Actuator fault tolerant control using adaptive RBFNN fuzzy sliding mode controller for coaxial octorotor UAV

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Research article

Actuator fault tolerant control using adaptive RBFNN fuzzy sliding mode controller for coaxial octorotor UAV Samir Zeghlachea, Hemza Mekkib, Abderrahmen Bouguerrab,∗, Ali Djeriouib a

LASS, Laboratoire d’Analyse des Signaux et Systèmes, Department of Electrical Engineering, Faculty of Technology, Mohamed Boudiaf University of Msila, BP 166 Ichbilia, Msila, Algeria b LGE, Laboratoire de Génie Electrique, Department of Electrical Engineering, Faculty of Technology, Mohamed Boudiaf University of Msila, BP 166 Ichbilia, Msila, Algeria

A R T I C LE I N FO

A B S T R A C T

Keywords: Coaxial octorotor Radial basis function neural network Fuzzy logic Sliding mode control Actuator fault tolerant control Lyapunov stability

In this paper, a robust controller for a Six Degrees of Freedom (6 DOF) coaxial octorotor helicopter control is proposed in presence of actuator faults. Radial Base Function Neural Network (RBFNN), Fuzzy Logic Control approach (FLC) and Sliding Mode Control (SMC) technique are used to design a controller, named Fault Tolerant Control (FTC), for each subsystem of the octorotor helicopter. The proposed FTC scheme allows avoiding difficult modeling, attenuating the chattering effect of the SMC, reducing the rules number of the fuzzy controller, and guaranteeing the stability and the robustness of the system. The simulation results show that the proposed FTC can greatly alleviate the chattering effect, good tracking in presence of actuator faults.

1. Introduction Helicopters are dynamically unstable and therefore suitable control methods are needed to stabilize them. In order to be able to optimize the operation of the control loop in terms of stability, precision and reaction time, it is essential to know the dynamic behavior of the process which can be established by a representative mathematical model. The robust control methods proposed in the literature are generally based on the inherent structure of quadrotor that leads to the deficiency in driving capability. A novel coaxial eight-rotor UAV model proposed in Ref. [1] has been used in this paper. It is designed with eight rotors that are arranged as four counter rotating offset pairs mounted at the ends of four carbon fiber arms in a cruciform configuration. The four sets of matched counter rotating rotor blades provide differential thrust from four equally spaced points, which allows the eight-rotor UAV to maneuver with higher agility. Besides that, it offers the advantage of markedly increased drive capability and greater ability to resist the disturbances of 8-rotor helicopter hybridizing tow quadrotors. In addition, the coaxial eight-rotor UAV has nearly twice overall thrust than a quad-rotor UAV without doubling the weight, and keeps the same floor space as a quad-rotor UAV. Obviously, the proportion of high coefficient between thrust and gravity, as well as the higher payload capacity, are provided by the eight-engine UAVs compared to a shelf UAV for the same type of engines and rotors [1]. Furthermore, the coaxial octorotor UAV can be designed to accommodate the case of motors failure keeping the helicopter flying.

∗

The contribution of this paper is that it provides very simple and effective fault detection, reconstruction and tolerance strategy for the coaxial octorotor helicopter. 2. Problem formulation Past research of the universal approximation theorem [2,3] show that any nonlinear function over a compact set with arbitrary accuracy can be approximated by the RBFNN. There have been significant research efforts on the RBFNN control for nonlinear systems [4]. In the present work an adaptive neural sliding mode control algorithm is proposed for a class of continuous time unknown nonlinear systems. This is in contrast to the existing sliding mode control design where the presence of hitting control may introduce problems to the controlled systems. These unknown nonlinearities are approximated by the RBFNN whose weight value parameters are adjusted on-line according to some adaptive laws. The purpose of controlling the output of the nonlinear system is to track a given trajectory. Based on the RBF model, the Lyapunov synthesis approach is used to develop an adaptive control algorithm. The chattering action is attenuated and a robust performance can be ensured. The stability analysis for the proposed control algorithm is provided. The RBFNN is used to approximate the unknown part of octorotor helicopter dynamic equation. This does not require modeling. Also, the approximation error, disturbance and the effects of faults can be compensated by the sliding mode control. The control problem considered in this work is to perform

Corresponding author. E-mail addresses: [email protected] (S. Zeghlache), [email protected] (H. Mekki), [email protected] (A. Bouguerra), [email protected] (A. Djerioui).

https://doi.org/10.1016/j.isatra.2018.06.003 Received 19 January 2018; Received in revised form 24 March 2018; Accepted 1 June 2018 0019-0578/ © 2018 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Zeghlache, S., ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.06.003

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asymptotic position and attitude tracking of the coaxial octorotor by designing flight controllers depended on FTC in presence of the actuator faults. To obtain (x , y, z ) → (x d , yd , z d ) and (φ , θ , ψ) → (φd , θd , ψd ) . Based on the dynamical model of the coaxial octorotor helicopter, the control system is divided into multiple subsystems (fully actuated subsystem and underactuated subsystem). 3. Prior work The multi rotors helicopters are underactuated systems with highly coupled states. Great efforts have been made to control multi rotors helicopter and some strategies have been developed to solve the path tracking problems for this type of systems. Compared to the existing works in the literature [5–37], the contributions of this paper are highlighted in the following aspects:

