Robust H8 Filtering
for Robot Manipulators Control
A. ABDESSAMEUD
University of Boumerdes
Laboratoire d’Automatique Appliquée, Process Control, Algeria
E-mail: Abdel_ssameud@hotmail.com
Abstract - This paper deals with the design problem of
an H8 filter for robot manipulators in the presence of
external perturbations. The proposed scheme is an
application of the H8 filter proposed by [De Souza et
al., 1993] to the class of robotic manipulator systems.
Using a stabilizing control law given in [Paden et al.,
1988], a stabilization result based on the weak
detectability of the system is used to demonstrate the
semi-global asymptotic stability of the equilibrium
point of the combined system (robot - H8 filter -
stabilizing controller). Simulation results on a two DO-
F robot manipulator show the asymptotic
convergence of the reconstruction and tracking error
vector.
I. INTRODUCTION
One of the fundamental problems in control systems
design is the estimation of the state variables of a dynamic
system in the presence of external perturbations and
system parametric uncertainties. Over the past three
decades, considerable interest has been devoted to state
estimation methods based on the minimization of the
variance of the estimation error. It has been shown that
these types of estimators may not be robust with respect
to system uncertainties in the dynamical model of the
system [De Souza et al., 1993]. The need to handle
uncertainty in filtering problems has motivated the use of
a new measure of performance – The H 8 norm -. In the
H8 estimation, the estimator is required to produce an
estimate of a given linear combination of the state
variables based on the available measurements. In
addition, it is designed to guarantee that the operator
relating the noise signals to the resulting estimation error
should possess an H8 norm less than a prescribed positive
value. This is equivalent to imposing an upper bound on
the maximum gain of the estimation error over all
frequencies.
Actually, H8 control is considered as the outcome of
several main lines of research, independently developed
from the work of [Zames 1981], [Doyle et al., 1989],
[Safonov et al., 1989], [Francis 1987] and many others.
They have expressed in mathematical terms some aspects
of mono and multivariable control such as performance
and robustness.
For linear systems, the H8 control theory has been
revealed to be a powerful tool to solve the disturbance
attenuation problem. This can be seen from the work of
[Doyle et al., 1989], [Glover et et. 1988] and [Sampei et
al., 1990].
The successful development of this theory is to a large
extent due to the use of time domain methods. In addition,
significant advances in the theory depend on results
concerning dynamic differential games, Riccati equations
the bounded real lemma, [James et al., 1995].
The development of a systematic analysis of the nonlinear
equivalent to the H8 control problem was initiated by the
important contribution of [Basar et al., 1990] and [Van
der Schaft 1991].
For nonlinear H8 control, an approach based on
Hamilton-Jaccobi equalities was introduced in [Van der
Schaft 1992] and [Isidori et al., 1992] to develop an H8
state feedback controller and an H8 output feedback
controller respectively. In the nonlinear case, the notion of
L2-gain generalizes the H8 norm of a linear transfer
function. Note that, in these works, the controller is
obtained by solving equations of Riccati (linear case) or
Hamilton-Jaccobi (nonlinear case) types, while
disturbance attenuation requires only to solve inequalities.
The robust estimation problem has also been considered
in the work of [Grimble 1988] and [Xie et al., 1993].
Recently, the problem of robust filtering for uncertain
systems was considered. In [Fu et al., 1992], linear
systems with norm-bounded parameter uncertainty have
been considered and a robust H8 filter with a guaranteed
performance in an H8 sense has been developed. Also, a
robust linear filter for a class of uncertain nonlinear
systems has been proposed in [Xie et al., 1992], where the
nonlinearities are introduced in the form of nonlinear
perturbation.
In [De Souza et al., 1993], the problem of H8 filtering for
a class of uncertain continuous-time nonlinear systems
was considered. The addressed problem is the design of a
nonlinear filter that guarantees both robust stability and a
prescribed H8 performance for the filtering error
dynamics for the whole set of admissible systems. A
Riccati equation approach is proposed to solve the robust
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