Fault Detection and Isolation using an Adaptive Unscented Kalman Filter

Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India

Fault Detection and Isolation using an Adaptive Unscented Kalman Filter Manasi Das*. Smita Sadhu** Tapan Kumar Ghoshal*** 

*Electrical Engineering Department, Jadavpur University India (Tel: +919804122571; e-mail: [email protected]). ** Electrical Engineering Department, Jadavpur University India (e-mail: [email protected]) *** Electrical Engineering Department, Jadavpur University, Kolkata, India, (e-mail: [email protected])} Abstract: An enhanced Fault Detection and Isolation (FDI) technique based on ‘t’ statistical test is presented here employing a residual based R-Adaptive Unscented Kalman Filter (AUKF). Although the use of AUKF is common for estimation problem, employing AUKF for FDI duty is quite rare. Adaptive Unscented Kalman filtering framework is chosen here due to its derivative free calculations in the algorithm and reportedly better accuracy. Monitoring the ‘t’ statistic of the innovation sequence, measurement faults are detected and corresponding alarm signals are generated here. The presented ‘t’ statistic based FDI technique comprising of consistency checking and abruptness checking is straight forward, demands less computational burden and minimizes false alarms. The superiority of the proposed AUKF based FDI method compared to non adaptive, nonlinear filtering based FDI techniques is demonstrated here by an extensive simulation study executed on a nonlinear LEO satellite planar model. The use of AUKF has provided faster convergence speed and is also found to be advantageous for the situations where the measurement noise statistics are unknown. Keywords: Adaptive filters, Fault detection, Fault isolation, Nonlinear systems, Unscented Kalman filter. 

1. INTRODUCTION

To surmount the difficulties of the EKF based FDI schemes various UKF based FDI methods may also be found in literature [Xiong2007, Bae2010, Bae2011, Cornejo2010]. In [Xiong2007] the advantages of the UKF approaches over the EKF approaches are described and a chi-square test on the normalized residual sequences, generated by the UKF, within a fixed length sliding window is shown to detect the sensor faults in the measurement of satellite attitude. Instead of monitoring the residual sequences a sensitivity factor is utilized to detect and isolate fault in [Bae2010, Bae2011], where a federated structure consisting of several UKFs is employed. The cumulative sum of the residual sequences generated by UKF, are monitored to detect fault in [Cornejo2010].

Three stages of model based FDI techniques [Gertler1988] are: (i) residual/ innovation generation (ii) statistical testing and (iii) logical analysis. In this paper an AUKF is employed for innovation generation purpose. Though adaptive filters are getting increasing attentions for their promising performances, involving them for FDI duties are still very rare in literature. Among the existing model based FDI schemes, Extended Kalman filtering (EKF) based schemes have the most prevalent use [Hajiyev2009, Mehra1995, Pirmoradi2005, Pirmoradi2007, Pirmoradi2009]. In [Hajiyev2009] FDI technique for a LEO satellite is shown based on the mathematical expectation of the spectral norm of the normalized innovation matrix provided by the EKF. In [Mehra1995] a multi-hypothesis EKF (MEKF) approach is developed for the detection and identification of faults, where several EKFs, each based on a particular hypothesis, run in parallel. A multi-stage FDI scheme for a spacecraft attitude determination system depending on a bank of EKF has been shown in [Pirmoradi2005, Pirmoradi2007, Pirmoradi2009]. Although EKF approaches are the mostly used techniques, the problems associated with the EKF based schemes are the possibility of divergence for strongly nonlinear systems even when fault is not present and the complexity of the Jacobian calculations.

978-3-902823-60-1 © 2014 IFAC

The advantages of AUKF over EKF and UKF in estimation problems are shown in [Das2013], from where it may be found that AUKF can perform satisfactorily even when the priori knowledge about the noise statistics are wrong or not available at all. But the possibilities of AUKF for FDI are still to be explored. [Hajiyev2013] & [Soken2010] are the only contributions regarding this area. However, both of this work is concerned mainly about fault tolerance not fault detection and isolation. Only when the innovation sequences fail in chi-square test in consequence of any fault, the noise covariances are changed adaptively in these works, so that the fault can be tolerated. Although the technique used in [Hajiyev2013] can identify whether there is any process fault or measurement fault, but this is not exactly the Fault 326

10.3182/20140313-3-IN-3024.00075

2014 ACODS March 13-15, 2014. Kanpur, India

Isolation/ Identification in true sense. Moreover, as in the healthy mode (fault free mode) these AUKFs work as UKFs, the performances of these filters may degrade if the prior knowledge about the noise covariances is wrong in any healthy mode.

