Real Time Embedded Kalman Filter Estimators for Fault Detection in a

International Journal of Computer Science and Applications, Technomathematics Research Foundation Vol. 8, No. 1, pp. 83 – 109, 2011

Real Time Embedded Kalman Filter Estimators for Fault Detection in a Satellite’s Dynamics 1 2

NICOLAE TUDOROIU

EHSAN SOBHANI-TEHRANI 3

KASH KHORASANI 4

ADINA ASTILEAN

5 6

TIBERIU LETIA

ROXANA-ELENA TUDOROIU

The main idea of this paper is the real-time implementation of the Fault Detection Kalman Filter Estimators (FDKFE) in a satellite’s Reaction Wheels during its scientific mission. We assume that the satellite’s reaction wheels are subjected to several failures due to the abnormal changes in power distribution, motor torque, windings current as well as the temperature caused by a motor current increase or friction. The proposed realtime FDKFE strategies consist of two embedded multiple model bank of nonlinear Kalman Filter (extended and unscented) estimators. This research work is based on our previous results in this field and now we are interested only in real-time implementation of some of these FDKFE strategies (FDDM_EKF and FDDM_UKF). Furthermore we will construct a benchmark to compare their results to have an overall image how perform these strategies. Keywords: Satellite’s Dynamics, Faulty Modes Detection and Diagnosis Extended Kalman Filter (FMDD_EKF), Faulty Modes Detection and Diagnosis Unscented Kalman Filter (FMDDM_UKF), Multiple Model Kalman Filters Bank (MMKFB)

1. Introduction The most important faults in the aerospace satellite’s systems are the result of unexpected failures, interferences as well as the age of their crucial components. Also the defective measurement and control loops equipment, in particular some of the sensors and actuators should be considered. Whenever these critical situations come out the satellite’s systems could lose the control, require much more energy, and could operate harmfully. Therefore to operate at high energy efficiency and to guarantee the equipment safety and reliability it is important to develop several FMDD strategies able to detect

[1Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, QC, Canada, H3G 1MB [email protected] 2 ” McGill University, 805 Street Sherbrooke Ouest, Montreal, Quebec, Canada [email protected] 3 Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, Quebec, Canada [email protected] 4,5,6

“Technical University of Cluj-Napoca, 15 Constantin Daicoviciu Street, Cluj-Napoca, Romania [email protected],[email protected], [email protected] ]

84 Tudoroiu et al.

and diagnose any time every faulty satellite’s system components and consequently corrective and reconfiguration actions should be initiated promptly. Up to date for the majority part of the satellites, the onboard measurements data acquisition is sent to the land stations relatively frequently. However the operators still remain implicated to monitor some of the telemetry measurements and to diagnose the inconsistency of the operating equipment [6]-[9]. Effectively the existing methods to identify and to adjust the equipment failures are mostly labor-intensive task, and consequently sustained, rhythmic and error-prone. In the majority situations the operators inspect telemetry plots manually to determine the satellite’s reaction wheels health motors current. Also they use statistical evaluation and comparison models, but these evaluations still necessitate considerable human knowledge, consequently error-prone that could generate harshly equipment operation [6]-[9]. In these circumstances the problem of satellite monitoring and fault diagnosis becomes a critical issue, very complex that need to be implemented in mainframe environment using more sophisticated control systems and artificial intelligent strategies [1]-[2],[6]-[9], [15]-[16]. Therefore the objective of our research is to develop more proficient, accurate and reliable FMDD strategies based on the nonlinear estimation techniques [1]-[2], [9]. An advanced linear and nonlinear FMDD strategies were developed in our research and excellent preliminary results have been obtained and published in [6]-[9]. In this paper the proposed FDDM strategy is model-based and is capable to properly monitor the telemetry data corresponding to satellite’s reaction wheels using a reliable nonlinear model [3]. This strategy assumes that there is a collection of fairly accurate models that could characterize achievable system healthy and faulty modes. These models are integrated into a Multi Model Kalman Filters Bank (MMKFB) nonlinear estimators that operate simultaneously at each time instance [1]-[2], [4]-[5], [6]-[11]. Each of these filters behaves according to a specific model from the model collection such that the global state estimate is their superposition state estimates. Also we assume that the jump between these models could be considered as a transition defined in a probabilistic approach. In simulation environment, based on the preliminary results published in [6]-[9], in Section 6 we will depict and we will explain the performance of the proposed FMDD nonlinear state estimators strategies following more or less the same scenarios developed in [6]-[8]. The comparison results incorporated in Table 1 of the same Section 6 reveal that the FMDD_UKF strategy is highly reliable and capable of detecting and diagnosing faulty satellite’s reaction wheel actuators more rapidly.

