On the Solution Sets for Uncertain Systems with Phase Constraints

Working Paper

ON THE SOLUTION SETS FOR UNCERTAIN SYSTEMS WITH PHASE CONSTRAINTS

A.B. Kurzhanskii February 1986 WP-86-11

International institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

NOT FOR QUOTATION WITHOUT P E R M I S S I O N O F THE AUTHOR

ON THE SOLUTION S E T S FOR UNCERTAIN SYSTEMS WITH PHASE CONSTRAINTS

A . B . K u r z h a n s k i i F e b r u a r y 1 9 8 6 WP-86-11

W o r k i n g P a p e r s are

i n t e r i m

r e p o r t s

o n w o r k of t h e

I n t e r n a t i o n a l I n s t i t u t e

f o r A p p l i e d Systems A n a l y s i s

a n d have r e c e i v e d

o n l y l i m i t e d r e v i e w . V i e w s

o r o p i n i o n s expressed

h e r e i n do

n o t n e c e s s a r i l y

repre- s e n t t h o s e of t h e I n s t i t u t e

o r

of

i t s

N a t i o n a l M e m b e r O r g a n i z a t i o n s .

INTERNATIONAL I N S T I T U T E FOR A P P L I E D SYSTEMS ANALYSIS A - 2 3 6 1 L a x e n b u r g , A u s t r i a

One of the means of modelling a system with an uncertainty in the parameters or in the inputs is to consider a multistage inclusion or a differential inclusion. These types of models may serve to describe an uncertainty for which the only avail- able data is a set-membership description of the admissible constraints on the unknown parameters.

A problem under discussion here deals with the specification

of the "tube" of all solutions to a nonlinear multistage inclu-

sion that arise from a given set and also satisfy an additional

phase constraint. The description of this "solution tube" is

important for solving problems of guaranteed estimation of the

dynamics of uncertain systems as well as for the solution of

other "viability" problems for systems described by equations

involving multivalued maps.

On the Solution Sets for

Uncertain Systems with Phase Constraints

A. B. K u r z h a n s k i i

INTRODUCTION

This p a p e r d e a l s with multistage inclusions t h a t d e s c r i b e a system with uncer- tainty in t h e model o r in t h e inputs [1,2]. In particular t h i s may b e a d i f f e r e n c e scheme f o r a d i f f e r e n t i a l inclusion 131. The solution t o t h e s e inclusions i s a mul- tivalued function whose cross-section at a s p e c l f i c instant of time i s t h e "admissi- ble domain" f o r t h e inclusion.

The problem c o n s i d e r e d h e r e is t o specify a s u b s e t of solutions t h a t c o n s i s t s of t h o s e " t r a j e c t o r i e s " . ~ h i c h satisfy a n additional p h a s e c o n s t r a i n t . These solu- tions are said to b e "viable" with r e s p e c t t o t h e p h a s e constraint 131. The Gross section of t h e set of a l l vlable solutions i s t h e attainablllty domain u n d e r t h e state c o n s t r a i n t . The d e r i v a t i o n of evolution equations f o r t h e l a t t e r domain i s t h e o b j e c t i v e of t h l s p a p e r .

The problem posed h e r e i s p u r e l y deterministic. However, t h e techniques applied t o i t s solution Involve some s t o c h a s t i c schemes. These schemes follow a n analogy between some formulae of convex analysis [4,5] and t h o s e f o r calculating conditlonal mean values f o r specific t y p e s of s t o c h a s t i c systems [6,7] which w a s pointed o u t in C8.91.

A s p e c i a l application of t h e r e s u l t s of t h i s p a p e r could b e t h e d e r l v a t i o n of solving r e l a t i o n s f o r nonlinear f i l t e r i n g u n d e r set-membership c o n s t r a i n t s o n t h e

"noise" and t h e d e s c r i p t i o n of t h e analogies between t h e t h e o r i e s of "guaranteed"

and s t o c h a s t i c filtering.

1. Discrete-time Uncertain Systems

Consider a multistage p r o c e s s described by a n n-dimensional r e c u r r e n t inclu- sion

where k E N , z ( k ) € E n , F ( k . z ( k ) ) i s a given multivalued map from IN XiRn into compiRn (IN i s t h e set of n a t u r a l numbers, complRn i s t h e set of a l l compact sub- sets of l R n ) .

