Pergamon

PII:S0967-0661(96)00193-1

ControlEng. Practice, Vol. 4, No. 12, pp. 1747-1752, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0967-0661/96 $15.00 + 0.00

OPTIMAL DESIGN OF ROBUST ANALYTICAL REDUNDANCY FOR A REDUNDANT STRAPDOWN INERTIAL NAVIGATION SYSTEM

Yan Dong* and Zhang Hongyue**

*Department of Precision Instrument, Division of Navigation and Control, Tsinghua University, Beijing 100084, P.R. China (yandong.bbs @ captain.net.tsinghua.edu.cn)

**Department of Automatic Control, Division 301, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R. China

(Received March 1996; in final form September 1996)

Abstract: In this paper, a new method of calculating the parity vector is proposed.

A performance criterion for robust fault detection is given. By using the theory of

generalized eigenstructure, the criterion can be solved to obtain the parity. The new

method was used for the failure detection and isolation of a redundant strapdown

inertial navigation system. By comparison with the classical Potter method, the new

parity is more sensitive to failures, giving rise to the conclusion that the optimal de-

sign of robust analytical redundancy is efficient and practical.

Keywords: Robust analytical redundancy, failure detection and isolation, redundant

strapdown inertial navigation system

1 . I N T R O D U C T I O N

A method of calculating parity vectors was pro- posed by Potter and Sunman (1977), and has been extensively used (Daly, et al, 1979; Desai and Ray, 1981; Chow and Willsky, 1984; Lou, et al, 1986; Chen Jie and Zhang Hongyue, 1990; Chen, et al, 1990). However, this method is derived under the condition that the sensors axe mounted in a spe- cial configuration. Under this configuration, the failure has the same influence on failure decision function. In actual systems, however, the spe- cially symmetric configuration cannot be satis- fied. Even though an ideal symmetric mounting configuration is designed, the implementation will not be perfect on account of the existence of sen- sor errors such as input lnisalignment errors, scale factor errors, or bias errors. Therefore, the appli- cation of this method is limited.

In this paper, a new method of calculating the par- ity vector is studied. A performance criterion for

robust fault detection is found by using the theory of generalized eigenstructure. Due to this perfor- mance criterion, the parity equation is highly sen- sitive to the failures and insensitive to the states and s)~stem noises and modeling errors.

In order to reduce the influence of sensor errors, the parity vector can be compensated by estimat- ing sensor errors, using non-linear filtering.

By comparison with the generalized likelihood test method (GLT), it can be concluded that the op- timal design of robust analytical redundancy is more efficient and practical in engineering sys- terns.

2. G E N E R A L I Z E D L I K E L I H O O D T E S T M E T H O D (GLT)

Suppose that there are n sensors without sensor errors; then the measurement equation is as fol- low s:

1747

1748 Yah Dong and Zhang Hongyue

m = H ~ , + ¢ , (1)

where w is the state vector [angular velocity], H !Ls the geometric matrix of sensors, and T/is a Gaus- sian white-noise sequence.

For detecting failure, the following parity equa- tions

p = ~.~ (2)

are formed, where p is an (n - 3)-dimensional vector, and p is defined as parity vector. The (n - 3) x n projection matrix v is determined such that

vH : o (a)

and

vT v = In -- H ( H T H ) - I HT (4)

~ v r = / . _ ~ . (~)

The rows of v are orthogonal, v is a left null- space of H. When the failure (hi) happens, the measurement equation is changed to:

(6) rn = Hw + bl + r I.

Now the parity vector is changed to:

p = ~b s + r e . (7)

It can be seen that the parity vector reflects the inconsistency between the outputs of the sensors; p is independent of the state variables.

