Reliable Computing 7: 219–230, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
Robust Control Using Interval Analysis
VLADIMIR N. SHASHIKHIN
Department of Technical Cybernetics, St.Petersburg State Technical University,
Politekhnicheskaya 29, 195251, Saint-Petersburg, Russia,
(Received: 22 December 1998; accepted: 30 July 2000)
Abstract. The synthesis procedure of a control law that guarantees properties of robust stability with
respect to structured parameter perturbations is proposed. The solution of the considered problem is
based on the Razumikhin’s method for functional differential equations generalized for parameter
perturbation systems with time delay. The extension is obtained by using interval Lyapunov functions.
The robust control law is represented through a solution of an interval matrix Riccati type equation.
Design of systems with robust properties is one of the most important problem in
control theory. Over the past two decades, there has been a great deal of interest in
the problem of robust control using matrix Riccati equations and inequalities ,
, . The ﬁeld may be divided into two main areas—robust controller design
based on H
control theory ,  and principles of linear quadratic
optimal control .
) method, that uses two Riccati approaches , considers unstructured
uncertainty, where no assumptions are made about the likely nature of uncertainties.
Recently, design methods have been developed that include the assumption that the
uncertainty has got a certain structure and can be bounded. Such uncertainties
are referred as parametric uncertainties. In the area of robust controller design for
systems with parametric uncertainties, early methods ,  have been developed
 and have become known as robust linear quadratic optimal control (RLQR).
These methods produce a controllers that guarantee both the asymptotic stability
and the minimized performance bound over all possible parameter perturbations
within the prescribed ranges.
Linear differential systems with the coefﬁcient matrix, that only known to sat-
isfy certain bounds, is very large class of systems with uncertain parameters. Such
systems are referred to as interval systems. Methods of robust control for interval
speciﬁed objects are based upon necessary and sufﬁcient conditions for quadrat-
ic stability. In Lyapunov direct method one optimal Lyapunov function  or
parameter-depend Lyapunov function  are used to solve robust stability problem
or robust performance problem. In works ,  to solve this problem it is used