Suppose t h e initial s t a t e z ( k , )

=

z 0 of t h e system i s confined t o a preassigned s e t :

w h e r e X 0 i s given in advance. A t r a j e c t o r y solution of system ( 1 . 1 ) t h a t starts from point z 0 at instant k o will b e denoted as z ( k l k o . z O ) . The set of a l l solutions f o r ( 1 . 1 ) t h a t start from z O at instant k O will b e denoted as X(k i k , , z O ) ( k E N , k l k O ) with f u r t h e r notation

Let Q ( k ) b e a multivalued map from IN into complRm and G ( k ) b e a single- valued map from N t o t h e set of m Xn-matrices. The p a i r G ( k ) , Q ( k ) , i n t r o d u c e s a state c o n s t r a i n t

o n t h e solutions of system ( 1 . 1 ) .

The s u b s e t of IRn t h a t consists of all t h e points of IRn t h r o u g h which at s t a g e s € [ k o , ~ ]

= I

k : k o S k S T

I

t h e r e p a s s e s at l e a s t one of t h e t r a j e c t o r i e s z ( k l k o , z o ) , t h a t s a t i s f y c o n s t r a i n t ( 1 . 3 ) f o r k € [ k 0 , r ] will b e denoted as X(s l ~ , k ~ . z O ) .

The aim of t h i s p a p e r is f i r s t t o study t h e sets X ( T I T , ~ ~ . X O )

=

X ( T , ~ ~ , X O ) and t h e l r evolution in "time" T .

In o t h e r words, if a t r a j e c t o r y z ( k l k o . z O ) of equation ( 1 . 1 ) t h a t s a t i s f i e s t h e c o n s t r a i n t ( 1 . 3 ) f o r a l l k E [ k o , s ] i s named "viable until instant T" ("relative t o c o n s t r a i n t ( 1 . 3 ) " ) , t h e n o u r o b j e c t i v e will b e t o d e s c r i b e t h e evolution of t h e set of a l l viable t r a j e c t o r i e s of ( 1 . 1 ) . H e r e at e a c h instant k > k O t h e c o n s t r a i n t ( 1 . 3 ) may "cut off" a p a r t of X ( k / k O . z O ) reducing i t t h u s t o t h e set x ( k , k O , z O ) .

The sets X(k . k D , z O ) may a l s o b e i n t e r p r e t e d as "attainability domains" f o r system ( 1 . 1 ) u n d e r t h e s t a t e s p a c e c o n s t r a i n t ( 1 . 3 ) . The o b j e c t i v e is to d e s c r i b e evolution of t h e s e domains.

A f u r t h e r o b j e c t i v e will b e t o d e s c r i b e t h e sets X ( s I r , k o . z O ) and t h e i r evelu- tion.

2. The A t t a i n a b i l i t y Domains

From the definition o f sets X(s I ~ , k ~ . z ~ ) it follows that the following proper- ties are true.

Lemma _2.1.

Whatever are the i n s t a n t s t , s , k , ( t a s > k 2 0 ) and the set IF Ecomp

I R n ,

thefollowing relation i s t r u e

Lemma

2.2.

Whatever are the i n s t a n t s s ,t , ~ , k

,l

( t a s

2 1 ;

~ a a k ; t > T ) and l the set F Ecomp

IR

" t h e following relation i s t r u e

Relation (2.1) shows that sets

X ( ~ , T . X )

satisfy a semigroup property which allows t o define a generalized dynamic system in the space 2"" o f all subsets of

I R " .

In general the sets X ( s ! t ,k .F

⁾

need not be either convex or connected. How- ever, it is obvious that the following is true

Lemma 2.3.

Assume that the map F i s Linear

in z:

where PEconvIR". Then for a n y set

F

EconvlR" each of the sets

X ( S ~ , ~ , I F ) E C O ~ V I R " ( ~

a s a k r o ) .

Here conv

IR"

stands f o r the set o f all convex compact subsets o f

IR"

.

3. The Onestage P r o b l e m

Consider the system

z E F ( z ) ,

GZ E Q .

z

E X .

where z

E I R

"

, X E

comp

IR" , Q E

conv

IR m ,

F ( K ) is a multivalued map from

IR"

into conv

IR

" .

^G

is a linear (single-valued) map from

IR

" into IR

^{m .}

I t is obvious that the sets F ( X ) = I u F ( z ) z

E X

1 need not be convex.

Let

Z , Z'

respectively denote the sets o f al solutions f o r the following sys-

tems:

(a)

z

€ F ( X ) , Gz E Q , (b) z ' ~ c o F ( X ) , G Z ' E Q ,

where c o F s t a n d s f o r t h e closed convex hull of F(X).

