Conditions for asymptotic stability of discrete minimum variance estimation formula

Deyst, J. J., Jr.

Price, C. F.

English

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EXPERIMENTAL ASTRONOMY LABORATORY 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
C A M B R I D G E  39, M A S S A C H U S E T T S  
https://ntrs.nasa.gov/search.jsp?R=19680012556 2020-03-12T09:04:42+00:00Z
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RE -41 
CONDITIONS FOR ASYMPTOTIC STABILITY OF THE 
DISCRETE, MINIMUM VARIANCE, LINEAR, ESTIMATOR 
John J. DeySt, Jr. and Charles F. Price 
March 1968 
I 
EXPERImNTAL ASTRONOMY LABORATORY 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
CAMBRIDGE, MASSACHUSETTS, 02139 
Approved : co( f7Lud&+ 
I 
Director 
Experimental Astronomy Laboratory 
I -  
ACKNOWLEDGEMENT 
This  report was prepared under DSR Contract  70344 spon-  
sored by t h e  Nat ional  Aero a u t i c s  and Space Administration 
through research  g r a n t  Ne6 22-009-207. 
N d  
The p u b l i c a t i o n  of t h i s  r e p o r t  does n o t  c o n s t i t u t e  ap  
proval  by t h e  Nat ional  Aeronautics and Space Adminis t ra t ion 
of t h e  f i n d i n g s  o r  t h e  conclusions contained t h e r e i n .  It  
i s  published only for  t h e  exchange and s t imu la t ion  of i d e a s .  
ABSTRACT 
stability of the discrete, homogeneous, linear, minimum 
variance estimation formulas is investigated. 
ditions for uniform asymptotic stability in the large are de- 
rived. The conditions, if satisfied, also imply stochastic 
controllability and observability of the plant. 
Sufficient con- 
I .  
CONDITIONS FOR ASYMPTOTIC STABILITY OF THE 
DISCRETE, MINIMUM VARIANCE, LINEAR,  ESTIMATOR 
BY 
John J. Deyst, Jr. and Charles  F. Price 
In t roduc t ion  
S t a b i l i t y  of t h e  minimum variance l i n e a r  e s t ima t ion  for-  
mulas i s  e s s e n t i a l  i n  cases  for  which t h e  e s t ima t ion  t a k e s  
p l a c e  over extended pe r iods  of t i m e .  For continuous systems, 
Kalman [2] has  provided condi t ions  which i n s u r e  s t a b i l i t y  of 
t h e  homogeneous f i l t e r  equat ions.  It appears  however, t h a t  
a s i m i l a r  t rea tment  of the  d i s c r e t e  e s t ima t ion  formulas i s  no t  
a v a i l a b l e  i n  t h e  c o n t r o l  l i t e r a t u r e .  I n  a r e c e n t  paper ,  
Sorenson [ 4 ]  d e r i v e s  upper and lower bounds on t h e  d i s c r e t e  
e r r o r  covariance mat r ix ,  b u t  does not  consider  s t a b i l i t y .  
I t  i s  t h e  purpose of t h i s  paper t o  d e r i v e  t h e  cond i t ions  which 
i n s u r e  t h a t  t h e  discrete minimum var iance  e s t ima t ions  formulas 
a r e  uniformly, asymptot ica l ly  s t a b l e  i n  t he  l a r g e .  
d e f i n e  c e r t a i n  n o t a t i o n .  The matrix norm 1 1 A \  1 i s  A A.  Matrix 
I i s  t h e  u n i t  mat r ix .  Given t w o  symmetric ma t r i ces  B and C,  of 
equa l  dimension, t h e  inequa l i ty  BLC impl i e s  t h a t  the  d i f f e r e n c e  
B-C i s  non-negative d e f i n i t e .  S imi l a r ly  B>C impl i e s  t h a t  B-C i s  
p o s i t i v e  d e f i n i t e .  F i n a l l y ,  i t  i s  t a c i t l y  assumed t h a t  all 
m a t r i c e s  a r e  bounded from above i n  norm. 
