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Accepted ManuscriptAdaptive Attitude Controller for a Satellite Based on Neural Network in thePresence of Unknown External Disturbances and Actuator FaultsAli Reza Fazlyab, Farhad Fani Saberi, Mansour Kabganian

PII: DOI: Reference: To appear in: Received Date: Revised Date: Accepted Date: S0273-1177(15)00745-0http://dx.doi.org/10.1016/j.asr.2015.10.026JASR 12481Advances in Space Research18 June 201518 October 201520 October 2015

Please cite this article as: Fazlyab, A.R., Saberi, F.F., Kabganian, M., Adaptive Attitude Controller for a SatelliteBased on Neural Network in the Presence of Unknown External Disturbances and Actuator Faults, Advances inSpace Research (2015), doi: http://dx.doi.org/10.1016/j.asr.2015.10.026This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.1Adaptive Attitude Controller for a Satellite Based on Neural Network in thePresence of Unknown External Disturbances and Actuator Faults1Ali Reza Fazlyab, 2Farhad Fani Saberi and 3Mansour Kabganian,1) Department of Mechanical Engineering, Amirkabir University of Technology, 15875-4413,Tehran, Iran, afazlyab@aut.ac.ir2) Space Science and Technology Institute, Amirkabir University of Technology, 15875-4413,Tehran, Iran, f.sabery@aut.ac.ir3) Department of Mechanical Engineering, Amirkabir University of Technology, 15875-4413,Tehran, Iran, kabgan@aut.ac.irAbstractIn this paper, an adaptive attitude control algorithm is developed based on neural network for a satellite. Theproposed attitude control is based on nonlinear modified Rodrigues parameters feedback control in the presence ofunknown terms like external disturbances and actuator faults. In order to eliminate the effect of the uncertainties, amultilayer neural network with a new learning rule will be designed appropriately. In this method, asymptoticstability of the proposed algorithm has been proven in the presence of unknown terms based on Lyapunov stabilitytheorem. Finally, the performance of the designed attitude controller is investigated by simulations.KeywordsActuator faults; Adaptive control; Attitude; External disturbances; Neural network; Satellite.1 IntroductionAttitude determination and control system (ADCS) is an important subsystem of a satellite. ADCSis responsible for stabilizing and pointing satellite to a specified target in space missions. Thissubsystem has different parts including, sensors, actuators, control algorithm and electronic controlboard. The control algorithm is an important part of ADCS that provides commands for actuators.2 A. Fazlyab et al.Difficult problems in the design of control algorithm for such a complex and nonlinear dynamicsystem are due to the inherent nonlinearities of the model because of large angle maneuvers,uncertainties and unknown environmental disturbances. Although many classical controllers forsatellite attitude control have already been proposed (Moradi, 2013; Hu & Xiao, 2012; Shahravi, etal., 2006), neural networks are promising tools for control applications because they are capable ofapproximating any well-behaved nonlinear function to any desired accuracy.Always, uncertainty in complicated system is an important problem. Adaptive control is one of theimportant controller to overcome this problem. In (Slotine & DiBenedetto, 1990) and (Junkins, et al.,1997; Sheen & Bishop, 1994) an adaptive controller was developed to estimate the model uncertaintyand uncertain moment of inertia based on the feedback linearization. Most of the previous proposedadaptive control methods deal with the unknown parameters. Furthermore, the attitude control ofsatellite based on the feedback linearization is dependent on the initial conditions due to singularityproblem (Paynter & Bishop, 1997). However, in the satellite, the uncertainties are nonlinear caused bydifferent sources (Leeghim, et al., 2009; Lee, et al., 2002). In recent years, the use of intelligentsystems such as neural networks, fuzzy systems, fuzzy-neural network systems and etc. have beenproposed to estimate unknown variables and parameters of the satellite dynamic. A lot of works hasbeen performed in estimation of system states by the neural network. However, minor works has beenperformed in estimation of the unknown parameters of the system using neural network. In (Zhenning& Balakrishnan, 2005) and (Atenica, et al., 2005), a Hopfield neural network is designed to estimatethe unknown parameters of the nonlinear system of an aircraft and robotic arm. This method wasbased on the linearization of the system dynamic model.Neural networks has many advantages over common adaptive controllers in providing desiredsystem performance. Neural networks are used in various applications, especially pattern recognition,identification, estimation, and control of dynamic systems (Yesildirek & Lewis, 2001; Lee & Kim,2001). For example, different methods have been proposed in the control and estimation of aircraftsand helicopters using neural network (Hovakimyan, et al., 2002; Kim & Calise, 1997; Leitner, et al.,31997). Adaptive output feedback control using a high-gain observer and radial basis function neuralnetwork was proposed for nonlinear dynamic equations represented by input–output models (Khalil,1996). Also, a nonlinear adaptive flight control system was designed by backstepping and neuralnetwork controller (Lee & Kim, 2001). Lewis et al. designed a multilayer neural network robotcontroller with a novel on-line weighted tuning algorithm by filtered error approach to estimate theunknown dynamic model of a robot (Lewis, et al., 1999; Lewis, et al., 1996). Weighted tuningalgorithms, including correction terms to the delta rule plus an added robustifying signal, guaranteebounded tracking errors as well as bounded neural network weights.Most pervious works have capability of parameters estimation but not function