2. The Satellite’s model For the reader information purpose we will reiterate briefly in this section some of the ideas explained already in [6]-[9], and well documented in [3]. The satellite considered in our research is assumed to be hypothetical having a scientific mission and communicates with the land stations providing permanently the collection of data acquisition. In each of all three-axes of the satellite operates independently an ITHACO high inertia rotor reaction wheel having a highly nonlinear dynamics given in [3]. The

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ITHACO reaction wheel has a torque constant gain and the current driver is assumed to be proportional to the torque. Examining the block scheme of the ITHACO’s reaction wheel [3], represented in our paper in Figure 1, we distinguish several internal and external closed control loops that control the electromotive force torque effect, the temperature sensitivity to the viscous and columbic friction, the reaction wheel speed, the motor torque, and the torque noise disturbance. Each one of these loops plays an important role and consequently their effects should be incorporated to get a high fidelity model capable to capture all the satellite’s system dynamics. From faults root-cause perspective we mention that the potential generating fault sources could be the bus voltage, Vbus , that keeps the motor torque inside the range of operating harmless speeds, the bearing lubricant viscous friction that has a high sensitivity with regard to the temperature, T, the motor torque effectiveness, kt , and the unusual spiky changes of motor current, im (t ) . In the same figure we also notice the presence of the Heave aside conditional blocks [3] that introduce supplementary constraints for the electromotive force torque and reaction wheel speeds. The satellite’s reaction wheel model represented in Figure 1 is more realistic compared to the generic literature models because it is of high complexity and high fidelity nonlinear dynamics capable to incorporate the effects of all these control loops. We don’t intend now to repeat all the model equations that derive from each loop as soon as we accomplished this task in [9], but for the development purpose of the FDDM strategy we need only to write in state-space representation the general form of the overall model, such as (i) State Equation: xj,1(k 1)     f j (xjk ,ujk ) wjk xj,2(k 1)

(1)

 y j ,1 (k )     h j ( x jk , u jk )  v jk  y j ,2 (k ) 

(2)

(ii) Output equation:

where the mode j-state vector representing the motor current I m, j (k ) and the reaction wheel angular speed w, j (k ) , respectively is defined as:

 x j ,1 (k )  Im, j (k)  x j (k ) := x jk :=    =  x j , 2 (k ) w, j (k)

(3)

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The input command u jk  Vcom could be the reaction wheel torque but in our research is was considered such a voltage generated by one of the three-axis satellite’s PID or PI-controller, representing the command voltage applied to the reaction wheel motor axis. The vectors w jk and  jk incorporate the effects of the process and measurement noises. Almost in all Kalman Filter estimation techniques they are considered random independent vectors with Gaussian distribution, white, zero mean and having the covariance matrices Qw j , Rvj . The discontinuous Heave aside H s , H b , H f blocks from Figure 1 are approximated by the nonlinear analytical approximations functions represented in Figure 2. In the following two sections we will reiterate and reformulate briefly the EKF and UKF nonlinear estimators embedded in the Multiple Model FMDD architecture well developed in [1]-[2], [6]-[11].

Fig.1 A detailed block diagram of a high fidelity reaction wheel model

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E M F t o rq u e lim it in g b lo c k a n a lit ic a l a p p ro x im a t io n , f1 (u ) y=f1(u)

1 0.5 0 -1 0 0

-8 0

-6 0

-4 0

-2 0

0 20 40 60 u S p e e d L im it e r b lo c k a n a lit ic a l a p p ro x im a t io n , f3 (u )

80

100

-8 0

-6 0

-4 0

-2 0

80

100

-8 0

-6 0

-4 0

-2 0

80

100

y=f3(u)

1 0.5 0 -1 0 0

0 20 40 60 u S ig n (. ) fu n c t io n b lo c k a n a lit ic a l a p p ro x im a t io n , f4 (u )

y=f4(u)

1 0 -1 -1 0 0

0 u

20

40

60

Figure 2: The nonlinear analytical approximations of the sign (.) and Heaviside blocks functions

3. Extended Kalman Filter (EKF) Estimator – The Module 1 of the integrated FDDM strategy The dynamics of this estimator is described by the following discrete state-space linear stochastic system obtained by linearizing the original highly nonlinear system represented by (1)-(2): x j , k 1  A jk x j , k  B jk u jk  w jk

(4)

y j , k  C jk x j , k  v jk

(5)

where A jk , B jk , C jk represent the Jacobean matrices. We notice that x j , k  R n -the model state could be sufficiently approximated by a Gaussian random vector of xˆ j , k mean and covariance matrix Pj , k |k . The EKF state estimator of the dynamical system (1)-(2) consists of two phases, namely prediction and correction phases. 1.1 Prediction Phase:

In this phase the current state estimate, xˆ j , k |k and the state

estimate covariance, Pj , k |k are projected forward, one step ahead, according to

xˆ j,k 1|k  f j (xˆ j,k|k , u jk ) Pj,k 1|k  Ajk Pj,k|k AjkT  Qwj

(6)

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and initialized by xˆ j ,0|0 , Pj ,0|0 . 1.2 Correction Phase. Is a recursive computational process that consists of following steps 1.2.1 Determine the gain of the filter

K 1.2.2

j , k 1

 P j , k  1| k C

T jk

(C

jk

P j , k  1| k C

T jk

 R j )  1

(7)

Once a new measurement y j , k 1 is available, correct the vector estimate

xˆ j , k |k and the matrix elements of Pj , k |k

xˆ j , k  1| k  1  xˆ j , k  1| k  K

j , k 1 ( y j , k 1

 h j ( xˆ j , k 1| k , u jk ))

Pj , k 1|k 1  ( I  K j , k 1C jk ) Pj , k 1|k

(8) (9)

The EKF estimator has only three main tuning parameters, namely P0|0 , Qwj and Rvj . The matrices Qwj and Rvj are determined by trials and errors, and incorporate the uncertainty in the tracking data. They have to be fine tuned because they have a significant contribution to make the EKF nonlinear estimator more robust. The covariance matrix Pj , k |k gives information about the uncertainty in the state estimate and provides an excellent criterion for the error bound.

