Deterministic feedback linearization, Girsanov ... - Science Direct

Systems & Control Letters 13 (1989) 81-92 North-Holland

81

Deterministic feedback linearization, Girsanov transformations and finite-dimensional filters M. COHEN DE LARA Cergrene, Ecole Nationale des Ponts et Chauss~es, Le Service La Courtine, 93167 Noisy Le Grand, France

J. L]~VINE Centre d'Automatique et Informatique, Section Automatique, Ecole Nationale SupJrieure des Mines de Paris, 35 rue St. HonorO, 77305 Fontainebleau, France

Received 10 October 1988 Revised 12 February and 5 April 1989 Abstract: The problem of extending state-feedback linearization methods of deterministic control theory to stochastic systems is

addressed. For Stratonovitch stochastic differential equations with smooth vector fields, necessary and sufficient geometric conditions for local and global linearization by diffeomorphism and absolutely continuous change of probability law are obtained, using the interpretation of Girsanov transformations as state-feedback on Brownian motions. For stochastic systems with single-input (or one-dimensional Brownian motion) and single-output (or one-dimensional observation process), necessary and sufficient geometric conditions to transform the Duncan-Mortensen-Zakai (DMZ) equation of filtering into that of an affine prime system are obtained, as well as interpretation of gauge transformation as Girsanov change of probability law. Keywords: Girsanov theorem; state-feedback linearization; stochastic system; diffeomorphism; gauge transformation; finite-dimen-

sional filter.

1. Introduction I n the s t u d y of S t r a t o n o v i t c h stochastic d i f f e r e n t i a l e q u a t i o n s with s m o o t h vector fields, or, briefly, stochastic systems, S t r a t o n o v i t c h calculus allows us to h a n d l e t h e m as if they were d e t e r m i n i s t i c differential equations. B r o w n i a n m o t i o n s are thus r e l a t e d to c o n t r o l functions, o r inputs, as it a p p e a r s in the s u p p o r t t h e o r e m of S t r o o c k a n d V a r a d h a n [12, p. 429]. I n d e t e r m i n i s t i c c o n t r o l theory, s t a t e - f e e d b a c k l i n e a r i z a t i o n m e t h o d s p r o v i d e g e o m e t r i c c h a r a c t e r i z a t i o n o f n o n l i n e a r c o n t r o l systems that can be t r a n s f o r m e d into a linear, c o n t r o l l a b l e one b y m e a n s of d i f f e o m o r p h i s m a n d static state-feedback. W e show how such results m a y be used either to linearize stochastic d i f f e r e n t i a l e q u a t i o n s , or to c o m p u t e finite d i m e n s i o n a l filters. T h e p a p e r is o r g a n i z e d as follows. Section 2 is d e v o t e d to b r i e f recalls on state-space linearization methods. In Section 3, we focus on the c o n n e c t i o n b e t w e e n G i r s a n o v t r a n s f o r m a t i o n on B r o w n i a n m o t i o n driving a stochastic system a n d f e e d b a c k on the i n p u t of a c o n t r o l system, as was n o t i c e d in [5,8]. A f t e r briefly recalling G i r s a n o v ' s theorem, we give a d e f i n i t i o n a n d a g e o m e t r i c c h a r a c t e r i z a t i o n for local a n d g l o b a l G i r s a n o v linearization o f stochastic systems: such a l i n e a r i z a t i o n is p e r f o r m e d b y m e a n s of a d i f f e o m o r p h i s m on the state space a n d an a b s o l u t e l y c o n t i n u o u s c h a n g e of l a w on the p r o b a b i l i t y space. T h r o u g h o u t the p a p e r , the results are stated in the case of o n e - d i m e n s i o n a l B r o w n i a n motion, since, in the m u l t i - d i m e n s i o n a l case, global results are n o t yet established. In Section 4, the links b e t w e e n filtering a n d s t a t e - s p a c e l i n e a r i z a t i o n are studied. It is well k n o w n that linear systems a d m i t f i n i t e - d i m e n s i o n a l filters a n d this p r o p e r t y is also p r e s e r v e d b y d i f f e o m o r p h i s m a n d gauge t r a n s f o r m a t i o n on the D M Z equation. T h a n k s to f e e d b a c k a n d l i n e a r i z a t i o n m e t h o d s , we give a g e o m e t r i c c h a r a c t e r i z a t i o n of a class of systems h a v i n g f i n i t e - d i m e n s i o n a l filters explicitly o b t a i n e d from a K a l m a n - B u c y filter. G a u g e t r a n s f o r m a t i o n s are also shown to b e r e l a t e d to G i r s a n o v t r a n s f o r m a t i o n s of p r o b a b i l i t y law. 0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

