Robust on

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Pergamon ConwoiEn&. Practice, VoL 2 No. 2 pp. 201-209 1994 Co~ © l~41~ Scie~ L~ Pd~diaOn~Bdu~All~n~w~d 0967.0~I/94 (~00 + 0.00 ROBUST ON LINE DIFFERENTIATION TECHNIQUES WITH AN APPLICATION TO UNDERGROUND COAL MINING 1 S. Crisafulll ,2 and T.P. Medhurst *Cooperative Research C entre for Robust and Adaptive Systems, Australian National U niversity, Canberra, ACT 0200, Australia **CSlRO Di¢isi on of F.xploration and Mining, P.O . Box 883, Kenmore, Q LD 4069, Australia Abstract: In this paper an algorithm for robust on-line rate estimation of noisy digital signals is developed. The algorithm is based on Kalman filtering techniques and consequently has the capability of coping with noise and signal discontinuities. This algorithm is then successfully applied to hydraulic leg pressure data obtained from mobile mechanised roof supports (breaker line supports) in an underground coal mine. The data is collected by a monitoring system (BLSm on) which allows real-time data processing to aid in determining the onset of roof caving events. Keywords: digital differentiation, rate estimation, underground coal mining insta- bility, Kalman filtering 1. INTRODUCTION The Australian Commonwealth Scientific and In- dustrial Research Organisation (CSIRO) has re- cently initiated a program for the Australian un- derground coal mining industry to utilise auto- mated computer controlled instrumentation for geomechanical applications (FoUington and Med- burst, 1992). The primary focus of this program is to conduct research into the engineering re- sponse of the rock mass to mining activities by employing key measurement and analysis strate- gies. A study of roof and pillar stability in under- ground coal mines was undertaken as part of this program (Follington et aL 1992). In conjunction with this, a breaker line support monitoring sys- tem (BLSmon) was also developed (Hutchinson and Medhurst, 1992). The BLSmon records hy- draulic leg pressures and canopy position of the mechanised roof supports (breaker line supports (BLS)) and relays this information to the surface in real time. At the surface, the information is collected by a laptop computer, thus providing IA n earlier version of this paper was presented at the IFAC W orld Congress, Sydney, Australi a, July 1993 2 The author wishes to acknowledge the funding of the activities of the Cooperative P~esearch Centre for Robust and Adaptive Systems by the Australian Government un- der the Cooperative Research Centres Program valuable information for observation and analy- sis. Recent studies (Maleki, 1990; Follington et al. 1992) have identified a possible signature of the onset of instability in underground coal mines. Under favourable mining conditions, the rate of change of leg loading was negative throughout most of the duration of an individual cutting cy- cle or lift. However, when the mining conditions were unfavourable, eg. an unstable roof, negative rates of change of pressure were reduced through- out the lift in both duration and magnitude, and positive rates of change were much greater. Thus on-line calculation of leg loading rates was per- ceived to be valuable in establishing roof stability criteria and minimising production losses result- ing from roof caving events. Simple approaches to taking the time derivative, such as taking the difference between two consec- utive points, are not appropriate for most prac- tical applications including this leg loading rate estimation problem. The main reason for this is the lack of robustness to noise and signal discon- tinuities. An ideal derivative function amplifies high-frequency noise resulting in an output signal that is not very useful. The introduction of ad hoc fixes such as clipping, low-pass filtering, etc usually results in loss of information and inher-

Transcript of Robust on

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Pe r g a mo nConwoiEn&.Practice,VoL 2 No. 2 pp. 201-209 1994

C o ~ © l ~ 4 1 ~ S c ie ~ L ~P d ~ d i a O n ~ B d u ~ A l l ~ n ~ w ~ d

0967.0~I/94 (~00 + 0.00

R O B U S T O N L IN E D I F F E R E N T I A T I O N T E C H N I Q U E S W I T H

A N A P P L I C A T I O N T O U N D E R G R O U N D C O A L M I N I N G 1

S . C r i s a f u l l l , 2 a n d T . P . M e d h u r s t

*Cooperative Research C entre for Rob ust and Adaptive Systems, Australian National U niversity,

Canberra, ACT 0200, Australia

**CSlRO Di¢is ion of F.xploration and M ining, P.O . Box 883, Kenmore, Q LD 4069, Austral ia

A b s t r a c t : I n t h i s p a p e r a n a l g o r i t h m f o r r o b u s t o n - li n e r a t e e s t i ma t i o n o f n o i s y

d i g i t a l s i g n a l s i s d e v e l o p e d . T h e a l g o r i t h m i s b a s e d o n K a l ma n f i l t e r i n g t e c h n i q u e s

a n d c o n s e q u e n t l y h a s t h e c a p a b i l i t y o f c o p i n g w i t h n o i s e a n d s i g n al d i s c o n ti n u i ti e s .T h i s a l g o r i t h m i s t h e n s u c c e s s f u l ly a p p l i e d t o h y d r a u l i c le g p r e s s u r e d a t a o b t a i n e d

f r o m m o b i le m e c h a n i s e d r o o f s u p p o r t s ( b r e a k e r li ne s u p p o r t s ) i n a n u n d e r g r o u n d c o a l

mi n e . T h e d a t a i s c o l l e c t e d b y a mo n i t o r i n g s y s t e m ( B L Sm o n ) w h i c h al lo w s r e a l - t i me

d a t a p r o c e s s i n g t o a i d i n d e t e r m i n i n g t h e o n s e t o f r o o f c a v i n g e v e n t s .

K e y w o r d s : d i g it a l di f fe r e n ti a ti o n , r a te e s t i m a t i o n , u n d e r g r o u n d c o a l m i n i ng i n s ta -

b i l i t y , Kalman f i l t e r ing

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

T h e A u s t r a l i a n C o m m o n w e a l t h S c ie n ti fi c a n d I n -d u s t r i a l R e s e a r c h O r g a n i s a t i o n ( C S I R O ) h a s r e -

c e n t l y i n i t i a t e d a p r o g r a m f o r t h e A u s t r a l i a n u n -

d e r g r o u n d c o a l m i n i n g i n d u s t r y t o u t i l i s e a u t o -

m a t e d c o m p u t e r c o n t r o l l e d i n s t r u m e n t a t i o n f o r

g e o me c h a n i c a l a p p l i c a t i o n s ( Fo U i n g t o n a n d Me d -

b u r s t , 1 9 92 ) . T h e p r i m a r y f o c u s o f t h i s p r o g r a m

i s t o c o n d u c t r e s e a r c h i n t o t h e e n g i n e e r i n g r e -

s p o n s e o f t h e r o c k ma s s t o mi n i n g a c t i v i t i e s b y

e m p l o y i n g k e y m e a s u r e m e n t a n d a n a l y s i s s t r a t e -

gies.

