In this, the final chapter of the thesis, we synthesise various ideas developed in the previous chapters in order to investigate several points raised by the following example
The underlying deterministic structure of this system is that of a
so it has associated with it the algebraic properties of such systems as described in Chapters I and II. Moreover, the input vector fields
where h is the output function h(x) ■ x^, so appearing to comply
with the necessary condition for finite dimensionality of the estimation
algebra, except, of course, <g1»82}L.A. is n0t abelian- Ic might there_ fore be expected that the filtering properties of this system should be 'nice'. This indeed turns out to be the case, but not in the positive sense to be desired.
The problem is that, as shown in Hazewinkel-Marcus Cl] (where the example was first studied from this point of view), the estimation algebra of (4.0.1) is W2> the Weyl algebra on 2-generators, where in
dx
(4.0.1)
* X j d t + X j d w 2dy « x 2dt + dv
graded polynomial form on 1R2 with gradation IR ® IR n ] n 2,nj » n2
8 8.
general we shall assume that is the faithful representation of the abstract Weyl algebra on n-generators given by
Thus, for our purposes, is the Lie algebra of differential operators
□
As an immediate corollary of Brocketts Principle we therefore deduce that there are no nontrivial f.d.c. statistics of any process whose
applies to (4.0.1) so it is in the sense of non-existence that, despite the rich algebraic structure already established, the process has 'nice' filtering properties. This result is therefore quite surprising, not only for the reasons already described but also because (4.0.1) is one of the simplest of nonlinear systems. It is natural to ask then, how general this behaviour is and, in the sequel, by limiting attention to
g.p. forms we go some way towards answering this point with the construction of a class of systems having estimation algebras isomorphic to W^.
First, though, in §4.1 the estimation algebra for an arbitrary minimal system Z in g.p.f. on IRn is studied particularly with regard to the general containment condition derived in Thm (3.1.6). Due to the polynomial nature of 1 it is obvious that A(E) <= £1(1) e Wn and using the on IRn with polynomial coefficients. For algebraic estimation the significance of this calculation lies in the following result. THEOREM 4,0.1 (Hazewinkel and Marcus [l ])
(i) As a Lie algebra, W is generated by
; i - l,..,n, j n- 1 }.
(ii) There are no nontrivial homomorphisms from into either
0j oo #
T (TM) or T (TM) for any finite dimensional manifold M.
estimation algebra is isomorphic to W^. In particular, this observation
results of Chapter I it is not difficult to show that if E is also in ninimal g.p.f. then 0(E) = li(E ) = (Recall that E is the system obtained from E » {f,g,h} by the perturbation f-*-f = f-j g, and 0(E),0°(E) are tensor spaces). However, whilst the graded structure of
A
Z can be readily shown to be preserved, minimality of E is not guaranteed in general. Conditions are derived in Thm (4.1.1) under which this will be the case.
As the next step in our construction of the required class of systems, we adapt a strong observability concept, introduced by Gauthier and B°rnard [I] to develop a canonical form for certain single input-single output systems. The representation thus obtained
except, of course, (4.0.1) is seen to closely resemble the structure of (4.0.l)^has two input
channels. However it can be shown through direct and tedious calculations that the system
dXj ” dw
dx2 - X|dt + xj dw dy - x 2dt + dv
still has A ■ W 2> so (4.0.1) remains our 'inspiration'. The full computations required to show this are omitted as they form the basis for the analysis of §4.3 in which we finally obtain our class of systems satisfying ASW . The results obtained are still unsatisfactory since we have to assume that certain generators have already been established as elements of the estimation algebra, and further work is required to weaken these hypotheses. On the positive side, however, our theorem only requires that 3 elements be found compared with the (4n— 2) of Thm (4.0.1(1)).
§4.1 Graded Polynomial Forms, Algebraic Estimation and Wq
At the end of §3.2 we proved a general containment condition
clearly demonstrating the polynomialnature of F. Equation (4.1.1) can therefore be reduced to
In some sense this is not surprising since both generators of
A
will be elements of W and consequently it is obvious that A c W . What we haveof course, we should like to be able to show that £ is minimal and in g.p.f. <“> £ is in g.p.f. and minimal, but whilst the graded structure can be shown to carry over quite readily, minimality does present some problems.
The first point to notice is that since £ is assumed to be in g.p.f. then £ is in g.p.f. with respect to the same gradation of the state space as that of £. (In particular, from the results of 92.2, both £ and £ are realisations of stationary finite Volterra series although these input- output maps may be different). The initial claim that £ is in g.p.f. is quite easy to prove. Indeed, if we denote by g' the vector field determined by it is readily seen that the iC^ component of g* is
However, by definition of the g.p.f., geV^ and using the decomposition of Vj given above, we get
A
Cn°(£) - w
n
n n
actually achieved here is the demonstration that il°(£) - which is clearly non-trivial, and also shortens the (trivially proved) chain
A c !5°(£) <= W . For the remainder of this section we intend to investigate n
how true equation (4.1.2) remains if we only assume £ is minimal and in g.p.f. In other words, we are asking the question how does the ltd correction term, J ^£g, affect the structure of the system £?. Ideally,
x - Lg (g.)(x) ‘ P i _ i g* e ® L (QJ j- 1 * c 9 Q^ ~ 2 A V. 2 92.
and, hence + V 2 c V o _ n .
Thus, £ is a linear analytic system defined on the graded space 9 IR J and satisfying feVQ , 8£V | and heQr , this last fact also following from the assumption that £ is in g.p.f. In other words, Z is in g.p.f.
Let us now turn our attention to the problem of determining when minimality of Z implies minimality of £. In general, this will not be true as the system
2 X lu
defined on the graded space IR IR 9 1RU 9 IK shows. For this example we find that
1 3x_
3 2 3 r<: n 3