estimation.Moreover, some of them use linearization to simplify the dynamic model, and some of themencountered chattering problems. In this paper we will consider external disturbances, dynamic modeluncertainty and actuator fault as an uncertain function to be estimated. So, we have developed anadaptive neural network controller for attitude control of a satellite in the presence of unknownexternal disturbances or dynamic model uncertainty. For this purpose, the well-known modifiedRodrigues parameters feedback law will be developed with the three-layered neural network based on(Lewis, et al., 1996) to estimate the unknown function represent the external disturbances or dynamicmodel uncertainty. The proposed adaptive neural network is based on modified Rodrigues parameterscontroller using new weight updates rule. Advantages of using modified Rodrigues parametersrepresentation include: (1) rotations of up to 360° are possible, (2) the parameters form a minimalparameterization (Crassidis & Markley, 1996) and (3) by using modified Rodrigues parameters, wecan rewrite the dynamic equations to develop the adaptive-neural network controller. By usingmodified Rodrigues parameters and nonlinear model of dynamic, singularity and chattering problemhave been solved and the stability of the proposed adaptive neural network controller has been provedbased on Lyapunov stability theorem. Finally, the performance of proposed controller will beevaluated by simulation results.4 A. Fazlyab et al.2 Dynamics and Kinematics of satelliteThe kinematic and dynamic equations of a rigid satellite can be modeled as follow (Wertz, 1978):σ& = F (σ )ω (1)[ ] 1( ) )1 2(1 2F(σ ) = I3×3 + σ × + σ σ T – + σ Tσ I3×3 (2)Jω& + [ω ×]Jω = u + ∆ (3)Here, the modified Rodrigues Parameter vector σ represents the attitude orientation of thesatellite, J ∈ R3×3 represents the symmetric inertia matrix of the satellite, ω ∈ R3 is the angularvelocity of the satellite, u∈R3 is the control torque, ∆ denotes the uncertain term including externaldisturbances and 3I3×3 ∈ R is the 3 × 3 identity matrix. [a×] is an operator on any vectora =[a1 a2 a3] T such that:[ ]–––× =0002 13 13 2a aa aa aa(4)By differentiating Eq. (1) and using ω = F -1(σ )σ& , the dynamic equation of the system can bewritten in the standard form,σ σ + σ σ σ = τH ( ) && C ( , & ) & (5)whereH(σ) = J F -1(σ ) (6)C(σ ,σ&) = – J F -1(σ ) F& (σ ,σ&) F -1(σ ) +[Jω ×]F -1(σ ) (7)τ = u + ∆ (8)Note that F -1(σ ) is well defined for all σ (Schaub & Junkins, 2004). These equations (Eq. (5) toEq. (8)) will be considered to develop the adaptive neural network controller. In this method, a three5layered neural network has been designed to estimate the uncertain term ( ∆ ). Then the estimatedfunction will be used in Eq. (8) to compute the control input of the system by removing the unknownterms and external disturbances. Then, the system stability will be proved in the presence of theproposed controller in the following sections.3 Adaptive-neural network control designAdaptive control covers a set of techniques which provide a systematic approach for automaticadjustment of controllers in real time, in order to achieve or to maintain a desired level of controlsystem performance (Landau, et al., 2011). The concept and principles of adaptive control areexplained by details in (Astrom & Witenmark, 2008; Slotine & DiBenedetto, 1990). Adaptive controlneeds an adjustment mechanism and we have used a multi-layer neural network as an uncertainfunction estimator of adaptive control. In machine learning and cognitive science, artificial neuralnetworks (ANNs) are a family of statistical learning which are usually used to estimate unknownfunctions . Artificial neural networks are generally presented as systems of interconnected “neurons”which exchange messages between each other. The connections have numeric weights that can betuned based on experience, making neural nets adaptive to inputs and capable of learning. Theconcept and principles of neural network (types of neural networks, types of activation functions,learning rules, applications and etc.) are explained by details in (Haykin, 2008; Zurada, 2006).For the adaptive-neural network control design, the following steps have been done.1) Define a sliding surface in the state space form.2) Rewrite satellite dynamic according to sliding surface.3) Design the control algorithm based on Lyapunov stability theorem.4) Design neural network in order to estimates the external disturbances.5) Provide weights update law of neural network and prove the stability of the controller.6 A. Fazlyab et al.3.1 Define filtered errorIn order to design an adaptive neural network controller, we define parameter s , in the state spacee and e& as follow:s = e& + Λ e (9)where Λ > 0 is a positive definite designing parameter matrix and (te ) is the tracking error vectorsdefine as:= σ -σe d (10)σ d is desired modified Rodrigues Parameter. Now the satellite dynamic is expressed in term ofthe filtered error as:H (σ )s& = – C(σ ,σ& )s + f (x)+ τ (11)where the nonlinear satellite function is defined as:f (x)= H (σ )(σ&& d + Λe&)+ C(σ ,σ& )(σ& d + Λe) (12)3.2. Controller designA general approximation-based controller is define as follow:τ = – f – Kv s (13)According to Eq. (8), τ is the combination of control torque u and external disturbance torque∆ , so:u = – f – Kvs – ∆ˆ (14)where ∆)is an estimate of ∆ and K s K e K ev = v & + v Λ is corresponding to outer proportionalderivative (PD) tracking loop. By using this controller the closed loop error dynamic is:H s = -C s – K s + ∆~v& (15)where the function approximation error is given by∆~ = ∆ – ∆ˆ (16)7Now according to the universal approximation property of NN there is a two layer NN such that:∆ = W T Ξ (V T x)+ ε (17)where ε represents the function reconstruction error and x is the input vector to NN and is definedas:

x = [eT e&T σ σ& σ&& d T ] (18)

d T d T V and W are the first and second layer weight matrices respectively and are defined as follow:==m mllw wln nmmv vmv vv vb bWv vv vb bVLM O ML LLM O ML L111 11111 11,(19)In order to prove the stability of the adaptive neural network controller in section 3.4, weconsider the following assumption:Assumption 1: The desired trajectory is bounded in the sense, for instance, thatdd d d≤ Φσσσ&&&(20)where RΦ d ∈ is a known constant. (Lewis, et al., 1996)Corollary 1: By using the first assumption for each time t, x(t) in Eq. (18) is bounded byx c c s≤ 1 Φd + 2 (21)We define Z the matrix of all the weights as follow for notational convenience:=VWZ00 (22)Assumption 2: There are ultimately converged ideal weighting matrices at the end of the learningprocess so that V F ≤ VM , W F ≤ WM , orZ F ≤ ZM (23)whereVM , WM and ZM are the known upper bounds.Now, suppose a NN estimate of ∆ be given by:8 A. Fazlyab et al.∆ˆ = Wˆ T Ξ (Vˆ T x) (24)with Vˆ and Wˆ the actual values of the NN weights given by the tuning algorithm to be specified. Sothe weight estimation errors and subsequently the hidden layer output error for a given x define as:

V~ V Vˆ , W~ W Wˆ (25)= – = –

Ξ= Ξ – Ξ≡ Ξ – Ξ ~ ˆ (V T x)(Vˆ T x) (26)

The Taylor series expansion of Ξ (x) for a given x may be written asΞ (V T x)= Ξ(Vˆ T x)+ Ξ′(Vˆ T x)V~T x + O(V~T x)2 (27)( ) ( )dz x zd zzˆˆ=ΞΞ′ ≡(28)The Jacobian matrix and O(z)2 denoting terms of order two. Denoting Ξˆ ′ = Ξ′(VˆT x), we have:Ξ~ = Ξ′(Vˆ T x)V~T x + O(V~T x)2 = Ξˆ ′V~T x + O(V~T x)2 (29)Lemma 1: For sigmoid, RBF and tanh activation functions, the higher-order terms in the Taylorseries are bounded by

O(V x) ~T c c Q V d ~ 4 F 3

c V sF~52≤ + + (30)whereci are computable positive constants (Lewis, et al., 1996).3.3. Error system dynamicsControl input is chosen as:u = – f – Kvs -υ – ∆ˆ (31)where υ is a robust function against higher-order terms in the Taylor series. In Eq. (31) bysubstituting ∆ˆ with Eq. (24) the control input is given by:u = – f – Kvs -υ -Wˆ T Ξ(Vˆ T x) (32)The proposed NN control structure is shown in figure 1.9Figure 1. Adaptive neural network control structureBy using this controller the closed loop error dynamic become:H s& = -(C + Kv )s -Wˆ T Ξ(Vˆ T x)+W T Ξ(V T x)+ ε -υ (33)Adding and subtracting W T Ξˆ yieldsH s& = -(C + Kv )s +W~ T Ξˆ +W T Ξ~ + ε -υ (34)with Ξˆ and Ξ~ defined in Eq. (26). Adding and subtracting now Wˆ T Ξ~ yields:H s& = -(C + Kv )s +W~T Ξˆ +Wˆ T Ξ~ -W~T Ξ~ + ε -υ (35)Now, we use the Taylor series approximation (Eq. (29)) for Ξ~ . So the closed loop error systembecomes:

H s& = -(C + Kv )s +W~T Ξˆ +Wˆ T Ξˆ ′V~T x + δ1 -υ (36)

where the disturbance terms are:

δ1( ) t =W~ T Ξˆ ′V~T x + W T O(V~T x)2 + ε (37)

By using Eq. (36), the stability of the satellite cannot be proved by Lyapunov stability theorem.So, the error system equation is changed as follow:

H s& = -(C + Kv )s +W~T (Ξˆ – Ξˆ ′VˆT x)+Wˆ T Ξˆ ′V~T x + δ -υ (38)

where the disturbance term (δ ) is10 A. Fazlyab et al.

δ (t) = WT Ξˆ ′V T x + W T O(VT x)2 + ε (39)~ ~

Corollary 2: By using Lemma 1 and some standard norm inequalities the disturbance term in Eq.(39) is bounded according to(t) ( b c Z ) c Z Z c Z Z sN d M M F M F~ ~δ ≤ ε + + 3 + 6 + 7 (40)(t) c c Z C Z sF F~ ~δ ≤ 0 + 1 + 2 (41)whereci are computable positive constants.3.4. Weight update rule and stability proofIn this situation, the problem is boundedness of the weights which is need to verify theboundedness of the control input τ(t). It can never be shown that Lyapunov derivative L& is negativefor all (ts ) and values of neural network weights, because of the NN reconstruction error ε , and thenonlinearity of satellite dynamic. So, it is just proved that Lyapunov derivative L& is negative outsidea compact set in the state space. However, boundedness of the tracking error and the neural netweights can be deduced. (Lewis, et al., 1996)..In order to prove the stability of the satellite dynamic equations in the presence of the proposedadaptive neural network controller, theorem 3.2 in (Lewis, et al., 1996) has been modified byconsidering a new control law (Eq. (13)) and satellite dynamic equation (Eq. (5)) as follow:Theorem 1: Let us consider the desired trajectory be bounded by Eq. (20). Take the control inputfor Eq. (5) as Eq. (13) with define the robust term and gain as follow:(t) K (Z Z M )sυ = – z ˆ F + (42)Kz >C2 (43)with C2 the known constant in Eq. (41). Let NN weight tuning be provided byWˆ & = M Ξˆ sT – M Ξˆ ′Vˆ T rx T – kM r Wˆ (44)11