4. Unscented Kalman Filter (UKF) Estimator – The Module 2 of the integrated FMDD strategy The nonlinear UKF estimator approach is a suitable estimator choice compared to EKF estimator. It is conceived to estimate the state of the nonlinear systems whose dynamics is given by the system representation (1) - (2). The critical issue of the Kalman Filters is the projection of the vector xˆ j , k through the nonlinear equations (1)-(2). The drawback of the EKF estimator is the propagation of the vector xˆ j , k through the linear system dynamics given by (4)-(5). By the linearization of the nonlinear equations (1)-(2) could be introduced large approximation errors that affect the true values of the mean and covariance for xˆ j ,k 1|k and Pj , k 1|k as well as the optimal value of the performance. Furthermore there are a small number of critical situations when the EKF estimator could diverge. The UKF estimator resolves this important issue deterministically using a finite collection of a spread weighted points (SWP), called sigma points, to capture completely and with high accuracy degree the mean and the covariance of the vector xˆ j ,k . Furthermore the UKF estimator propagates the vector xˆ j , k through the true nonlinear

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system dynamics (1)-(2). In [2], [4], and [6]-[11] it is shown that the vector xˆ j , k |k whose mean is x j ,k |k and the covariance are Pj , k |k could be covered by a set of 2n  1 SWP, incorporated in the matrix X j , k |k ,i , i  0,...., 2n and given by: X j , k |k ,0  x j , k |k X j ,k |k ,i  x j , k |k  ( ( n   ) Pj , k |k )i , i  1, n

(10)

X j , k |k ,i  n  x j ,k |k  ( ( n   ) Pj , k |k )i , i  1, n where

   2 (n  k )  n k and



(11)

are two main “fine tuning parameters” [4], that usually are set to k = 0

and 0.0001    1 . The vectors ( (n   ) Pj ,k |k )i represent the column or the row vectors of index i from the matrix ( (n   ) Pj ,k |k ) . The sigma point’s collection is propagated through the nonlinear system dynamics getting another sigma points collection X j , k 1|k  f j ( X j ,k |k , u jk ) . Their distribution captures the true value of the mean “given by the weighted average of these transformed sigma points” [4]: 2n

x j ,k 1|k 

W

( m)

i

X j ,k 1|k ,i

(12)

i 0

Similar, the true value of the covariance “is given by the weighted dot product of the transformed sigma points” [4]: 2n

Pj , k 1|k 

W

i

(c )

( X j , k 1|k ,i  x j ,k 1|k )( X j ,k 1|k ,i  x j , k 1|k )T

(13)

i 0

where: W0(m) 

 (n   ) 2

1

1

2

, Wi( m ) 

0.5 (n   ) 2

, i  1,.....2n , W0(c)  W0(m)  (1   2   ) ,

(14)

and the parameter  incorporates any prior knowledge regarding the distribution of the vector xˆk 1|k , and its value is set to   2 if it is a Gaussian distribution. Analogous, the columns of X k 1|k will be propagated through the output nonlinear system dynamics and we obtain:

90 Tudoroiu et al.

Y j , k 1 k  g j (k , X j ,k 1 k , u jk )

(15) and its mean is y j ,k 1|k : 2n

y j , k 1|k 

W

(m)

i

Y j , k 1 k ,i

(16)

i 0

Compared to EKF performance the UKF estimator is more appropriate for the majority of nonlinear process models, since the new statistics variables introduced (mean and the covariance) are easily calculated using simple vector and matrix algebra manipulations. Also it could be mention that UKF estimator implementation is extremely fast since it does not require calculation of Jacobean matrices, an important source of numerical complications and difficulties. Furthermore the sigma point’s collection captures the statistics (mean and the covariance) of the random variables independently of the choice of matrix ( (n   ) Pj ,k |k ) . Concluding the UKF estimator calculate a collection of SWP related to the system state xk and to the process and measurement noises

w jk  W and v jk  V , stored as columns or rows into L  (2 L  1) matrix

X j ,k |k , where L is the dimension of the state space vector, for an un-augmented UKF

estimator, or the dimension of the state space vector augmented by the measurement and process noises space dimensions, for an augmented UKF estimator. Indifferent of their architectures, the UKF estimators consists of the same two phases, namely prediction and correction phases.