•

• Compared with the passive fault tolerant control presented in Refs. • •

• • •

•

•

•

[5–9] which requires the fault estimation. This design method becomes very difficult in case of a large or severe fault/damage. In this work the designed FTC require a simple RBFNN for fault detection and reconstruction. The FTC scheme proposed in this paper (based SMC, RBFNN and fuzzy inference system) has certain advantages compared to [10–16], which are based on backstepping and traditional SMC methods for multi rotors helicopter. In Refs. [17,18] authors propose a very complex FTC structure based on projection method, where, this later needs a switching block (to switch between two controls strategies). In this work we propose a simple FTC structure which does not need switching block and uses only one control strategy based on SMC, RBFNN and fuzzy inference system. In Refs. [19,20] the authors propose a FTC based sliding mode control for nonlinear longitudinal model of a Boeing 747-100/200 airplane in presence the modeling uncertainties and actuator faults. Compared with [21–23], where authors used non linear observers for fault detection and reconstruction. In this paper the RBFNN is used to detect, reconstruct the faults. In Refs. [24,25] a complex FTC scheme applied to the aeronautical system based on appropriate combination of SMC and control allocation has been proposed. In this work the proposed FTC approach is based on combination between simple SMC, RBFNN and fuzzy inference system the proposed combined scheme could be interesting and this approach can achieve tolerance to a wide class of total system failures. In Refs. [26,27], in those papers authors propose respectively a novel combination between fault reconstruction and fault tolerant attitude control for aircraft benchmark under sensor failure and for over-activated spacecraft with reaction wheels under actuator failure based sliding mode observer and optimal control method. Indeed, the proposed algorithm in our case is based on RBFNN and robust fuzzy sliding mode control method for coaxial octorotor helicopter under actuator failure. A fuzzy integral sliding mode-backstepping controller for an autonomous quadrotor helicopter (X4-flayer) is proposed in Ref. [28], in which the integral action is replaced by an inference fuzzy system to eliminate the static error. Moreover, a boundary layer (“sat” function) is used to reduce the chattering phenomenon. However, the fuzzy logic inference mechanism is not employed in the stability analysis. By against in the proposed work the fuzzy inference system mechanism is employed in the stability analysis using Lyaponov theorem. An intelligent fuzzy controller was applied in several papers such as [29–33] to control the position and orientation of a multi rotor unmanned aerial vehicles with good response in simulation. However, a major limitation of this works was the trial and error approach for tuning of input variables. In Ref. [29] a fuzzy controller

•

based on on-line optimization of a zero order Takagi-Sugeno fuzzy inference system, which is tuned by a back propagation algorithm, is applied to a quadrotor where some uncertainties are considered. The work presented in Refs. [30–32] proposes an adaptive neural network control to stabilize a multi rotor unmanned aerial vehicles against modeling error and wind disturbance. Intelligent controls algorithm based on neural networks proposed in Ref. [33] handle better among the coupling of the states, the uncertainty of model interference but it can't compensate the actuator faults. In this work a non linear control based on robust controller is proposed to achieve position and angle finite time convergence in presences the uncertainty of model due to actuator failure. Adaptive global sliding mode controllers were employed in a class of non linear systems based on RBF neural network, double loop recurrent neural network and fuzzy compensator to handle model uncertainties and external disturbances [34–36]. This algorithm presents a quick response, not need a precise model, and not sensitive to external disturbances but the faults are not taken into account. In this work an adaptive sliding mode based on RBFNN and fuzzy inference system is proposed in the presence of model uncertainties, external disturbance and actuator failure. As stated in Ref. [37], A Recent approach for dealing with uncertainties is based on the use of sliding mode method to enhance the robustness of FTCs. However, the problem of FTC based on SMC schemes is still in its early stage of development, and a few results have been reported in the literature. In this contest and considering the existing results, the systemic design of the FTC scheme in this paper focuses on very simple structure, more sensitive detection and a quicker compensation procedure.

3.1. Contribution The main contribution of this paper is to propose an adaptive and robust control approach in presence of actuator faults were the proposed method consists on the combination of RBFNN, which is used to estimate the control low of the SMC without knowing the inverse dynamics and a fuzzy inference system to attenuate the chattering effect caused by the discontinues control part of the sliding mode controller for coaxial octorotor helicopter. By using Lyapunov's method, the tracking errors of the closed loop are proven to be exponentially stable where the fuzzy logic inference mechanism is employed in the stability analysis. Compared to the techniques available in the literature the main contributions of the paper can be summarized as follows: 1) A new coaxial octorotor helicopter robust control methodology is introduced based on hybrid control such as fuzzy sliding mode control and RBFNN; 2) In contrast to [38,39] where stability is not proven, in the proposed approach an exponential tracking error convergence is guaranteed employing the fuzzy inference mechanism for the coaxial octorotor helicopter in its nonlinear coupled configuration and in the presence of the error modeling, disturbance and actuators faults; 3) The proposed control strategy does not need a decoupling procedure, nor use local linear sub-models; 4) The proposed FTC controller can eliminate the chattering phenomenon which characterizes the sliding mode control; 5) A suitable adaptive parameter-tuning law is suggested to dominate the perturbations of the system and the actuator faults without the information of their upper bounds. In order to proof the effectiveness of the proposed approach, simulation studies of the control scheme are carried out using Matlab/ Simulink and the coaxial octorotor helicopter dynamical model. The proposed control scheme allows us to avoid the nonlinear modeling problems, guarantee the stability of the helicopter in closed loop, to provide the robustness and to obtain the desired trajectory tracking 2

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rotors and are considered as the real control inputs to the dynamical system, b denotes the lift coefficient; d denotes the force to moment scaling factor. Assumption 1. The roll, the pitch and the yaw angles (φ , θ , ψ) are −π π −π π bounded as follows: roll angle by 2 < φ < 2 ; pitch angle, 2 < φ < 2 ; and yaw angle, − π < ψ < π . 4.2. Dynamical modeling of the coaxial octorotor in faulty operation The dynamic model of the octorotor helicopter in faulty operation is presented by: Fig. 1. Coaxial octorotor configuration.