The following statement is t r u e

Lemma 3.1. The s e t s Z , c o Z , Z' s a t i s f y t h e f i l l o w i n g i n c l u s i o n s

L e t p(l Z )

=

s u p { l'z

z

€ 2

I

d e n o t e t h e s u p p o r t function [4] of set Z. Also denote

Then t h e function O(1 ,p , q ) may b e used t o d e s c r i b e t h e sets c o Z . Z 0 . Lemma 3.2. T h e f i l l o w i n g r e l a t i o n s a r e t r u e

p ( l Z )

=

p ( l l c o ~ )

=

s u p inf O(l,p , q )

,

q EF(X), p c I R m

I P (3.2)

p(l

123 =

inf sup O ( l , p , q ) , q E F ( X ) , p E!R

P v (3.3)

I t i s not difficuIt to give a n exampIe of a nonlinear map F ( z ) f o r which Z i s nonconvex and t h e f u n c t i o n s p(l ICOZ), p ( l 2 3 d o not coincide, s o t h a t t h e inclu- sions Z C c o Z , c o Z c Z b a r e s t r i c t .

Indeed, assume X

=

[ 0 j , z E R'

Then

The set F ( 0 ) i s a nonconvex polyhedron 0 K D L in Figure 1 while set Y i s a s t r i p e . H e r e , obviously, set Z which i s t h e i n t e r s e c t i o n of F ( 0 ) and Y , t u r n s to b e a nonconvex polyhedron 0 A

B D

L , while sets co Z , Z ' a r e corlvex polyhedrons 0 A B L and 0 A C L r e s p e c t i v e l y ( s e e F i g u r e s 2 , 3). The corresponding points h a v e t h e c o o r d i n a t e s

A

=

(0

.

^{2 ) , B}

⁼

^{(1/2 , 2 ), C}

⁼

^{(1 , 2 ) , D}

⁼

^{(3/7 , 3/7), K}

⁼

^{(0 , 3 ) , L}

⁼

^{( 3}

.

^O),

0

=

( 0 , 0).

Clearly Z c co Z C Z'

This example may also serve to illustrate the existence of a gap between (3.2) and (3.3).

?

F i g u r e 1

F i g u r e 2 '7-

1

F i g u r e 3

F o r a l i n e a r - c o n v e x map F ( z )

=

^Az

+ P

( P E c o n v IR " ) t h e r e is n o d i s t i n c t i o n between Z , c o Z , a n d 2':

Lemma 3.3 S u p p o s e F ( z )

=

Az

+

P w h e r e P E c o n v IR ", A is a l i n e a r m a p p r o m IRn i n to IRn. T h e n Z

=

c o Z

=

2'.

4. The One S t e ~ e Problem

-

An Alternative Approach.

The d e s c r i p t i o n of Z , coZ. Z ' m a y b e given i n a n a l t e r n a t i v e f o r m which, how- e v e r , allows to p r e s e n t all of t h e s e sets as t h e i n t e r s e c t i o n s of some v a r i e t i e s of c o n v e x multivalued maps.

I n d e e d , w h a t e v e r are t h e v e c t o r s 1 .p (1 #0) I t i s p o s s i b l e to p r e s e n t p =ML w h e r e M b e l o n g s to t h e s p a c e lh4 m x n of real m a t r i c e s of dimension m X n . Then, obviously,

p ( l 1 ~ )

=

s u p inf O(l,ML,q)

=

p ( ~ C O Z ) , q EF(X),

MEW.^^".

v Y

p(L Iz')

=

inf s u p #(L , M L , ~ ) q EF(X). M E I M " ' ~ "

hJ g

o r

w h e r e

p ( L 2 3

=

inf

1

O(1.M)

I

M ~lh4""" ) ,

From (4.1) I t follows t h a t

Z r

u n

R ( M , q ) C n

u

R ( M . ~ ) , M c N r n X n

v<F(XJ Y M ~ E F ( X J

w h e r e

S i m i l a r l y (4.2) y i e l d s

Moreover a stronger assertion holds.

Theorem 4.1. Thefollowing relations are t r u e

where M E I M " ~ " .

Obviously f o r F ( z )

=

.4Y + P , ( X , P E C O R " ) we have F ( X )

=

c o F ( X ) and Z

=

Z'

=

c o z .

This first scheme o f relations may serve t o be a basis f o r constructing recurrent procedures. Another recurrent procedure could be derived from t h e following second scheme. Consider t h e system

f o r which we are t o determine t h e set Z o f all vectors z consistent with inclusions (4.7), (4.8). Namely, we a r e t o determine the restriction F r ( z ) o f F ( z ) t o set Y . Here wehave

F ( z ) i f z EY F r ( z )

= I ⁴

^{i f z E Y}

where as b e f o r e Y

=

[ z : G z E Q

1.

