Research on Fault Diagnosis of Satellite Attitude

Mei-ling WANG*, Hua SONG, Chun-ling WEI

Research on Fault Diagnosis of Satellite Attitude Control System based on the Dedicated Observers Abstract: In this paper, the design principle of disturbance decoupling for nonlinear unknown input observer is introduced. And then a bank of unknown input observers are designed for the satellite attitude control system, and each observer is decoupled to the flywheel fault of a particular axis, which is sensitive to the flywheel faults of the other axes. The observers generate a structured residual set, and the flywheel fault isolation is realized by fault separation logic. Finally, the feasibility of the method is verified by simulation analysis. Keywords: Satellite attitude control system; unknown input observer; fault diagnosis

1 Introduction Satellite attitude control system, one of the core systems of the the satellite for normal operation, its reliability and stability play a very important role in the normal operation of the satellite. Due to the problem of increasingly high requirements on the satellite performance, the complex flight environment, and the limited resources on satellite, the complex satellite attitude control system is inevitable showed different types of faults, which lead to system performance degradation and system fault, and resulting in serious losses. Thus, the study of the fault diagnosis of the satellite attitude control system is very important [1,2]. Recent years, the fault diagnosis method based on state estimation has been widely used and achieved remarkable results [3]. However, in the current study, the fault diagnosis of linear systems is studied deeply, the nonlinear systems are not mature due to their own complexity, especially for the modern systems which are often accompanied by disturbances and uncertainties, the fault diagnosis of nonlinear systems becomes more difficult [4]. Reference [5] used an sliding mode observer, but this method exists a certain time lag phenomenon; Reference [6] used a robust fault diagnosis method for satellite actuator based on discrete proportional integral observer. This paper studied the fault detection and isolation of nonlinear systems based on the state estimation method, and then the dedicated observers thought

*Corresponding author: Mei-ling WANG, school of automation science and electrical engineering, Beihang University, Beijing, China, E-mail: [email protected] Hua SONG, school of automation science and electrical engineering, Beihang University, Collaborative Innovation Center of Advanced Aero-Engine, Beijing, China Chun-ling WEI, National Key Laboratory of Space Intelligent Control Technology, Beijing, China

Unauthenticated Download Date | 12/31/17 6:01 PM

Research on Fault Diagnosis of Satellite Attitude Control System

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is used to design a set of unknown input observers, such that each observer is not sensitive to a specific fault, and generating a set of structured residuals. Finally, realized the flywheel fault detection and isolation of the satellite attitude control system.

2 Fault diagnosis method based on nonlinear unknown input observers For the following nonlinear system:

x =A( x, u ) + K ( x, u ) f + E ( x)d (1) y = c( x) n Where x(t ) ∈ R represents the state variable of the nonlinear system; A( x, u ), K ( x, u ) represents the nonlinear functions of the system; u ( x) ∈ R m represents the input n n vector of the system; f ∈ R represents the fault vector, d ∈ R represents the p unknown disturbance vector; y (t ) ∈ R represents the output of the system and E ( x) represents the interference distribution matrix. Rewrite the model (1) using the following state transformation: Γ =T ( x) (2) Then, the model after transformation is described by

( x) T=

∂T ( x) ( A( x, u ) + K ( x, u ) f + E ( x)d ) ∂x

(3)

The transformation must be selected to ensure that it can remain unaffected by the unknown inputs but still reflect actuator faults. And it is easy to see from Eq. (3) that the transformation needs to satisfy the following equation:

∂T ( x) E ( x) = 0 ∂x

(4)

In this way, the transformation model can be obtained by Eq. (4), and the system after transformation is:

∂T ( x) = Γ T= ( x) ( A( x, u ) + K ( x, u ) f ) ∂x

(5)

This model is called disturbance decoupling because it has nothing to do with the interference d . Assume rank ( E ( x)) = p , then according to the Frobenius theorem, if the interference distribution matrix E ( x) meets the following sufficient and necessary conditions:

rank ( E ( x)[ei ( x), e j ( x)]) = p,

i,= j 1, 2, ⋅⋅⋅, n − p

(6)


472


th where ei ( x) is the i column of E ( x) , and satisfies

[e= i ( x ), e j ( x )] then Eq. (4) exists

∂e j ( x) ∂x

n− p

ei ( x) −

∂ei ( x) e j ( x) ∂x

(7)

independent solutions, [7]

Γ = Ti ( x),= i 1, 2, ⋅⋅⋅, n − p i

(8)

And then, the complete transformation is

 T1 ( x)    ( x)    = Γ T= Tn − p ( x)   

(9)

It can be seen from Eq. (9) that the state order of the transformed system is n − p < n , and the output information of the original system is needed when using the reduced order state Γ to restore the original state of the system:

y * = c* ( y )

(10)

where c* ( y ) is a specific transformation of y = c( x) and satisfies:

 ∂T ( x)    ∂x =n rank  *  ∂c (y) y = c ( x )    ∂x  

(11)

According to Γ and y* , the inverse function Ψ 0 satisfies

x =Ψ 0 (Γ, y* )