Before proceeding t o  t h e  d e r i v a t i o n ,  i t  i s  necessary t o  
T 
-1 - 
I1 F i l t e r  Equations and Conditions f o r  S t a b i l i t y  
Consider a d i s c r e t e  t i m e  s t o c h a s t i c  p rocess  whose s t a t e  
vec tor  x ( k )  s a t i s f i e s  t h e  recurs ion  r e l a t i o n  
I x (k+ l )  = Q(k+l ,k)x(k)  + G(k)v(k)  ; 119(k+l,k) 1 I 1 1 > 0  
I (2-1) where v ( k )  i s  a white  random vector  sequence with 
E[v (k ) l  = 0 Q ( k )  1 E21X (2-2) T E[v(k)v (j)] = AkjQ(k) , 
and A i s  t h e  Kronecker d e l t a .  Linear measurement vec to r s  z ( k )  
a r e  a v a i l a b l e  a s  ou tpu t s  and t h e  z (k)  a r e  def ined a s  
k j  
z (k)  = H(k)x(k)+w(k) (2-3) 
where w(k) i s  a white  random vector  sequence with 
For t h e  system (2-1) through (2-4) t h e  discrete minimum variance 
l i n e a r  es t imate  of x ( k )  i s  determined by equat ions 
A 
G(k) = P (k)  [ P '  (k)'l@(k,k-l)~(k-l)+€IT(k)R(k)-lz(k)] ; x(O)=E[x(O)l 
(2-5) 
-1 T P ( k )  = [ P '  (k )  +H (k)R(k) - lH(k) I - l  P(0)=E[x(O)xT(O)] (2-6) 
P '  ( k+ l )  = Q(k+l,k)p (k)eT(k+l,k)ffi(k)Q(k)GT(k) (2-7) 
where $(k)  i s  t h e  minimum variance es t imate ,  P (k )  i s  t h e  esti-  
mation error covariance mat r ix  a f t e r  processing t h e  measurement 
z ( k )  and p '  (k+l)  i s  t h e  extrapolated error covariance matr ix .  
Now t h e  system 
y ( k )  = P (k)P '  (k)- '@(k,k-l)y(k-l)  (2-8) 
r e p r e s e n t s  t h e  homogeneous p a r t  of (2-5).  I f  t h e r e  ex is t  r e a l  
s c a l a r  func t ions  V ( y ( k ) , k ) ,  y l ( I I ~ ~ ~ ) l I ~ ,  v 2 ( l l y ( k ) l l )  and 
Y,( 1 l y ( k )  1 1 )  such t h a t  f o r  some f i n i t e  N 2 0 * 
* 
Defining these  condi t ions  over N s t e p s  i s  equiva len t  t o  Kalman's 
d e f i n i t i o n s ,  over a f i n i t e  i n t e r v a l ,  i n  the continuous case [ 2 ]  
-2- 
t 
= o  l i m  y L ( p ) =  Q) 
P *Q) (2 -11)  
then t h e  system (2-8) i s  uniformly asymptot ical ly  s t a b l e  i n  
t h e  l a r g e  [ 3 I .  
numbers %,a2, f3,,B2 such t h a t  t he  condi t ions  
I n  the  sequel it w i l l  be shown t h a t  i f  t h e r e  a r e  r e a l  
k-1 
i=k-N 
512 c @(k,i+l)G(i)Q(i)GT(i)@T(k,i+l)~a21 O<s, %<a, (2-12) 
k 
f31E c 
i=k-N 
eT (i , k)  HT (i) R ( i ) - l H  (i) 9( i , k) (B21 OCB10 (2-13) 
hold for a l l  k 2 N,' then  t h e  funct ion V ( y ( k ) , k )  def ined a s  
P 
(2-14) 
s a t i s f i e s  condi t ions  (2-9),  (2-10) and (2-11) . Thus i f  t h e  
system (2-1) through (2-4) s a t i s f i e s  condi t ions  (2-12) and 
(2-13),  t h e  homogeneous system (2-8) i s  uniformly asymptot ical ly  
s t a b l e  i n  t h e  l a r g e .  N o t e  t h a t  (2-12) and (2-13) imply t h a t  t h e  
system is  s t o c h a s t i c a l l y  c o n t r o l l a b l e  and observable [I 1 . 
111. A Lower Bound 
Since G(k) given by (2-5,6,7) is the minimum variance 
l i n e a r  es t imate ,  t h e  estimate G(k)  defined a s  * 
g ( k )  = I k  C (aT (i , k)HT (i) R ( i )  ( i ) d h (  i , k)  
kzN (3-1) 
-1 
k>N (3-2) 
* It can be shown t h a t  z ( k )  i s  the maximum l ike l ihood  es t imate  
for t h e  x (k )  system with no process noise  (Q(k)=O) 8 and ignoring 
a l l  d a t a  before  k-N. 