Lemma 4.1 A s s u m e F ( z ) EcompIRn for a n y z a n d Q E c o n v R

"'.

Then

over all n x m matrices L , (2 € I M n

Denote t h e null vectors and matrices as [ D i m E R m , [ O l m , n € R m x n , the ( n x n ) unit matrix as En and the ( n x m ) matrix Lmn as

Suppose

z

^€ Y . Then f O j m E Q

-

Gz and f o r any ( n x m ) -matrix L we have ) O l n ^E L (Q

-

G z )

.

Then it follows that f o r z E Y .

On t h e o t h e r h a n d , s u p p o s e

z

? Y.

L e t u s d e m o n s t r a t e t h a t in t h i s case

n

I F @ ) + L ( Q - G z )

1 = 4

L

Denote A = F ( z ) , B

=

Q - G z . F o r a n y A > O we t h e n h a v e

S i n c e

f

0

I,?B

we h a v e

f

0 jn ?L,B. T h e r e f o r e t h e r e e x i s t s a v e c t o r L SIRn, L # O a n d a n u m b e r y > O s u c h t h a t

( ~ , z ) z y > O f o r a n y

z

€ L , B , D e n o t e

L

=

f z : ( L , z ) > y j T h e n L 2L,B a n d

( A + X L , B ) r , ( A - X L , B ) C ( A + U ) n ( A - U ) S e t A being bounded t h e r e e x i s t s a A

>

0 s u c h t h a t

( A + A L ) n ( A - A L )

= 4 .

H e n c e

a n d t h e Lemma i s p r o v e d .

5. S t a t i s t i c a l U n c e r t a i n t y . The Elementary Problem.

C o n s i d e r t h e system

z

=

^q + ( , Cz = v + 7 ) ,

w h e r e

and

t.7

are independent gaussian random v e c t o r s with z e r o means ( E t = O , E v = O ) and with v a r i a n c e s E t t '

=

R . E v v '

=

N , where R > O n N > O ( R EIM,. N E N ,).

Assuming at f i r s t t h a t t h e p a i r h

=

[ q . v

1

i s fixed, l e t us find t h e conditional mean E ( z y =0, h = h a ) u n d e r t h e condition t h a t after one r e a l i z a t i o n of t h e values

t.7

t h e r e l a t i o n s

a r e satisfied. A f t e r a s t a n d a r d calculation we have

where

P-' =

R-I

+

G'N-lG.

A f t e r applying a well-known matrix transformation [6]

we have

The matrix of conditional v a r i a n c e s i s

I t does not depend upon h and i s determined only by q , v and t h e element A

=

RG'K-'G. T h e r e f o r e i t makes s e n s e t o c o n s i d e r t h e sets

and

of conditional mean values. Each of t h e elements of t h e s e sets h a s one and t h e same v a r i a n c e

P,.

The sets W.(A) and W ( A , q ) are obviously convex while W ( A ) may not b e convex.

Lemma 5.1 T h e f o l l o w i n g i n c l u s i o n s a r e t r u e ( Z c Z 7

z

C W ( A ) ,

z'c w'(A),

W ( A ) C W ' ( A ) ^, (5.2)

I t c a n b e s e e n t h a t W ( h , q ) h a s e x a c t l y t h e same s t r u c t u r e as R ( M , q ) of (4.3) (with only h s u b s t i t u t e d by M). Hence f o r t h e same r e a s o n as b e f o r e w e have

w h e r e t h e i n t e r s e c t i o n s a r e t a k e n o v e r t h e c l a s s D of a l l possible p a i r s D

=

f R,N

j

of nonnegative matrices R,N of r e s p e c t i v e dimensions. However, a p r o p e r t y similar Lo t h a t of Lemma 4.1 h a p p e n s t o b e t r u e . Namely if by D ( a , @ ) w e denote t h e c l a s s of p a i r s [ R,N

j

where R

=

aE, , N

= @ E m , a >

0 , @

>

0 , t h e n t h e element X will depend only upon two p a r a m e t e r s a , @ .

Theorem 5.1 Suppose m a t r i z G i s o f f u l l r a n k m .

h e n

the following equali- t i e s are t r u e

H e r e i t s u f f i c e s Lo t a k e t h e i n t e r s e c t i o n s only o v e r a one-parametric v a r i e t y D e D ( 1 , B ) .

T h e r e are some s p e c i f i c d i f f e r e n c e s between t h i s scheme and t h e one of 54.

These could b e t r a c e d more explicitly when we p a s s Lo t h e calculation of s u p p o r t functions p(l

I z ) ,

p ( l / Z ' ) f o r Z , Z ' .