A s t u d y o f r o o f a n d p i l l a r s t a b i l it y i n u n d e r -

g r o u n d c o a l m i n e s w a s u n d e r t a k e n a s p a r t o f th i s

p r o g r a m ( Fo l l i n g t o n e t aL 1 9 9 2 ) . I n c o n j u n c t i o n

w i t h t h i s , a b r e a k e r l i n e s u p p o r t mo n i t o r i n g s y s -

t e m ( B L S m o n ) w a s a l s o d e v e l o p e d ( H u t c h i n s o n

a n d M e d h u r s t , 1 9 92 ). T h e B L S m o n re c o r d s h y -

d r a u l i c l e g p r e s s u r e s a n d c a n o p y p o s i t i o n o f t h e

m e c h a n i s e d r o o f s u p p o r t s ( b r e a k e r li ne s u p p o r t s

( BL S) ) a n d r e l a y s t h i s i n f o r ma t i o n t o t h e s u r f a c e

i n r e a l t i me . A t t h e s u r f a c e , t h e i n f o r ma t i o n i sc o l l e c t e d b y a l a p t o p c o m p u t e r , t h u s p r o v i d i n g

IA n ear l ier vers ion o f th is paper was presented a t the

IFAC W orld Congress, Sydney, Australia, July 19932T h e a u t h o r w i s h e s t o a c k n o w l e d g e t h e f u n d i n g o f t h e

act iv i t ie s o f the Coo perat iv e P~esearch Centre for Rob ust

a n d A d a p t i v e Systems by t h e A u s t r a l i a n G o v e r n m e n t u n -

d e r t h e C o o p e r a t i v e R e s e a r c h C e n t r e s Program

v a l u a b l e i n f o r ma t i o n f o r o b s e r v a t i o n a n d a n a l y -

sis.

Re cen t s tud ies (M alek i , 1990 ; Fo l l ing ton e t a l .

1992) have iden t i f i ed a poss ib le s igna tu re o f the

o n s e t o f i n s t a b i l i ty i n u n d e r g r o u n d c o a l m i n e s.

U n d e r f a v o u r a b l e mi n i n g c o n d i t i o n s , t h e r a t e o f

c h a n g e o f l e g l o a d i n g w a s n e g a t i v e t h r o u g h o u t

m o s t o f t h e d u r a t i o n o f a n i n d i v i d u a l c u t t i n g c y -

c le o r l if t. H o w e v e r , w h e n t h e m i n i n g c o n d i t i o n s

w e r e u n f a v o u r a b l e , e g. a n u n s t a b l e r o o f , n e g a t i v e

r a t e s o f c h a n g e o f p r e s s u r e w e r e r e d u c e d t h r o u g h -

o u t t h e li ft in b o t h d u r a t i o n a n d m a g n i t u d e , a n d

p o s i t iv e r a t e s o f c h a n g e w e r e mu c h g r e a t e r . T h u s

on- l ine ca l cu la t ion o f leg load ing ra t es w as per -

c e i v e d t o b e v a l u a b l e i n e s t a b l i s h i n g r o o f s t a b i l i t y

c r i t e r i a a n d mi n i mi s i n g p r o d u c t i o n l o s s e s r e s u l t -

i n g f r o m r o o f c a v in g e v e n t s .

S i mp l e a p p r o a c h e s t o t a k i n g t h e t i me d e r i v a t i v e ,

s u c h a s t a k i n g t h e d i f f e r e n c e b e t w e e n t w o c o n s e c -

u t i v e p o i n t s , a r e n o t a p p r o p r i a t e f o r mo s t p r a c -

t i c a l a p p l i c a t i o n s i n c l u d i n g t h i s l e g l o a d i n g r a t e

e s t i ma t i o n p r o b l e m . T h e ma i n r e a s o n f o r t h i s is

t h e l a c k o f r o b u s t n e s s t o n o i s e a n d s i g n a l d i s c o n -

t inu i t i es . An idea l der iva t ive func t ion amp l i f i es

h i g h - f r e q u e n c y n o i s e r e s u l t in g i n a n o u t p u t s i g n a l

t h a t i s n o t v e r y u s e fu l . T h e i n t r o d u c t i o n o f a d

h o c f ixes such as c l ipp ing , l ow-pass f i l t e r ing , e t c

u s u a l l y r e s u l t s i n l o s s o f i n f o r ma t i o n a n d i n h e r -

201

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202 S. Crisafulli and T.P. Med hurst

e n t l a r g e p h a s e d e l a y s . T h u s m o r e - s o p h i s t i c a t e d

t e c h n i q u e s a r e r e q u i r e d f o r a n i m p l e m e n t a t i o n o f

a r o b u s t o n - l i n e a l g o r i t h m .

t i v e l y s t r a i g h t f o r w a r d t o d e s i g n a K a l m a n f i l t e r

a l g o r i t h m o n c e a s ig n a l m o d e l h a s b e e n s p e c i fi e d

a l o n g w i t h t h e c o r r e s p o n d i n g s t a t i s t i c s.