-3- 
has an error covariance matrix satisfying the inequality 
cov{G(k)-x(k)/ e ~I[~(k)-x(k)l[~(k)-x(k)]~j 2 P(k) (3-3) 
Now, with (2-1) and (2-3), the measurement z(i) may be written 
as 
z (i) =H (i) @(i , k) x (k) +w (i) -H (i) @(i , k) 
k-1 
j =i 
@(k, j+l) G (1.) v (1) (3-4) 
where, as usual, the state transition matrix has the properties 
@(k,i)=@(k,k-l)@(k-l,k-2) .. . .@(i+l,i) 
@(i,k)=@(k,i) , @(k,k)=I (3-5) -1 . 
Combining (3-4), (3-3) and (3-1) yields 
cov fx(k)-z(k)\ =cov P(k)C GT (i , k) HT (i) R (i) -’w (i ) (3-6) i- i=k-N 
k k-1 
j =i 
c QT(i,k)HT(i)R(i)-’H(i)Q(i,k) C @(k,j+l)G(j)v(j) 
( i=k-N 
Adding non-negative definite terms on the riqht of (3-6), by 
alterinq the lower limit of j ,  produces the inequality 
k 
(3-7) x(k)-g(k)‘ 1. cov I- P(k) C QT(i,k)HT(i)R(i)-lw(i) 
i=k-N j=k-N i 
5 1 i&-N 
k k-1 
+cov’s(k) C qT(i,k)HT(i)R(i)-’H(i)@(i,k) C @(k, j+l)G(j)v(j;[ 
k-1 
+ c  O(k, i+l) G (i) Q (i ) GT (i) aT (k, i+l) 
i=k-N 
Applying (2-12), (2-13) and (3-3) obtains an upper bound on p(k) 
1 k-1 
iuk-N 
+ Z: @(k,i+l)G(i)Q(i)GT(i)QT(k,i+l) 5 ( q+ %)I 
and thus a lower bound on v (y(k),k) is 
P 
-4- 
IV. An Upper Bound. 
Define the inverse of P(k) as W(k) and from (2-6)  
-1 T 
W(k)*(k)-l=P' (k) +H (k)R(k)-%(k) (4-1 1 
wl(k) = W(k1-H (k)R(k)"H(k) = p'(k)-' 
W; (k+l) = aT (k+l, k) -lW(k) @(k+l ,k) -' 
Similarly, define Wl(k) and W1' (k) as 
T 
(4-2 1 
(4-3) 
and with the help of (2-6) and ( 2 - 7 ) ,  (4-2) and (4-3) become 
W1 (k) =[W; (k) -l+G (k-l)Q (k-1) GT (k-1) 1-l 
W;(k+l) = QT(k+l,k)-lW1 (k)@(k+l,k) -1 +@ T (k+l,k) -1 H T (k)R(k) -1 . 
(4-4 1 
H (k) @(k+l , k) -' (4-5 1 
By noting the similarity between (4-4), (4-5) and (2 -6 ) ,  (2-7) 
it is seen that w (k) may be interpreted as the estimation error 
covariance matrix for a system described by the equations 
1 
(4-6 1 x(k+l) = @T(k+l,k)-1x(k)+4T(k+l,k) -1 H T (k)s(k) 
z (k) = GT(k-l)x(k)+m(k-1) (4-7) 
Applying the results of Section I11 to this system, it is clear that 
-1 -1 k Wl(k) 5 [ C ~(i,k)-lG(i-l)Q(i-l)GT(i-l)~T(i,k) 1 
i=k-N (4-9) 
-1 T -1 T k-1 + QT(k,i+l) @ (i+l,i) H (i)R(i)-lH(i)@(i+l,i)-l~(k,i+l)-l 
i=k-N 
-5 - 
o r  
k-1 
. Wl(k) 5 [ @( k, i+l) G (i) Q(i) GT (i) aT (k ,  i+l) 1 -' 
i-k-N-1 
k-1 
+ c aT (i , k)HT (i) R (i) -lH (i) @( i , k )  
i-k-N 
and according t o  (4-2) 
k-1 
w(k) 5 [ C @ ( k , i + l ) G ( i ) Q ( i ) G T ( i ) b ; r ( k . i + l ) ] - '  - . .  
i=k-N 
Therefore an upper bound on V ( y ( k ) , k )  i s  
P 
(4-10) 
(4-11) 
2 5 Y 2  ( I  lY(k) I I )  
(4-12) 
To t h i s  p o i n t ,  it has  been shown t h a t  V ( y ( k ) , k )  s a t i s f i e s  
cond i t ions  (2-9) and (2-11) .  Condition (2-10) i s  y e t  t o  be 
s a t i s f i e d .  