1.1 Prediction Phase: Project one step ahead the state and the estimated covariance according to X j ,k 1 k  f j (k , X j , k |k , u jk ) 2n

x j ,k 1|k 

W

(m)

i

X j ,k 1 k ,i

i 0

2n

Pj , k 1|k 

W

(c)

i

( X j , k 1|k ,i  x j ,k 1|k )( X j ,k 1|k ,i  x j ,k 1|k )T  Qwj (17)

i 0 2n

y j ,k 1|k 

W

i

i 0

(m)

Y j ,k 1 k ,i

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where the initial values for the state estimate and estimated matrix covariance xˆ j ,0|0 , Pj ,0|0 are assumed as known and the matrix attached to the state vector sigma points X k 1

is given by

X k 1  [ xˆk 1 Z k1 Z k1]

Z k1

k  {1,..., }

for

 xˆk 1 *[111...1]    n   *

Pk 1 , Z k1

dim(W )

 xˆk 1 *[111...1]    n   * Pk 1

(18) (19)

dim(W )

2.2 Correction Phase. Is a recursive computational process that 2.2.1 Calculate the filter gain

K j , k 1  Pxjˆ ,k 1|k , yˆ j ,k 1|k ( Pyˆ j ,k 1|k , yˆ j ,k 1|k )1

(20)

2n

Pyˆ j ,k 1|k yˆ j ,k 1|k 

W

Pxˆ j ,k 1|k , yˆ j ,k 1|k 

W

(c )

i

[(Y j ,k 1 k ,i  y j ,k 1|k )(Y j ,k 1 k ,i  y j ,k 1|k )T ]  Rvj

(21)

[( X j ,k 1 k ,i  xˆ j ,k 1|k )(Y j ,k 1 k ,i  yˆ j ,k 1|k )T ]

(22)

i 0 2n

i

(c )

i 0

2.2.2 Update the state estimate and estimate covariance, once a new measurement z j ,k 1 is available:

xˆ j , k 1| k 1  xˆ j , k 1| k  K

j , k 1 ( z j , k 1

 h j ( xˆ j , k 1| k , u jk ))

Pj , k 1|k 1  ( I  K j , k 1C jk ) Pj , k 1|k 5.

(23) (24)

Design of the FMDD Strategy-The Embedded Multiple Model Kalman Filters Bank (MMKFB)

The proposed FMDD strategy is built based on the embedded multiple model (MM) approach with the EKF and UKF nonlinear estimators described in the previous two sections. Detailed procedures for this strategy we developed in [6]-[11] based on the references [1]-[2] and [4]. The main assumption of this approach is that the nonlinear model system dynamics (1)-(2) could be accurately approximated by a first-order “Markov hybrid nonlinear system” [1], such that “the system modes (normal or faulty) at time k is selected by a discrete process m j (j denoting the quantities pertaining to system mode m j ) and modeled as a s-state” [1] S  {m1 , m2 ,..., mN } “with the transition

92 Tudoroiu et al.

probabilities matrix   { ij } -representing an important design parameter, and defined

as

 ij (k )  P{m j (k  1) | mi (k )}, mi , m j  S subjected to the following constraints” [4]:



0   ij ( k )  1, i  1, N , j  1, N ,

ij ( k )

(25)  1, i  1, N

(26)

j

If we consider that the mode

mj

is active at the instance time k

then m j (k )  {m(k )  m j } . The collection of the models S is supposed to envelop all potential pattern models of the system behavior. The objective of the FMDD strategy is to identify which mode from the collection S  {m1 , m2 ,..., mN } is active at each time instance k . The critical issue of the FMDD approach is an open problem-dependent design that concerns the methodology design of the covering models set S. Furthermore this methodology design is appropriate if there is a good understanding and enough information about all possible faulty modes in the system (1)-(2). The implementation of the FMDD strategy needs to accomplish the same tasks from [1], [6]-[11], and we will reiterate them very briefly: 1. Developing the suitable set of the models set design based on the enough information about all possible faulty modes in the system (1)-(2). 2. Filter selection to select a model-based recursive filter bank such in FMDD_EKF module 1 or FMDD_UKF module 2, described in Sections 3 and 4 that estimates the state of the healthy or faulty system modes derived from (1)(2). 3. Estimate fusion that combines the Multiple Model Filters Bank (MMFB) estimates to get the entire estimate. 4. Filter re-initialization to reinitialize the filter models during each cycle of the recursive process that is a critical issue for multi model FMDD bank filter approach. For the proposed FMDD strategy the filter states are reinitialized using the previous values of the state estimates and covariance matrices without any performance degrade when parameters system structure or its modes change. The embedded interacting multiple model FMDD filter bank integrated architecture runs each parallel filter bank recursively, cycle after cycle. During a cycle each matching filter model bank has the input u jk , and its past information statistics are accurately captured by the state estimate xˆ 0j , k |k and the covariance matrix Pj0,k |k . The dynamics of the selected Kalman Filters bank (EKF or UKF) associated with each system mode (healthy or faulty) derivates from the equations (1)-(2) and the probability of a given model matches uniquely the corresponding system mode, and consequently provides an important indication about the active (in operation) reaction wheel actuator fault. The mode probability evolutions, in tight combination with the