S t a n d a r d a p p r o a c h e s f o r i m p r o v i n g o n s i m p l e

t e c h n i q u e s u s u a l l y r e q u i r e t h e i n t r o d u c t i o n o f

k n o w l e d g e o f t h e s i g n a l c h a r a c t e r i s t i c s a n d n o i s e

s t a t i s t i c s a p r i o r i i n f o r m a t i o n ) . I n t h i s c a se ,

n o i s e r e f e r s t o a n y u n c e r t a i n t i e s i n s i g n a l s d u e

to mo d e l e r r o r s p r o c e s s n o is e ) o r s e n s o r e rr o r s

m e a s u r e m e n t n o i s e) o r a n y o t h e r e r ro r s t h a t c a n -

n o t b e a c c o u n t e d f o r . I n t h i s c a s e , t h e b e s t t h a t

c a n b e a c h i e v e d i s t o s p e c i f y t h e s t a t i s t i c s o f t h e s e

e r r o r s a n d t h e n m a k e u s e o f t h e k n o w l e d g e of

t h e s e s t a t i s t i c s i n o r d e r t o c o p e w i t h t h e s e e r -

r o r s i n t h e b e s t p o s s ib l e w a y . D ig i t a l d i f f e r e n t i a -

t i o n t e c h n i q u e s a r e a v a i l a b le w h i c h y i e l d o p t i m a lp e r f o r m a n c e u n d e r c e r t a i n c o n d i t i o n s C a r l s s o n ,

1989; Car lsson et al . , 1 9 9 1 ). H o w e v e r , t h e d r a w -

b a c k s o f s u c h a p p r o a c h e s a r e c o n c e p t u a l c o m p l e x -

i t y a n d t h e a m o u n t o f e f fo r t r e q u i r e d i n o b t a i n i n g

e x p l i c i t s i g n a l m o d e l s , q u a n t i f y i n g n o i s e s t a t i s -

t ic s , etc. A ls o , a c e r t a in d e g r e e o f e x p e r t i s e i s

r e q u i r e d f o r s u c h a n a p p r o a c h w h i c h i s o f t e n n o t

i m m e d i a t e l y a v ai l a b le t o t h e p r a c t i t i o n e r .

T h e d e v e l o p m e n t o f t h e s i g n a l m o d e l i s f i rs t a d -

d r e s s e d . C o n s id e r a d i s c r e t e t ime s i g n a l Y k, w h e r e

k i s t h e d i s c r e t e t ime i n s t a n t . A s t a t i s t i c a l d e -

s c r i p t io n o f t h e d y n a m i c a l b e h a v i o u r c a n b e o b -

t a i n e d b y a t h e s t o c h a s t i c s i g na l m o d e l o f t h e

f o r m

z k = G q - 1 ) w k 1)

w h e r e w k i s a z e r o m e a n w h i t e n o i s e s e q u e n ce a n d

G q - 1 ) i s a d i s c r e t e - t im e l i n e a r t r a n s f e r f u n c t i o n

i n t e r m s o f t h e b a c k w a r d s h i f t o p e r a t o r q - 1 . S y s -

t e m i d e n t i f i c a t i o n t e c h n i q u e s L j u n g , 1 98 7 ) c a n

t h e n b e u s e d t o f in d t h e b e s t s e t o f p a r a m e t e r s

for G q - 1 ) a n d t h e b e s t s t a t i s t i c s f o r w k s u c h

th a t z k i s cl o s e t o Y k i n s o me s e n s e u s u a l l y l e a s t

s q u a r e s ) .

T h e a b o v e m o d e l 1 ) c a n a l so b e w r i t t e n i n s t a t e -

s p a c e f o r m a s

_xk+l = F_xk + _W 2)

z~ = I=ITxk 3)

I n t h i s p a p e r a n o v e l d ig i t a l d i f f e r e n t i a t o r i s d e v e l -

o p e d w h i c h is b o t h e a s y to i m p l e m e n t a n d u t i l i se si n f o r m a t i o n f r o m a s i g n a l m o d e l a n d s t a t i s t i c s .

T h e K a l m a n f i l te r b as i s o f t h i s d i f f e r e n t i a t o r al -

l o w s t h e i n c o r p o r a t i o n o f a s i g n a l m o d e l a n d n o i s e

s t a t i s t i c s i n a l o g i c a l f a s h io n . H o w e v e r , u n l i k e

t h e o p t i m a l t e c h n i q u e s o f C a r l s s o n et al. 1991)

w h i c h r e q u i r e e x p l i ci t i n f o r m a t i o n o f t h e s i g n a l

m o d e l a n d n o i s e s t a t i s t i c s , t h i s n e w s c h e m e u s e s

t h i s i n f o r m a t i o n i m p l i c i t l y i n it s o p e r a t i o n . I t i s

n o t t h e o p t i m a l i t y o f t h e K a l m a n f i l te r t h a t i s

i m p o r t a n t i n t h i s n e w s c h e m e , b u t r a t h e r t h e d e-

s i g n a p p r o a c h w h i c h h a s f e a t u r e s t h a t l e a d t o a

p r a c t i c a l s o l u t i o n .

2. K A L M A N F IL T E R I N G : B A C K G R O U N D

I N F O R M A T I O N

S o m e r e l ev a n t b a c k g r o u n d i n f o r m a t i o n o n K a l m a n

f i l t e r i n g is f i r s t o f a l l p r e s e n t e d i n t h i s s e c t i o n .

T h e K a l m a n f i l t e r i s a w e ll k n o w n a n d w i d e l y

u s e d a l g o r i t h m . I t h a s b e e n u s e d in m a n y d i v e rs e

f i e ld s s u c h a s p r o c e s s i n d u s t r i e s , a e r o s p a c e , e c o -n o m i c s , t e l e c o m m u n i c a t i o n s e t a A n d e r s o n a n d

Moore , 1979; Sorenson , 1985; Ge lb , 1986; Mu-

r a k a m i , 1 9 9 1) . A s u c c e ss f u l a p p l i c a t i o n o f K a l m a n

f i l t e r i n g re l i e s h e a v i l y o n t h e s e l e c t i o n o f a n a p -

p r o p r i a t e s t a t e - s p a c e s i g n al m o d e l . I t i s r e la -

w h e r e x k i s t h e s t a t e v e c t o r o f t h e s y s t e m , F i s

t h e s y s t e m m a t r i x , I=I i s t h e o u t p u t v e c t o r a n d W_k

i s t h e p r o c e s s no i s e v e c t o r . T h e u n d e r b a r i s u s e d

t o d e n o t e a c o l u m n v e c t o r , b o l d - t y p e d e n o t e s a

m a t r i x a n d t h e s u p e r sc r i p t T d e n o t e s t h e m a t r i x

t r a n s p o s e o p e r a t i o n . G i v e n G q - 1 ) a n d w k in

1 ) , t h e r e a r e ma n y c h o i c es r e a l i s a t i o n s ) o f F , H

a n d W k i n 2 ) - 3 ) w h ic h g iv e r i s e t o i d e n t i c a l z k

i n b o t h s y s t e m s . A n i n t e r m e d i a t e q u a n t i t y x k i s

n o w p r e s e n t i n 2 ) - 3 ) a n d t h i s i s k n o w n a s t h e

s t a t e .