Hi : failure happens, E(p) # 0, E ( I ( p - p ) ( p -

Because p is a white-noise sequence, the probabil- ity density function under the two hypotheses can be written as follows:

g o :

1 ~pT(vvT)_ lp) I Io = V~--~lvvTl½e=pi-

Hi:

(9)

f l = V~I1) I )T[~ .O " 2

(1o) The logarithm likelihood ratio test of the proba- bility density is as follows:

1 ^(p) = l . ( i~ / /o ) = ~[p(,~,,r)-9

- ( p - D)T (vvT) - I (p -- p)]. (II)

From the likelihood ratio function, the failure de- cision function (DFD) can be obtained as:

D F D = pT(t~uT)-lp = pTp.

The decision rule is as follows:

(12)

D F D >__ TD failure happens (13) D F D < TD no failure.

By a similar process, the isolation function (DFIi) can be calculated as follows:

The choice of v is the key point of the FDI method. v can be calculated as follows (Potter and Sun- man, 1977):

w = I - H ( H T H ) - I H r = [w( i , ] ) ] v2(1, 1) = w(l, 1) v(l,]) = w(1, ] ) /v(1, 1) ( y = 2 , 3 , , .)

i - i

k = l ( i = 2 , 3 , , . - a )

i--1

,,(i, y) = [,~(~, y) - ~ ,,(k, i)~,(k, Y)II,,(~, i) k=-I

(i = 2, 3...n -- 3 ; j = i + 1, i + 2...n;y < i) ,,(~,y) = o

(s) Suppose the noises of the sensors are independent, and their covariances are cr 2, two hypotheses can be drawn:

Hypothesis H0: no failure, E(p) = O, E(pp T) := ~ T ~ 2

D E Ii = [PT ( VVT ) - l v' ]2 -- [PT V' ]2

where vi is the i-th column of matrix v. If

(14)

DFI~ = max DFII, (15) l<i<•

then sensor k is the most probable failed sensor.

3. M E A N V A L U E T E S T M E T H O D ( M V T )

In this section, a new mean value test method is introduced (Yan Dong and Zhang Hongyue, 1995). Suppose that xl, x2,..., x , are samples of the normal space X. X has a N(/~0, a0) distribu- tion. MVT can be used to detect the hypdthesis H0(p --- Po); the formula for the mean value test is as follows:

u - x - ~o v ~ , ( l ~ ) (TO

where

Optimal Design of Robust Analytical Redundancy 1749

n

= i E Xi. (17) n

i = 1

5 . P A R I T Y V E C T O R C O M P E N S A T I O N U S I N G N O N L I N E A R F I L T E R I N G

If Ho(# = #0) is true, U has a N(0,1) distribu- tion. When H0 is not true, U has a N{*, 1) dis- tribution; * is a nonzero value. This means that when H0 is not true, U will exceed some thresh- olds. The threshold can be determined using the normal distribution table, given a confidence level a . The decision function is as follows:

U > U~ failure happens ' (18)

U <_ U~ no failure.

Then p can be detected statistically as follows:

u = (19) O'v.

where ~r.. is the root mean value of v., k is the k - th mumber of sampling points, and

k 1

= ~ Ep(k). (20) i = 1

Failure can be detected using U (shown in (19)) and the rules in (18).

Considering the sensor errors, the measurement equation can be described as follows:

r n = (I+H~)(H+H.~)w+b+'7, (24)

where H is an n x 3-dimensional geometric ma- trix, Hse is an n x n-dimensionai scale factor error matrix, Hm~ is an n × 3- dimensional input mis- alignment error matrix, b is an n × 1-dimensional bias vector, and '7 is an n × 1-dimensional white- noise sequence.

Different configurations give different//me. It can be described as :

Hrnel,1 Hmel,2 Hme2,1 Hme2,2

Hl'ltbe

Hmen,1 Hmen,2

H,~ can be expressed as:

Hmel,3 Hme2,s

HFrt e f t , 3

(25)

4 . O P T I M A L D E S I G N O F R O B U S T

A N A L Y T I C A L R E D U N D A N C Y

A new method of calculating the parity vector has been proposed (Zhang Hongyue and Patton, 1992). From formula (6), it is known that, if the failure happens, the measurement equation is changed to:

m = H~o + B6! + '7. (21)

If the k-th sensor has failed, the k - th diagonal of B is 1, and other elements of B axe 0.