V& G x (ˆ T Wˆ s)T – kG r Vˆ ˆ = Ξ′

(45)with M = M T > 0 , G = GT > 0 are constant matrices, and k > 0 . By choosing large enough controlgain Kv , the filtered tracking error s(t) and NN weight estimates (Vˆ and Wˆ ) are uniform ultimateboundedness. Note that the practical bounds are given by the Eq. (50) and Eq. (51). Also, byincreasing the gains Kv , in Eq. (32), the tracking error can be kept as small as desired. (Lewis, et al.,1996)Proof: According to (Lewis, et al., 1996) let the corollary 1 hold with a given accuracy εN forall x in the compact set U x ≡ {x x ≤ bx} with bx > c1Qd in Eq. (21).DefineUr = {s s ≤ (bx – c1Qd ) c2}. Let s(0)∈Ur . Then the approximation property holds.Now, define a Lyapunov function candidate as:L(s W V ) sT H ( )s tr{W T M W } tr{V~T G V~}1 2~ ~1 21 2~,~,-1 -1= σ + +(46)By differentiating Lyapunov function and substituting it from Eq. (38), the Lyapunov derivativebecomes:( ) { ( )}

+ { ( + –T ~ ~ 1&Ξ′)}+ ( ) δ – υT T Tˆ ˆ–1T T T T Tv

= – + – + + Ξ – Ξ′Ttr V G V x s W sL s K s s H C s tr W M W s V xs2 ~ ~ ˆ ˆ ˆ1 2 & & & (47)By using the tuning rules in Eq. (44) and Eq. (45) give( ) ( ) ( )( ) ( ) δ υϖ υ= – + – + –= – + – + – + +T TvTT T TvTs K s k s tr Z Z Z sL s K s k s trW W W k s trV V V s~ ~& ~ ~ ~ ~ (48)Since ~ ( ~) ~, ~ 2 ~ ~ 2F F F F Ftr Z T Z – Z = Z Z – Z ≤ Z Z – Z , there results( ) ( )( ) ( )[ ( ) ]

v min M 0 1F F F

Mv F M F Z FMv F M F Z Fs K s k Z Z Z C C ZK s k s Z Z Z K Z Z sL K s k s Z Z Z K Z Z s s~ ~ ~~

~ 2 2min2 2min+ + – – –≤ – + – – +& ≤ – + – – + + ϖ (49)where Kvmin is the minimum singular value of Kv and the last inequality comes from Eq. (43). Now,12 A. Fazlyab et al.it must be shown that the term in bracket is positive and hence L& becomes negative.Defining C3 = ZM + C1 k and completing the square yields( )( ) 3 2 3 2 min 0min 2 0~ 2 4~ ~k Z C k C K s CK s k Z Z C CvFv F F= – – + –+ – –which is guaranteed positive as long as eitherrvbKk C Cs ≡+>min023 4 (50)orZFZ > C + C + C k ≡ b02~ 3 2 3 4 (51)whereC Z C k3 = M + 1 (52)Thus, Lyapunov derivative L& is negative outside a compact set and large enough control gainKv , in the right hand of Eq. (50) results br < (bx -c1Qd ) c2 . Then, any trajectory s(t) beginning inUr , remains completely in Ur . Now, According to a standard Lyapunov theorem extension (Lewis, etal., 1996; Narendra & Annaswamy, 1987), uniform ultimate boundedness of both the filtered trackingerror s and NN weight approximation errorF~Zcan be deduced.In the last paragraph, the stability of the satellite has been proved. Note that, in ideal case whichdisturbance term is equal to zero, and the control input be given by Eq. (32) with υ = 0 , theasymptotic stability for satellite dynamic can be proved similarly, by using Lyapunov stabilitytheorem and Eq. (36). In ideal case, weight tuning of neural network becomes:TWˆ & = M Ξˆ s , Vˆ & = G x (Ξˆ ′T Wˆ s)T (53)Finally, the design procedures of the proposed controller have been illustrated in figure 2 as follow:13Input of NN(Eq.(10))