S ig n a l s a r e n e v e r me a s u r e d p r e c i s e ly i n p r a c t i c e

d u e t o m e a s u r e m e n t e rr o r s s e n s o r e r r o rs , p o o r

m e a s u r e m e n t t e ch n i q u es , eta). T h e a b o v e m o d -

e ls c a n b e e x t e n d e d t o a c c o u n t f o r t h i s d e f ic i e n c y

b y t h e i n t r o d u c t i o n o f a n a d d i t i v e t e r m w h i c h a c-

c o u n t s f o r t h e s e e r r o r s s t a t i s t i c a l l y . T h i s c a n b e

w r i t t e n a s

Yk = z~ + nk 4)

w h e r e n k i s t h e m e a s u r e m e n t n o i s e w h i c h i s m o d -

e l le d b y z e r o m e a n w h i t e n o i s e a l so a s s u m e d t o

b e i n d e p e n d e n t o f W k) a n d Yk a r e n o i s y m e a s u r e -

m e n t s o f z k. A s t a t e - s p a c e m o d e l is o b t a i n e d b y

c o m b i n i n g 2 ) - 3 ) a n d 4 ) ,

x _ k + l = F x _ k + _ W k 5 )

y ~ - - H _ T x _ k+ n k . 6 )

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Rob ust On-Line D ifferentiation Techniques

A c o n c i s e d e s c r i p t i o n o f K a l m a n f i l t e r o p e r a t i o n

based on s igna l model (5 ) - (6 ) i s as fo l lows : Given

t h e n o i s y y ~ m e a s u r e m e n t s a n d t h e p r e c is e k n o w l-

e d g e o f r , H a n d t h e s t a t i s t i c s o f W k a n d n ~ ,

t h e K a l ma n f i l t e r p r o v i d e s t h e b e s t l i n e a r ( l e a s t

s q u a r e s ) e s t i m a t e o f t h e s t a t e v e c t o r (k _~) ( t h e h a t

d e n o t e s a n e s t i m a t e ) . F u r t h e r m o r e , i f t h e n o is e

d i s t r i b u t i o n s a r e G a u s s i a n , t h e n t h e e s t i ma t e i sK k =

o p t i ma l ( l e a s t s q u a r e s ) , i . e . t he bes t over a l l l i n -

ear and non l ine ar es t im ato r s . -xk =

R e m a r k : I n p r a c t ic e , t h e p a r a m e t e r s a n d s t a ti s - k _~+1 =

t i c s a r e s e l d o m k n o w n e xa ~ :t ly b u t t h e K a l ma n i ll - P~ + I =

t e r u s u a l l y s t i l l p r o v i d e s v e r y g o o d p e r f o r ma n c e

d u e i t s i n h e r e n t r o b u s t n e s s t o m o d e l u n c e r t a i n -

t ies .

The alman filter algorithm

* O b t a i n v a l u e s f o r ~ a n d ~

* Set in i t ial condi t ions x_0 and P0

* I t e r a t e t h e f o l l o w in g e q u a t i o n s

T h e K a l m a n f i l t e r e q u a t i o n s a r e n o w p r e s e n t e d .T h e s t a t e e s t i m a t e i s g i v e n b y

= + - 7 )

x^~+l = F ~ (S)

w h e r e ~ i s a o n e - s t e p - a h e a d p r e d i c t i o n o f t h e

s t a t e a n d K ~ is t h e K a l m a n g a in . T h e K a l m a n

gain i s g iven by

K ~ = P ~ H H r P k H + a ~ ) - 1 9)

2 is t h e v a r ia n c e o f n ~ . T h e m a t r i x P k i sh e r e ~ .t h e e s t i m a t i o n e r r o r c o v a x i a n ce d e f i n e d b y

P k = E [ x _ ~ ~ ) ~ k - ~ ) T ] i 0 )

where E[ . ] d e n o t e s t h e e x p e c t e d value m e a n ) .

T h i s m a t r i x c a n b e o b t a i n e d r e c u rs i v e ly b y s o lv -

i n g t h e m a t r i x R i c c a ti d i f fe r e nc e e q u a t i o n R D E )

g i v e n b y

P/~+l =

r + H T p

+ 1 1 )

w h e r e ~w i s t h e p r o c e s s n o i s e c o v a r i a n c e g i v e n

by ]E~ = E [W_kW_~] . T h e i n i t ia l c o n d i t io n s P0

a n d _ ~ n e e d t o b e in c l u d e d . T h e r e e x i s t a p p r o -

p r i a t e c h o i c e s o f t h e s e i n i t i a l c o n d i t i o n s w h i c h

r e s u l t in u n b i a s e d e s t i ma t e s fr o m k = 0 . H o w -

ever , i n p rac t i ce the in i t i a l f i l t e r t rans ien t i s usu -

a l l y n o t o f g r e a t i n t e r e s t s o t h e i n i ti a l c o n d i -

t i o n s c a n b e s e t t o a r b i t r a r y v a l u e s . T h e e f f e c t s

o f w r o n g i n i t ia l c o n d i t i o n s a r e q u i c k l y d i s s i p a t e d

u n d e r s t a n d a r d r e a s o n a b l e c o n d i t i o n s ( A n d e r s o n

a n d Mo o r e , 1 9 7 9 ) .

T h e a l g o r i t h m w i l l n o w b e r e w r i t t e n i n a n a l t e r -

n a t i v e c o mp a c t w h i c h i s s u i t a b l e f o r c o d i n g o n a

c o m p u t e r .