2n

W



Pk 

    T {( Xˆ k  xˆ k )( Xˆ k  xˆ k ) }  Q k

c i

i

i0

Correction (Measurement Update)  Calculate the sigma points of xˆ k as:

In our work an AUKF [Das2013] which is quite different in structural viewpoint from the AUKFs mentioned in [Hajiyev2013] & [Soken2010], is employed for FDI purpose. In the healthy mode, even when the prior knowledge about the noise covariances is wrong, this AUKF algorithm does not degrade its performance. T statistics of the innovation sequences, generated by this AUKF, are monitored here to detect and isolate faults. T statistical test is a ‘direct parallel test’ which is relatively easier to execute compared to the ‘compound scalar tests’ (like chi square test) and it yields distinct binary signature for each of the measurement variables which in turn helps to isolate the faults also [Gertler1988]. So like the multi-hypothesis fault isolation technique this ‘t’ statistics based fault isolation technique doesn’t require any bank of filter and consequently needs less computational efforts. Furthermore the simulation results have shown that, inspite of the presence of abrupt faults, the estimation errors of the AUKF don’t increase drastically. This feature provides enough fault recovery time without any estimation malfunction.

  Xˆ 1k  [ xˆ k

 xˆ k ] n( 2 n 1) 

.....

2n

W

Pz 

c i

i

i0

2n

W

Pxz 

c i

i

Calculate the Kalman gain Kk as: K k  Pxz Pz 1

(10)

Find the innovations as: inn  ( z k  zˆ )

(11)

Estimate the state as: xˆ k  xˆ  K k * inn Calculate the estimated state error covariance as: Pk  Pk  K k Pz K kT

(12)

Calculate the sigma points of xˆ k as: Xˆ k  [ xˆ k

xˆ k ] n( 2 n 1) 

.....

( n   ) [ zeros ( n  1)

Transform the sigma points

( Xˆ k



Pk

Pk ]

2n

W

m i

 Zˆ k

i

Find the residual as: res k  z k  zˆ k Calculate the residual covariance as: Pres 

1 kw

i

)( res i )

(16) (17)

k

 ( res

(14)

) through the function h as: (15) i0

(18)

T

i  k  k w 1

(If, ' ws ' is the window

size then, k w  k

if, k  ws

& k w  ws

if, k  ws )

Estimate the diagonal elements of Rk as: Rˆ k



Pk 1 ]

2n

d ia g

c     T  diag ( Pres )  diag [  W i {( Zˆ k  zˆ k )( Zˆ k  zˆ k ) }]

(1)

i

i0

Estimate Rk as: Rˆ k  diag ( Rˆ k

diag

)

(19)

i

(20)

Obtaining new sigma points Xˆ k as shown in (14) do not add

k 1

(2)

computational load as these would be used in the next cycle of the UKF algorithm.

Project the state ahead as:  Xˆ k

(13)

Adaptation of Rk

Transform the sigma points ( Xˆ ) through the function fd as:

m

(9)

i

 k

Calculate the sigma points of xˆ k 1 as:

i

(8)

i

 k

Prediction (Time Update)

i0

(7)

i

    T {( Xˆ k  xˆ k )( Zˆ k  zˆ k ) }

i0

( xˆ k 1 ) & Measurement noise covariance ( Rˆ k 1 ).

Xˆ k  f d ( Xˆ k  1 )

 Zˆ k

Calculate the cross covariance between the predicted state and measurement as:

Initialize, State error covariance (Pk-1), Estimated states



m i

    T {( Zˆ k  zˆ k )( Zˆ k  zˆ k ) }  Rˆ k 1

Initialization

Pk 1

(5)

Calculate the predicted measurement’s covariance as:

Table 1. Residual based R adaptive UKF

2n

W i0

The residual based AUKF as proposed in [Das2013] by the present authors is employed here for FDI. The AUKF algorithm is shown in Table 1.