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2. Recalls on state-feedback linearization

Let us consider the following single-input n o n l i n e a r c o n t r o l system of the f o r m

2,--f(x,)+g(x,)~

(1)

where x ~ N" is the state, v is the i n p u t a n d f , g are s m o o t h vector fields on N". S t a t e - f e e d b a c k linearization u n d e r u n i t a r y static s t a t e - f e e d b a c k a n d for single-input systems aims at t r a n s f o r m i n g the system (1) into a linear, c o n t r o l l a b l e one

~, = A ~ , + Bu

where r a n k { B , A B . . . . . A " - ~ B }

= n

(2)

by means of d i f f e o m o r p h i s m s ( = (b(x) a n d u n i t a r y s t a t e - f e e d b a c k c o n t r o l s of the form v = c~(x) + u. Definition 2.1. Let f , g be two s m o o t h vector fields on N" a n d x 0 ~ N~. T h e c o n t r o l system (1), with initial c o n d i t i o n x 0, is said to be locally linearizable ( u n d e r u n i t a r y state s t a t e - f e e d b a c k ) if there exist an o p e n n e i g h b o u r h o o d W of x 0, a d i f f e o m o r p h i s m q~ d e f i n e d on W a n d a s m o o t h f u n c t i o n c~ on W such that the feedback c o n t r o l ~, = a ( x ) + u a n d the d i f f e o m o r p h i s m ~ = ~ ( x ) t r a n s f o r m system (1) into a linear c o n t r o l l a b l e one (2). The control system (1), with initial c o n d i t i o n x 0, is said to be globally linearizable if it is locally linearizable, with W = R " a n d q~ global d i f f e o m o r p h i s m of R ", that is qb is a o n e - t o - o n e d i f f e o m o r p h i s m from R" to R". Remark 2.1. W e state the definition of s t a t e - f e e d b a c k l i n e a r i z a t i o n only in the single-input a n d u n i t a r y static feedback case but, in the sequel, all the local results stated for o n e - d i m e n s i o n a l B r o w n i a n m o t i o n s can easily be e x t e n d e d to m u l t i - d i m e n s i o n a l B r o w n i a n motions. However, this extension for global results is not clear. The restriction to u n i t a r y f e e d b a c k is m o t i v a t e d by the fact that we use B r o w n i a n m o t i o n s in place of controls a n d that a general f e e d b a c k of the form v = a ( x ) + / 3 ( x ) u , w i t h / 3 everywhere invertible, p r o d u c e s a semi-martingale which is no longer a B r o w n i a n m o t i o n u n d e r an equivalent p r o b a b i l i t y law.

W e briefly recall the geometric tools n e e d e d for solving the s t a t e - f e e d b a c k linearization p r o b l e m (for b a c k g r o u n d m a t e r i a l in differential g e o m e t r y see [11,3]). Let h be a s m o o t h function on R ", X a s m o o t h vector field on R n, so X = E7=1 X,(O/Ox~) in local coordinates. W e d e n o t e b y L x h the Lie derivative of h a l o n g X: this is a s m o o t h function on R" expressed in local c o o r d i n a t e s by L x h ( x ) -- E"i= l Xi( x )( Oh / O x , )( x ). By i n d u c t i o n L kx~ l h = L x ( L kxh ), k ~ ~ . The Lie b r a c k e t of two s m o o t h vector fields X a n d Y on R ~ is the s m o o t h vector field [X, Y] on R" defined by L i x . r l h = L x L r h - Lr, L x h , for any s m o o t h function h on N ~, or in local c o o r d i n a t e s