V. A Relevant Control P rob lem 
Consider t h e  system 
P 
where u ( k )  i s  a con t ro l  i npu t .  If t h e  cost func t ion  J i s  de- 
f ined  as 
k 
J = [yT(i)HT(i)R(i)-lH(i)y(i)+uT(i)P' ( i ) - ' u ( i ) ]  (5-2) 
i=k-N 
then  it w i l l  be use fu l  t o  determine t h e  opt imal  con t ro l  sequence 
Define l a r g e  vec to r s  Y and U a s  u*(k) which minimizes (5 -2) .  
Y =  u =  
and matrices M,B,L t h u s  
0 
R(k-1) 
0 
'R(k-N) 
M 3 :  
- 
- 
(5-3) 
(5-4) 
P' (k)  0 1  0 
P' (k-1) . 1 ;L=i"' H(k-l! 0 i o  P ' i k N )  H (k-N ) ; B. 
-6 - 
so t h e  c o s t  J may be w r i t t e n  as 
T T -1 T -1 J = Y L M  L Y + U B  U 
Fur ther ,  i f  mat r ices  C and D are 
r (k ,k-N-1)  1 9(k,k-1) 
@ (k-1 , k-N-l ) I 
- 
@(k,k-2) . . @(k,k-N) 
@(k-l ,k-2) .  @(k-1,k-N) 
I . 
- 1  
. . . 
0 
- 
and t h e  i n i t i a l  condi t ion  i s  defined a s  
yo = y(k-N-1) 
then  
Y = Cy0 + DU 
and 
J = [ U  T T  D + yoC T T  I L  T - 1  M LICyo+DUI + U T B -1 U 
The f i r s t  and second d e r i v a t i v e s  of J with r e s p e c t  t o  U(gradient  
vector and hess ian  matr ix)  are 
(5 -10) T T  T T  T - 1  T -1 - =  aJ a U  2 [ U  D +yoC ]L M LD+2U B 
Controls  u ( k )  are unconstrained so (5-10) and (5-11) provide 
necessary and s u f f i c i e n t  condi t ions f o r  t h e  optimal control 
sequence u*. S e t t i n g  (5-10) equal t o  zero obtains 
T T -1 -1 -1 T T -1 U* = - [D L M LD+B 1 D L M LCyo 
and s u b s t i t u t i n g  i n t o  (5-9) y i e l d s  t h e  m i n i m u m  cost 
J*=Y~C L [M -M L[L M L+D B D 3 L M I L C Y ~  T T T -1 -1 T -1 T-1 -1 -1 -' T -1 
(5-11) 
(5-12) 
T T T  T -1 = y 0 C L [M+LDBDL 1 LCyo (5-13) 
Now, from t h e  d e f i n i t i o n s  of L ,  C,  M and condi t ion (2-13) 
T T -1 
plI 5 C L M LC 5 P21 (5-14) 
Because R(k) i n  Eq. (2-4) i s  bounded below, from (5-14) 
t h e r e  ex is t s  a r e a l  number B, such t h a t  
(5-15) 
-7- 
Also, because t h e  norms of R ( k )  I @(k,k- l ) ,  P '  ( k ) ,  and H(k) a r e  
bounded above, it fo l lows  t h a t  t h e r e  are r e a l  numbers f.3, and 
p5 such t h a t  
p41 5 M+LDBDL~ 5 p51 0 < p48 p5 < a 
and t h e r e f o r e  
(5-16) 
(5-17) 
-' > [M+LDBDL T 1 -1 P5-lI  
B4 1 - 
Equations (5-13), (5-15) and (5-17) combine t o  e s t a b l i s h  t h a t  
t h e  minimum cost i s  p o s i t i v e  i f  t h e  i n i t i a l  s t a t e  i s  non-zero 
2 -1 -1 J* = yoC T T T  L [M+LDBDLT] LCyo P,p5 I ly(k-N-1) 1 l 2  (5-18) 
VI. S t a b i l i t y  
I n  order  t o  use t h e  r e s u l t s  of  Sec t ion  111 t o  prove s ta-  
b i l i t y ,  t h e  homogeneous equat ion (2-8) i s  w r i t t e n  a s  two equat ions  
y ( k )  = y ' (k )+u(k )  (6-1) 
y '  (k)  = @(k,k- l )y(k- l )  (6-2) 
where u (k )  i s  considered as  a con t ro l  i npu t  and 
u ( k )  = [P(k)P '  ( k ) - I I y ' ( k )  (6-3) 