Lemma 5.2 Thefollowing i n e q u a l i t y i s t r u e

~ ( 1 1 2 3

= f " ( l ) s f ( l )

=

inf [ + ( l , h ) l D € D ( l , @ ) , @ > O j (5.6) where f "(1 ⁾ i s the second conjugate to f ( 1 ) in t h e sense ofFenche1 [ 4 ] .

Moreover if we s u b s t i t u t e D ( 1 , @ ) in (5.6) f o r a b r o a d e r c l a s s D t h e n a n e x a c t equality will b e a t t a i n e d , i.e.

~ ( 1 1 ~ 3

= f " ( 1 )

=

inf ftJ(1,h)jD E D

j

^(5.7) More p r e c i s e l y , w e come t o

Theorem 5.2 Suppose m a t r i z G is of f u l l r a n k m . Then e q u a l i t y (5.7) w i l l be t r u e together w i t h t h e f o l l o w i n g r e l a t i o n

p(l

I z )

= p ( l l c o ~ ) = s u p inf

I

tJ(l . h . q ) d

E D I ~

e F ( X )

I

( 5 . 0 )

Problems (5.73, (5.8) a r e "stochastically dual" t o ( 3 . 3 . ( 3 . 2 ) .

The r e s u l t s of t h e a b o v e may now b e applied t o o u r basic problem f o r multis- t a g e systems.

6. Solution t o the Basic Problem

Returning t o system ( 1 . 1 ) - ( 1 . 3 ) we will s e e k f o r t h e sequence of sets X [ s ]

=

X ( s , k , . ~ ~ ) t o g e t h e r wlth two o t h e r s e q u e n c e s of sets. These a r e

-

t h e solution set of t h e system

zk.1 ~ c o F ( k . X 0 [ k ] ) , X ' [ k 0 1

=

X' G ( k + 1 ) E Q ( k + l ) , k a k ,

and X . [ s ]

=

X.(s . k , . ~ ' ) which i s obtained d u e t o t h e following relations:

X . [ s ]

=

C O Z [ S ] where Z [ k +1] i s t h e solution set f o r t h e system

z ( k + 1 ) E F ( k , c o Z [ k I ) , Z [ k o ] = X u .

The sets X.[T]. X T T ] a r e obviously convex. They satisfy t h e inclusions X [ T ] ~ X . [ T ] c X ' [ T ]

w h e r e e a c h of t h e sets X [ T ] , X.[T]. X * [ T ] lies within

Y ( T )

=

[ z : G ( T ) z E Q ( T ) I , ~ > k , + l ,

The set x ' [ T ] may t h e r e f o r e b e obtained f o r example by e i t h e r solving a s e q u e n c e of problems ( 6 . 1 ) , ( 6 . 2 ) ( f o r e v e r y k E [ k , . s -11 with X t k o ]

=

x u ) ( t h e f i r s t scheme of 54) o r by finding a l l t h e solutions z [ k ] = E ( k . k O , z o ) of t h e equation

t h a t could b e prolongated until t h e i n s t a n t T

+

1 and finding t h e r e l a t i o n of t h i s set LO X [ T ] , X . [ T ] , and X.[T].

Following t h e f i r s t scheme of 54 w e may t h e r e f o r e c o n s i d e r t h e r e c u r r e n t sys- t e m

z ( k + I )

=

( I , - M ( k + l ) G ( k + l ) ) F " ( k ,S ( k )) + M ( k + l ) Q ( k + 1 ) ( 6 . 4 )

where M ( k + 1 ) E IR m X " .

From Theorem 4.1 w e may now d e d u c e t h e r e s u l t

Theorem 6.1 The s o l v i n g r e l a t i o n s por X [ s

1,

X.[s], X*[s ] a r e a s pollows

~ [ s ]

=

s ( s ) f o r ~ O ( k , s ( k ) ) = F ( k . S ( k ) ) ( 6 . 6 ) x 0 [ s ]

=

~ ( s ) f o r ~ ' ( k , S ( k ) )

=

c o F ( k , S ( k ) ) ( 6 . 7 )

x.[s] =

c o S ( s ) f o r ~ O ( k , s ( k ) = F ( k , c o S ( k ) ) . ( 6 . 8 )

~t is obvious thrlt X [ T ] i s t h e e x a c t solution while X.[T], X'[T] are convex majorants f o r X [ T ] . Clearly by interchanging and combining r e l a t i o n s ( 6 . 7 ) , ( 6 . 8 ) from s t a g e t o s t a g e i t i s possible t o c o n s t r u c t a v a r i e t y of o t h e r convex majorants f o r X [ T ] . However for t h e l i n e a r case they a l l coincide with X [ T ] .