i=l

j=l

By i n d u c t i o n we define a d ° Y = Y, adlxY = [X, Y] . . . . , a o- kx+ l . r. = [ X , adkxg], k ~ N . If d x ~ , . . . , d x , , is the dual basis of the s t a n d a r d basis 3 / 3 x ~ . . . . . 3 / 3 x n of T(R ") = Nn, a 1-form o~ on R" is a linear c o m b i n a t i o n of the d x , with s m o o t h coefficients, that is ~o = ~27_~ ~0, d x , . 0~ is said to be closed if for all i, j, Oog/~xj = O(oJOx,. o~ is said to be exact if there exists a s m o o t h function h such that ~0 = d h , that is: ~0 = E ~ , = l ( 3 h / 3 x , ) d x , . A n exact 1-form is always closed a n d b y Poincar6's t h e o r e m a closed form on a simply c o n n e c t e d o p e n set is exact. If w is a 1-form on R n and X a s m o o t h vector field o n R", we d e n o t e w ( X ) the s m o o t h function on R" defined by ~o(X) = S,','_ 1 ~iXi. In p a r t i c u l a r if ~0 = d h , then ~0(X) = L x h . If 0 is a s m o o t h injective m a p p i n g from R" to R " , we d e n o t e b y O , ( X ) the s m o o t h vector field on 0 ( R " ) which is the image of the s m o o t h vector field X on R ".

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Remark 2.2. It follows from this last definition that system (1) is locally linearizable if and only if there exist an open neighbourhood W of x 0, a diffeomorphism • defined on W and a smooth function a on W such that (a) ~ . ( g ) is a constant vector field, identified with a column matrix B, (b) ~ . ( f + a g ) is a linear vector field ~ ~ A~, with A a square matrix of size n, (c) the pair (A, B) is controllable. The state-feedback linearization problem in the analytic case of a single-input control system has been solved by Brockett [5], and for multi-inputs by Jakubczyk and Respondek [13] (see also [22,10]). The following theorem is stated in the smooth single-input case [9, Th. 4.1]. Theorem 2.1. Let f , g be two smooth vector fields on R n. The single-input system (1), with initial condition x o, is locally linearizable if and only if (i) g( x o ), a d f g ( x 0) . . . . . adT- lg( x o) are linearly independent vectors, (ii) the 1-form ~ defined by ~(ad)-ag) = ( - 1~, ~- 18c,, i = 1 . . . . . n, is closed in a neighbourhood of x o.

Remark 2.3 If conditions (i), (ii) of Theorem 2.1 are satisfied, the construction of • and ct is as follows. Since the 1-form o~ is closed in a simply connected neighbourhood of x 0, there is a C a function h such that o~ = dh on this neighbourhood. The feedback control v = u - L T - l o ~ ( f ) ( x ) , or ct = - L 7 l w ( f ) , and diffeomorphism qb = (41 . . . . . ~ ) " = ( h, Lyh . . . . . L T - lh ) ', where ' denotes transposition, provide the pair (A, B) in Brunovsky canonical form [14], that is, 0 0

1 0

0 1

...

A=

0 0

0 0

0

0

...

B=

1 0

i/.

(3)

Conversely, given an arbitrary controllable pair (A, B), it is well known from linear control theory [14] that by linear change of coordinates and linear feedback, one may always transform A and B into the Brunovsky canonical form. Therefore, assuming that system (1) is locally linearizable, there is no loss of generality to consider that (A, B) is in Brunovsky canonical form. In this case it can be proved that necessarily:

t ~ = ( t ~ l , t f ~ 1. . . . . t T - l ~ l ) ',

~=d~

1 and

a=-L~-lo(f).

(4)

These formulas will be useful in the sequel. Global versions of Theorem 2.1 are much more restrictive [4,21,8,9]. The following version is borrowed from [8, Th. 4]: Theorem 2.2. Let f , g be two smooth vector fields on R n. The single-input system (1) is globally finearizable if and only if: (i) g ( x ) , a d f g ( x ) . . . . . adT-~g(x) are linearly independent vectors for all x ~ R n, (ii) the 1-form w defined by w(ad)-lg) = ( - 1)n-ari,n, i = 1, . . . . n, is closed in ff~', (iii) the vector fields g, a d / g . . . . , adT-- lg are complete, with f = f - ( L 7 - l~o( f ) ) g.

