Exact finite-dimensional filters via systems realization for a class of discrete-time nonlinear systems *. J. LEVINE and G. PIGNIE. Centre d'A ulonwlique et d'In/ ...
Systems % Control North-Holland
Letters
5 (1985) 403-412
May 1985
Exact finite-dimensional filters via systems realization for a class of discrete-time nonlinear systems * J. LEVINE
and G. PIGNIE
Centre d’A ulonwlique FRlllLT
et d’In/ormotique
de I’Ecole
Received 2 July 1984 Revised 11 October 1984 and 14 January
Nationale
Supkieure
da
Mines
de Paris,
3Ss rue Sf - Honor&,
77305
Fontoinebleau,
1985
We obtain necessary and sufficient conditions for the existence of a finite-dimensional filter for the discrete-time nonlinear system by the absence of noise in the dynamics and by y& =h(x,)+q(x,)o,, k = 0, l,.... This system is distinguished (2’) x*+1 =/(x/o, the correlation between the state and the intensity of noise in the observations. The necessary and sufficient condition provides an explicit formula for the minimal filter and various system-theoretic properties of (2) and of the minimal filter. Keywords;
Finite-dimensional
filtering,
Systems
realization,
discrete-time
nonlinear
systems.
Introduction Until now, most of the works dealing with finite-dimensional filters (FDF) have been in the setting of continuous time, continuous-state systems, more precisely for partially observed diffusions (see [1,2,3,5] and others). In the opposite situation, the case of discrete-time, discrete-state systems gives rise to naturally finite-dimensional filters when the number of states is finite. In the intermediate case of discrete-time, continuous-state systems, very general work has been done in [14] for one-dimensional filters, leading to abstract conditions that remain very difficult to exploit; another type of result can be found in [ll] where sufficient conditions of existence of an FDF in terms of finite Volterra series are derived for very particular systems. We consider here systems of the form
(2)
Xk.+l=f(xkh i yk=h(Xk)+~(Xk)Uk, k=0319*..3
characterized by the absence of noise in the dynamics and by a correlation between state and noise in the observations. The FDF problem for (X) is studied with Systems Realization methods [6,13,15], in the spirit of [2,5]. When the noises are Gaussian, a necessary and sufficient condition for the existence of an FDF is obtained in Section 3: an FDF exists if the functional space &#=Span
((qof”)(~of”)3,!,
5
* This
work has been supported
0167-6911/85/%3.30
((II”r”)(llo~“)‘),~:(hjof”)I1ci,
j9P,Vkao
j=l
i
by the French
6 1985, Elsevier
Army
Science Publishers
1
under contract
DRET.83.34.177.
B.V. (North-Holland)
403
Volume 5, Number 6
SYSTEMS & CONTROL LETTERS
May 1985
(where fk is the k-th iterate of f and where ’ denotes transposition) is finite dimensional. This condition provides an explicit formula for the minimal realization of the filter (Section 4). In Section 5, the links between existence of an FDF and the immersion (see [4,12]) of (2) in a system having a specific algebraic form (of the filtering class) is studied. It results that (2) can be immersed in a linear system if it admits an FDF and n 0 fk = 17,Vk. Section 6 is devoted to simple examples. 1. Model and assumptions In the system (1) above, we assume that X~ evolves in a pure paracompact connected C’ manifold X of dimension n, the index k playing the role of time. The initial state x,, is a random vector in X having a probability density P,, with respect to pa, a given Lebesgue measure on X, and P,, E C”( X, R +), with supp PO= x. The observation vector yk and the noise realization uk at time k are in R J’, Vk a 1. (Hl) f is a Cl-diffeomorphism from X to X. (H2) h E C”( X; IwP), 17E C’( x; R PxP) and (det v)-‘(O) A {x E X ]det n(x) = 0) is a C’ manifold of dimension at most n - 1. (H3) The random