Lemma 6.1 A s s u m e F O ( k , s ( k ) )

=

A ( k ) S ( k ) + P ( k ) w i t h P ( k ) , X O b e i n g closed a n d compact. T h e n X [ k ]

=

X'[k ]

=

X.[k ] por a n y k r k 0.

Consider t h e system

z ( k + I )

=

( I , - M ( k + l ) G ( k + 1 ) ) F 0 ( k , Z ( k ) ) -M ( k + l ) Q ( k + l ) , Z ( k d

= x O ,

( 6 . 9 ) denoting i t s solution as

z ( k ; M k ( . ) ) f o r F o ( k , Z ) = F ( k , Z ) z.(k .Mk(.)) for F O ( k , Z )

=

F ( k , c o Z ) Z * ( k . M k ( . ) ) f o r F O ( k

, z ) =

c o F ( k , Z ) Then t h e p r e v i o u s suggestions yield t h e following conclusion

Theorem 6.2 Whatever i s t h e s e q u e n c e hi,(.), t h e p o l l o w i n g s o l v i n g i n c l u s i o n s a r e t r u e

with Z ( S ,M.(-)) ~ Z . ( S , M a ( - ) ) CZ.(s.M,(.)) Hence we a l s o h a v e

o v e r a l l M, ( s )

However a question a r i s e s which i s w h e t h e r (6.11)-(6.13) could t u r n into e x a c t equalities.

Lemma 6.2 A s s u m e t h e s y s t e m (Z.Z), to be l i n e a r : F ( k , z ) = A ( k ) z + P ( k ) w i t h s e t s P ( k ) , Q ( k ) c o n v e z a n d compact. T h e n t h e i n c l u s i o n s (6.11)-(6.13)

turn

i n t o t h e e q u a l i t y

Hence in t h i s c a s e t h e i n t e r s e c t i o n s o v e r M ( k ) could b e t a k e n e i t h e r in e a c h s t a g e as in Theorem 6.1 ( s e e ( 6 . 6 ) , ( 6 . 7 ) ) o r at t h e final s t a g e as in (6.14).

Let u s now follow t h e second scheme of 54, considering t h e equation

and denoting t h e set of i t s solutions t h a t starts at

z 0

EX' as ~ ' ( k , k O , z 0 ) with

(, ~ z ~ ( k , k ~ , z ~ ) l z o ~ ~ ~ j = X o ( k , k O , x o ) = X o [ k ]

.

According t o Lemma 4.1 w e s u b s t i t u t e (6.15) by t h e equation

z ( k + l ) E n ( P ' ( k , Z ( k ) ) - L G ( k ) z ( k ) + L Y ( k ) ) ,

z 0

E X ' , L

and t h e calculation of X O [ k ] should t h e n c e follow t h e p r o c e d u r e

Denote t h e whole solution "tube" f o r k o 5 k S s as

fro[.].

Then t h e following a s s e r t i o n will b e t r u e .

Theorem 6.3 A s s u m e Z o [ k

I

t o be t h e c r o s s - s e c t i o n of t h e t u b e

ei;,[.]

a t i n s t a n t k . h e n

H e r e 3 z o [ s ] 2%;' [ s ] and t h e set f ; , [ s ] may not l i e totally within Y ( s )

The solution of equation (6.16) i s equivalent t o finding a l l t h e solutions f o r t h e inclusion

~ (+ I ) k E

n

^{( P ( k}

, z )

- L G ( ~ ) z + L Q ( ~ ) ) ,

z

( k , ) E X , (6.17) L

Equation (6.17) may b e substituted by a system of "simpler" inclusions

f o r e a c h of which t h e solution s e t f o r k o S k S s will b e denoted as

Theorem 6.4 The set Yt.,[..L

(.)I

of viable s o l u t i o n s to the i n c l u s i o n

i s the r e s t r i c t i o n of set

defined for stages [ k g ,

. . .

. s + l ] to the stages [ k , ,

...,

s ] . The i n t e r s e c t i o n i s t a k e n here over all constant matrices L .

However a question a r i s e s , w h e t h e r this scheme allows a l s o t o c a l c u l a t e

&',

[ s

1.

Obviously

o v e r a l l s e q u e n c e s L [ a ]

=

L ( k ,). L ( k , + l ) , . . .,L ( s +1) j . Moreover t h e following p r o p o r t i o n i s t r u e .