In the next section we shall try to extend this deterministic approach to stochastic systems, by diffeomorphisms on the state-space and transformations of the Brownian motion.

84

M. Cohen de Lara, J. L~vine / Deterministicfeedback linearization

3. Girsanov linearization

The stochastic counterpart of system (1) is given by the following Stratonovitch stochastic differential equation, or stochastic system, dx t =f(x,)

dt + g ( x , ) o dr,,

(5)

where f , g are two smooth vector fields on Rn, vt is a one-dimensional Brownian motion on a probability space (~2, d , P ) with its natural filtration Y = ( 4 ) , >_0- The Stratonovitch increment o dr, of v, will play the role of v, dt in Equation (1). If dr, is the It6 increment, we recall the conversion formula for any s m o o t h function h: (6)

h ( x , ) o dr, = h ( x , ) d r , + ~( L , h ) ( ~ , ) dt.

By making use of Stratonovitch calculus, it is clear that if the control system (1) is locally linearizable, we can transform the stochastic system (5) into a linear, controllable one, at least up to a stopping time ~-, d~,=A~ tdt+B

odu t

whererank(B,

A B . . . . . A"

1B} = n ,

(7)

with a new stochastic input u~ = v t - fd ~ "a(x,) ds instead of the Brownian m o t i o n vt. Of course, u t is no longer a Brownian motion under the original probability law P, but possibly under a new probability law Q by Girsanov's theorem [17, Ch. 6, §3]. With our notations, we state a version of this theorem: T h e o r e m 3.1. Let x, be a solution of Eq. (5), defined for all time t >_ 0, ~- an o~-stopping time and a a smooth function on R". I f the Girsanov exponential M,=exp

~(x~)dv~-Sj

°

~tx~)

ds

(8)

is an o~-martingale, then the process u~ = v~ - f~ A "~(x,) ds is a Brownian motion under a new probability law Q, equivalent to P on . ~ , with R a d o n - N i k o d y m derivative

dpdQ ,~, = Mt"

(9)

General conditions ensuring that M t is a martingale can be found in [17]. This is the case, for instance, when c~ is bounded.

Let us recall that two probability laws P and Q are said to be equivalent on ~ if they are mutually absolutely continuous on each o~,. T o fix notation, the expectation of a r a n d o m variable Z under probability law P is denoted ~ e ( Z ) . Therefore, by Girsanov linearization, we mean that the stochastic system (5) is transformed into a linear controllable one not only by d i f f e o m o r p h i s m ~t = ¢b(xt), but also by a change of the original probability law P. In this way, we relate the law of the process x, to the one of a G a u s s i a n process, solution of Eq. (7). 3.1. Local Girsanov linearization

We define the counterpart of state-feedback linearization as follows: Definition 3.1. Let f, g be two smooth vector fields on R ~, x0 ~ R n, and suppose that the stochastic differential equation (5) has a solution x t starting from the deterministic initial condition x 0, which is defined for all time t _> 0. The stochastic system (5) is said to be locally G i r s a n o v linearizable if there exist an open neighbourhood V of x o, with associated exit time T (~ is an o~--stopping time), a d i f f e o m o r p h i s m • defined on V and

M. Cohen de Lara, J. I~vine / Deterministic feedback linearizat~on

85

a smooth function a on V, called feedback drift, such that the process u t = v t - f d ^ ' a ( x ~ ) d s is a Brownian motion under a new probability law Q, equivalent to P on ~-, and that the process ~, A• = O(x, ^ ,) satisfies Q-a.s. for all t > 0, ~,A,-

O(Xo) = fo'^~ A~ s ds + fot^~ B o du s

(10)

with (A, B) a Controllable pair. The stochastic counterpart of Theorem 2.1 is: Theorem 3.2. The stochastic system (5) is locally Girsanov linearizable if and only if: (i) g( xo), a d f g ( xo), . . . , adT-lg(xo) are linearly independent vectors, (ii) the 1-form oo defined by w(ad~-ag) = ( - , i,,, i = 1 . . . . . n, is closed in a neighbourhood of xo. That is, the stochastic system (5) is locally Girsanov linearizable if and only if the associated deterministic counterpart (1) is locally linearizable. Proof. Suppose that the stochastic system (5) is locally Girsanov linearizable. Introducing the notation T~O for the tangent mapping to diffeomorphism O at point x, the It6-Stratonovitch formula provides P-a.s. for all t > 0 the following decomposition for ~t A~:

s ds

,o

=~,A~- O(x0) =

o du s fot A "r

TxO(f)(xs)dS+Jo

= f o t A ~ ' c ~ , ( f ) ( ~ S ) ds-~- f o t A ' r O . ( g ) ( ~ s )

f t A "r

TxO(g)(Xs)O

° do s

t A "r

=fo

dv~

r t A "r

(°*(f)Ui')+a°O-l(~s)O*(g)(~s))dS+Jo

*(g)(~')°dus

= f o t A ' O . ( f + ag)(lis) ds + f o t A ' O . ( g ) ( ~ , ) o dus and since P and Q are equivalent on ~-, we identify Q-a.s. Vt 0, (Xo) = fo'

as + [tArB "0

o du~

where ~- is the exit time from V and where we have written u t = v , - f d ^ ~ ( x , ) d s . Now since a is bounded, it follows from Girsanov's theorem that M,, given by Eq. (8), is a martingale and that formula

M. Cohende Lara, J. L~vine/ Deterministicfeedback linearization

86

(9) defines a probability law Q equivalent to P on ~ , under which the process u, is a Brownian motion. [] In fact x, usually escapes from V with strictly positive probability and it would be nice to have results when V-- R ' . This is investigated in the next section.

3.2. Global Girsanov linearization In accordance with the deterministic situation, we should say that system (1) is globally Girsanov linearizable if it is locally Girsanov linearizable, with V and • of Definition 3.1 such that V = R" and global diffeomorphism of R ' . Nevertheless, we have to be more specific about the growth of the different vector fields and functions involved, because the Girsanov exponential (8) is not always a martingale. Therefore, we shall assume that the smooth vector fields f , g of (5) satisfy: (H1) f is globally Lipschitz and g is bounded, as well as its first and second partial derivatives. Then, the stochastic differential equation (5), with any deterministic initial condition x 0 ~ R ' , has a solution x t defined for all time t >_ 0, since the It6 drift and g are globally Lipschitz [17, Th. 4.6, p. 128].

Definition 3.2. Let assumption (H1) be satisfied. The stochastic system (5) is said to be globally Girsanov linearizable if there exist a global diffeomorphism q) of R" and a globally Lipschitz feedback drift a on R" such that the process u, = v, - fga(x,) ds is a Brownian motion under a new probability law Q, equivalent to P on o~, and that the process 4, = q~(x,) satisfies Q-a.s. for all t >_ 0

~,-~(Xo)=

fo'Atis d s + £tB o du,

(11)

with (A, B) a controllable pair. Theorem 3.3. Let assumption (H1) be satisfied. The stochastic system (5) is globally Girsanov hnearizable if

and only if: (i) g(x), adfg(x) .... , a d T - l g ( x ) are linearly independent vectors for all x ~ R ' , (ii) the 1-form ~o defined by ¢0(ad)-lg) = ( _ 1 ) ' - 18i,,, i = 1,.. ., n, is closed in R ", (iii) LT-~w( f ) is globally Lipschitz and the vector fields, g, ad fg . . . . . adT-~g are complete, with f = f ( LT-lw(f))g. That is, the stochastic system (5) is globally Girsanov linearizable if and only if the associated deterministic counterpart (1) is globally linearizable, with globally Lipschitz feedback control. Proof. The necessary part is a straightforward application of Theorems 2.2 and 3.2 when we notice that the feedback drift and - L T - l o ~ ( f ) are equal up to a linear function (see Remark 2.3). The sufficient part is also a straightforward application of Theorems 2.2 and 3.2 if we show that the Girsanov exponential (8), where a = - L T - l ~ o ( f ) and r = + re, is a martingale. This happens to be the case when [17, Ex. 3, p. 220]: V T > 0, q6 > 0, sup,~to, rl E e ( e x p ( S a ( x , ) 2) < + ao. Now this condition is satisfied because a is assumed globally Lipschitz and because, by the growth assumptions on f and g (in particular g bounded), it may be proved that V T > 0, 38 > 0, supt~[0,r] R:e(exp(8 [I x, I[ 2 ) < + m, by an easy extension of [17, Th. 4.7, p. 137]. []