When calculating the par i ty vector, the residual should be insensitive to the state value, so vH should be made as small as possible. Meanwhile, in order to make the residual sensitive to the fail- ure, the v B should be made as large as possible. Therefore, the performance criterion can be de- signed as follows:

HvHII (22) ~ n J = ~m ilvB[i.

By using the theory of generalized eigenstructure, the criterion can be solved to obtain an opt imal redundancy relation for failure detection. The so- lution of equation (22) can be wri t ten as follows:

H H T v = ABBT v, (23)

where A is the eigenvalue of matr ix H H T - A B B T , and v is the corresponding eigenvector.

/ L e =

-[tsel 0 -- - 0 0

0 H'se2 .-. 0 0 .- • . . . : •

0 0 "'" 0 H~e.

(26)

b = [ 6 1 , 6 2 , . . . , 6 . 1 T , ' 7 = [ ' 7 1 , ' 7 2 , ' " , ~ n l T

Suppose Hsei, bl, Hmei,y(i = 1. . .n,] = 1, 2, 3) axe random constants; the s tate variable vector x is chosen as:

[(~)1, ~)2, &~3, Ssel, Sse2,'", H .... Hmel, 1, Hmel,2,

Hn, el,3,"', Sn, en, 1, Hmen,2, Hme,,,3, 61,62,-- ", bn ]T (27)

It is easy to see tha t equation (24) is non-lineax.

The system equation can be described as:

zk+1 = xk + gk, (28)

where 9k = [91k, g2k , g3k, 0,' "', 0] T, gik({ = 1, 2, 3) axe Gaussian white noises.

For detecting failure, it is desired tha t the decision function should only be related to the failures and the noises. Therefore, the par i ty vector should be compensated as follows (Yan Dong and Zhang Hongyue, 1994):

p* = p -- v([Ime + l~I.e121.ne + ISl.eH)& - vb (29)

1750 Yan Dongand ZhangHongyue

~ ( H s ~ H ~ - & ~ , ~ o ) + ~(b - £) + ~n. (3o)

From formula (30), it can be seen that if c~

w, ~/se ~ Hse, I~rne ~ IIme, b ~ b, t hen p* "~ vt 1. In th is case, when the fa i lure happens ,

p* ~ ttbf -t- vr/. (31)

Therefore, the failure can be detected and isolated using the compensated parity, p*.

Hse5x 5 ~-~

bl b2

b6xi = b3 b4 bs be

Hsei 0 0 0 0

~6xl~---

r]l W2 r/3

W4 W5 ~6

0 0 0 0

I'ts e 2 0 0 0 0 / /sea 0 0 0 0 Hse4 0 0 0 0 HseS

6. A P P L I C A T I O N A N D S I M U L A T I O N

Six gyros which are m o u n t e d on a r egu la r po ly- hedron wi th 12 faces (specific s y m m e t r i c configu- r a t ion ) , and five gyros which are m o u n t e d on an even cone face (not sa t i s fy ing the specific s y m m e t - ric conf igura t ion) are s imula ted . The geomet r ic ma t r i ces of the two conf igura t ions are as follows:

bl ~1 b2 r/2

b s x l : b3 r / sx l = "/3 b4 ~4 b5 ~75

The moun t ing e r ror m a t r i x of the six sensors can be der ived as:

~ X 3 =

sin0 0 cosO -sinO 0 cosO cosO sinO 0 cosO -sinO 0

0 cosO sinO 0 cosO - s in0

H s x a =

sina 0 cosa sinacos~ sinasin~ cosa

- s inacos~ s inas in~ cosa -s inacos~ - s inas in~ 2 cosa sinacos~ - s inas in~ cosa

where 0 = 31.72 ° , sinc= : ~ /~ , /~ = 72 ° , 0,c=,/~

are the idea l m o u n t i n g angles. The t rue m o u n t i n g w

angles are desc r ibed as:

0 i = 0 + 6 0 ~ , i = 1 , 2 , . - ' 6 a ~ = a + 6 a l , i = 1 , 2 , . - . 5 /~ = / ~ + 6 / ~ , i = 1 , 2 , - . , 5

where 60 i , i = 1, 2 , . . . , 6, 6o~, 6/~ii = 1, 2, ..., 5 axe the misa l ignments .