203

PkH _ (_I=ITpk_I=I+ cry ) -1 (12)

+ K _ - 1 3 )

_ k 1 4 )

F (P~ - K kH Tp ~) F T + E~o (15)

3. K A L M A N F I L T E R B A S E D

D I F F E R E N T IA T O R : T H E O R Y

T h e p r o b l e m t h a t i s a d d r e s s e d i s t h i s s e c t i o n i s

t o o b t a i n a n e s t i m a t e o f t h e t i m e d e r i v a t i v e o f a

g iven no i sy d i scre te - t im e s igna l , y~ , which i s gen-

e r a t e d f r o m s o m e p h y s i c a l so u r c e . I f t h e s i g n a l is

n o t c o r r u p t e d b y n o i s e i n a n y f o r m , s i mp l e t e c h -

n i q u e s s u c h a s t a k i n g t h e d i f f e r e n c e b e t w e e n t w o

s u c c e s s i v e p o i n t s w o u l d b e s u i t a b l e o r e v e n mo r e

s o p h i s t i c a t e d a p p r o a c h e s a s d e s c r i b e d i n /R . a b i ne r

and Ste ig l i tz 1970) . These t echn iques do no t re-

q u i r e t h e u s e o f a n y e p r i o r i k n o w l e d g e o f t h e

u n d e r l y i n g p r o c e s s w h i ch g e n e r a t e s t h e s ig n a l.H o w e v e r , i n a l mo s t a l l p r a c t i c a l s i t u a t i o n s , t h e

m e a s u r e d s ig n a l i s c o n t a m i n a t e d b y s o m e f o r m o f

n o i se . T h e a b o v e s i mp l e te c h n i q u e s a r e o f t e n u s e -

l es s due to the fac t tha t an idea l d i f fe ren t i a to r in -

h e r e n t l y a mp l i f ie s h i g h - f re q u e n c y n o i s e . T h e a p -

p r o a c h t h a t i s t r a d i t i o n a l l y a d o p t e d i n t h i s c a s e

is to ut i l i se the e p r i o r i k n o w l e d g e o f t h e s i g n a l

mo d e l a n d n o i se s t a t i s t i c s a s i n ( Ca r l ss o n et al .

1991) and(Car l s son , 1989) . Th i s l eads to a so -

l u t io n w h i c h r e q u i r e s a c e r t a i n d e g r e e o f e x p e r -

t i s e t o i mp l e me n t a n d t h u s i s o f t e n f o r mi d a b l e i n

p r a c t i c e .

T h e a p p r o a c h t a k e n i n t h i s p a p e r t o s o l v e t h e

p r o b l e m d o e s n o t u s e i n f o r ma t i o n o f t h e s i g n al

mo d e l a n d t h e n o i s e s t a t i s t i c s e x p l i c i t l y . Ra t h e r ,

they a re impl i c i t i n the des ign so i t i s no t imper-

a t i v e f o r t h e u s e r t o h a v e a n i n t i ma t e k n o w l e d g e

o f K a l ma n f i l t e r i n g o r s y s t e ms i d e n t i f i c a t i o n i n

o r d e r to u s e t h e a l g o r i th m. T h e d e v e l o p m e n t o f

t h e a l g o r i t h m s t a r t s b y b u i l d i n g a s u i t a b l e s i g n a l

mo d e l o f t h e f o r m ( 5 ) -( 6 ) . T h e a p p r o a c h t a k e n

h e r e d o e s n o t u s e s t a n d a r d s y s t e m i d e n t i f i c a t i o n

t e c h n i q u e s b u t r a t h e r a mo d e l s t r u c t u r e i s p o s -

t u l a t e d b a s e d o n t h e d e s i g n e r s i n t u i t io n o f w h a t

c h a r a c t e r i s t i c s t h e mo d e l i s e x p e c t e d t o p o s s e s s .

T h e r e s u lt in g m o d e l h a s o n ly tw o p a r a m e t e r s a n d

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204 S. Crisafulli and T.P. M edhurst

t h e s e a r e t r e a t e d a s u s e r - s p e c i f i e d d e s i g n p a r a m -

e t e r s .

A d e r i v a t i v e e s t i m a t e t h a t i s v e r y l ow - p a s s i n n a -

t u r e i s d e s i r e d f o r t h i s l eg l o a d i n g r a t e a p p l i c a t i o n

d e t a i l s c o n t a i n e d in n e x t s e c t io n ) . T h u s a m o d e l

f o r t h e e v o l u t i o n o f t h e d e r i v a t i v e t h a t h a s s u c h a

c h a r a c t e r i s t i c i s p o s t u l a t e d . A s i m p l e s t o c h a s t i c

m o d e l w h i c h a c h i ev e s t h i s i s

/ )k = i - q - ~ ) 2 w k; 0 < c ~ _ < 1 1 6)

w h e r e , a s i n t h e p r e v i o u s s e c t i o n , w k is z e ro m e a n

w h i t e n o is e , 1 / 1 - q - l a ) 2 i s t h e t r a n s f e r f u n c t io n

c o r r e s p o n d i n g t o G q - 1 ) a n d d ~ i s t h e d e r iv a t i v e

v a l u e. T h i s m o d e l f o r d~ c o n s i s ts o f w h i t e n o i s e

f i l t e r e d b y a l o w - p a s s f i l t e r . T h e f i l t e r h a s a r e a ld o u b le p o l e a t z = a i n t h e z -p l a n e . T h e d e s i r e d

s l ow t r a n s i e n t r e s p o n s e o f t h e d e r i v a t iv e c a n b e

o b t a i n e d b y c h o o s i n g a n e a r o r e q u a l o n e . T h i s

m o d e l c a n a l s o b e c o n s i d e r e d a s a l e a k y d o u b l e

i n t e g r a t o r a p o l e a t z = 1 i s a n i n t e g r a t o r , a

p o l e a t z = a ; a < 1 i s a l e a k y i n t e g r a to r ) . A n

e q u i v a l e n t r e p r e s e n t a t i o n o f t h i s m o d e l i s

d k = 2 a d k - 1 - a 2 d k - 2 + w k 17)

0 1 0 )

F = - a 2 2 a 0 .

0 ~- 1

U s i n g t h e a b o v e d e f i n i ti o n s , t h e s y s t e m s p e c if i ed

b y 1 8 ) - 2 1 ) c a n b e e q u iv a l e n t l y w r i t t e n a s

_xk+1 = F x k + w k 22)

ya = t:I_Tx_~+ nk 23)

dk = 8T_xk. 24)

N o te t h a t 2 2 ) - 2 3 ) is i d e n t i c a l t o 5 ) - 6 ) . I t is

n o w s t r a i g h t f o r w a r d t o c o n s t r u c t a K a l m a n f i l t e r

b a se d o n 2 2 ) - 2 3 ) s i n c e i t i s i n s t a n d a r d f o r m .