W

2n

 Estimate the measurement as: zˆ k 

( n   ) [ zeros ( n  1)



Pk ]

(6)

 ˆ ) Zˆ k  h ( X k

xˆ k 1 ] n( 2 n 1) 



Pk

) through the function h as:

Predict the measurement as: zˆ k 

2.1 AUKF Algorithm

 xˆ k 

ˆ 1 X k

  Zˆ k  h ( Xˆ 1k )

2. PROPOSED AUKF BASED FDI TECHNIQUE

.....



( n   ) [ zeros ( n  1)

Transform the sigma points (

The organization of rest of the paper is as follows. Section 2 briefly describes the proposed AUKF based FDI technique. In section 3 a case study on a LEO satellite planar model and the simulation results are shown. Section 4 provides the discussions and section 5 draws the conclusions.

Xˆ k 1  [ xˆ k 1

(4)

i

(3)

i

Project the state error covariance ahead as: 327

2014 ACODS March 13-15, 2014. Kanpur, India

2.2 ‘t’ Statistics of Innovation Sequences

windows, as shown in (24). The absolute difference between the two means is then thresholded as shown in (25).

The ‘t’ statistic test is done on each of the two innovations generated by the adaptive filter to detect and isolate the faults.

diff

mean

If, diff

For ith (i may be either 1 or 2, as there are two innovation sequences corresponding to two measurement variables) innovation ( inn i ), the t statistic is calculated for consecutive

mean



1 L

k 1

t

k

i

( j) 

jk L

t

i j   k  L 1

( j ) ;

 T 2  Abrupt jump.

(24) (25)

Where, L = Window size for abruptness checking and T 2 = Threshold for abruptness checking (empirically chosen as 10 and 0.35 respectively for the case study).

N samples [Pirmoradi2005] at kth time instant as: ___ _

t i(k) 

inn i S/

.

(21)

N

____

Where, inn i = Sample mean of the innovation sequence in the sliding window 

k

1 N

 inn

i

(22)

( j)

j  k  N 1

1 N 1

k

 ( inn

____

i

( j )  inn i )

2

When any significant abrupt jump is detected in the computed ‘t’ statistic, we should also check the consistency of the abrupt jump (i.e., the number of time instants for which the ‘t’ statistic persists with its higher value) before generating any alarm for fault. This checking is indispensable to reduce the false alarms specially at the presence of any abrupt fault. It is done here as shown in (26). If,  j  1 to M  1, t i k  j   T1  Consistent jump.

S = Sample standard deviation of the innovation sequence 

(2) Consistency Checking:

(23)

j  k  N 1

N= suitably chosen window size to calculate t statistic = 30, for the case study.

(26)

Where, M = Window size for consistency checking and T1 = Threshold for consistency checking (chosen as 5 and 3 respectively for the case study). Note that T1 should be chosen from the ‘t’ statistic table for a given confidence level and degree of freedom.

(For both these summations, the lower bound is unity when k
The parameter values for these two above stated tests should be chosen carefully because the thresholds provide the trade off between the False Alarms and Missed Detection and the window sizes provide the trade off between the False Alarms and Detection Delay.

The used ‘t’ test indeed is a test to detect any change in the sample mean of the innovation sequences. The calculated ‘t’ statistic implies to the normalized difference between the sample mean and the expected population mean (which is zero in our case) as can be seen from (21). Whenever this sample mean rises significantly from the expected zero mean value, the ‘t’ statistic crosses its threshold declaring a fault. Therefore employing ‘t’ statistic test for detecting any fault is quite justified.

An alarm is generated after executing these two tests in sequence as shown in the flow chart in Fig. 1.

2.3 Proposed Logical Analysis of the Calculated ‘t’ Statistic for Alarm Generation The calculated ‘t’ statistic is then processed logically to generate the alarms. The logical analysis step is devised here to reduce false alarms. It consists of two test named as: (1) abruptness checking and (2) consistency checking. The flow chart for generating the alarms through these tests is shown in Fig.1. (1) Abruptness Checking: To detect any fault the alarm should be generated when the tstatistic of innovation sequence changes abruptly and significantly. This is checked here by comparing the two means of the t-statistic sequence with in two overlapping

Fig. 1. Flow chart of Alarm Generation. 328

2014 ACODS March 13-15, 2014. Kanpur, India

The above stated analysis of the calculated ‘t’ statistic can generate alarms for each of the two measurement variables individually which not only detects the faults but also can isolate the faults.

z k 1  X r  r cos   x k 1 cos x k 3  v k 1

(31)

z k 2  Z r  r sin   x k 1 sin x k 3  v k 2

(32)

Where, vk1 and vk2 are the measurement noises with standard deviation 100 m and 10 m respectively. The measurement equation can be written as z k  h ( x k )  v d . Measurement

3. CASE STUDY To compare the performances of the proposed adaptive filter based FDI scheme with the non adaptive EKF, UKF based schemes; a Low Earth Orbit space vehicle planar model is considered. The simulation results here show the superiorities of the AUKF over the UKF and EKF for detecting and isolating the faults.

noise covariance would be denoted by the symbol R k .