sequence ( uk ) k L c is stationary and independent. Furthermore, ok has a probability density V E C’(R P; W+) with respect to the Lebesgue measure of R P, Vk = 0, 1,. . . . We shall talk about the C” (resp. analytic, or shortly Cw) case when X, P,,, j, h, q and V are C” (resp. P). A system (2) will also be denoted (X) = (X, f, h, 71,[wp, P,), or (2) = ( X, f, h, 9, 08p, xc) when the initial state x0 plays a particular role. We shall denote f” for the k-th iterate of f, namely f”(x) =f 0 f”-‘(x) Vk 2 1, with f’(x) = x, and yk) the sequence of all the observations up to time k, Y, being the empty sequence. Y,=(y1,..., Finally we denote k = X - U ka c(det(q 0 f”))-‘(O). Obviously llo( X - 2’) = 0. To ensure that X is an open dense subset of X, we also assume that: (H4) Elk, > 0 such that for every neighborhood N of U$eo(det(n 0 j”))-‘(O) one can find k, such that Vk >, k,, (det(q 0 fk))-‘(0) c N. Remark. The filtering problem for a system (8) where 9 depends on x has never been studied, up to the authors’ knowledge, in the discrete time case. For the continuous-time case see [8]. The following result is an easy generalization of Bayes’ formula on a manifold: 1. The conditional unnormaIized measure of xk knowing Y, has a density Pk(. 1Y,) E C”( X; R +) with respect to po, given by
Theorem
P,(XlY/,.)=ldet
~(~)~-1~(~-1(~)(Yk-h(~)))J;‘(f’(~))Pk--l(f’(X)~Yk-l)
Po(xlYo)=Po(x)
“a
‘1
(1)
VXEZ
where J, is the Jacobian off relative to X. 2. Filters and realization Definition 1. We call a jilter (resp. local filter) for (X), a realization (resp. local realization), with unspecified dimension, of (1) of the form: ak+l
=
@ktak9
Pk+l(X~Yk+I)=uk+l(ak+l~
404
Yk+l), x
)
VxEZ(resp.VxE
U,=f”(V))
(2)
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for k 3 0, for U an open subset of 2, and Pk(. 1Yk) satisfying (1). In (2), ak (a,. ) is supposed continuous on k (resp on Uk) for each a, and a, is chosen according to u&a,, x) = P,(x) Vx E k (resp. Vx E U). The space A generated by every ak in (2) is called the space of the filter’s parameters. 2. (i) We say that (2) is an FDF (finite-dimensional filter) if one can choose (A, Qk, uk) such that A is a C’ finite-dimensional manifold, Gin-E C’( A X R “; A), a, E C’(A; C”( 2; R,)), Vk 3 0 (resp. C’(A; C’(U,; R,)) in the local case). If dim A = r, the corresponding filter will be called an r-dimensional filter. (ii) We say that (2) is stationary if Dp,= @ Vk > 0 and if ~,(a, x) = o,(x)p(a, x) Vk 2 0, Vx E 2 (resp. Vx E U,). (iii) We say that (2) is analytic if A is C”, if a, E C”(A x lRp; A), and if uk E C”(A; C”(& R,)) (resp. C”( A; C”( Uk,; Iw+))). (iv) We say that (2) is inuertible if Gk( ., y) is a C’-diffeomorphism from A to A Vk 2 0 for each y (and C”-diffeomorphism in the analytic case). Definition
Proposition
1. To each system (X) there corresponds a stationary filter of the form (2) (but generally infinite
dimensional ). In Definition 2, the FDF is the analogue for discrete-time systems of the notion of ‘universal finite-dimensional filter’ (see [3]). In the linear-Gaussian case, one obtains the unnormalized Kalman filter (see [9]). Finally, the filter parameters uk are sufficient statistics of Yk, and our definition is slightly more restrictive than the one of [14] because of the smoothness of A, ap,, ulr in (i).
Remark.
3. Existence
of an FDF in the case of Gaussian
noises
Throughout the sequel, we shall assume that the noises are Gaussian: (H5) V(o) = exp - +u’o. In (l), one must thus replace V(.rl-‘(x)(y, -h(x))) by
exp- i[(vk - h(x))‘(ll(x)?‘(x))-‘(yk
-h(x))]
=exp-~[y~(ll(x)l)'(x))-'y,-2y~(q(x)~'(r))-'h(x)+h'(x)(ll(x)q'(x))-'h(x)]
= exp - $
t [ HcfJ3(x)y:v,B - 2HJx)y:] i ol,p=1
+fh(x)
(3)
where we have denoted
kp(4=
M+/‘b)>n.‘,
( theeement ’
(cy, p) of thematrix (7(x)$(x))-‘),
Ha(x) = 5 f-L,pb)hpb)~ p=1
va,p=1,...,
p.