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conditional logic requirement to reach the setting fault detection threshold value [1], [5][11], represent the core for achieving simultaneously the faulty modes detection and the diagnosis. Furthermore the most benefit of this embedded multi model FMDD Kalman filter bank strategy is to provide supplementary information about the sensor or actuator type, their location, the fault severity, and its event time. The general steps of the embedded IMM algorithm are presented according to the references mentioned at the beginning of this section. Step 1: Prediction phase: Calculate the prediction estimate of the model probability ˆ (k  1 | k ) , the mixing probability at time k,  ij (k ) the fusion state estimate and its covariance xˆ j 0 ( k | k ) , Pj 0 ( k | k ) using the following relationships [1], [9]:



ˆ j ( k  1 | k )   ij ( k ) 

N 1

 (i, j )  i ( k ),

j  1, 2 , ..., N

(27)

 (i, j )   i ( k ) , i , j  1 , 2 , ...., N ˆ j ( k  1 | k )

(28)

 xˆ (k | k )   (k ), i, j  1, 2,...., N P (k | k)   [P(k | k) (xˆ (k | k)  xˆ (k))(xˆ (k | k)  xˆ (k | k)) ] (k), j 1, 2,...., N N

xˆ j 0 (k | k ) 

i

1

ij

N

j0

T

i

1

j0

i

j0

i

ij

(29) (30)

Step 2: Filter selection based on the EKF or UKF estimators to estimate the healthy and faulty modes states. In this step we use the equations integrated inside the FMDD_EKF or FMDD_UKF modules (1 or 2), according to the filter selection option. Step 3: Correction phase: Adjust the mode probability [2] based on the new information available in the step 2, at k+1,  (k  1)

 j (k  1) 

 j (k  1| k ) L j (k  1) N

  (k  1| k ) L (k  1) j

, j  1, 2,.........., N

(31)

j

1

where L j (k  1) represents the likelihood function [2] at time k+1, given by Lj (k 1) 

1

1 1 exp[  jT (k 1)(Pj, yy  (k 1| k))  j (k 1)], 2 | (2 )Pj, yy  (k 1| k) |

j  1, 2,........., N (32)

Step 4: Fault detection decision making is the conditional logic decision that compares the maximum probability of the set modes design  FMDD max to the probability mode threshold value, Threshold : if

 FMDD max  Threshold then a fault occurrs

otherwiswe no fault occurrs

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At this stage we are ready to apply FMDD_EKF and FMDD_UKF strategies for faulty mode detection and diagnosis of the reaction wheel faults for a highly fidelity and highly nonlinear satellite’s model. These results and our comparative studies with [6]-[9] are presented in the next section. A simplified flowchart representing the integrated structure of the satellite’s reaction wheels and the FMDD integrated block scheme could be seen in Figure 3. The simplified embedded FMDD_EKF/ FMDD_UKF block scheme is well represented Figure 4.

 (k )

Vsp w( k ) PID Controller

Reaction Wheel

-

z (k )  y (

Vcom



FMDD EKF/UKF

xˆ (k  1 | k  P(k  1| k 

z1

 (k  1 FMDD Logic Condition

Fault Detection

Figure 3: The satellite’s and FMDD integrated structure

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z (k ) xˆ1 (k | k )

xˆ10 (k | k )

P1 (k | k ) xˆ2 (k | k )

Interaction (Mixing)

P10 (k | k )

1 (k  1)

xˆ2 (k  1 | k  1

EKF (UKF) Estimator mode 2

P2 (k | k )

P2 (k  1| k  1 xˆ N (k  1| k  1

xˆ N (k | k ) PN (k | k )

xˆ1 (k  1| k  1 P1 (k  1| k  1

EKF (UKF) Estimator mode 1

xˆ N 0 (k | k ) PN 0 (k | k )

EKF (UKF) Estimator mode N

1 (k  1)

PN (k  1| k  1

 N (k  1)

1 (k  1) Estimate combination

Model probabilities update

 N (k  1)

 N (k  1) 1 (k  1)

 N (k  1) FMDD Logic Condition

xˆ ( k  1 | k  1 P (k  1 | k  1

Faulty mode detection

Figure 4 The FMDD strategy block scheme

6. Real-time control and FDKF strategy implementation The control system literature rarely includes extensively the real-time software and hardware implementation aspects, and it doesn’t pay attention beyond algorithms and sampling time selection. Normally the implementation aspect and real-time system design are connected together but in the most cases this connection is always ignored. Moreover the real-time system design is treated from control perspective ignoring the implementation aspects of the control algorithms. However in the last years the real-time implementation and design aspects get more transparency to control engineering field due to advent of software tools like MATLAB/SIMULINK with its RTW (Real-Time Workshop) and the RTWT (Real-Time Windows Target) Toolboxes. Definitely these new real-time platforms do the implementation of real-time experiments easier and save much time but on the other hand they have some drawbacks regarding a good insight view of the real-life problems that could emerge during the real-time implementation of the control system. Building a real-time ACS requires normally two stages: PID controller and estimators design as well as their digital implementations. At controller and estimators design stage some performance indices are defined and the PID controller and EKF and UKF estimators are designed to optimize these indices.