Therefore, HoesxO can be described as follows:

Hsez 0 0 0 0 0 0 Hoe2 0 0 0 0 0 0 H.,e3 0 0 0 0 0 0 Hse4 0 0 0 0 0 0 HseS 0 0 0 0 0 0 Hses

Hme6x3 :

C089601 0 --8in0~01 --C0S0602 0 --sinO&02 --sine6ea cos0603 0 --sin9604 --cos0604 0

0 -sinO60s cosO60s 0 -sinO6Os -cos960s

Because the expression of Hmesxa is very com- plex (Yan Dong and Zhang Hongyue, 1994), it is omitted here. In the expressions of Hmee×3 and Hmesx3, 6ai and 6~i,i = 1,2 .... ,5 axe input misalignments, their mean-square value is 0.018 °, the mean-square value of the scale-factor errors Hsei(i = 1, 2...5) is 0.001, the mean-squaxe value of the biases bi(i = 1, 2, ..., 5) is 0 .00004°/second, and the covariance of gi(i = 1, 2, 3) and tIi(i = 1, 2, ..., 5) is 1 . 5 5 E - 11(°/second) 2. The first sen- sor failed at 1.25 seconds, and the value of the failure is O.OOOO15 °/secong. The sampling time is 0.05 seconds.

(a) No sensor e r ror case

The m i n i m a l fa i lures t h a t can be de t ec t ed by the G L T m e t h o d for six sensors using the pa r i ty vec tor ca l cu la t ed by the p roposed me thod , and by the P o t t e r m e t h o d , are jus t the same.

3.6e -- 5 3.6e -- 5 3.9e -- 5 2.7e -- 5 9.0e -- 6 3.9e -- 5

The m i n i m a l fa i lure t h a t can be de t ec t ed by the

Optimal Design of Robust Analytical Redundancy

G L T and M V T m e t h o d s for tlve sensors is as fol- lows:

(GLT) m e t h o d

3.1e -- 5 2.3e -- 5 2.3e -- 5 2.5e -- 5 2.2e -- 6

( M V T ) m e t h o d

2 . 6 e - 5 2.4e - 5 2.4e - 5 2.6e - 5 2.2e - 6

It can be seen f rom the above resu l t s t h a t the a c -

c u r a c y of the M V T m e t h o d us ing a p a r i t y vec to r ca l cu l a t ed by the new m e t h o d is b e t t e r t h a n t h a t of t he G L T m e t h o d us ing a p a r i t y vec to r ca lcu- l a t ed by the P o t t e r m e t h o d .

1751

2E-010 1 ~E-010 1.6E-010 1.4E-010

~ 1.2E-010 I 1E..010

8E-011 6E-011 4E-011

2E-011

0 0 10 20 30 40 50 60 70 80 90100

lme (" secor )

Fig. 1. Curves of the de t ec t ion value of the G L T m e t h o d

The de t ec t ion and i so la t ion curves of the M V T m e t h o d are shown in Fig.2

(b) Sensor er rors case

The m i n i m a l fa i lure t h a t can be d e t e c t e d by the G L T and M V T m e t h o d s is as follows:

(GLT) m e t h o d

3.1e - 5 1 . 5 e - 5

1.6e - 5 1.6e -- 5 1.9e -- 6 2.2e - 5

( M V T ) m e t h o d

3 .0e - 5

6e - 6

8e - 6 1 .9e - 5

1 .9e - 6

6e - 6

F r o m the above formulae , i t can be seen t ha t the accuracy of M V T m e t h o d us ing the p a r i t y calcu- l a t ed by the new m e t h o d is h igher t h a n t h a t of G L T m e t h o d using the p a r i t y ca l cu l a t ed by the P o t t e r m e t h o d .