Filtered error(Eq.(9))Lyapunov theorem forstability proof (Eq(46))Proof sectionDesign section

Satellite dynamic in termsof filtered error (Eq.(11)&(12))Weights updateof NN (Eq.(53))Estimation of Externaldisturbances by NN(Eq.(24))Robust term(Eq.(42))Control law (Eq.(14))Closed loop errordynamic (Eq.(15))PD term ofcontrol lawInput of NN(Eq.(10))Weights updateof NN (Eq.(53))Figure 2. Control Design procedures block diagram4 Simulation and resultsIn this section, in order to demonstrate the performance of the designed adaptive controller to trackthe desired trajectory in the presence of uncertainty, two different scenarios have been proposed andin each scenario, two different kinds of uncertainty have been investigated: environmentaldisturbances and actuators fault.1-A) In this scenario, we use PD controller in the presence of environmental disturbances to showthat PD controller performance without neural network estimator in stabilizing the system.1-B) In this scenario, we use PD controller and the actuator fault as an unknown function to showthat PD controller cannot stabilize the satellite without neural network estimator.2-A) In this scenario, we use adaptive neural network controller in the presence of environmentaldisturbances to show the advantages of proposed controller.14 A. Fazlyab et al.2-B) In this scenario, we use adaptive neural network controller in the presence of actuator fault toshow that the proposed controller can stabilize the system unlike PD controller.The simulation scenarios and satellite dynamics and controller parameters are summarized in table1 and table 2, respectively.Table 1. Simulation scenarios

A B1 PD controller + environmental disturbances PD controller + actuator fault2 Adaptive neural network controller +environmental disturbancesAdaptive neural network controller +actuator fault

Table 2. Simulation parametersparameters Quantity DefinitionR 30 Number of neurons of hidden layerTanh Activation function of hidden layerM 500 × I ( R + )1 ×( R + )1 Constant matrixG500×I16×16 Constant matrixW0 randc(R + )3,1 Initial second layer weight matric of NNV0 randc 16( , R) Initial first layer weight matric of NNKv5×I3×3 PD controller gainK 1000 Positive constantKz 1 Gain of robust termΛ15× I3×3 Positive definite designing parameter15matrix[J x J y J z ] [1000 500 700]Kg.m2 Inertia matrix of satellite[φd θd ψ d ] [0 0 0]deg Desired Euler angles of the satellite[ωx0 ωy0 ωz0 ] [0 0 0]deg/ s Initial angular velocity of the satellite[φ0 θ0 ψ 0 ] [0 – 53 – 53]deg Initial Euler angles of the satellitepointing accuracy 0.5 Deg. for Yaw0.1 Deg. for pitch and rollFor the external disturbances in simulations,A) The environmental disturbances have been considered as:2( 3sin( ) 3cos( )), 10 ( )1( 2 sin( ) 2 cos( )), 10 ( )2( 3sin( ) 2 cos( )), 10 ( )40 040 040 0T N t N t NmT N t N t NmT N t N t Nmdzdydx–– –= – + ×= + – ×= – + ×where N0 is the mean orbital rate of the satellite.B) The actuator faults have been considered in such a way that:Actuator of x axis can generate only half of the control torque from 1 to 10 seconds and actuatorof z axis cannot generate control torque from 3 to 20 seconds.1) Figure 3, illustrates the result of applying the controller in the presence of external disturbanceswithout using the neural network adaptive estimation mechanism. The results show that, the PDcontroller can stabilize the system when there is only environmental disturbances during 30 seconds,but by considering actuator fault as external disturbance, the attitude error is high. Hence, this attitudecontrol algorithm does not satisfy the attitude control requirements. Figure 4, figure 5 and figure 616 A. Fazlyab et al.show the control torque (u ), applied torque (control torque u + environmental disturbances andactuator fault effects ∆ ) and angular velocity of the satellite respectively.2) Figure 7 shows the result of applying the neural network adaptive controller in the presence ofdisturbances torques. The results show that the neural network adaptive controller improves thesystem performance and can stabilize the system by estimating the external disturbances in about 15second even when there is actuator fault in the system. Figure 8 to figure11 show the control torque,applied torque, angular velocity of the system and exact estimation of disturbances torquerespectively.According to the scenario of part B, when the faults occur in actuators, the magnitude of controltorque decreased. In figure 11 part B, the neural network estimates external disturbances include:environmental disturbances and decreased amount of control torque due to actuator faults. Since, theamount of decreased control torque due to actuator faults is very much greater than the amount ofenvironmental disturbances and the control torques have surges, thus the external disturbancesestimation by the neural network have surge, too.(A) (B)Figure 3. Attitude of the satellite (scenario #1)0 5 10 15 20 25 30 35 40 45-2000200Time (second)ψ (Deg.)0 5 10 15 20 25 30 35 40 45-50050Time (second)θ (Deg.)0 5 10 15 20 25 30 35 40 45-1000100Time (second)φ (Deg.)