(4)
Ho(x) = i li,(x)h,(x), a-1 3. The canonical space, denoted E-U,is the functional space generated by the linear combinatioris with constant coefficients of the family H,, B 0 f k, H, 0 f k, namely:
Definition
W =Span{H,,pof
k, H,ofkJa,j3=l
,..., p,k&O}.
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A basis of W is called a canonical basis. We have the following fundamental result: Theorem
2. The three following
properties
are equivalent:
(i) (2) has an FDF (resp. local FDF). (ii) W is finite dimensional (resp. 3U, open subset of 2, such that W 1u is finite dimensional). (iii) 3r E N, Se,,. , . , B,EW,~REGL(~,R),~)\‘,.~,CL,,~EIW,VLU,P=~ ,..., p,Vi=l,..., rwithX,,, = x’a. L-0such that Vx E X (resp. Vx E U),
Sketch of proof. (i) - (ii). If (X) has an FDF given by (2) of dimension r K + co, then, for every k and Y,:
Pk(XIYJ=ak(@k-,(
..-(@lJ(%
Y,), YZ)3...>Yk)Y x)=%h
x).
(6) Since ok and 3, j < k, are assumed differentiable with respect to y, Pk(. 1Y,) is differentiable with respect to CY~,...,Y~) and z(xlyk)=
i
J
a-1
~;,~.~(a,-,,
yj ,..., ~~)$$(a~,
x),
Vi=l,...,p,V~=ll,...,k,
(7)
with ATjk’ I ,
ES$(a,-,,
2 i ,,....
yk).
. . F(aj,
yj+,)T(aj-,,
iA-,-
Yj>-
J
On the other hand, using (1) and (3), it is easily seen that p~(xlY~)=~~(x)exp-~~~~~~~[H.,~(f’-*(x))~~~~-1H.(f7-*(x))y~]
(8)
I -
where a,(x) is independent of Y, (see (13) below)). Thus, differentiating (!), we obtain: z(x,q)=
- i [
J
H,~i(fj-k(X))y;+Hi(fj-~(x))
P,(xlY,).
a-l
(9)
I
Now, let us choose p + 1 observation paths Y,, Ir,. . . , YP+,. k in IwPAsuch that the (p + 1, p + 1) matrices zj=
1 ;
-y;.j
...
-Yf.
j
1 -Yp+l.1 j a** -Y,41t j 1 i : are invertible Vj = 1 ,...,k.Sincerl(x).IWP=[WPVxE~,by(H2),wealsohave{h(x)}+Il(x).RP=IWP VXE~-T~WVXE*,
‘{~nz.j)t~s-~,.._,
y,,,.j=h(f’-k(x))+q(fj-k(x))u,,,,j,
406
p+l:j-I,...,
k such that
Vj=l,...,
k,Vm=l,...,
p+l.
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Thus the choice of such Y,, k,. . . , Y,,,. k is always possible. Then, putting together (7) and (9) for these p + 1 observation paths, 1
(l
-Y!,.j
...
-YeP;.j)
Hi
of'-"(x)
Hl.i"fJ-k(x)
l,..., p, Vj = 1,. __, k, Vm = 1,. . . , p + 1, Vx E 2. Remark that u~(u~~,~,X) + 0 Vx E 2 since supp Pk(. I Y,) 3 2. Thus, noting
vi=
we have on ,?
Pi.1 H,.i
dp
= (z,:’
.Xijk)
i Hp.i
“’
PP.: I
Vi=1 ,..., p,Vj=l,...,
; P:..r
...
k,
(11)
P,p.+Y’ 4
of-
where Xiip is the matrix of coefficients xy.j, k(u ,,,, k, Y,,,.k), (Y= l,..., proves that SupdimSpan{H,,pofj-k,
H-of’-“(a,/3=1,...,
p, j=l,...,
r, m = l,...,
p + 1. However, (11)
k}
k. Y,
Q SupdimSpan{p’;.‘..Icu=l,..., r,m=l,..., p+l}a. P-l.... p is positive definite Vz E Z; N E C’( Z; IwPxp), with I??an open dense subset of Z, and satisfies N(z)?+“(z) = (L(z))-’ Vz E Z; finally, there exists a matrix ME Rpxr such that +(z)=@(z))-‘Mz, 408
x(z)=N(r)
VZE~.