3.1 LEO Satellite Planar Model (1) Process Model This is a classic example of nonlinear problem. Consider a near earth satellite in a nearly circular orbit [Dan2006], [Grover1983 pp. 298] as shown below in Fig. 2.

Fig. 2. LEO satellite planar model. 3.2 Simulation Data and Parameters

With reference to the Fig. 2, the system can be modelled as: x1  r  x 2

(27) 2

GM

2

r GM

x 2  r  r    x1 x 4 

2

x1

2

 w

(28)  w

x 3    x 4

(29)

2 x2 x4 2 r x 4       r x1

(30)

Where, r is the distance of the satellite from the center of the earth, θ is the angular position of the satellite in its orbit. G (=6.6742 x 10-11 m3/kg/s2 ) is the universal gravitational constant, M (=5.98 x 1024 kg) is the mass of the earth and w  is the random noise due to space debris, atmospheric drag, out gassing etc. and act here as the process noise. The above non linear system equations are in continuous time domain in the form, x  f ( x )  w . Where, x=[x1; x2; x3; x4] & w=[0; w  ; 0; 0]. It needs to be discretized for filter implementation. The discretized form can be represented as, x k  f d ( x k 1 )  w d . Where, fd is the discretized nonlinear function, wd is the discretized process noise and xk is the discretized state. The discrete process noise covariance is denoted by Qk. (2) Measurement Model The projection of the satellite radius along X and Z axes as can be seen in Fig. 2, are taken as the measurement variables and are represented as:

Extensive simulation studies have been carried out in MATLAB for the comparative studies of the proposed adaptive filter based scheme with the non adaptive filter based schemes. The relevant parameters for the filters and FDI algorithm are provided in Table 2. Table 2. Simulation Parameters A. Filter Parameters Symbol

Value

Comment/description

α

0.6

Determines the spread of sigma points.

β

2

Incorporates prior knowledge about the noise distribution.

σwd

0.001 m./sec2.

Standard deviation (S.d.) of the discrete process noise.

Qk

Diag([0,σwd2,0,0])

Discrete process noise covariance

σv1

100 m.

S.d. of the discrete measurement noise1.

σv2

10 m.

S.d. of the discrete measurement noise2.

R_true

Diag([σv12, σv22])

True value of discrete measurement noise covariance.

x0

[6.57x106 m; 0 m/sec; 0 rad;

True value of the initial states.

GM r 329

3

rad./sec.]

2014 ACODS March 13-15, 2014. Kanpur, India

xˆ 0

x0

Initial state estimation of all the employed filters.

P0

Diag([0 0 0 0])

Initial error covariance of all the employed filters.

wS

150

Window size for adaptation

Ts

0.1 sec.

Sampling time.

Fig.4 shows the ‘t’ statistics of the innovation sequences generated by the AUKF. It may be observed from this figure that only the ‘t’ statistics of the corresponding innovation sequence in which measurement the fault has occurred (i.e., innovation(2) of the corresponding measurement(2)), rises above the threshold declaring an occurrence of a fault. So, it may be claimed that the proposed ‘t’ statistic based technique can detect as well as isolate the fault at the same time.

B. FDI Parameters Symbol

Value

Comment/description

N

30

Window size to calculate t statistic

L

10

Window size for abruptness checking

M

5

Window size for consistency checking

T1

3

Threshold for consistency checking

T2

0.35

Threshold for abruptness checking

In Fig.5 the second innovation sequence by AUKF, UKF and EKF are plotted which clearly shows the superiority of the AUKF over the UKF and EKF. It may be observed that the rise of ‘t’ statistics in AUKF is more compared to the others and also it persists for longer duration above the threshold. This property of AUKF is in turn helpful to easily isolate the faults. The plot of the second diagonal element of the measurement noise covariance as shown in Fig 6, illustrates that the measurement fault is immersed in the measurement noise covariance by the AUKF which leads to satisfactory estimation performance even after the occurrence of fault.