Actual ψψ

Desired

Actual θDesired θ

Actual φDesired φ

0 10 20 30 40 50 60-2000200Time (second)ψ (Deg.)0 10 20 30 40 50 60-50050Time (second)θ (deg.)0 10 20 30 40 50 60-1000100200Time (second)φ (Deg.)

Actual ψDesired ψ

Actual θDesired θ

Actual φDesired φ

17(A) (B)Figure 4. Control torque (scenario #1)(A) (B)Figure 5. Applied torque to the satellite (scenario #1)0 10 20 30 40 50-200-150-100-50050Time (seconds)Control torque (N.m)

ControControl torque of x l torque of y axisaxisControl torque of z axis

0 10 20 30 40 50 60 70-250-200-150-100-50050100Time (seconds)Control torque (N.m)

ContrContrControl torque ol torque ol torque of x axisof y axisof z axis

0 10 20 30 40 50-200-150-100-50050Time (seconds)Applied torque (N.m)

Applied torque of x axisApplied torqApplied torue of y axisque of z axis

0 10 20 30 40 50 60 70-250-200-150-100-50050100Time (seconds)Applied torque (N.m)

ApplieApplieApplied torque d torque d torque of x axisof y axisof z axis

18 A. Fazlyab et al.(A) (B)Figure 6. Angular velocity of the satellite (scenario #1)(A) (B)Figure 7. Attitude of the satellite (scenario #2)0 10 20 30 40 50 60-60-40-200204060Time (seconds)Angular velocity (deg/second)

Angular velocity of x axisAngAngular velocitular velocity of y axisy of z axis

0 10 20 30 40 50 60-60-40-20020406080Time (seconds)Angular velocity (deg/second)

AngAngular velocitular velocity of x axisy of y axisAngular velocity of z axis

0 5 10 15 20 25-2000200Time (second)ψ (Deg.)0 5 10 15 20 25-50050Time (second)θ (Deg.)0 5 10 15 20 25-150-100-500Time (second)φ (Deg.)

Actual ψDesired ψ

Actual θDesired θ

Actual φDesired φ

0 5 10 15 20 25-2000200Time (second)ψ (Deg.)0 5 10 15 20 25-50050Time (second)θ (Deg.)0 5 10 15 20 25-150-100-500Time (second)φ (Deg.)

Actual ψDesired ψ

Actual θDesired θ

Actual φDesired φ

19(A) (B)Figure 8. Control torque (scenario #2)(A) (B)Figure 9. Applied torque to the satellite (scenario #2)(A) (B)Figure 10. Angular velocity of the satellite (scenario #2)0 10 20 30 40 50-200-150-100-50050100Time (seconds)Control torque (N.m)

Control torque of x axisControl torqControl torue of y axisque of z axis

0 10 20 30 40 50-200-150-100-50050100150200Time (seconds)Control torque (N.m)

Control torque of x axisControl torqControl torque of y axisue of z axis

0 10 20 30 40 50-200-150-100-50050100Time (seconds)Applied torque (N.m)