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Otherwise stated, (Z’) belongs to the filtering class if it is described by zk+, = RI,,
t-w
i ~k=(L(zk))-‘MZk+N(*k)uk,
with N(z)N’(z)=
(L(z))-‘,
Qz E 2, and L(z) as in (ii).
Theorem 4. (2) admits an FDF iff there exists (2’) of the filtering class such that (2) is immersed Furthermore, (2) can be immersed in a linear system iff (2) admits an FDF and q 0 fk = q, Qk. Sketch of proof.
in (IT).
If B is an immersion from (X) to (.S’), we have
h(fk(x))+~(fk(x))~~=(L(Rk8(x))-‘MRk8(x)+N(R~’B(x))vk
Qk,Qx,Qu,.
(*)
Both sides being affine functions with respect to uk, their coefficients are equal and it is not difficult to deduce, using the properties of systems of the filtering class, that H,,p(fk(x))=Xb.BR~(x),
%(fk(x))
= UWx),
which proves that dim W $ dim Span{ 0,, . _. ,0,} < r. Conversely, if (8) has an FDF, using (iii) and setting zk = 0(x,)= e( fk(x)) = RkB(xO), we have ‘k+l = Rz,, and it is easily seen that (*) holds true with L, M, N as in Definition 5. Finally, one can easily check that (8’) is linear iff n( f k(x)) = q(x) Vk, Qx, which achieves the proof. Remark. The concept of immersion has been firstly introduced by Fliess and Kupka [4] for continuous-time
nonlinear systems, and by Monaco and Normand-Cyrot [12] for discrete-time nonlinear systems. The definitions differ slightly from these references to take account of the fact that in (X), the controls are the noises vkr appearing only in the observation. Theorem 4 shows that the FDF property is equivalent to the existence of a nonlinear change of variable for which, at least, the transformed dynamical equation is linear. Moreover the wholetransformed system is linear iff n is right-invariant by f.
6. Examples
Three academic simple examples are presented. For a really applied problem the reader is referred to
WI. (6.1) X =]O, l[, f(x) system is thus xk+l
=- l-x, 1+x,’
= (1 - x)/(1 + x), h = 0, n an arbitrary scalar function from IO, l[ to IO, m[. The
(15)
Yk=?(xk)Uk.
Applying Theorem 2, we easily find that a canonical basis for (15) is two-dimensional with 8,(x) = (q(~))-~, &(x) = (~((1 - x)/(1 + x)))-r, and a minimal realization of the filter is
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with 6, given by
62k(X) =P,(,,(sbh( s))-“? 4*+,(x)=2Po(~)(l+~)-2(rl(X))-~-l(’l(~))-~. Finally, according to Theorem 4, the immersion obtained by the change of variables z’ = 0,(x), zz = LIZ(x), puts the system (15) in its ‘filtering’ form
Z
(17)
y,= (zky2v
k,
and, by Theorem 4, (15) or (17) cannot be obtained from a linear system by immersion. (6.2)
Linear
Gaussian filters
xk+l = Fx,, yk=HXk+JVk.
(18)
Applying the preceding results, one obtains the minimal realization in the form of the ‘information filter’ (see [7]). These calculations are left to the reader. For more details, see [9]. (6.3)
A polynomial
x k+l =Fxk, yk = hx[ +
Sensor problem (19
qX,QV,,
X=10, w[, F > 0, h, q # 0, p, q E N - (0}, yk scalar. An immediate application of Theorem 2 shows that for p f 24, a two-dimensional filter exists (if p = 2q the minimal filter reduces to a scalar one), and the minimal realization is:
(20) Pk(xIYk)=5,(x)
exp(aix-24+aixP-2q),
with
Also, (19) can be set in its filtering form by the immersion z’ = xmzq, z2 = xP-29, zk+l=
F-2q
0
0
Fp-29
1
'k,
(21)
,$=h(t:)-'Z;+(Z;)-"2"k,
and no linear system equivalent (in the input-output 410
sense) to (19) or (21) exists.
