Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.1 Multiple Inputs, Multiple Outputs Case
This chapter contains several additional and complementary considerations, which are
related to adaptive high-gain observers. The first section provides some insight into the
multiple outputs case. A generalization of the observer of Chapter 3 is also presented. In the
second section, we develop an observer for a continuous-discrete system.
For these two cases, we define the requirements necessary to achieve exponential convergence.
5.1 Multiple Inputs, Multiple Outputs Case
From a theoretical point of view, the multiple outputs case is harder to handle than the case
of a single output. Indeed, there is no unique observability form (see [19, 20, 27, 51, 57, 63, 84]
and the references herein). The multiple outputs case is also more complex in practice since
the various normal forms lead to different definitions of the observer. In the section below,
we propose a generalization of the normal form (3.2) together with the definition of the
corresponding observer. The proof of the convergence of this observer is given in Subsection
5.1.3.
Although in a more compact form than in Chapter 3, we keep the proof self contained,
which implies that we will repeat ourselves to some extent. The modifications of the proof,
which are specific to the Multiple Inputs/Multiple Outputs(MIMO) case, are denoted by a
thin vertical line in the left margin. The single output case, which was presented before, is
included in this generalized version.
5.1.1 System Under Consideration
We focus on a blockwise generalization of the multiple input, single output form (3.2) of
Chapter 3. A similar form has been used in the Ph.D. thesis of F. Viel, [113] for a high-gain
extended Kalman filter. Another choice has been made in [57] for an even dimension state
vector.
− The state variable x(t) resides, as before, within a compact subset χ ⊂ R n ,
− The input variable u(t) resides within a subset Uadm ⊂ R nu ,
− The output vector y(t) is within R ny where ny ≤ 1.
The system is of the form:
and the state variable is decomposed as
�
dx
dt = A (u) x + b (x, u)
y = C (u) x.
x(t) = � x1(t), ..., xny(t) �′
,
, (5.1)
where for any i, i ∈ {1, ..., ny}: xi ∈ χi ⊂ Rni ny �
, χi compact. Therefore n = ni, and each
element xi(t) is such that:
xi(t) = � x 1 i (t),x 2 i (t), ..., x ni
i (t)�′ .
93
i=1