(12)

When there is no fault, the system inevitably has the state estimation error, in order to ensure that the designed observer is asymptotically convergence, a feedback quantity is needed to be introduced:

R(T ( x), c( x)) = 0

(13)

And if the Eq. (13) exists, the following condition must be satisfied:

 ∂T ( x)   ∂x  rank   < n− p+m  ∂c( x)   ∂x 

(14)



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where m is the independent outputs number of the system. As seen from Eq. (11) :

 ∂c* (y)   ∂T ( x)  y =c( x)  = rank  rank n +    ∂x  ∂x    And from Eq. (15):  ∂T ( x)   ∂x  rank  =n  ∂c( x)   ∂x 

(15)

(16)

Then it can be obtained by Eq. (14) and Eq. (16) that p < m , that is, the number of independent measurement signals required is greater than that of independent unknown disturbance inputs. In addition to ensuring the fault detection observer stable, Eq. (13) can also be used as a residual to determine whether the system is fault, that is because once the fault occurs, the Eq.(13) will no longer be established. So the reduced order observer can be designed as [9]

∂T ( xˆ ) = zˆ A( xˆ , u ) + H ( x, u ) R(Γˆ , y ) ∂x xˆ = Ψ 0 ( Γˆ , y* )

(17)

The state estimation error e and residual r of the observer is defined as:

e = Γˆ − Γ

(18)

= r R(Γˆ , y )

(19)

The residual vector r can be used for fault detection. And the Taylor series expansion of the differential equation governing the dynamics of the estimation error e is:

∂T ( x) K ( x, u ) f ∂x ∂T ( x) K ( x, u ) f = F (t )e + ο (e 2 ) − ∂x

e ρ (e, t ) − =

F (t ) =

(20)

∂Ψ 0 (Γ, y* ) ∂ ∂T ( x) A( x, u )) ( ∂Ψ 0 ∂x ∂Γ *

∂R (Γ, y )

2 + H (Ψ 0 (Γ, y ), u ) Where , and ο (e ) is the second and higher ∂Γ order terms of e . In order to make the state estimation error uniformly bounded, we should select the appropriate feedback matrix H ( xˆ , u ) to make the e = F (t )e asymptotic stable. Generally, H ( xˆ , u ) can be obtained by pole placement or simulation analysis [8]. In addition, in order to ensure that all faults affecting the original system can be reflected


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in the estimation error of the observer, according to the Eq. (20), the transformation matrix T ( x) should also satisfy the following equation

 ∂T ( x)  (21) �K ( x, u )  = rank ( K ( x, u )) rank   ∂x  Based on the above analysis, the robust fault detection observer is shown on Figure 1. f

d

y

Nonlinear dynamic system

u

y

C * ( y)

*

Γˆ

Disturbance decoupling model

u

xˆ

R(Γˆ , y )

γ

H ( xˆ , u ) ⋅ γ

Figure 1. Sketch map of robust fault detection

In the picture, u represents the input vector of the system; d represents the unknown disturbance vector; f represents the fault vector; y represents the output of the * system; c ( y ) is a specific transformation of y = c( x) and y* = c* ( y ) ; R (Γˆ ( x), y ) is the feedback quantity; γ is the residual error, and H ( xˆ , u ) is the feedback matrix.

3 Dedicated Observers Design For Flywheel Fault Isolation In this section, this paper analyzed the actuator fault diagnosis of satellite attitude control system based on unknown input observers, then designed a bank of unknown input observers which are respectively decoupled with the three axis flywheel faults so as to achieve the purpose of fault isolation. Firstly, the dynamic equation of the satellite attitude control system is [10]:  I y − Iz  1 wy wz    I  x   Ix  w x     Iz − Ix  w  =  wx wz  +  0  y  I   y  w z     I − I y  x  0 w w x y  I    z 

Where,

0 1 Iy 0

 0  M + T  x dx  0   M y + Tdy     M z + Tdz  1 I z 

(22)

represents three axis angular velocity of satellite, wz  T represents the actuator input control torque; Tdx Tdy Tdz  represents the space environmental disturbance torque; and  I x I y I z T represents the satellite inertia moment.  M x

My

 wx

M z 

T

wy

T



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When the actuator fails, the space form of the satellite dynamics equation can be written as Eq.(23):

x (t ) = Φ ( x, t ) + Bu f (t ) + Bd y = Cx(t )

(23)

(t ) u (t ) + f (t ) is the flywheel output when a fault occurs, where, u f = T f (t ) = [ f1 (t ) f 2 (t ) f 3 (t ) ] is fault vector function, f i (t ), i = 1, 2,3 is respectively corresponding to the x, y, z flywheel fault. Use u (t ) + f (t ) to replace u f (t ) in Eq. (23), and the Eq.(23) can be rewritten as:

x (t ) = Φ ( x, t ) + Bu (t ) + Bf (t ) + Bd y = Cx(t )

(24)