Applying (6-1) and (6 -2) ,  the func t ion  V ( y ( k ) , k )  becomes P 
=y ' (k)P (k )  -'y' (k)  -yT (k)HT(k) R(k) - l H  (k)  y (k)  
- [ y (k) -y ' (k)  1 TP ' (k 1 [ y (k) -y ' (k)  I 
-1 
=y ' (k)  [ @(k , k-1) P (k-1) (PT (k,  k-1) ffi (k-1) Q (k-1) GT (k-1) ] y (k)  
-yT (k) HT (k) R (k) -l€I (k)  y (k) -UT (k) P I (k)  - lu  (k)  
=yT(k-l) [ P (k- l )+@(k- l  k) G(k-1) Q (k-1) GT (k-1) (PT (k-1, k )  ] -'y (k-1) 
-yT (k)HT (k)  R (k) - l H  (k)  y (k) -UT (k)  P I (k)  - lu  (k)  
-8- 
* - < yT (k-l)P (k-1) -'y (k-1) -yT (k) HT (k)  R (k)  - l H  (k)  y (k)  -UT (k)  P (k) - l u  (k)  
(6-4) 
hence 
yT (k) P (k) -'y (k)  -yT (k-1) P (k-1) -'y (k-1 ) 
<-yT (k) HT (k )  R (k)  - l H  (k) y (k) -UT (k) P a (k) -lq (k )  (6-5) - 
whence 
yT ( k ) P  (k)  -'y (k) -yT(k-N)P (k-N) -'y (k-N) 
and from d e f i n i t i o n s  (2-14) and (5-2) 
Now d e f i n e  a matr ix  t ransformation 8(k,k-N-l)  as 
8 (k,  k-N-1)  =[P (k)P  (k) -'@(k,k-l) 1 [ P ( k - l ) P  a (k-1) -'@(k-l, k-2) 3 e . .  
From (2-8) and t h e  f a c t  t h a t  a l l  mat r ices  i n  t h e  product 
on the  r i g h t  of (6-8) a r e  non-singular and bounded below i n  norm 
I I y (k-N-l)J I=LJe (k,k-N-l) -'y (k) 1 I,>@, I ! y (k) I I 
F i n a l l y  (6-8) y i e l d s  
Vp(y(k) .k ) -Vp(y(k-N) ,k -N) i -B385~61  2 -  l y (k )  1 I 2 s V 3 ( 1  l y (k )  1 1 )  (6-10) 
and s i n c e  y3( I ly (k)  1 1  ) i s  negat ive f o r  a l l  non-zero vec to r s  y ( k ) ,  
t h e  system (2-8) i s  uniformly asymptotically s t a b l e  i n  the l a r g e .  
Conclusions 
I t  has  been shown t h a t  i f  a l i n e a r  s t o c h a s t i c  system i s  
s t o c h a s t i c a l l y  observable  and con t ro l l ab le ,  then  the corresponding 
discrete homogeneous minimum variance es t imat ion  equat ions  a r e  
uniformly asymptot ical ly  s t a b l e  i n  t h e  l a r g e .  
-9- 
1. 
2.  
3 .  
4.  
References 
Aoki, M., Optimization of S tochas t i c  Systems, Academic Press, 
New York, 1967, pp. 212, 217. 
Kalman, R.  E.,  "New Methods i n  Wiener F i l t e r i n g  Theory", Proceedinqs 
of  t h e  F i r s t  Symposium on Enqineerinq Applicat ions of Random 
Function Theory and P robab i l i t y ,  E d i t e d  by Bogdanoff, J. L., 
and Kozin, F., John Wiley and Sons, New York, 1963. 
Kalman, R. E . ,  and Bertram, J. E., "Control System Analysis 
and Design V i a  t h e  Second Method of Lyapunov, I1 Discrete- 
Time Systems", Transact ions of t h e  ASME, Journa l  of Basic 
Engineering, June 1960, p.  394. 
Sorenson, H. W 0 8  "On t h e  Error Behavior i n  Linear M i n i m u m  
Variance Estimation ProblemsH, IEEE Transact ions on Auto- 
mat ic  Control,  Vol. AC-12, N o .  5 ,  O c t o  1967. 


Conditions for asymptotic stability of the discrete, minimum variance, linear estimator

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