Applied torque of x axisApplied torqApplied torque of y axisue of z axis

0 10 20 30 40 50-250-200-150-100-50050100150200Time (seconds)Applied torque (N.m)

Applied torque of x axisApplied torqApplied torque of y axisue of z axis

0 5 10 15 20 25 30-40-30-20-10010203040506070Time (seconds)Angular velocity (Deg/second)

Angy of x axisular velocitAngAngular velocitular velocity of y axisy of z axis

0 5 10 15 20 25 30 35-40-30-20-10010203040506070Time (seconds)Angular velocity (Deg/second)

Angular velocity of x axisAngulaAngular velocity r velocity of y axisof z axis

20 A. Fazlyab et al.(A) (B)Figure 11. Estimation of external disturbances by adaptive controller (scenario #2)5 ConclusionIn this paper, an adaptive attitude control algorithm has been developed based on neural networkfor a satellite in the presence of unknown external disturbances or dynamic model uncertainty. In thismethod, we have considered environmental disturbances and actuator fault as an uncertain functionto be estimated. For this purpose, the well-known modified Rodrigues parameters feedback law hasbeen developed with the three-layered neural network. Then, the system stability has been proved inthe presence of the proposed controller based on Lyapunov stability theorem. Finally, attitude controlperformance has been investigated by simulations in the presence of actuator faults and externaldisturbances. Therefore, two different scenarios have been proposed and in each scenario, twodifferent kinds of uncertainty have been investigated: environmental disturbances and actuators fault.The results show that, the PD controller can stabilize the system during 30 seconds when we onlyconsider the environmental disturbances, but by considering of the actuator faults, it cannot satisfy theattitude control requirements. Moreover, the results show that the adaptive neural network controllerhas improved the system performance and it can stabilize the system in about 15 second by estimatingthe external disturbances and the actuator faults simultaneously.Nomenclature0 5 10 15 20 25-20020Time (seconds)N.m0 5 10 15 20 25-1000100Time (seconds)N.m0 5 10 15 20 25-50050Time (seconds)N.m

External disturbances of z aNeural network output for z xisaxis

ENxternal distureural network bances of y aoutput for y xisaxis

External disturbances of x axisNeural network output for x axis0 5 10 15 20 25 30 35 40-50050Time (seconds)N.m0 5 10 15 20 25 30 35 40-40-20020Time (seconds)N.m0 5 10 15 20 25 30 35 40-5000500Time (seconds)N.m

External disturbances of z axisNeural network output for z axis

ExtNeernal disural netwturbanceork outps of y aut for y xisaxis

External disturbances of x axis

Neural network output for x axis21

Symbols Definitions Symbols DefinitionsCi (i = )3,2,1,0 Computable positive knownconstantsZ matrix of all NN weightse Tracking error ∆ Uncertain term (externaldisturbances)G Constant matrix ∆ˆ Estimation of uncertain termI Identity matrix ∆~ Approximation error of uncertaintermJ Inertia matrix of satellite δ Disturbance term of error systemequationK Positive constant ε function reconstruction errorKv PD controller gain υ Robust termK z Gain of robust term Ξ activation functionM Constant matrix σ Modified Rodrigues parametersN0 mean orbital rate of the satellite σ d Desired modified RodriguesparametersO( )2 Terms of order 2 Φ d Known constants Filtered error [φ θ ψ ] Euler angles of the satelliteu Control torque [φd θd ψd ] Desired Euler angles of thesatelliteV Ideal first layer weight matric ofNN[φ0 θ0 ψ0 ] Initial Euler angles of the satelliteW Ideal second layer weight matricof NNω Angular velocity of the satelliteVˆ Actual value of first layer weightmatric of NNω0 Initial angular velocity of thesatelliteWˆ Actual value of second layerweight matric of NNΛ Positive definite designingparameter matrix

22 A. Fazlyab et al.

~VApproximation error of first layerweight matric of NN)( F Converged ideal matrix~WApproximation error of secondlayer weight matric of NN)( M Upper bound of matrixx Input vector to NN

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