All the window sizes are empirically chosen here as shown in Table 2. The process noise covariance (Qk) is assumed to be constant and same as the true value for all the filters. The true measurement noise covariance (R_true) is given in Table 2. The non adaptive filters measurement noise covariance values are same as R_true, whereas the AUKF is only initialized by R_true. 3.3 Simulation Results The fault detection and identification efficacy of the proposed AUKF based FDI scheme is examined against the standard algorithm like EKF and UKF based schemes in two simulation scenarios: (1) at the presence of abrupt fault and (2) at the presence of transient fault.

Fig. 3. RMS errors of ‘r’by the AUKF, UKF and EKF at the presence of fault.

(1) At the Presence of Abrupt Fault: From the extensive simulation studies carried out, it is found that the proposed AUKF based technique performs prominently better than the non adaptive filter based techniques especially at the presence of large abrupt faults. A large abrupt fault of 100m is introduced in measurement 2 (Zk2) on 200 sec. In Fig.3 the RMS errors of state1 (r) by the filters with 100 Monte Carlo runs have been shown when the fault has occurred. It may be observed from this figure that the error by the AUKF doesn’t increase rapidly and drastically even on the occurrence of large measurement fault. This phenomenon provides sufficient time to the user to recover the fault.

Fig. 4. ‘t’ statistics of the innovation sequences by the AUKF. (inn(1) & inn(2) are respectively the innovations corresponding to first and second measurements)

330

2014 ACODS March 13-15, 2014. Kanpur, India

202.2 202.3

1 1

1 1

0 1

It may be found from the above table that type1 error i.e., False Alarms are generated by both EKF and UKF at 57.2 sec. and 108.3 sec. but AUKF does not generate any False Alarms due to these transient faults. In that context it should be noted that when any alarm signal goes from its 0 value to 1, that time instant is considered as the fault detection time. It may also be observed from Table 3 that the detection delay is almost same for the proposed AUKF, UKF and EKF. Fig. 5. ‘t’ statistics of the innovation of second measurement only.

4. DISCISSIONS 

The t-statistic based proposed FDI method is found to be straightforward and simple, with less computational burden.



After the generation of ‘t’ statistic, the logical analysis step comprising of consistency checking and abruptness checking presented here reduces the false alarms.



The use of AUKF here instead of UKF/ EKF to generate the innovation sequences has the following advantages:

Fig. 6. Second diagonal element of the measurement noise covariance (R).



The change of amplitude in the t-statistic when fault is encountered is more in the AUKF compared to the UKF and EKF, i.e., AUKF is more sensitive to the faults which in turn may be advantageous in fault detection.



At the presence of abrupt faults, the alarm signals persist for longer durations in the AUKF based scheme. For which, it can detect and isolate the faults more easily leading to reduced false alarms and quick identification.



At the presence of transient fault also, it is observed that the EKF and UKF based standard algorithm generate more false alarms compared to proposed AUKF based scheme.



Use of AUKF provides longer time to recover the fault compared to the UKF and EKF, with in which the estimation results remain satisfactory inspite of the presence of fault.

(2) At the Presence of Transient Fault: Transient faults of magnitude 50 m, 80 m and 100 m are introduced in measurement 2 (z2) respectively on 50 sec., 100 sec. and 200 sec. The representative alarm signals for these faults are presented in Table3. Table 3. Generated Alarm Signals for Transient Faults Time (sec.) 52.6 52.7

Alarms for measurement 2 EKF UKF AUKF 0 0 0 1 1 1

57.1 57.2

0 1

0 1

0 0

101.3 101.4

0 0

0 0

0 1

102.4 102.5

0 1

0 1

1 1

108.2 108.3

0 1

0 1

0 0

202.1

0

0

0



The residual based R adaptive UKF is chosen here due to its inherent positive definiteness property.



The ‘t’ statistics of both the residual and the innovation sequences may be applied for FDI purpose. However, in this particular problem no significant difference is discerned between these two options and hence only innovation based ‘t’ statistics results are shown.



The following assumptions should be kept in mind to apply the proposed scheme: (i) the process noise covariance should be constant, (ii) while the measurement noise covariance (in fault free case) may either be constant or may vary slowly.