Theorem 6.5 Assume F ( k ,z) to be linear-convez: F ( k . 2 )

=

A ( k ) z + P ( k ), w i t h P ( k ) , Q ( k ) convez a n d compact. Then (6.19) t u r n s to be a n e q u a l i t y .

7 . S o l u t i o n t o the Basic Problem. " S t o c h a s t i c " Approximations.

The calculation of X [ s ] , X . [ s ] , X ' [ s ] may b e a l s o p e r f o r m e d o n t h e basis of t h e r e s u l t s of 55. Namely system (6.61, (6.7) should now b e s u b s t i t u t e d by t h e following

z ( k + l )

=

( I , - ~ ( k + I ) G ( ~ + I ) ) F ~ ( ~ , ~ ( k )) - ~ ( k + l ) ~ ( k + l ) (7.1)

T h e o r e m 7.1 Assume t h a t i n Theorem 6.1 S ( k )

i s

s u b s t i t u t e d b y H ( k ) a n d M ( k ) by F ( k ). Then the r e s u l t of t h i s theorem r e m a i n s t r u e .

If set Q ( k ) of (1.3) i s of s p e c i f i c t y p e

where y ( k ) a n d 6 ( k ) a r e given, t h e n (1.3) is t r a n s f o r m e d into

which could b e t r e a t e d as a n equation of o b s e r v a t i o n s f o r t h e u n c e r t a i n system 7 S e t s X [ s ] . X . [ s ] . X ' [ s ] t h e r e f o r e give us t h e g u a r a n t e e d e s t i m a t e s of t h e unknown s t a t e of system (1.1) o n t h e basis of a n o b s e r v a t i o n of v e c t o r y ( k ) . k E [ k , , s ] due t o equation (7.4). The r e s u l t of Theorem 7 . 1 t h e n means t h a t t h e solution of t h i s problem may b e o b t a i n e d via equations (7.1)-(7.3), a c c o r d i n g t o formulae (6.8)-(6.10) with M ( k ) , S ( k ) substituted r e s p e c t i v e l y by F ( k ) , H ( k ) . The deterministic problem of nonlinear " g u a r a n t e e d " f i l t e r i n g i s h e n c e approximated by r e l a t i o n s o b t a i n e d t h r o u g h a "stochastic f i l t e r i n g " approximation scheme.

8. T h e S e t X ( s

1

t , k , I F ) .

Assume t h a t t h e s e q u e n c e y [ k ,

t ]

i s fixed. Let u s discuss t h e means of con- s t r u c t i n g sets X(z

! t

, k , I F ) , with s E [ k , t ] . From t h e r e s p e c t i v e definition one may d e d u c e t h e a s s e r t i o n

Lemma 8.1 The following equality i s t r u e

H e r e t h e symbol X(s

I

s ,

t

, IF), t a k e n f o r s 5 t , s t a n d s f o r t h e set of states z ( s ) t h a t s e r v e as s t a r t i n g points f o r a l l t h e solutions z (k , s

.

^z ( s ) ) t h a t s a t i s f y t h e r e l a t i o n s

C o r o l l a r y 8.1 Formula (8.1) may b e substituted f o r

where R is any s u b s e t of W" t h a t includes X(t

.

^{k , IF).}

Thus t h e set X(s

1

t , k , IF) i s d e s c r i b e d through t h e solutions of two p r o b - lems t h e f i r s t of which is t o d e f i n e X(s , k , IF) (along t h e techniques of t h e above) and t h e second is t o define X(s

I

s

.

^{t , R).} The solution of t h e second problem will b e f u r t h e r specified f o r IF 6 compRn and f o r a closcd convex Y.

The underlying elementary o p e r a t i o n is t o d e s c r i b e X*

-

t h e set of a l l t h e vec- t o r s z E W" t h a t satisfy t h e system

In view of Lemma 4.1 w e come t o

Lemma 8.2 Th set X' may b e d e s c r i b e d as

From h e r e i t follows:

T h e o r e m 8.1

The set X(s

1

s ,

t

, R) may b e d e s c r i b e d as t h e solution of t h e r e c u r r e n t system (in backward "time")

where

Finally w e will specify t h e solution for t h e l i n e a r c a s e

z ( k

+ I )

E ~ ( k ) z ( k ) + ~ ( k ) ~ ( k )

=

: G ( ~ ) z E Q ( ~ ) I Assume

w h e r e A E i M n X n , G E

IM

_{, P ,} Q , Z are convex and compact.

