Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
A.1 Resolvent of a System
A.1 Resolvent of a System
In this subsection, we recall basic concepts from the theory of linear differential equations.
Details can be found in [106].
A first order system of linear differential equations is given by:
where
1. t ∈ I, an interval of R,
2. x ∈ R n ,
dx
dt
3. A(t) is a t dependent matrix of dimension (n × n),
4. b(t) is a t dependent vector field of dimension n.
= A(t)x(t)+b(t) (A.1)
First, remind, that provided that the applications A(t) and b(t) are continuous then for
all s ∈ I and for all x0 ∈ R n , this equation has a unique solution on I such that x(s) =x0.
We now consider the associated homogenous equation:
dx
dt
= A(t)x(t). (A.2)
Let us denote by x(t, s, x0) the solution of (A.2) at time t with initial condition x(s, s, x0) =x0.
We define the application:
ξ ↦→ x(t, s,ξ ), (A.3)
that associates to any element ξ of R n the solution of (A.2) starting from ξ.
Let c1,c2 be two positive scalars and ξ1, ξ2 two elements of R n . Consider the trajectory
c1x(t, s,ξ 1)+c2x(t, s,ξ 2). For t = s, we have:
c1x(s, s,ξ 1)+c2x(s, s,ξ 2) =c1ξ1 + c2ξ2 = x(s, s, c1ξ1 + c2ξ2).
From the unicity of solutions, we conclude that
c1x(t, s,ξ 1)+c2x(t, s,ξ 2) =x(t, s, c1ξ1 + c2ξ2).
That is to say that for all t and s in I, (A.3) is linear. Therefore there is a t and s dependent
matrix such that:
x(t, s, x0) =φ(t, s)x0,
with φ(s, s) = Id. This matrix is called the resolvent 1 of (A.2). The resolvent has the
following properties:
Theorem 69
1. φ(t, s) is linear w.r.t. the variables s and t,
1 φ(s, s) =Id is also said to be the resolvent of the equation (A.1).
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