T h e so lu t i o n is g iv e n b y 7 ) - 1 1 ) . T h e d e r iv a t i v e

e s t i m a t e f ol lo w s d i r e c t l y fr o m 2 4 ) a n d i s g iv e n

b y

~ k = _ r 3T _ ~ k . ( 2 5 )

T h e q u a n t i t y ] ~ h a s t h e f o r m

z ~ = ~ [ w _ k w ~ ]

= E 0

0 0 0 )= 0 ~ 0

0 0 0

~, 0 ]

( 2 6 )

o r i n s t a t e - s p a c e f o r m i t c a n b e g i v e n b y 2 i s t h e v a r i a n c e o f w ~ .he re a w

0v -k + 1 = - a 2 2 a v _ k + 1 8 )

W k

d k = ( 0 1 ) _ ~ k ( 1 9 )

w h e r e v k is t h e s t a t e v e c to r . S in c e t h e a b o v e i s a

m o d e l f o r t h e d e r i v a t i v e o f t h e s i g n a l, t h e a c t u a l

s i g n a l P k ) c a n b e m o d e l l e d b y i n t e g r a t i o n o f t h i s

q u a n t i t y . I n d i s c r e t e - t i m e t h i s ca n b e ac h i ev e d

s i m p l y b y t h e r e c u r s i o n

P k + l = P k + d k r . 20)

w h e r e r i s t h e s a m p l i n g p e r i o d . T h e q u a n t i t y P k

c a n n o t b e m e a s u r e d d i r e c t ly d u e t o m e a s u r e m e n t

n o i s e , t h e a c tu a l s i g n a l Yk a n o i sy v e r s io n o f P k )

i s m e a s u r e d ,

Y ~ = P k + n k 21)

w h e r e n t i s o n c e a g a i n ze r o m e a n w h i t e m e a s u r e -

m e n t n o i s e . T h i s n o w c o m p l e t e s t h e s ig n a l m o d e l .

T h i s m o d e l c a n b e r e p r e s e n t ed i n a c o m p a c t f o r m

b y d e f i n i n g c e r t a i n q u a n t i t i e s . D e f i n e

w.= o.)H r = o o 1 ) _ I 3 r= o 1 o )

T h e a b o v e a l g o r i t h m c a n b e s p e c i f ie d b y o n l y t w o

p a r a m e t e r s , a t h e d o u b l e p o le p o s it i o n a n d 7 t h e

r a t i o o f t h e p r o c e s s n o i se t o t h e m e a s u r e m e n t

n o i s e w h ic h i s d e f in e d a s

2A O to

3 = ~ . ( 2 7)

T h e t r ad i ti o na l a p p r o a c h t o K a l m a n f il te ri ng i c -

t a t es t h a t t h e s e t w o p a r a m e t e r s b e e v a l u a t e d v i a

s y s t e m s i d e nt i f ic a t i on t e c h n i q u e s a n d t h u s t h e

p r o b l e m i s f u l l y s p e c i f ie d i n a d v a n c e . T h i s a p -

p r o a c h t r e a ts t h e s e t w o p a r a m e t e r s a s u s er - s pe c -

i fi ed d e s i g n p a r a m e t e r s a n d i t i s u p t o t h e u s e r

t o t u n e t h e s e t o o b t a i n t h e d e s i r e d t y p e o f p e r -

f o r m a n ce . T h e s i g n a l m o d e l a n d n o i s e s ta ti s-

t i cs a r e t h u s i m p l i c it i n t h e a l g o r i t h m . I n o t h e r

w o r d s , t h e i r c h o i c e s a r e d i c t a t e d b y s u b j e c ti v e -

t y p e p e r f o r m a n c e m e a s u r e s r a t h e r t h a n t h e t ra -

d i t i o n a l o b j e c t i v e c r i t er i a u c h a s p r e d i c t i o n e r r o r

v a r ia n c e . T h e p a r a m e t e r s c a n b e c o n s i d e re d t o

h a v e s o m e f o r m o f a l g o r i th m i c si g ni f ic a nc e . T h e

p a r a m e t e r 3' c a n b e i n t e r p re t e d a s a l t e r i n g t h e

b a n d w i d t h o v e r w h i c h t h e di f fe r en t ia t io n s t a k i n g

p l a c e a n d t h e a a d j u s t s t h e f o r g e t t i n g f a c t o r o f

t h e a l g o r i t h m . T h e s e p a r a m e t e r s a l s o h a v e s ig ni f-

i c a n c e f r o m a m i n i n g v i e w p o i n t . I t i s l i ke l y t h a t

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Robust On-Line D ifferentiationTechniques 205

Fig. 1.

600

500

400

300

2OO

1000

L

100l i i i i L i

O O00 300 400 500 600 700 800

time

i 0 0 0

G

C t }0

2

0

2

i

4 0 i 0 0 2 0 0 600 7000 ~ 0 500 900 1000

time

Syn the s i se d no i se - f re e da t a t op) a nd th e de r iva t ive e s t im a te f r om Ka lm a n f i lt e r ba se d d i f f e re n-

t i a t o r b o t t o m )

I

F i g . 2 . P l a n v i e w o f m i n i n g b y s p li t a n d l if t m e t h o d

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206 S. Crisafulli and T.P. Medhurst

the choice of these parameters will ultimately be

determined by these mining-based criteria.

The other main feature of this algorithm is its

ability to handle signal discontinuities and bad

data . A linear model often models a physical pro-

cess quite well over the region of interest . How-

ever, difficulties can be encounte red when anoma-

lies occur, such as discontinuities in the data or

when some other atypical behaviour occurs (eg.

a mining cycle delay in the leg loading rate ap-

plication). Provided these conditions can be de-

tected, the Kalma n filter can cope with these ab-

normal behaviours by setting the measurement

2 to a high value. In other words,oise variance a n

no confidence is given to the measurements Yk

and they are treated as being extremely noisy.Also, the occurrence of an anomaly may result in

the past dat a being useless and so the memory of

the algorithm should be reset. This can be easily

performed by re-initialising the Riccati equation

solution Pk and, if necessary, the Kalman filter

state.