The performance of the FDI depends mildly on the window size chosen for the ‘t’ test. Too large window may slowed 331

2014 ACODS March 13-15, 2014. Kanpur, India

down the response whereas too small window may give rise to false alarms. The window size is chosen empirically here.

Grover B. R. (1983). Introduction to random signal analysis and Kalman filtering. John Wiley & Sons Inc., New York. Hajiyev Chingiz, (2009). Innovation approach based sensor FDI in LEO satellite attitude determination and control system, the book: Kalman Filter: Recent Advances and Applications, I-Tech Education and Publishing KG, Vienna, Austria, 347-374. Hajiyev C., Soken H. E. (2013). Robust adaptive unscented Kalman filter for attitude estimation of pico satellites. International Journal of Adaptive Control and Signal Processing. Mehra R., Seereeram S., Bayard D. & Hadaegh F. (1995). Adaptive Kalman filtering, failure detection and identification for spacecraft attitude estimation. In Control Applications, 1995., Proceedings of the 4th IEEE Conference on (pp. 176-181). IEEE. Pirmoradi F., (2005). Fault detection and diagnosis in a spacecraft attitude determination system, M.A.Sc. dissertation. Dept. Mech. Eng., University of British Columbia, Vancouver. Pirmoradi F., Sassani F. and de Silva C.W. (2007). An efficient algorithm for health monitoring and fault diagnosis in a spacecraft attitude determination system. In Systems, Man and Cybernetics, 2007. ISIC. IEEE International Conference on (pp. 4024-4030). IEEE. Pirmoradi F., Sassani F. and de Silva C.W. (2009). Fault detection and diagnosis in a spacecraft attitude determination system. Acta Astronautica, 65(5), 710729. Soken H. E., Hajiyev C. (2010). Pico satellite attitude estimation via robust unscented Kalman filter in the presence of measurement faults. ISA transactions, 49(3), 249-256. Xiong K., Chan C. W. & Zhang H. Y. (2007). Detection of satellite attitude sensor faults using the UKF. Aerospace and Electronic Systems, IEEE Transactions on, 43(2), 480-491.

5. CONCLUSIONS A Fault Detection and Isolation (FDI) scheme to detect and isolate faults for a highly nonlinear system using R-adaptive UKF has been proposed, elaborated and exemplified here. The residual based adaptation of the measurement noise covariance matrix is recommended due to its inherent positive definiteness. The fault detection and isolation capability of the proposed AUKF based scheme is found to be noticeably superior compared to the UKF and EKF based schemes by a through simulation study on a LEO satellite planar model. The presented logical analysis of the ‘t’ statistics comprising of consistency checking and abruptness checking is found to minimize the false alarms. The fault recovery time provided by this scheme is also found to be more as the estimation results do not degrade drastically on the occurrence of any abrupt fault. Moreover, the computation burden of the proposed simple straightforward FDI scheme is less as it does not require any bank of filters. The results presented here are expected to encourage further studies on the proposed adaptive filter based FDI scheme in more relevant practical problems. ACKNOWLEDGEMENT The first author would like to thank the Council of Scientific and Industrial Research, India, for financial support. Facilities for the research have been provided by the Head of the Electrical Engineering Department and the Coordinator, Centre for Knowledge Based Systems, Jadavpur University, Kolkata, India. REFERENCES Bae J. & Kim Y. (2010). Attitude estimation for satellite fault tolerant system using federated unscented Kalman filter. International Journal of Aeronautical and Space Sciences, 11(2), 80-86. Bae J., Yoon, S. & Kim Y. (2011). Fault-Tolerant Attitude Estimation for Satellite using Federated Unscented Kalman Filter., the book: Advances in Spacecraft Technologies, InTech, 213-232. Cornejo N. E., Amini R., and Gaydadjiev G. (2010). Modelbased fault detection for the Delfi-n3Xt attitude determination System. In Aerospace Conference, 2010 IEEE (pp. 1-8). IEEE. Dan Simon, (2006). Optimal state estimation, Kalman, H-α and nonlinear approaches, Chapter 14. John Wiley & Sons Inc., Hoboken, New Jersey. Das M., Smita Sadhu and T. K. Ghoshal (2013). An adaptive sigma point filter for nonlinear filtering problems. International Journal of Electrical, Electronics and Computer Engineering, 2(2), 13-19. Gertler J. J. (1988). Survey of model-based failure detection and isolation in complex plants. Control Systems Magazine, IEEE, 8(6), 3-11.

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