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At implementation stage, multiple control tasks should be scheduled to run the FPGA (Field-Programmable Gate Array) onboard microprocessor or controller module. We have to take care about task scheduling taking into account the limited available computing resources. Firstly to select the sampling time Ts , it should take into account the limited computation time provided by the hardware such that to avoid the conflict with its computation time delay (control latency). Since the control latency is typically affected by control jitter, delay and loss can occur alternatively in the system at different instants of time. The loss of the ACS control signal u (k ) leads to the controller computer failure, unable to update its output during any one sampling interval, and accordingly the one step delayed input u (k  1) it will be applied again. Since this situation may well occur accidentally at any instant of time, the failure to deliver a control signal can be treated as a casual ACS input disturbance u p (k ) . Anyway this interaction between control performance and task scheduling will be investigated in the future work. 7. Simulation Results for Fault Detection and Diagnosis The embedded multiple model (MM) algorithms based on the EKF and UKF estimators are now applied for faulty mode detection and diagnosis (FMDD) of a high nonlinear dynamics of the satellite’s reaction wheels in the uncertainty environment. For comparison purpose of the Kalman Filter estimators performance we will consider several scenarios developed for MM based on linear standard Kalman Filter [KF] in [6][8] and MM based on EKF and UKF nonlinear estimators is well developed in [9]. We consider the same sources of the faults injected to the reaction wheels from [8]-[9], namely an unexpected viscous friction change generating anomalies in the temperature T, an abnormal change in the bus voltage Vbus , and the loss of the effectiveness of the motor torque kt . We use the same notations such in [6]-[9] and we designate by F1  ( z1 , u1 ), F2  ( z2 , u2 ), F3  ( z3 , u3 ), F4  ( z4 , u4 ) the measurement observations and

the inputs of the nonlinear Kalman Filter estimators bank for the normal mode ( F1 ) as well

as

for

the

faulty

modes

( F2 , F2 F3 ).

We

also

designate

by

F5  ( z23 , u23 ), F6  ( z24 , u24 ), F7  ( z34 , u34 ) the measurement observations and the

inputs of the nonlinear Kalman Filter estimators bank for three simultaneous faulty modes ( z2 z3 , u23 ), ( z2 z4 , u24 ), ( z3 z4 , u34 ) . All these modes are labeled from F1 to F7 as an alternative to a mode probability graphical representation. In this way it is easier to identify the occurrence and the dynamic evolution of fault modes within each window horizon. These modes have the following signification:

a.

The healthy mode F1 is generated by the nonlinear equations (1)-(2) for the nominal values of the parameters: T  30 [ 0C ] , kt  0.029 , and Vbus  24[V ] .

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b.

The faulty mode F2 derivates from (1)-(2) by an abnormal increase of

c.

temperature, similar to [8]-[9]. The faulty mode F3 derivates from (1)-(2) by a sudden change of Vbus , let say at

d.

the half of the initial value. The faulty mode F4 derivates from (1)-(2) by an abnormal change in the motor current corresponding to a loss of effectiveness of kt , similar to [8]-[9].

e.

The faulty mode F5 derivates from (1)-(2) by mixing the conditions (ii)-(iii).

f.

The faulty mode F6 derivates from (1)-(2) by mixing the conditions (ii)-(iv).

g.

The faulty mode F7 derivates from (1)-(2) by mixing the conditions (iii)-(iv).

Similarly to [8]-[9] the following five scenarios for the highly nonlinear dynamics of the satellite’s ACS reaction wheels are now considered for further investigation in this section: 1. First scenario (single faulty mode) The normal mode states ( x1 , x2 ) are represented by the motor current I m ( x1 ) and the angular speed of the wheel by  w ( x2 ) for each axis, and their dynamics is described by the equations (1)-(2). Similarly we could represent the dynamics of the faulty mode ( x1 j , x2 j ), j  1, N according to the same equations (1)-(2), and the root-cause faulty

parameters are represented by T , kt ,Vbus , taking the values specified from ( i)-(vii). For this case we consider only the first four modes: F1  F4 . The simulation results are depicted in the Figures 4-5. These results disclose very good fault mode detection and classification with very fast transient response for the nonlinear EKF estimator, and a slow transient response for the nonlinear UKF estimator.

98 Tudoroiu et al. First Scenario 1.1

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2. Second scenario (sequence of multiple fault modes) We consider the following sequence of the modes: F1  F2  F1  F3  F1  F4 , where the reaction wheel faults F2 , F3 F3 occur at the beginning of different window horizons and persist within the window for the same duration given by the window length. The specific parameters, level of process and measurement noise, and initialization settings here are the same as that of the first scenario. The results for this scenario in terms of the mode probabilities and their index are depicted in Figures 6(a) and 6(b). It should be noted that FMDD_ EKF algorithm (Figure 6(a)) performs with short transient response unlike the FMDD_UKF algorithm (Figure 6(b)) that performs with a long transient response in some of the window horizons.