⎧ φ¨ = ⎪ ⎪ θ¨ = ⎪ ⎪ ψ¨ =

1 ˙ ˙ {θψ (Iy Ix

− Iz ) − Kfax φ˙ 2 − JH Ω θ˙ + U2f }

1 ˙ ˙ (Iz {φψ Iy

− Ix ) − Kfax θ˙ − JH Ω φ˙ + U3f }

1 ˙ ˙ (Ix {φθ Iz

− Iy ) − Kfax ψ˙ + U4f }

4. Dynamical model

⎨ ⎪ x¨ = ⎪ ⎪ y¨ = ⎪ z¨ = ⎩

1 {(cos φ sin θ cos ψ + sin φ sin ψ) U1f m 1 {(cos φ sin θ sin ψ − sin φ cos ψ) U1f m 1 {(cos φ cos θ) U1f − Kftz z˙} − g m

4.1. Dynamical modeling of the coaxial octorotor in healthy operation

The fault in each rotor is modeled with a bias which multiplies to the forces in healthy operation shown in equation (5)

with best performances in presence of actuator faults. The rest of paper is organized as follow. In section 4, nonlinear dynamic model of the quadrotor aircraft is introduced in healthy and faulty operations. The octorotor UAV flight controller design based on FTC is presented in section 5, simulation results are given in section 6. Finally conclusions on the present paper are driven.

⎨ ⎪ x¨ = ⎪ ⎪ y¨ = ⎪ z¨ = ⎩

1 ˙ ˙ {θψ (Iy Ix

− Iz ) − K1 φ˙ 2 − JH Ω θ˙ + U2}

1 ˙ ˙ (Iz {φψ Iy

− Ix ) − K2 θ˙ − JH Ω φ˙ + U3}

1 ˙ ˙ (Ix {φθ Iz

− Iy ) − K3 ψ2 + U4}

(6)

⎧ φ¨ = ⎪ ⎪ θ¨ = ⎪ ⎪ ψ¨ =

1 ˙ ˙ {θψ (Iy Ix

− Iz ) − Kfax φ˙ 2 − JH Ω θ˙ + U2} + dφ

1 ˙ ˙ (Iz {φψ Iy

− Ix ) − Kfax θ˙ − JH Ω φ˙ + U3} + dθ

1 ˙ ˙ (Ix {φθ Iz

− Iy ) − Kfax ψ˙ + U4} + d ψ

⎨ ⎪ x¨ = ⎪ ⎪ y¨ = ⎪ z¨ = ⎩

1 {(cos φ sin θ cos ψ m 1 {(cos φ sin θ sin ψ m 1 {(cos φ cos θ) U1 − m

2

2

+ sin φ sin ψ) U1 − Kftx x˙ } + d x − sin φ cos ψ) U1 − Kfty y˙ } + d y Kftz z˙} − g + d z

(7)

⎧ dφ = − 2I2 l (β7 F7 + β8 F8 + β5 F5 + β6 F6 − β3 F3 − β4 F4 − β1 F1 − β2 F2) x ⎪ ⎪ dθ = − 2 l (β F3 + β F4 + β F5 + β F6 − β F7 − β F8 − β F1 − β F2) 3 4 5 6 7 8 1 2 2Iy ⎪ ⎪ d ⎪ d ψ = − b (β2 F2 + β3 F3 + β6 F6 + β7 F7 − β1 F1 − β4 F4 − β5 F5 − β8 F8) ⎪ ⎪ d x = − cos φ sin θ cos ψ + sin φ sin ψ (β F1 + β F2 + β F3 + β F4 + β F5 + β F6 1 2 3 4 5 6 m ⎨ ) β F β F + + ⎪ 7 7 8 8 ⎪ cos φ sin θ sin ψ − sin φ cos ψ (β1 F1 + β2 F2 + β3 F3 + β4 F4 + β5 F5 + β6 F6 d = − y ⎪ m ⎪ + β7 F7 + β8 F8) ⎪ ⎪ cos φ cos θ (β1 F1 + β2 F2 + β3 F3 + β4 F4 + β5 F5 + β6 F6 + β7 F7 + β8 F8) ⎪ dz = − m ⎩ (8)

(2)

⎧U1 = F1 + F2 + F3 + F4 + F5 + F6 + F7 + F8 ⎪U = l 2 (F + F + F + F − F − F − F − F ) 7 8 5 6 3 4 1 2 2 ⎪ 2

where Fi = b

(5)

Where the unknown resultant actuator faults are given by:

(1)

Ω i ; stand for the angular speed of the propeller i with i = 1, ..., 8.Ix , Iy , Iz represent the inertias of the coaxial octorotor; JH denotes the inertia of the propeller; U1denotes the total thrust on the body in the z-axis; U2 and U3 represent the roll and pitch inputs, respectively; U4 denotes a yawing moment [42].

Ω i2

(4)

If we replace equation (6) in (4) we obtain

where m denotes the total mass. g represents the acceleration of gravity. l denotes the distance from the center of each rotor to the center of gravity; Ki denote the drag coefficients and positive constants.