4. Application to finite-dimensional filtering We now consider a stochastic system with signal x t and one-dimensional observation y,, given by

dx,=f(x,) dt+g(x,)o dy, = h ( x , ) dt + dw,.

dr,,

(12a) (12b)

M. Cohen de Lara, J. I~vine / Deterministic feedback linearization

87

We assume that x t ~ R " , y t ~ R ; x o is an n-dimensional deterministic vector, y 0 = 0 ; (v t, wt) is a two-dimensional Brownian motion. To fix notation, we denote £2 = C°(N+, R 2) and P the associated Wiener measure. Its filtration ~ = ( ~ ) t _>0 is that generated by the Brownian motion. We further assume that f , g, h are smooth, that assumption (H1) is satisfied as well as: (H2)

Vt>0,

lFp( fdh(xs) ' 2 ds) < + oc.

So the processes x t and Yt are defined for all t >_ 0 and the formula

d,

as)

1= e x p ( - foth(xs) d ys +

dP0

defines a new probability law P0 (note that we are not actually focusing attention on this particular Girsanov transformation). Under Po, xt and Yt are independent processes and the process x t has the same marginal law as under P. We recall that the filtering problem consists of computing the conditional law of the state x t knowing all the past observations up to time t, denoted by Yt = ( Ys 10 _< s < t }, for every t. This is usually performed under the probability law Po via the Kallianpur-Striebel-Kushner formula: ~:p(~(x,) [ r,,)

(Or' (Pt, ~) 1) '

where 0, (Pt, (½L2B + L(A+BK',I~+'-- ½(C~)2)qb)

~Lo,¢x )+L~,~/+(c.x)~)

_

½(h

o

q)_~)2

,)),>,

+(.o

By the controllability and observability of a prime system [71, ~, has a strictly positive density for all t > 0, so that: 1 2 :LB+LtA+BK,)~+

_ ½ ( C ~ ) 2 = 5L~.(g) l 2

+

L4a.(f+(Lgx)g ) -

(½h 2-

=h

(20) (21)

Equation (21) can immediately be rewritten as V x ~ R ~,

h(x)=C~(x)=e~(x)

and if q~.(g) =

E~=~G,(()~/a(~, then

i~l

j:l

(22)

t, : 1

.

so that Eq. (20) provides G i ( ~ ) G j ( ( ) = O if ira n or j=~ n and G 2 ( ( ) = 1, Therefore G j ( ~ ) = 0 for j = 1 . . . . . n - 1 and by continuity of Gn, there exists e = +1 such that Gn(() = e. Thus q ) , ( g ) ( ( ) = eB and ~ , ( f + ( L g x ) g ) ( ( ) = (A + B K ' ) ( + l, which can be rewritten as: V~R",

~.(g)(()=eB

and

q~.(f+(L~x-K'cb-/)g)(~)=A(.

(23)

M. Cohen de Lara, J. Lrvine / Deterministic feedback linearization By Remark 2.2, canonical form = (h, L / h , . . . , On the other

89

this means that the system ~, = f ( x t ) + e g ( x t ) v is globally linearizable into Brunovsky by means of the diffeomorphism ~. But t/i1= h, by (22), and therefore (4) provides L T - l h ) ' , so that condition (i) is proven. hand, it follows from feedback uniqueness (see R e m a r k 2.3 and (4)), that

L~gX - K ' ~ - l = - L T h ,

(24)

and Eqs. (20) and (22) provide H o ~ - 1 ___0, that is, Lf X+I

z

+X=0.