Lemma 8.3 The s e t X may b e defined as

P(L

I ^{X )} =

inf lp(X

I

P ) + P(X

I

Z ) + P @

I

Q ) I

o v e r a l l t h e v e c t o r s X € R n , p € R m t h a t s a t i s f y t h e equality t

=

A' X

+

^{G ' p .}

The l a t t e r r e l a t i o n yields:

Lemma 8.4 The set X may b e defined as

X

r L ' ( Z

+

P )

+

M ' Q

=

H ( L , M ) ( 8 . 5 )

whatever are t h e m a t r i c e s L € i M n X n and M E

IM

m x n t h a t s a t i s f y t h e equality L ' A

+

M ' G = E n . Moreover t h e following equalities are t r u e

o v e r a l l L E i M n X n , M E h( m X n

Corollary 8.2 Suppose I A

/

# 0 . Then conditions ( 8 . 5 ) , ( 8 . 6 ) may b e s u b s t i t u t e d f o r

X

r (En

-

M ' G ) A-' ( Z

+

P )

+

M ' Q

=

H ( M ) ,

where

M E ~ M ~ ~ .

The l a t t e r r e l a t i o n s may b e used f o r r e c u r r e n t p r o c e d u r e s . These are e i t h e r X [ k ] = n ( H k ( L , M ) L A ( k ) + M G ( k ) = E n I , ( 8 . 7 )

with

or

with

-

18

-

Hk(L , M ) = L ' ( X [ k

+

11 + P ( k ) ) + M ' Q ( k ) XLt1

= Y [ t l

³

s s k s t

where

Theorem 8.2

The set

X ( s

1

s

.

^{t , Y )}

may be derived due t o either equations

( 8 . 7 )

-

( 8 . 9 )

o r

( 8 . 1 0 ) , ( 8 . 1 1 ) .

Remark

As mentioned in the sequel t o 5

7 ,

set

Y ( t )

may be generated due t o a measurement equation

where

6 ( k )

is the restriction on the "noise" in t h e observations. Then each o f the sets

X ( T , k , XO)

gives a

"

guaranteed" estimate for the unknown state o f the system

( 1 . 1 )

on t h e basis o f the available measurement

y (.)

=

( y ( k o ) ,

. . .

, y ( k ) )

obtained due t o equation

Thus sets

X ( T , k , ,

xO) solve the "filtering" problem, whilst

X ( s

1

^T , k o , X u )

gives the solution o f either the interpolation ("refinement") problem ( i f

k ,

s

s

s

^{T )}

or t h e extrapolation problem ( i f

k , 5 T S s ) .

In

$ 7

the approximation o f

X ( T

.

^{k g ,}

^x') was given through stochastic filter-

ing procedures. The same approach may be propagated t o give an alternative

approximation scheme f o r sets X ( s

1

s , k D , XD).

The schemes of t h i s p a p e r allow to treat nonlinear systems. However in t h e l i n e a r case t h e y d o not coincide with t h e p r o c e d u r e s given in [2.10] f o r solving g u a r a n t e e d estimation problems with set-membership instantaneous c o n s t r a i n t s .

References

[I] Krasovskii. N.N. Control u n d e r Incomplete Information and Differential Games, P r o c . I n t e r n . Congress of Mathematicians, Helslnki, 1978.

[21 Kurzhanskii. A.B. Control a n d O b s e r v a t i o n u n d e r C o n d i t i o n s of U n c e r - t a i n t y , Nauka, Moscow, 1977.

[31

Aubin, J.-P., and A. Cellina. LXfferential I n c l u s i o n s . Springer-Verlag.

Heidelberg, 1984.

[41 R o c k a f e l l a r , R.T. C o n v e z A n a l y s i s . P r i n c e t o n University P r e s s , 1970,

[51

Ekeland, I. and R. Teman. Analyse Convexe et problemes Variationelles.

Dunod, P a r i s , 1974.

[6] Albert, A. R e g r e s s i o n a n d t h e Moore-Penrose P s e u d o - I n v e r s e . (Mathematics in S c i e n c e & Engineering S e r . : Vol. 94). Acad. P r e s s 1972

[7] Davis. M. L i n e a r E s t i m a t i o n a n d Control T h e o r y . London Chapman-Hall.

1977.

[8] Kurzhanskii, A.B. Evolution Equations in Estimation Problems f o r Systems with Uncertainty. IIASA Working P a p e r WP-82-49, 1982.

[9] Koscheev, A.S. and A.B. Kurzhanski. Adaptive Estimation of t h e Evolution of Multisgate Uncertain Systems. Ivestia Akad. Nauk. SSSR Teh. Kibernetika ("Engineering Cybernetics"). No. 2. 1983.

1101 Schweppe,

F.

Uncertain Dynamic Systems. Prentice-Hall Inc., Englewood Cliffs, N e w J e r s e y , 1973.