The ideal case performance is now demon-

strated by presenting simulation results based on

noise-free dat a. Fig. 1 shows a plot of syn-

thesised ramp signals (top) and the derivativeestimate (bot tom) . Notice the derivative esti-

mate exhibits critically damped-like transitions

between levels with a minimum amount of un-

dershoot, overshoot and ringing. This perfor-

mance was obtained by tuning the parameters

to the following settings: a) double pole position

a = 0.93 b) process noise to measurement noise

rat io 7 = 0.0001. These settings are also used on

the leg loading rate est imation application in the

next section.

4. KALMAN FILTER BASED

DIFFERENTIATOR: APPLICATION

A brief overview of pillar extraction coal min-

ing is first presented to provide readers without

coal mining expertise with sufficient background

knowledge in order to appreciate the application.

This will be followed by some geomechanical de-

tails of the application and then t he results of

applying the algorithm to the problem.

The fu ndame ntal concept of bord and pillar meth-

ods of mining is that the coal seam is divided

into a regular block-like array by driving pri-

mar y headings through it which are intersected at

regular intervals by connecting cutthrough head-

ings. The pillars support the overlying st rat a dur-

ing the first workings as the headings are driven.

They may be extracted systematically on subse-

quent second workings, depending on t he scheme

adopted.

A popular scheme for extracting coal pillars is

the split and lift method. Fig. 2 provides a plan

view of this mining sequence. Afte r first workings

the coal pillars can be mined by first driving a

heading through the pillar, a split, and then sub-

sequently removing the remaining coal or fender.

Mining of the fender is undertaken by driving into

the coal with a continuous miner (CM) (see Fig.

2) in such a manner as to retreat away from the

unsupported area. After extraction the area col-lapses or caves to form an area devoid of coal

called the goal A more comprehensive descrip-

tion is presented in(Sleeman, 1986).

Up to 90% extr action of the n s tu coal can be ob-

tained in pillar extraction mining and has distinct

advantages over longwall mining methods where

highly faulted deposits exist. Typically, produc-

tion rates of 800 tonnes per unit production shift

are achieved with costs per tonne ranging from

AUD$15.00 to AUD$35.00 depending on the op-eration.

Numerous studies have shown tha t the success of

pillar extraction methods depend on establishing

the correct fender width for the particular min-

ing conditions. The fender width should be suffi-

cient to allow it to yield when it is formed, thus

providing a destressed condition for subsequent

extraction. If the fender is carelessly formed, of

wrong dimensions or varying width, problems oc-

cur during its extraction and coal is left behind

as stooks (see Fig. 3), causing severe roof control

problems during later stages. Such problems can

result in loss of production for periods of days

and can sometimes cascade periodically through-

out the entire panel.

Breaker Line Supports (BLSs) are remote-con-

trolled mechanised roof supports that provide a

supporting (hydraulic powered) force by push-

ing up against the roof during the mining op-

eration (item 1 and 2 in Figs. 2 and 3). They

are used for roof control in bord and pillar panel

layouts, specifically during pillar ext raction min-

ing. BLSs have been operating in Australian un-

derground coal mines since 1987 (McCowan and

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208 S. Crisafulli and T.P. M edhurst

B r o w n , 1 9 9 0 ) .

F i g . 3 s h o w s t h e s t r a t a l o a d i n g re g i m e on t h e

B L S u n i t s a n d a s s o c i a t e d f e n d e r t o b e m i n e d .

B a s e d o n t h e a u t h o r s f i e ld o b s e r v a t i o n a n d d a t a

c o l l e ct i o n , t h e p r o p o s e d m e c h a n i s m s u g g e s t s t h a t

t h e l o w e r i n g o f t h e b r i d g i n g b e a m a b o v e t h e r o o f

s u p p o r t s i s c o n t r o l l e d p r i m a r i l y b y s u p p o r t o f -

f e r ed b y t h e y i e l d i n g f en d e r . T h e e f fe c t s o f ro c k

s t r u c t u r e a n d g e o l og i c a n o m a l i e s c a n b e a c t i v e in

m o v e m e n t a n d c h a r a c t e r o f t h e r o o f b e a m b u t d u e

t o t h e r e l a t iv e l y h i g h s t i f fn e s s o f t h e B L S u n i t s

a n d t h e a r e a t h e y c o v e r w h e n a c t i v e l y p r e s s e d

a g a i n s t t h e r o o f , l e g l o a d i n g m e a s u r e m e n t s a r e

s u g g e s t e d t o b e a k i n t o f e n d e r r e s p o n s e . I n g e n -

e r a l t e r m s , i f t h e f e n d e r p r o v i d e s a fi n i te a m o u n t

o f s u p p o r t , t h e n l o a d i n g r a t e s c a l c u l a t e d f r o mm e a s u r e m e n t s t a k e n a d j a c e n t to t h a t s u p p o r t c a n

h e l p d e l i n e a t e h o w m u c h a n d f o r h o w l o n g t h e

f e n d e r c a n s u r v i v e t o k e e p t h e t u n n e l f r o m c o l -

l a p s in g . O n e c o u ld e n v i s a g e a p o in t o f n o re -

t u r n i n t e r m s o f s u p p o r t o r a c r it i c a l l o a d i n g r a t e

w h e n t h e f e n d e r i s e x t r a c t e d . B a s e d o n t h e s e a r-

g u m e n t s , i t i s cl e a r t h a t o n - l in e e s t i m a t i o n o f t h e

B L S l e g l o a d i n g r a t e s a r e a n i n v a l u a b l e q u a n t i t y

in a s s e s s in g r o o f s t a b i l i t y . T h e t o p p lo t o f F ig s .