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3. Third scenario (sequence of simultaneous fault modes) We consider the following three faulty modes sequences: F1  F2  F3  F5  F1 3.1 3.2

F1  F2  F4  F6  F1

3.3

F1  F3  F4  F7  F1

The time horizon windows are partitioned as [0, 30], [30, 70], [70,100], [100,150], [150,200], [200,250], and [250,300] corresponding to the N=7 modes considered in this scenario. Figure 7 depicts the index of mode probabilities for all the above three cases. In Figure 7(a) the simulation results for the IMM_EKF algorithm reveal a very good classification for all the three cases with a very fast transient response in all window horizons for the last two cases and with a small transient response for the last two window horizons in the first case. In Figure 7(b) we note that the IMM_UKF algorithm performs very well in identifying and classifying the fault mode sequence with a very fast transient response for the last two cases and a small transient time within the first and last window horizons. 4. Fourth scenario (Sequence of simultaneous multiple fault modes) We consider the following z  [ z1 z23 z1 z24 z1 z34 ], u  [u1 u23 u1 u24 u1 u34 ] with the same

case settings and

occurrences as in the third scenario. The IMM UKF algorithm results as shown in Figure

Real-Time Embedded Kalman FilterEstimators 101

7(b) reveal a very good detection and classification accuracy, with only a short transient response for the sampling interval [0, 30]. From Figure 7(a) one can observe that the IMM_EKF algorithm performs inaccurately and misclassifies the fault modes that have occurred in the window horizons [70,150] and [200, 250]. This scenario clearly demonstrates the superiority of the IMM_UKF algorithm over the IMM_EKF algorithm. Third Scenario 3.1

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5. Fifth scenario (never seen fault cases) We are now interested in determining if the FDDI strategy will be capable of detecting “never seen” faults. For simulation purposes we consider the following cases: 5.1 A fault is generated as T=30 [C], kt=0.002, Vbus= 4 [V], that is close to the 7th mode. 5.2 A fault is generated as T=30 [C], kt=0.029, Vbus= 4 [V], that is close to the 4th mode. 5.3 A fault is generated as T=30 [C], kt=0.029, Vbus= 12 [V], that is close to the 6th mode. 5.4 A fault is generated as T=10 [C], kt=0.029, Vbus= 4 [V], that is far from any generated fault modes. All these situations are represented in Figures 8(a) and 8(b). From Figure 9(a) we observe that IMM_EKF algorithm identifies and classifies well in the first two cases 5.1 and 5.2, misclassifies the fault in the third case 5.3 and it is not capable to identify and classify the real fault generated in the last case 5.4. Compared to IMM_EKF algorithm the simulation results from figure 8(b) reveal the superiority of the IMM_UKF algorithm to accomplish this task. It performs very well in the first three cases and it is unable to find an appropriate interpretation of the fault in the last case. The fault is misclassified and interpreted such as fifth fault mode.

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104 Tudoroiu et al.

7.1 FMDD Strategy Analysis The originality of this FMDD approach is the implementation of two FMDD_EKF and FMDD_UKF embedded algorithms to successfully solve a specific problem of faulty modes detection and diagnosis in the satellite’s reaction wheels. Figures 7 and 8 show the robustness results of our proposed FDDM_EKF and FDDM_UKF strategies in terms of the noise levels and matrix transition probabilities (  ). For large changes in noise levels the simulation results reveal performance degrade for the both FMDD strategies. The fine tuning values of noise levels are determined by trials and errors selection process. Furthermore the threshold value Threshold is not drastically because the mode probability of an active faulty mode is monotonically increasing in the steady-state. The drawback of the proposed FMDD strategy is that the modes set design is a critical issue for this embedded structure that requires an enough understanding and a priori information of potential system faulty modes. However, this issue is highly problemdependent and could be more realistic approach if there are enough models to cover all possible system faulty modes at every time occurrence. Due to limited number of potential faulty modes in our specific application we assume a few different scenarios. These scenarios correspond only to the most likely fault events in a satellite’s reaction wheels, and therefore we not intend to consider all potential faulty modes that may take place in reality. However, the main objective of this paper is to express and confirm the applicability and the efficiency of the multi model FMDD_EKF and FMDD_UKF strategies to detect and diagnose the faulty modes in satellite’s reaction wheels. The software package used in this paper is MATLAB-version 6.5 and the SIMULINK Toolbox. 7.2 Comparison Results Overall the simulation results reveal a very good identification and classification of the sequence of fault modes using the both multi model FMDD_EKF and FMDD_UKF strategies with a slightly superiority of the FMDD_UKF strategy. The superiority of the FMDD_UKF strategy versus FMDD_EKF strategy concerning the identification and the classification of the fault mode sequence it is obviously in the fourth scenario. For this scenario the simulations of FMDD_UKF strategy from Figure 7(b) reveal a very good identification and classification accuracy, with only a short transient response of few samples in the first sampling time interval, when compared to the FMDD_ EKF strategy that performs with misclassification of the active faulty mode inside the window horizons [70,150], [200, 250] as shown in Figure 7(a). However concerning the FMDD_UKF strategy the simulation results from Figure 7(b) reveal for the first three scenarios a small delay of some samples in the same first sampling time interval [0, 30], by reason of the transient response of the Kalman filters. Slightly superiority of FMDD_UKF strategy versus FMDD_EKF strategy is observed for the second scenario depicted in the both figures 7 (a) and (b) that reveal a superior classification these two strategies but with long transient response of FMDD_EKF strategy for the window horizon [200, 250]. For the scenario 3.1 we observe a short transient of few samples inside the time intervals [0, 30] and [250,300]. For the scenarios 3.2 and 3.3 we notice the same transient response for the first two window horizons [0, 30] and [30, 70] due to the switching from one faulty mode to another one. The Figure 8(b) reveals that the IMM FMDD_UKF strategy is able to classify the majority of unmodeled faulty modes. Also we observe that the FMDD_UKF strategy is robust and very accurate for cases that have been not modeled

