Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
1.1 Context
1.1 Context
We can model any physical process by considering it as a system. In our framework, a
set of relevant variables describes the evolution of the state of the system with time. These
are called the state variables. The system interacts with the outside world in three different
ways:
− input, or control, variables are quantities that have an effect on the system behavior
and that can be set externally, they are denoted u(t),
− output variables, or measurements, are quantities that are monitored, generally they
are a subset or a transformation of the state variables, we denote them by y(t),
− perturbations are variables that have an effect on the system behavior and that cannot
be controlled; most of the time they cannot be measured 1 .
The state variables are represented by a multidimensional vector, denoted x(t). The evolution
of x(t) with time is accounted for by an ordinary differential equation 2 :
dx(t)
dt
= f(x(t),u(t),t).
The relation between state and output variables, i.e. the measurement step, is described by
an application:
y(t) =h(x(t),u(t),t).
A system is therefore defined as a set of two equations of the form:
� dx(t)
dt = f(x(t),u(t),t)
y(t) = h(x(t),u(t),t)
The state estimate rendered by the observer can be used for monitoring purposes, by
a control algorithm as schematized in Figure 1.1, or processed off-line (e.g. in prototyping
assessment as in [21], section 3.6).
For example, in the juggling experiment, the state variables are the 3D position and speed
of the balls. The input variables are the impulses the hands apply to the balls.
With eyes wide open, the output variables are the balls position. In the one eyed juggling
experiment the output variables are an imperfect knowledge of the balls position, e.g. because
of the hindered depth perception. In the blind juggling experiment, tactile signals are the
output variable.
In all these three cases the model is the same: the one of a falling apple.
1 From an observer’s point of view, measured perturbations and control are both inputs. The controller’s
point of view is different since it uses the controls to stabilize the system and reject perturbations.
2 We do not consider either partial differential equations or algebraic differential equations.
2