⎨U3 = l 2 (F3 + F4 + F5 + F6 − F7 − F8 − F1 − F2) 2 ⎪ ⎪U4 = d (F2 + F3 + F6 + F7 − F1 − F4 − F5 − F8) b ⎩

0 ≤ βi ≤ 1 (i = 1…8)

⎨U3f = l 2 (F3f + F4f + F5f + F6f − F7f − F8f − F1f − F2f ) 2 ⎪ ⎪U4f = d (F2f + F3f + F6f + F7f − F1f − F4f − F5f − F8f ) b ⎩

+ sin φ sin ψ) U1 − K 4 x˙ }

Ω = Ω2 + Ω3 + Ω6 + Ω7 − Ω1 − Ω4 − Ω5 − Ω8

− Kfty y˙ }

⎧U1f = F1f + F2f + F3f + F4f + F5f + F6f + F7f + F8f ⎪ 2 ⎪U2f = l 2 (F7f + F8f + F5f + F6f − F3f − F4f − F1f − F2f )

− sin φ cos ψ) U1 − K5 y} K 6 z˙} − g

− Kftx x˙ }

U1f , U2f , U3f and U4f are the control inputs of the system in faulty operation written by:

2

1 {(cos φ sin θ cos ψ m 1 {(cos φ sin θ sin ψ m 1 {(cos φ cos θ) U1 − m

2

Fif = (1 − βi ) Fi

The configuration of the octorotor is represented in Fig. 1. It is similar to a quadrotor with two coaxial counter-rotating motors at the ends of each arm. This configuration has several advantages compared to the star configuration used in the literature [40,41] in terms of stability and size. A classical star octorotor needs more arms, and each arm needs to be longer to guarantee adequate spacing among the rotors. We have adopted this configuration for its higher thrust to weight ratio. Consider first a body-fixed reference frame RB with the X, Y, Z axis originating at the center of mass of the vehicle. The X axis points to the front direction, the Y axis to the left direction and the Z axis upwards. Consider second an inertial frame RI fixed to the earth{o, x , y, z }. The equations governing the motion of the system are obtained using the Euler-Lagrange approach and give the commonly used model [42].

⎧ φ¨ = ⎪ ⎪ θ¨ = ⎪ ⎪ ψ¨ =

2

Assumption 2. The resultant of actuator faults related to the helicopter is bounded where:

(3)

dj ≤ d+j , j = φ , θ , ψ, x , y, z·d+j > 0

with i = 1, …, 8 denote the thrust generated by eight 3

(9)

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Fig. 2. Bloc diagram of the proposed FTC. Table 1 Rules base [43]. 3

5. The proposed FTC design for octorotor Helicopter

ufsq = flc (sq) =

This section mainly introduces the FTC method that is applied here to design the fault tolerant controller of the coaxial octorotor helicopter as shown in Fig. 2. In the case of our work in order to eliminate chattering, a FLC is used to approximate the discontinue control computed by:

ufsq = flc (sq)·q = z , ψ, φ , θ

sj ufsj

∑p = 1 αqp ωqp 3

∑p = 1 αqp

= αq1 ωq1 + αq2 ωq2 + αq3 ωq3·q = z , ψ, φ, θ

(11)

Rule1

Rule2

Rule3

P

Z

N

P

Z

N

(10)

ufsq = flc (sq) = ωq ·q = z , ψ, φ , θ

where ufsq,q = z , ψ, φ , θ is the outputs of the FLC, which is obtained by the sliding surfaces variables sq ,q = z,ψ,j,θ. The fuzzy membership functions of the input variables sq, q = z , ψ, φ , θ and output discontinuous control ufsq, q = z , ψ, x , y sets are presented by Fig. 3. All the membership functions of the fuzzy input variable are chosen to be triangular. The used labels of the fuzzy variable surfaces sq are: {negative (N), zero (Z) and positive (P)}. The fuzzy labels of the hitting control variables ufsq are a singletons membership functions decomposed into three levels represented by a set of linguistic variables: {negative (N), zero (Z) and positive (P)}. Table 1 presents the rules base 0≤ which contains 3 rules [43]:where αq1 ≤ 1,0 ≤ αq2 ≤ 1 and 0 ≤ αq3 ≤ 1, for q = z , ψ, φ , θ are the membership degree of rules 1,2 and 3 presented in Table 1, ωq1 = ωq , ωq2 = 0 and ωq3 = −ωq, are the center of output membership functions ufsq for q = z , ψ, φ , θ .The relation αq1 + αq2 + αq3 = 1 for q = z , ψ, φ , θ is valid according to the triangular membership functions. Moreover, the fuzzy hitting control can be further analyzed as the following four conditions [43], and only one of four conditions will occur for any value of sq, q = z , ψ, φ , θ according to Fig. 3. Condition 1 Only rule 1 is activated (sq > βq, αq1 = 1, αq2 = αq3 = 0)

Condition 2 Rule 1 and 2 are (0 < sq ≤ βq, 0 < αq1, αq2 ≤ 1, αq3 = 0)

(12) activated

simultaneously

ufsq = flc (sq) = αq1 ωq1 = αq1 ωq ·q = z , ψ, φ , θ Condition 3 Rule 2 and 3 are (−β < si ≤ 0i , 0 < αi2, αi3 ≤ 1, αi1 = 0)

activated

(13) simultaneously

ufsq = flc (sq) = αq3 ωq3 = −αq3 ωq ·q = z , ψ, x , y

(14)

Condition 4 Only rule 3 is activated (si ≤ −βi , αi1 = 1, αi2 = αi3 = 0)

ufsq = flc (sq) = ωq ·q = z , ψ, φ , θ

(15)

According to four possible conditions shown in (12)–(15)

ufsq = flc (sq) = ωq (αq1 − αq3)·q = z , ψ, φ , θ

(16)

And it can see that

sq ufsq = sq flc (sq) = ωq αq1 − αq3 sq ·q = z , ψ, φ , θ

Fig. 3. Membership functions of input variablessq and output ufsq for q = z , ψ, φ, θ 4

(17)

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where x z and x ψ are the inputs state of the network, k is the input number of the network, l is the number of hidden layer nodes in the network, Hz = [Hz1 Hz 2…Hzn]T and Hψ = [Hψ1 Hψ2…Hψn]T are the output of Gaussian function, Wz and Wψ are the neural network weights, εz and εψ are approximation errors of neural network, εz ≤ εzN and εψ ≤ εψN . RBFNN approximation of the functions fz and fψ is used. The network input is selected as x z = [ez e˙ z ]T and x ψ = [eψ e˙ ψ]T , and the output of RBFNN is:

5.1. Controller design for fully actuated subsystem A controller for the fully actuated subsystem of the coaxial octorotor is designed by using the proposed FTC. The objective is to ensure that the state variables(z , ψ) converge to the desired values (z d , ψd ) despite the presence of actuator faults. The altitude and yaw subsystems models can be rewritten by:

cos φ cos θ U1 − g + fz m

z¨ =

U ψ¨ = 4 + fψ Iz

(18)

ˆ zT Hz (x z ) fˆz (x z ) = W

(32)

(19)

ˆ ψT Hψ (x ψ) fˆψ (x ψ) = W

(33)

(20)

where Hz (x z ) and Hψ (x ψ) are the Gaussian function of neural network. The control input U1 and U4 in equations (26) and (27) are rewritten as:

With fz and fψ are unknown functions obtained by:

fz = d z −

fψ =

Kftz m

(Ix − Iy ) Iz

z˙

˙ ˙− φθ

Kfax Iz

2 ψ˙ + d ψ

U1 =

m (−fˆz (x z ) + g + z¨d + cz e˙ z + ηˆz flc (sz ) + μz sz ) cosφcosθ

(34)

U4 =

1 (−fˆψ (x ψ) + ψ¨d + c ψ e˙ ψ + ηˆψ flc (sψ) + μψ sψ) Iz

(35)

(21)

The tracking errors along altitude and yaw subsystems are presented by:

ez = z d − z

(22)

eψ = ψd − ψ

(23)

The time derivatives of the sliding surfaces in equations (24) and (25) are computed by:

s˙ z = z¨d −

The sliding surfaces are chosen by:

sz = e˙ z + cz ez

(24)

sψ = e˙ ψ + c ψ eψ

(25)

1 s˙ ψ = ψ¨d − U4 − fψ (x ψ) + c ψ e˙ ψ Iz

U1∗

m = (−fz + g + z¨d + cz e˙ z + ηˆz flc (sz ) + μz sz ) cosφ cosθ

(26)

U4∗

1 = (−fψ + ψ¨d + c ψ e˙ ψ + ηˆψ flc (sψ) + μψ sψ) Iz

(27)

fz =

WzT Hz (x z )

+ εz

fψ = WψT Hψ (x ψ) + εψ

(37)

s˙ z = −fz (x z ) + fˆz (x z ) − ηˆz flc (sz ) − μz sz

(38)

s˙ ψ = −fψ (x ψ) + fˆψ (x ψ) − ηˆψ flc (sψ) − μψ sψ

(39)

Let us define the approximation errors between the uncertainties functions fz (x z ), fψ (x ψ), and the estimated uncertainties functions fˆ (x z ), fˆ (x ψ) by:

With μ z and μ ψ are the positive constants, ηˆ z and ηˆ ψ are estimated online by suitable adaptation laws. RBF networks are adaptively used to approximate the unknown fz and fψ . The structures of RBFNN with receptive field units are shown in Fig. 4(a) and (b).

( x ψ − Cψkl 2 ) ⎞ Hψl = Gψ ⎜⎛ ⎟ 2 Bψl ⎝ ⎠

(36)

Submitting equation (34) into (36) and (35) into (37), we obtain

where the coefficient cz > 0 and c ψ > 0 The corresponding ideal law controls are given by:

( x z − Czkl 2 ) ⎞ Hzl = Gz ⎜⎛ ⎟ Bzl2 ⎝ ⎠

cos(φ)cos(θ) U1 + g − fz (x z ) + cz e˙ z m

z

ψ

fz͠ (x z ) = fz (x z ) − fˆz (x z )

(40)

fψ͠ (x ψ) = fψ (x ψ) − fˆψ (x ψ)

(41)

equations (38) and (39) are rewritten by:

(28)

(29)

s˙ z = −fz͠ (x z ) − ηˆz flc (sz ) − μ z sz

(42)

s˙ ψ = −fψ͠ (x ψ) − ηˆψ flc (sψ) − μψ sψ

(43)

By replacing (30), (32) in (40) and (31), (33) in (41), we get

(30)

ˆ zT Hz (x z ) fz͠ (x z ) = WzT Hz (x z ) + εz − W

(31)

Fig. 4. The structure of RBFNN in the (a) altitude channel (b) yaw channel. 5

(44)

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Fig. 5. The structure of RBFNN in the (a) roll channel (b) pitch channel.

∼T fψ͠ (x ψ) = Wψ Hψ (x ψ) + εψ

Table 2 Physical parameters of the coaxial octorotor. Symbol

Value

Ix Iy IZ Kp Kd

4.2 × 10−2 (Kg m2) 4.2 × 10−2 (Kg m2) 7.5 × 10−2 (Kg m2) 2.9842 × 10−5 (Ns2) 3.2320 × 10−7 (Nms2) 1.6 (Kg) 9.81(m/s2) 0.23 (m) 5.5670 × 10−4 (N/m/s) 5.5670 × 10−4 (N/m/s) 6.3540 × 10−4 (N/m/s) 5.5670 × 10−4 (N/rad/s) 5.5670 × 10−4 (N/rad/s) 6.3540 × 10−4 (N/rad/s)

m

g l Kftx Kfty Kftz Kfax Kfay Kfaz

(49)

5.2. Stability analysis The Lyapunov functions are defined by:

Vr =

1 2 1 ∼T ∼ 1 sr + βr Wr Wr + λr η͠ r2 ·r = z , ψ 2 2 2

(50)

where βr and λ r for r = z , ψ are a positives coefficients. Defining the switching gain adaptation error as:

η͠ r = ηr − ηˆr ·r = z , ψ

(51)

where ηr are the actual switching gains, the estimated values of the actual switching gains ηˆr will be estimated by using the adaptation law

ˆ z, W ˆ ψ as the weighting error vectors between the optimal Define W ˆ z, W ˆ ψ and the current estimated values W ˆ z, W ˆ ψ by: values W

1 ηˆ˙r = − ωr αr1 − αr 3 sr ·r = z , ψ λr (52) ∼ ∼ The time derivative of Wz , Wψ and η͠ r in equations (46), (47) and (51) are given by:

∼ ˆ zT Wz = WzT − W

(46)

T ∼ ˆ˙ z W˙ z = −W

(53)

∼ ˆ ψT Wψ = WψT − W

(47)

T ∼ ˆ˙ ψ W˙ ψ = −W

(54)

η˙͠ r = −ηˆ˙r ·r = z , ψ

(55)

ˆ ψT Hψ (x ψ) fψ͠ (x ψ) = WψT Hψ (x ψ) + εψ − W

(45)

Then equations (44) and (45) can be rewritten by:

fz͠ (x z ) =

∼T W z Hz (x z )

+ εz

The time derivatives of the Lyapunov functions in equation (50) are obtained by:

(48)

Fig. 6. Trajectory of the output variables: faults in five motors. 6

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Fig. 7. Trajectories of roll and pitch angles: faults in five motors.

Fig. 8. Control inputs: faults in five motors.

∼T ∼ V˙r = sr s˙r + βr Wr W˙ r + λr η͠ r η˙͠ r ·r = z , ψ

εr for r = z , ψ is sufficiently small in the design.

(56)

By substituting (42), (43), (48), (49), (52), (53), (54), and (55) into (56), which is followed by simplification, the time derivative of Lyapunov functions are

ηr ωr αr1 − αr 3 ≥ εr N ·r = z , ψ

∼T ˆ˙r ) − sr (εr + ηˆ flc (sr ) + μ sr ) + λr (ηˆ − η ) ηˆ˙ ·r V˙r = −Wr (sr Hr (x r ) + βr W r r r r r

V˙r = −ϑr sr − μr sr2 < 0 with ϑr > 0, μr > 0 for r = z , ψ

= z, ψ

We obtain

5.3. Controller design for underactuated subsystem A controller for the underactuated subsystem of the coaxial octorotor is designed by using the proposed FTC. The objective is to ensure that the state variables(x , θ) and (y, φ) converge to the desired values (x d , θd ) and (yd , φd ) despite the presence of actuator faults. The underactuated subsystem model in faulty operation can be rewritten as:

(58)

By replacing (58) in (57), we get

V˙r = −sr εr − ηr ωr αr1 − αr 3 sr − μr sr2·r = z , ψ

(61)

(57)

The parameters of the RBFNN are adjusted by the following laws:

ˆ˙r = − 1 sr Hr (x r )·r = z , ψ W βr

(60)

(59)

We get V˙r < 0 for r = z , ψ approximately as the approximation error 7

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U3∗ =

Iy l

(−fθ + c1θ e¨x + c2θ e˙ x + θ¨d + c3θ e˙ θ + ηˆθ flc (sθ ) + μθ sθ )

(74)

With μφ and μθ are the positive constants, ηˆφ and ηˆθ are estimated online by suitable adaptation laws. RBF networks are adaptively used to approximate the unknown fφ and fθ . The structures of RBFNN with receptive field units are shown in Fig. 5(a) and (b).

Fig. 9. System position in the 3D space: faults in five motors.

⎧ φ¨ = ⎪ ⎪ θ¨ = ⎨ x¨ = ⎪ ⎪ y¨ = ⎩

l U Ix 2

+ fφ

l U Iy 3

+ fθ

(cos φ sin θ cos ψ + sin φ sin ψ) U1 m (cos φ sin θ sin ψ − sin φ cos ψ) U1 m

(62)

With fφ , fθ , fx and f y are unknown functions obtained by

˙ K ( ) ˙ ˙ Iy − Iz − fax φ˙ 2 − JH Ω θ + dφ fφ = θψ Ix Ix Ix Kfax 2 (I − Ix ) J Ω φ˙ ˙ ˙ z fθ = φψ − θ˙ − H + dθ Iy Iy Iy

fx = d x − fy = d y −

Kftx m Kfty m

x˙

y˙

(76)

fφ = WφT Hφ (x φ) + εφ

(77)

fθ = WθT Hθ (x θ ) + εθ

(78)

(79)

(63)

ˆ θT Hθ (x θ ) fˆθ (x θ ) = W

(80)

(64)

where Hφ (x φ) and Hθ (x θ ) are the Gaussian function of neural network. The control input U2 and U3in equations (64) and (65) are rewritten as:

(65)

U2 =

(66)

U3 =

ex = x d − x

(67)

e y = yd − y

(68)

eφ = φd − φ

(69)

eθ = θd − θ

(70)

sφ = c1φ e˙ y + c2φ e y + e˙ φ + c3φ eφ

(71)

sθ = c1θ e˙ x + c2θ ex + e˙ θ + c3θ eθ

(72)

Ix ˆ (−fφ (x φ) + c1φ e¨y + c2φ e˙ y + φ¨d + c3φ e˙ φ + ηˆφ flc (sφ) + μφ sφ) l Iy l

(−fˆθ (x θ ) + c1θ e¨x + c2θ e˙ x + θ¨d + c3θ e˙ θ + ηˆθ flc (sθ ) + μθ sθ )

(81) (82)