Here, the i so la t ion func t ion DFIi of t he G L T m e t h o d for five sensors is as follows:

1 . 8 3 1 6 4 e - - 1 0

9.99124e -- 1 1

6.96662e -- 1 0 .

3.27748e - 1 1

1.41671e -- 1 1

DFI1 is the la rges t one, so sensor 1 c a n be iso-.

l a ted .

3 I /~ . [_ / ~.~U velue

2 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I: ?,..,,:],,^ ~ . t ,

i i . 5 I: I - s e n s o r 1

.~ t~: ~,~ ~, "4 r % ; 4 - - ~ D i ' l l ~ r 4

i-u* 0 0 10 20 3040 50 80 7080 00100

Tlrne (" 0~35 second )

Fig. 2. De tec t ion value of the M V T m e t h o d

7. C O N C L U S I O N

Af te r numerous s imula t ions , it can be conc luded t h a t non l inea r f i l ter ing can e s t i m a t e the sensor er- rors accura te ly . The p a r i t y vec tor c o m p e n s a t i o n a p p r o a c h can reduce the influence of sensor errors such as i npu t misa l ignmen t s , scale fac tor errors , and biases, and can allow the use of a cons t an t t h r e sho ld for fa i lure de tec t ion . If the re is no sen- sor er ror , and the specific s y m m e t r i c configura- t ion is sat isf ied, the accuracy for the M V T and G L T m e t h o d s is a b o u t the same, no m a t t e r w h a t m e t h o d is used to ca lcu la te v. But if the spe- cific s y m m e t r i c conf igura t ion is not be ing sat isf ied, the accuracy of the M V T m e t h o d using the par - i ty ca l cu la t ed by the new m e t h o d is b e t t e r t h a n t h a t of the G L T m e t h o d using the pa r i t y calcu- l a t ed by the P o t t e r m e t h o d . In the case of sensor errors , the accuracy of the M V T m e t h o d using

1752 Yan Dong and Zhang Hongyue

the parity calculated by the new method is much better than that of the GLT method using the parity calculated by the Potter method. There- fore, this proves that the optimal design of rc~- bust analytic redundancy is efficient and practical.

R E F E R E N C E S

Chen Jie, Zhang Hongyue, (1990). Parity vec- tor approach for detecting failures in dy- na~nic systems, Int. J. Systems Sci., Vol, 21, No.4, 765-770

Chen Jie, Zhang Hongyue and Yi Guangqu, (1990). Acta Aeronautica et Astronautic a sinica, Voi.11, No.S, March, 175-182

Chow, E. Y. and Willsky, A. S. (1984). Ana- lytical redundance and the design of robust failure detection systems. IEEE Trans. Aut. Control, AC-29, 603-614

Daly~ K. C:, Gai:E and Harrison, J. V. (19791). Generalized Likelihood Test for FDI in Re- dundant Sensor Configurations. Journal eft Guidance and Control, 2:(2), 9-17

Desai, M. and Ray, A. {1981). A fault detection

and isolation methodology. Proc. 20th Conf. on Decision and Control, 1363-1369

Lou, X. C., Willsky, A. S. and Verghese, G. L. (1986). Optimally robust redundancy rela- tions for failure detection in uncertain sys- tems. Automatica, 22,333-344

Potter, I. E. and Sunman, M. C. (1977). Thresh- olclless redundancy management with arrays of skewed instruments. Integrity in Electronic Flight Control Systems. Agardograph-224, 15-11 to 15-25

Yan Dong and Zhang Hongyue, (1994). Parity Vector Compensation Using Non-linear Fil- tering, IFAC Workshop on Safety, Reliabi- lity and Applications of Emerging Intelligent Control Technologies, December 12-14, Hong Kong, 177-181

Yan Dong and Zhang Hongyue, (1995}. Mean Value Test for FDI in Redundant Sensor Con- figurations, American Control Conference, June 21-23

Zhang Hongyue and R.J. Patton, (1992). Op- timal Design of Robust Analytical Redun- dancy for Uncertain Systems, IEEE TEN- CON'92, Beijing