0 0  1/ I x B =  0 1/ I y 0   0 0 1/ I z  is the flywheel fault distribution matrix, and can be Where

represented as B = [ B1 B2 B3 ] . Then the design principle of flywheel unknown input observer is as follows: Assume that the i axis flywheel fails, and then establish an unknown input observer i , use the i axis flywheel fault as the unknown inputs, and make the observer be robust to the i axis flywheel fault, while be sensitive to the other axis flywheel fault, so the observer is decoupled with i axis flywheel fault. Therefore, in order to achieve the flywheel fault isolation, three observers are needed and the three observers generate three residuals. Then the fault separation judgment logic is:

γ i ≤ εi γj

  ⇒ i axis flywheel fails ≥ ε j , ∀j ≠ i 

(25)

Where ε i is the fault detection threshold for i axis flywheel. In this paper, assume that the z axis flywheel is fault, and an unknown input observer is designed to decouple the z axis flywheel fault. When the z axis flywheel is fault, the attitude dynamics model is:

x (t ) = Φ ( x, t ) + Bu (t ) + B3 f (t ) + Bd y = Cx(t )

(26)

In order to make the transform be decoupled with the fault, it is needed to meet the following equation:

∂T ( x) B3 = 0 (27) ∂x Because rank ( B3 ) = 1 , the number of independent transformation is n − 1 = 3 − 1 = 2 , = Γi Ti = ( x1 , x2 ), i 1, 2 it is clear that the T ( x) consisted of any independent function can satisfies the Eq.(27). Choose Γ1 = T1 ( x)= x1 , Γ 2 = T2 ( x)= x2 , then


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x  = Γ T= ( x)  1   x2  (28) In addition, it is also required to use a measurement signal to restore the state x ,

x =Ψ 0 (Γ1 , Γ 2 , y* )

(29)

* y* c= ( y ) [ 0 0 1] y , then the observer decoupled with z axis flywheel Choose= fault can be designed as: *  ∂T (Ψ 0 (Γˆ , y )) = Γˆ (Φ (Ψ 0 (Γ, y* )) + Bu ) + H (Ψ 0 (Γ, y* ))γ (30) ∂* x T x , and the residual vector is Where ψ (Γˆ , y ) =Γ Γ

0

(

1

2

3

)

γ = y − η (Γˆ , y* )

(31)

* Where η (Γˆ , y ) = ( xˆ1 x2 ) . In simulation, selecting an appropriate feedback matrix H is needed to guarantee 1 the convergence of the state estimation error, and in this paper choose H = 1 .  Similarly, for the x and y axis flywheel fault, design two unknown input observers which are respectively decoupled with the x and y axis flywheel fault. Therefore, the residual set composed of three unknown input observers can be obtained, and then according to Eq. (25), the fault flywheel can be determined.

4 Simulation and analysis In this paper, assume that there is a constant bias in the flywheel output, and design unknown input observers to isolate the flywheel fault. In simulation, the satellite inertia moment is designed as: I x = 1849.3765,I y = 1435.234, I z = 2278.8824 , the space disturbance torque Ax × sin(ω0 t ), Td y = Ay × sin(ω0 t ) , and Ax = 1.4 *10( −5) , is designed as Td x = ( −5) ( −5) Ay = 1.5*10 , Az = 1.6 *10 , ω0 = 0.02rad / sec . When there is no failure with three axis flywheels, the residuals of the three observer residuals are as follows:



Figure 2. Residual of

x axis unknown input observer


y axis unknown input observer

477


478



z axis unknown input observer

Figure 2 - 4 show that when there is no fault with the flywheels, the three observers are able to track the system state very well, the residual errors caused by the unknown input disturbance is less than 0.2 × 10−16 , so the fault detection threshold value can be set as ε= ε= ε= 0.3 × 10−16 , and then the judgment logic for the flywheel fault 1 2 3 isolation is:

γ i ≤ 0.3 × 10−16 γ j ≥ 0.3 × 10

−16

  ⇒ i axis flywheel fails , ∀j ≠ i 

(32)

t ≤ 600 s  u uout =  in uin + ∆ t > 600 s , and When the x axis flywheel fails at 600s, its output is −5 ∆ = −6 × 10 N ⋅ m , the three observer residual are as follows:




x


y axis unknown input observer

479

axis unknown input observer


480



z axis unknown input observer

Figure 5 - 7 show that when x axis flywheel fails, the residual of x axis unknown input observer remains unchanged, while the other two observer residuals rises rapidly and exceeds the threshold after the failure (the bold line in figures), and according to the fault isolation logic it is easy to judge that the x axis flywheel fails.

5 Conclusion This paper has introduced the principle of unknown input observer and flexibly applied it in the satellite attitude control system, established a series of unknown input observers, and ensured that each observer is decoupled with a particular axis flywheel failure, while sensitive to the other axis failure. And then used the structured residuals set to isolate the flywheel fault. Finally, conducted simulation analysis on the condition of non fault and constant deviation of the flywheel, and the feasibility of this method is verified. And this method could be applied to fault detection and isolation of the flywheel in satellite attitude control system. Acknowledgment: This study was supported by the National Natural Science Foundation of China (No. 61573059).



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