The time derivatives of the sliding in equations (62) and (63) are computed by:

s˙ φ = φ¨d −

l U2 − fφ (x φ) + c1φ e¨y + c3φ e˙ φ + c2φ e˙ y Ix

l s˙θ = θ¨d − U3 − fθ (x θ ) + c1θ e¨x + c3θ e˙ θ + c2θ e˙ x Iy

(83)

(84)

Submitting equation (81) into (83) and (82) into (84), we obtain

The sliding surfaces are chosen as [44]:

s˙ φ = −fφ (x φ) + fˆφ (x φ) − ηˆφ flc (sφ) − μφ sφ

(85)

s˙θ = −fθ (x θ ) + fˆθ (x θ ) − ηˆθ flc (sθ ) − μθ sθ

(86)

Let us define the approximation errors between the uncertainties functions fφ (x φ), fθ (x θ ) and the estimated uncertainties functions fˆ (x φ), fˆ (x θ ) by:

where the coefficient cuθ > 0 and cuφ > 0 for (u = 1…3) The corresponding ideal law controls are given by:

Ix (−fφ + c1φ e¨y + c2φ e˙ y + φ¨d + c3φ e˙ φ + ηˆφ flc (sφ) + μφ sφ) l

( x − C 2) Hθl = Gθ ⎛⎜ θ 2 θkl ⎞⎟ Bθl ⎠ ⎝

ˆ φT Hφ (x φ) fˆφ (x φ) = W

The tracking errors along the underactuated subsystem are presented by:

U2∗ =

(75)

where x φ and x θ are the inputs state of the network, k is the input number of the network, l is the number of hidden layer nodes in the network, Hφ = [Hφ1 Hφ2…Hφn]T and Hθ = [Hθ1 Hθ2…Hθn]T are the output of Gaussian function, Wφ and Wθ are the neural network weights, εφ and εθ are approximation errors of neural network, εφ ≤ εφN and εθ ≤ εθN . RBF network approximation of the functions fφ and fθ is used. The network input is selected as x φ = [eφ e˙ φ]T and x θ = [eθ e˙ θ]T , and the output of RBFNN is

+ fx + fy

2 ⎛ ( x φ − Cφkl ) ⎞ Hφl = Gφ ⎜ ⎟ 2 Bφl ⎝ ⎠

φ

θ

fφ͠ (x φ) = fφ (x φ) − fˆφ (x φ)

(73)

(87) Fig. 10. Evolution of the parameters variations.

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Fig. 11. Trajectory of the output variables: parameters variations.

Fig. 12. Trajectories of roll and pitch angles: parameters variations.

fθ͠ (x θ ) = fθ (x θ ) − fˆθ (x θ )

η͠ v = ηv − ηˆv ·v = φ , θ

(88)

(98)

where ηv are the actual switching gains, the estimated values of the actual switching gains ηˆv will be estimated by using the adaptation law

equations (85) and (86) are rewritten by:

s˙ φ = −fφ͠ (x φ) − ηˆφ flc (sφ) − μφ sφ

(89)

s˙θ = −fθ͠ (x θ ) − ηˆθ flc (sθ ) − μθ sθ

(90)

1 ηˆ˙v = − ω v α v1 − α v3 s v ·v = φ , θ (99) λr ∼ ∼ The time derivative of Wz , Wψ ,and η͠ r in equations (93), (94) and (98) are given by:

(91)

T ∼ ˆ˙ z W˙ z = −W

(100)

(92) ∼ ∼ Define Wφ, Wθ as the weighting error vectors between the optimal ∼ ∼ ∼ ∼ values Wφ, Wθ and the current estimated values Wφ, Wθ by:

∼ W˙ ψ =

T ˆ˙ ψ −W

(101)

∼ ˆ φT Wφ = WφT − W

(93)

The time derivatives of the Lyapunov functions in the equation (50) are obtained by:

∼ ˆ θT Wθ = WθT − W

(94)

∼T ∼ V˙v = s v s˙ v + βv Wv W˙ v + λ v η͠ v η˙͠ v ·v = φ , θ

(95)

By substituting (89), (90), (95), (96), (99), (100), (101) and (102) into (103), which is followed by simplification, the time derivative of Lyapunov functions are

(96)

∼T ˆ˙ v ) − s v (ε v + ηˆ flc (s v ) + μ s v ) V˙v = −Wv (s v Hv (x v ) + βv W v v

By replacing (77), (79) in (87) and (78), (80) in (88), we get

ˆ φT Hφ (x φ) fφ͠ (x φ) = WφT Hφ (x φ) + εφ − W ˆ θT Hθ (x θ ) fθ͠ (x θ ) = WθT Hθ (x θ ) + εθ − W

η˙͠ v = −ηˆ˙v ·v = φ , θ

Then equations (91) and (92) can be rewritten by:

∼T fφ͠ (x φ) = Wφ Hφ (x φ) + εφ fθ͠ (x θ ) =

∼T Wθ Hθ (x θ )

+ εθ

+ λ v (ηˆv − ηv ) ηˆ˙v ·v = φ , θ

(103)

(104)

The parameters of the RBFNN are adjusted by the following laws:

5.4. Stability analysis

ˆ˙ v = − 1 s v W βv

The Lyapunov functions are defined by

1 1 ∼T ∼ 1 Vv = s v2 + βv Wv Wv + λ v η͠ v2 ·v = φ , θ 2 2 2

(102)

Hv (x v )·v = φ , θ

(105)

By replacing (105) in (104), we get

(97)

V˙v = −s v ε v − ηv ω v α v1 − α v3 s v − μ v s v2·v = φ , θ

where βv and λ v for v = φ , θ are a positives coefficients. Defining the switching gain adaptation error as:

(106)

We get V˙v < 0 for v = φ , θ approximately as the approximation error ε v for v = φ , θ is sufficiently small in the design 9

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Fig. 13. Control inputs: parameters variations.