+

(25)

Now, if we denote fll = - L T h + K'q~ + l, Eqs. (25) and (24) may be rewritten as L,gX = fla and L f X = - ½(L~gfl~ + f12) _ h. Let us prove by induction on i t h a t Lad,/-~eg X = fli" This is true for i = 1. Assuming that the relation holds true up to i - 1, we have, by definition of Lie bracket,

which proves the assertion. It follows that ~r = d X and therefore condition (ii) is satisfied. Sufficiency: Denoting ~ = (h, Lyh . . . . . LT-~h) ' and a = - L T h , we have ~ . ( e g ) ( ~ ) = B and q b . ( f + aeg)(~) = A~, where A, B are given by (3). Since the 1-form ~r is closed in R", there exists a function X satisfying X(Xo) = 0 and d x = ~r. Moreover, by condition (ii), X satisfies L~g X = fl~ and there exists a real number X such that L f X = - ½(Legfll + f12) _ ~k, that is, ct = L~g X - K ' ~ - l

and

H = L f X + ~1 / ~2x +

:1( L ~ x ) 2 + x = 0 .

Therefore, ~ . ( f + ( L ~ g x ) e g ) ( ~ ) = ( A +BK')~+I and, since h ( x ) = ~ ( x ) = Cqb(x), it follows from Equation (17) that I,, = ¢.(eX+Xt&) is solution of the D M Z equation associated with system (19). By uniqueness, v, is the unnormalized conditional law associated with system (19). [] Corollary 4.1. I f the assumptions (i) and (ii) of Theorem 4.1 hold, then Pt is the output of a finite dimensional filter Pt = ek' JV'(mt, Pt), where ,A/'(m, P ) is the Gaussian measure on R " with mean m and variance matrix P and kt, m t, Pt are given by dP, = ( ( A + B K ' ) P , + P , ( A + B K ' ) ' + B B ' - P t C C ' P t ) dt, dmt= ( A + BK'dkt=-½((Cmt)Z

P t C C ' ) m t dt + l dt + PtC' d y ,, +CPtCt)dt+Cmtdyl,

Po = O,

r n o = ~ o,

ko=O.

Remark 4.1. If the assumptions (i) and (ii) of Theorem 4.1 hold with K--- 0 and l = 0, then ~ . ( P t ) is the unnormalized conditional law associated with system (12), where f , g and h are replaced by ~ . ( f ) , ~ . ( g ) and h o ~ - 1 . But h o ~ - t ( ~ ) = C~, ~ . ( g ) ( ~ ) = B and ~ . ( f ) ( ~ ) = A ~ + ct(~)B, where ct = --LB( X o ~-1). Therefore, (qb.(pt), q~) = Epo(AtqJ(~t) lYe), where d ~ t = A ~ t dt + ct(~,)B dt + B o dvt, d y t = C~ t dt + d w t. On the other hand, ¢.(eX+Xtpt ) is the unnormalized conditional law associated with the following system:

(j0'o ,s ds)

d ~ , = A~ t d t + B o d v , = A~ t dt + a ( ~ t ) B dt + B o d vt d y , = C~ t dt + d w , .

Therefore, the gauge transformation e x+M, acting on the D M Z equation, has an effect on the original system: the Brownian motion v, is transformed into the process v~ + f ~ L e ( ~ o q~-~)(~s) ds. This is a

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M. Cohen de Lara, J. l~vine / Deterrninistic feedback linearization

state-feedback on Brownian m o t i o n and we shall now conclude by obtaining an interpretation of gauge transformation as Girsanov absolutely continuous change of probability law, specifying a remark in [23]. Proposition 4.1. Considering system (12), let ?t be a real number and X be a smooth function on ~ ,

such that Lg X is globally Lipschitz. The following assertions are equivalent: (i) if Ot is the unnormalized conditional law associated with system (12), then v, = eX+X~O, is the unnormalized conditional law associated with system (12) having new drift f -- f + ( L g x ) g , (ii) e x~~°) = 1 and P-a.s. for all t > 0, H ( x , ) = 0, where H is given by Eq. (18), (iii) the formula dQ ,~, dP

(26)

= exp(x(x,) + at)

defines a new probability law Q on (I2, o~-) for which the semi-martingale u, defined by u,=v,-

fotLgX(X~)