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remarks
In this paper, we solve in great details the FDF problem for a class of discrete-time nonlinear systems without dynamical noises and with Gaussian observation noises whose intensity is correlated to the state. Concerning the necessary and sufficient condition of Theorem 2, it is worth noting that it is a global condition in the sense that it is not obtained as the rank of some Jacobian at a given point, as in [6]. Another interesting point is the fact that the minimal realization of the filter is analytic even if the functions f, h, rl are only C’. But this is a consequence of the fact that the Gaussian density is analytic with respect to y. All these properties can be easily generalized for nonstationary systems (SE), and for observation functions h, and 7k depending on the past observations. In this case, the space W becomes
where Q ~f:~/,=-~(f~-,( ... (LO(x))) .*. ). Remark that the finite dimensionality of R-Ucan easily be tested by a computer program in a language like Macsyma, at least when the functions in W are polynomials, rational functions, or trigonometric functions. Finally, concerning a practical implementation, FDFs of low dimension can be computed in real time by microcomputer, as shown in the example of [ll], by the following procedure: since we want to compute a finite number of integrals of the form E(rp,(x,) ] Y,), . ., E(qN(xp) ] Y,) Vk 2 1, and since Pk(* ]Y,) depends on the parameters a:, . . . , a6 of the filter, we can precompute the above expectations E(cp,(xk) ] Y,) for ‘every’ possible value of uk, and memorize them. In a second step, the evolution of the parameters uk can be computed in real time by a microcomputer (the recursion (12) is very simple for low dimensions) and it suffices to search in the memorized file the current values of the expectations E((p, ] Y,) when the next parameters are available. To conclude, let us mention that the absence of noise in the dynamics might be quite restrictive in some applications, but the extension of our results to this case constitutes currently a widely open problem.
References [l] V.E. Ben&, Exact finite dimensional filters for certain diffusions with nonlinear drifts. Sroehosfics 5 (1981) 65-92. [2] R.W. Brockett, Remarks on finite dimensional nonlinear estimation, in: C. Lobry, Ed. Analyse des Sysc~mes, AstCrisque 75-76 (1980) 47-55. [3] M. Chaleyat-Maurel and D. Michel, Un theorbme de non existence de filtre de dimension finie, CR. Acad Sci. 2% (1983) 933-936. [4] M. Fliess and 1. Kupka, ,A finiteness criterion for nonlinear input-output differential systems, SIAM J. Control Optic. 21(5) (1983) 721-728. [5] M. Hazewinkel, S.I. Marcus and H.J. Sussmann, Nonexistence of exact finite-dimensional filters for conditional statistics of the cubic sensor problem, Sysrems Conrrol Lerr. 3 (1983) 331-340. [6] B. Jakubczyk, Invertible realizations of nonlinear discrete-time systems, Proc. Conf. on Information Sciences and Systems, Princeton (1980). [7] P.G. Kaminsky, A.E. Bryson Jr. and S.F. Schmidt, Discrete square-root filtering: a survey of current techniques, IEEE Trans. Automat. Control 16 (1971) 721-735. [8] H. Kunita, Nonlinear filtering for the system with general noise, in: M. Kohlmann, W. Vogel, Eds., Srochosfic Confrol Themy and Stochuslic Di/fereenria/ Sysfems, Lecture Notes Control and Information Sciences No. 16 (Springer, Berlin, 1979) 496-509. [9] J. Levine and G. Pignie, Exact finite dimensional filters for a class of nonlinear discrete-time systems, Submitted. [lo] J. Levine and G. Pig& The finite dimensional filtering problem for a class of nonlinear discrete-time systems, Proc. o/Pth IFAC World Congress, Budapest (1984). 411
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S.I. Marcus, Optimal nonlinear estimation for a class of discrete-time stochastic systems, IEEE Trans. Auromor. Conrrol 24 (1979) 297-302. [12] S. Monaco and D. Normand-Cyrot, On the immersion of a discrete-time polynomial analytic system into a polynomal affine one, sysfems Confrol L&r. 3 (1983) 83-90. [13] D. Normand-Cyrot,, Th&se, UniversitC Paris Sud (1983). 1141 G. Sawitzki, Finite dimensional filter systems in discrete-time, Skxhnsrics 5 (1981) 107-114. [lS] E. Sonlag, Polynomial Response Maps, Lecture Notes Control and Information Sciences No. 13 (Springer. Berlin. 1979). [II]
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