (Ducard, 2009; Cao, et al. 2018). Term
Value
Unit
Term
Value
Unit
Ixx Iyy
2.56 10.9
kg·m2 kg·m2
CLβ m
0.087 28
/ kg
Izz
11.3
kg·m2
b
3.1
m
Ixz
0.5
kg·m
c
0.58
m
Izx
0.5
kg·m2
CNβ
0.087
/
2
0
λ2 = 5, σ = 0.5 , E = 0 1 0 , C = 0 1 0 , 0 0 1 0 0 1 γ = 0.01
Proof: Define a Lyapunov function as
V2 (T ) − V2 (0) ≤ −
0.053
k = 0.5, λ1 = 5
where k is constant.
V2 =
CNδ
Fig.2 Vehicle attitude system fault and it’s detection
(30)
Cao Lijia et al. / IFAC PapersOnLine 52-24 (2019) 47–52
51
errors converge quickly in the presence of different reference attitudes; Combined with (27-29) and the fault detecting performance in Fig.2, it is concluded that, with fˆ and urc to detecting and compensating unknown faults, the favourable fault-tolerant performance can be fulfilled by controller (23). From the simulation results in Fig. 4, it is observed that the smoothness and continuity of aileron, elevator and rudder deflection under external disturbance can be guaranteed by using the method proposed in this paper. 6. CONCLUSIONS In this study, first, an observer and RBFNN based fault detection method is designed to detecting the unknow fault of vehicle’s attitude system, then, a robust controller and the output of RBFNN is applied to design the fault-tolerant controller; And the effective of the proposed fault detection and tolerant method is proved by Lyapunov theorem and simulations. In the end, this work is one of the first attempts in implementing observer and RBFNN for the unmanned aerial vehicle’s fault detection and tolerant control process; this study provides a new idea that combined the intelligence methods with the traditional control methods to solve the practical engineering problems.
Fig.3 Vehicle’s attitudes Tacking
ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation of China (Nos.61703409, 11705122), Sichuan Science and Technology Program (Nos. 2018JY0197, 19ZDZX0037); Nature Science Foundation of Sichuan University of Science & Engineering (Nos.2017RCL12, 2018RCL18);Sichuan Key Provincial Research Base of Intelligent Tourism (No.ZHZJ18-01); Aeronautical Science Foundation of China (No.201605U8002); Xi'an Scientific and Technological Program (No:201805048YD26CG32(2)); Research Foundation of Department of Education of Sichuan Province (No.17ZA0271). REFERENCES
Fig.4 Vehicle’s deflections of aileron, elevator and rudder From the simulation results in Fig.2, it can be seen that favourable fault detecting performance can be achieved by the proposed observer based RBFNN fault detection method. −1 So the term Δ= f +E r − fˆ satisfy
T 0
Δ 2 dt ≤ ∞ , combined
with (27-29), it can be concluded that lim s = 0 . t →∞
From the simulation results in Fig.3, it is seen that, compared to the controller (30), the controller (23) make the tracking
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