4 a n d 5 s h o w t y p i c a l B L S l e g p r e s s u r e d a t a d u r -

i n g f a v o u r a b l e a n d u n f a v o u r a b l e ( r o o f l o w e ri n g

p r o b l e m s ) m i n i n g c o n d i t i o n s r e s p ec t iv e l y . N o t i c e

t h e p r e d o m i n a n t l y n e g a t i v e a n d p o s i t i v e r a t e s f or

e a c h r e s p e c t i v e s e q u e n c e . C h a r a c t e r i s i n g t h e s e

o b s e r v a t i o n s f o r e a c h min e s i t e w i l l b e t h e k e y i n

a l e r t i n g m i n i n g p e r s o n n e l o f t h e o n s e t o f in s t a b i l-

i ty.

T h e K a l m a n f i lt e r b a s e d d i f f e r e n t i a t o r i s n o w a p -

p l ie d to t h e B L S le g p r e s s u r e d a t a . F r o m a n

a l g o r i t h m i c p o i n t o f vi e w , t h e m a i n o b j e c t i v e is

t o p r o v id e a r e l i a b l e o n - l in e l e g l o a d in g r a t e e s -

t i m a t e s i n s p i t e o f s i g n al n o i se a n d d i s c o n t in u -

i t ie s . I n m o s t p r o b l e m s o f t h i s t y p e , t h e s u c c es s

o f t h e a p p l i c a t i o n o f t e n re l ie s o n t h e i n t r o d u c t i o n

o f a p r / o r / k n o w l e d g e . I n t h i s p a r t i c u l a r a p p l i c a -

t i o n l a r g e d i s c o n t i n u i t i e s c a n b e o b s e r v e d i n t h e

l e g p r e s s u r e d a t a d u e t o t h e p e r i o d i c r e l o c a t i o n

a n d r e a d j u s t m e n t o f t h e B L S ( s e e t o p p l o t s o f

F ig s . 4 a n d 5 ) . A l s o n o t i c e h o w th e t r e n d i n

t h e r e g i o n o f i n t e r e s t i s r el a t i v e ly s m o o t h a n d s o

e v e n r e l a t i v e l y s m a l l d i s c o n t i n u i t i e s c a n b e c o n -

s i d e r e d as a n a n o m a l y . T h e s e o b s e r v a t i o n s c a n

b e i n c o r p o r a t e d i n t o t h e s c h e m e a s a pr ior i in-

f o r m a t i o n a n d w i l l r e s u l t i n m u c h b e t t e r p e r f o r -

m a n c e . F o r t h i s p a r t i c u l a r m i n e s it e , t h e d a t a

w a s d e e m e d t o b e u n r e l i a b l e i f t h e p r e s s u r e s w e r e

b e lo w 1 5 0 b a r o r i f t h e i n t e r s a m p le d e v i a t i o n w a s

mo r e t h a n 2 0 b a r . I n t h e s e c a s e s , o n e o r mo r e o f

t h e f o l l o w in g a c t i o n s c a n b e t a k e n i ) t h e m e a s u r e -

me a t n o i s e v a r i a n c e ( a ~ ) i s s e t t o a l a r g e v a lu e

i i ) t h e R ic c a t i e q u a t i o n s o lu t i o n ( e s t ima t io n e r -

r o r c o v a r i a n c e P k ) i s r e - i n i t ia l i s e d t o a l a r g e v a lu e

a n d i i i ) t h e K a l m a n f i l t e r s t a t e e s t i m a t e ( x ~ ) i s

r e - i n i ti a l i s e d . S p e c i fi c a ll y , t h e a c t i o n s t h a t w e r e

t a k e n i n t h i s a p p l i c a t i o n a r e :

i f yk _< 150 t h e n pe r form i ) , i i ) an d ii i )

i f lY~ - Yk - l l >__ 20 t h e n pe r for m i ) an d i i ) .

T h e b o t t o m p l o t s o f F i g s . 4 a n d 5 sh o w t h e

d e r i v a ti v e e s t i m a t e f r o m t h e K a l m a n f i l te r b a s e d

d i f f e re n t i a t o r. N o t i c e h o w s m o o t h t r e n d s a r e p r o-

v i d e d w i t h o u t e x c es s iv e p h a s e d e l a y s . T h i s h y -d r a u l i c le g p re s s u r e r a t e i n f o r m a t i o n c a n t h e n b e

u s e d t o e s t a b l i s h c r i t i c a l r a t e s , i n f e r min in g c o n -

d i t i o n s a n d a i d u n d e r g r o u n d c o a l m i n e r s i n d e ci d -

i n g w h e n t o w i t h d r a w f r o m u n s a f e o r u n p r o f i t a b l e

m i n i n g a r e as .

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

A n o v e l o n - l i n e d ig i t a l d i f f e r e n t i a t o r h a s b e e n d e -

v e lo p e d w h ic h c a n b e e a s i l y a n d r e l i a b ly u s e d i n

p r a c t ic e . T h e a l g o r i t h m c a n b e e a s il y t u n e d b y

t h e u s e r t o a c h ie v e t h e d e s i r e d p e r f o r m a n c e w i t h -

o u t r e q u i ri n g e x p e r t k n o w l e d g e o f K a l m a n f i lt e r

a n d s y s t e m s i d e n t if i c a t i o n t h e o r y . T h e a p p l ic a -

t i o n t o t h e B L S d a t a w a s s u c c e s s fu l a n d a l lo w s

r e a l - t i m e c a l c u l a t i o n o f k e y g e o m e c h a n i c a l p a -

r a m e t e r s w h i c h a i d i n a s s e s s m e n t r e g a r d i n g r o o f

s t a b i l i t y w h e r e m i n e r s a r e w o r k i n g i n h a z a r d o u s

a r e a s . R e s e a r c h i n t h i s a r e a is o n g o in g w i th t h e

p u r p o s e o f p r o v i d i n g i n f o r m a t i o n r e g a r d i n g r o o f

a n d p i l l a r l o a d i n g r e g i m e s a n d f o r e s t a b l i s h i n gmin e - b a s e d B L S l e g lo a d in g r a t e s f o r c r i ti c a l c o n -

d i t i o n s .

6 . R E F E R E N C E S

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500

400

300

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100 200 300 400 500 600

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700 800 900 1000

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time 10 see intervals)

F i g . 5 . Typi c a l B LS le g p re s sure da t a dur i ng unfa voura b l e mi n i ng c ondi t i ons t op) a nd t h e de r i va ti vee s t i ma t e f rom t he Ka l ma n f il te r ba se d d if fe re n ti a t o r bo t t om )

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