(27)

ds

is a Q-Brownian motion. Proof. (i) ~ (ii): Since vt = eX+X/pt is the unnormalized conditional law associated with system (12) having new drift f , it follows from Eq. (17), with q~ = Id, that necessarily (Pt, H g o ) = 0, for all go in Cff(Rn). Therefore, 0 = (v,, Hgo) = (p,, eX+X'Hgo) = ~_p,,(A, e x p ( x ( x t ) + M ) H ( x , ) g o ( x , ) [Y,) and thus A t e x p ( x ( x , ) + M ) H ( x 0 = 0, P0-a.s. for all t >__0. N o w since A, e x p ( x ( x , ) + a t ) > o, we get H ( x t ) = 0, P0-a.s. for all t >__0. But P0 and P are equivalent on ~" and x, is a continuous process, so that we have P-a.s. for all t >__O, H ( x , ) = 0. Of course, vo = 6xo implies that e x(''~ = 1. (ii) ~ (iii): Let us give another expression for e x p ( x ( x , ) + a t ) :

d ( × ( x , ) + at) = G x ( x , )

at + L

x(x,) o dr, + a dt

= a dt + L / x ( x , ) dt + ½( L2gX)(Xt) dt + L g x ( x , ) dr, 1

= - :(Lgx)

2

( x , ) dt + L g x ( x , ) dr,

(by (6))

because H ( x , ) = O.

Therefore, e x p ( x ( x , ) + a t ) = exp

L g x ( x ~ ) dvs -

is a Girsanov exponential, which is an ~ - m a r t i n g a l e because Lg X is globally Lipschitz (see T h e o r e m 3.3). We then know from Girsanov's T h e o r e m that formula (26) defines a new probability Q on (£2, o~'), for which u t defined by (27) is a Q-Brownian motion. (iii) ~ (i): We have: (v,, go) = (Or, eX+Xtgo) = ~-eo(At e x p ( x ( x , ) + at)Co(x,) l Y,). It can easily be checked that a new probability law Q0 can be defined in two equivalent ways: dQ0 ,~, dQ =A~I=

dPol dOo ~, "d~pp~, d P [~, *~ d P 0 = , =exp(x(xt)+at).

Now, by (27), we have d x , = f ( x , ) d t + g ( x t ) o d u , where f = f + ( L g x ) g and u, is a Q-Brownian motion. Therefore, it follows from the properties of P0 with respect to P, that under Q0, x, and Yt are independent processes and the process x t has the same marginal law as under Q. Since dQo/dPol.~- is a function of x, and thus independent of Yt under Q0, it is easily seen that the K a l l i a n p u r - S t r i e b e l - K u s h n e r

M. Cohen de Lara, J. LOvine / Deterministic feedback linearization

91

formula gives {v,, ep) = ffZQo(A;eo(x,)lY,). That is, vt is the unnormalized conditional law associated with system (12) having new drift f = f + ( L g X ) g . [] Thus, gauge transformations concern only the feedback change of Brownian m o t i o n in the state equation of system (12) and not in the observation equation as in (13).

5. Concluding remarks In one dimension (n = 1) with g ( x ) = 1 and h ( x ) = x, it is easily seen that there exist (k, l) ~ R 2 such that conditions (i) and (ii) of T h e o r e m 4.1 are satisfied if and only if f ' + f 2 is a quadratic function with positive leading term. Then, the diffeomorphism • is the identity on R and the gauge transformation is given by X ( X ) = f x ° f ( u ) du, as was already noticed in [2]. In fact, T h e o r e m 4.1 does not enlarge the class of systems having finite-dimensional filters but it gives a new geometric characterization of some of these systems. More precisely, it draws an exact picture of the complicated constraints existing between f , g and h to have a finite-dimensional filter related by gauge transformation and diffeomorphism to that of a simple affine system. O n the other hand, T h e o r e m 4.1 does not include the general case of [2] where it is proved that there exists a finite-dimensional filter if f ' + f 2 -t- x2 is a quadratic function with positive leading term. This results from the fact that vt = eXot satisfies the bilinear stochastic partial differential equation

(dv,,qS)=

1 d2 v,, 2 d x 2 q ~ - ½ ( f '

) +f2+x2)~

dt+(vt,

xqS) o d y t,

which is no longer the D M Z equation associated with a filtering problem, but which admits a finite-dimensional realization [16]. Nevertheless, the geometric m e t h o d s developed here can be extended to such a case by considering gauge transformations relating D M Z equations to more general stochastic partial differential equations.

Acknowledgements The authors are indebted to Prof. D.L. Elliott for introducing G i r s a n o v linearization to them and to Prof. R. Marino for very helpful conversations.

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