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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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C a r l s s o n , B . , A . A h l6 n a n d M. S t e r n a d ( 1 9 9 1 ) ,

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S ig n a l Mo d e l s , I E E E Tr a n s a c t i o n s o n S i g n a l

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300

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100 200 300 400 500 600

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700 800 900 1000

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100 200 300 400 500 600 700 800 900 1000

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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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