The simulations have been carried out under the many situations of losses control effectiveness with different magnitude in five motors. The faults is assumed to occur at t = 25 s, t = 30 s, t = 35 s, t = 40 s and t = 45 s with 20% loss of control effectiveness in the first motor, 30% loss in the second motor, 40% loss in the third motor, 50% loss in the fourth motor and 60% loss in the fifth motor, respectively. After occurrence of the fault, it is expected that the original trajectory tracking is still being maintained without significant degradation of the tracking performance. 6.1. Loss of control effectiveness in five motors with different magnitudes The obtained simulation results of the proposed FTC in faulty operation for three positions (x , y, z ) and three angles (φ , θ , ψ) are given in Figs. 6 and 7, control inputs (F1, F2, F3, F4, F5, F6, F7 and F8) are given in Fig. 8, the global trajectory in 3D is given in Fig. 9. According to this figures, there is a very good tracking of the desired trajectories for both control strategies at outset. However, the actual trajectories corresponding to the proposed FTC strategy converge to their desired trajectories after the transient peaks of low amplitudes. This test shown clearly demonstrates the good performance and robustness property of the proposed FTC.

Fig. 14. System position in the 3D space: parameters variations.

ηv ω v α v1 − α v3 ≥ ε vN ·v = φ , θ

(107)

We obtain

V˙v = −ϑv s v − μ v s v2 < 0 with ϑv > 0, μ v > 0 for v = φ , θ

(108)

6. Simulation results and discussion

6.2. Flight with uncertainties of the inertia and the masse

The proposed control strategy has been tested by simulation in order to check the effectiveness and the performance attained for the path following problem. The parameters of the coaxial octorotor system are given as [42] are presented in Table 2.

To demonstrate the robustness of the proposed control technique, the inertias and masse uncertainties are introduced like presented in Fig. 10. The obtained results are depicted in Figs. 11–14. Fig. 11 shows the trajectory of the output variables of the octorotor, when the 10

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Fig. 15. Comparison between the proposed FTC and the controllers proposed by Alessandro Freddi et al. [5], En-Hui Zheng et al. [44] and Hao Liu et al. in Ref. [45] in faulty operation.

explains the inefficiency of this control approach after occurrence the actuator faults. However, the actual trajectories corresponding to the proposed FTC strategy converge to their desired trajectories after the transient peaks of low amplitudes. This test shown clearly demonstrates the good performance and robustness property of the proposed FTC with respect to the feedback linearization controller and second order sliding mode controller. To make a quantitative comparison between the proposed FTC and the controllers proposed in Refs. [5,44,45], three well-known performance criteria are used [46]. These are integral of square error (ISE), integral of the absolute value of the error (IAE), and integral of the time multiplied by the absolute value of the error (ITAE). The obtained values for each criterion are summarized in Table 3.

Table 3 Values of the performance criteria of the proposed FTC and the control strategies proposed in Refs. [5,44,45] in faulty operation. Control Method

ISE

IAE

ITAE

Strategy of [5] Strategy of [44] Strategy of [45] Proposed FTC

4704.12 2780.19 2564.18 1685.52

6269.67 4824.54 4717.52 2736.18

349254 242955 214819 150813

parametric variations are introduced. The assigned navigational tasks are successfully achieved and the reference trajectories are tracked with high accuracy. Besides that, the controller succeeds in rejecting the external disturbances and shows a good immunity against parameters variation which demonstrates the robustness of the proposed control approach, as well as, the stability of the closed loop dynamics. It is also noticed that tilt angles (pitch and roll) are optimized (Fig. 12), and consequently the energy consumed is minimized. As regard the control signal presented by Fig. 13, it is obviously remarkable the smoothness of this signal which can be easily applied in real physical system. Fig. 14 shows the evolution of the octorotor position and orientation during its flight in 3D.

7. Conclusions In the present paper the problem of controlling a coaxial octorotor aerial vehicle when the actuators fails is proposed. First a dynamical model for the octorotor aerial vehicle in healthy and faulty operations is presented, then this models is exploited to build a fault tolerant controller based on the Radial Base Function Neural Network and Fuzzy Sliding Mode Control. The control law has finally been tested in simulation. The derived fault tolerant controller shows that despite a rotors failure, the octorotor can remain the flying and it can perform trajectory control in x, y and z perfectly and can also control yaw, roll and pitch angles. In addition the comparative study performed with other works developed in the literature, has shown the effectiveness of the proposed control approach in faulty operation. The future works should consider the active fault tolerant control strategy, using non linear adaptive observer to estimate the faults in the system, and should envisage the experimental implementation of the proposed control scheme.

6.3. Performance comparison To explore the effectiveness of the proposed FTC, a comparison between the tracking problems obtained using the proposed FTC, the feedback linearization controller, presented by Alessandro Freddi et al. in Ref. [5], the second order sliding mode controller presented by EnHui Zheng et al. in Ref. [44] and the backstepping controller, presented by Hao Liu et al. in Ref. [45], have been performed on the coaxial octorotor UAV. The performances comparison are investigated in faulty operation with loss of control effectiveness in all motors occurring at t = 60 s actually the comparison is obtained with a sampling period h = 0.01. The simulation results of the global trajectory in 3D tracking for both feedback linearization controller, second order sliding mode controller and the proposed FTC approach in faulty operation with actuators faults occurring at t = 60 s are respectively shown in Fig. 15. According to these figure, there is a very good tracking of the desired trajectories for both control strategies (feedback linearization controller, second order sliding mode controller and the proposed FTC) at outset, but after the occurrence of actuators faults (after t = 60 s), the actual trajectories along x position, y position, z position (Fig. 15) corresponding to the feedback linearization controller and second order sliding mode controller are deviated from their desired values, which

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