Real-Time Embedded Kalman FilterEstimators 105

but close to represented faults, especially the fault 5.1 very closed to the seventh faulty mode, the fault 5.2 is closed to the fourth faulty mode, and finally the fault 5.3 is slightly closed to the sixth faulty mode. Furthermore for the cases that are completely far from the fault, for example the fault 5.4, the IMM_UKF FDDI strategy is not capable to classify well these combinations; the fault is misclassified and interpreted such as fifth fault mode. Also we observe that the FMDD_UKF strategy is quite robust in terms of the noise levels. This is illustrated in 1 0  5 1 0  Figure 10(a) (with Qw  Rv  106   ), Figure 10(b) (with Qw  Rv  10  0 1 ), and 0 1     1 0  Figure 10(c) (with Qw  Rv  104   ). Furthermore, the simulation results depicted in 0 1 Figure 9 reveal that the FMDD_EKF strategy is not robust to the same level of process and measurement noises. Moreover for robustness comparison purpose we try to report the results from [8] of the FMDD_KF strategy based on the same measurement and process noise levels, and the same probability matrices, carried out in Figure 10. Furthermore, we remark that the FMDD_UKF strategy is also robust in terms of the transition probability matrix  , subjected to the limitations given by (25)-(26) [1], [6][11]. For comparison purpose we put together all these results obtained in this paper and the previous one of our research from [8]-[9] to create the benchmark from Table 1, including some of the representative performance of all three FMDD strategies, FMDD_KF, FMDD_EKF, and FMDD_UKF. From this benchmark we find that the FMDD_UKF strategy outperforms the FMDD_KF and FMDD_EKF strategies in terms of accuracy and robustness. Table 1: The performance comparison of the FMDD_KF, FMDD_EKF, and FMDD_UKF strategies

Item

1 2

3

4

Performance

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Small

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Good

Very Good

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Good

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8. Conclusions In this paper, we have studied the possibility of using two interactive multiple models based on Extended and Unscented Kalman Filter estimators to detect and diagnose the faults in reaction wheels of the attitude control system (ACS) in a satellite. The main contributions in our research are summarized briefly as follows: (a) Detection and identification of reaction wheel faults of the ACS due to a number of possible sources that generate soft and hard anomalies during a scientific mission of a satellite, (b) Implementation of a bank of real-time parallel Kalman filters for faulty modes covering a quite large set of the most likely and commonly possible scenarios of reaction wheel failures, (c) Detection and diagnosis of both partial and significant reaction wheel failures for different scenarios using the real-time EKF and UKF estimation algorithms, (d) Comparison of performance capabilities and advantages of real-time UKF estimation algorithm with respect to the real-time EKF estimation algorithm, and (e) Robustness analysis of both real-time estimation algorithms to the selection of model transition probabilities, modeling errors, and noise statistics under different scenarios. The approach proposed in this paper is probabilistic in nature and yields results that are more accurate and having good fault classification capabilities than the spectral analysis that is well studied in the literature. Based on fault identification analysis that is carried it can be observed that the real-time UKF estimation algorithm is robust to modeling uncertainties, and to statistics of noise measurements and process noise. The both real-time algorithms work similar to a real-time neural network estimator and classifier employed to perform satisfactory FDKFE. Compared to a real-time neural network estimator and classifier, the real-time UKF estimation algorithm doesn’t need an on-line training that takes an extensive amount of computational resources. Another similar approach uses observer banks of autoregressive time series models for fault diagnosis, based on Box-Jenkins linear autoregressive models, back propagation neural networks, and radial function networks [2], [11]-[12]. Compared to the above approaches and real-time EKF estimation algorithm, the real-time UKF estimation algorithm performs better and has a much less computational burden and complexity, and it furthermore operates much faster. Perhaps the biggest drawback of predictive modelbased approaches is the need for a suitable quantity of data for training and testing the system during the development phase [10], [13]. Moreover, stability of the real-time UKF estimation algorithm still remains an open question, which needs further investigation. Also, the behavior of the real-time UKF estimation algorithm for diagnosis in a fast or rapidly changing process dynamics needs to be explored further.

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