Hoo SYNTHESIS FOR TIME DELAY SYSTEMS – AN LMI APPROACH
Alexandru Serbanescu1, Corneliu Popeea2
1ARS T&TT, Oranje Straat 7, 2514 JB Den Haag, The Netherlands
2University Politehnica Bucharest, Automatic Control and Computers Faculty, Splaiul Independentei 313,
Bucharest VI, Romania
Keywords: Time Delay Systems, 
 H synthesis, Linear
Matrix Inequalities (LMI), Semidefinite Programming,
Projection Lemma.
ABSTRACT
The problems arising in Time Delay Systems (TDS)
have increased complexity with respect to those related
to Linear Delay Free Systems. Linear Matrix Inequalities
(LMI) proved to be a useful instrument for solving
several control problems. The present paper is
approaching the output feedback 
 H control problem for
a set of TDS for whom both the state dynamics and the
output depends on the delayed states. The characteristics
of this class of systems require a specific approach. Two
new procedures are proposed to solve the optimal and the
sub-optimal 
 H control problems.
1. INTRODUCTION
Starting with the beginning of the 1980’s, the 
 H control
problem has been extensively studied. For the delay free
systems several results approaching the problem of output 
 H
feedback synthesis have been published in the 1990’s. For a
general theoretic framework for this problem and other related
issues see [Ionescu 19981]. [Gahinet 1996] and [Iwasaki 1994]
extended the general control problem using the Bounded Real
Lemma (BRL) and linear matrix inequalities (LMI). Necessary
and sufficient conditions for the existence of an 
 H controller
were given in terms of three LMI’s.
The delay is frequently a source of instability and
encountered in various engineering systems such as chemical
processes, hydraulics, power plants or combustion engines. As
a result the stability of TDS received much attention in the last
years. Since these systems include perturbations, it is important
to study the 
 H problem for this class of systems. There have
been published several results in the last years regarding the
control for TDS, see [Park 2000], [Dugard 1998], [Niculescu
1998], [Li 1996], [Esfahani 1998] and [Shaked 1998]. For an
analytical approach, see [Marshall 1992]. Most of the
approaches that have numerical relevance (can be applied in
general situations), are Lyapunov based methods. As a general
pattern a Lyapunov functional* is defined for the system and
the final results are expressed in terms of LMI’s. The resulting
LMI’s are dependent on the way the Lyapunov functional is
chosen.
The purpose of this paper is to offer necessary and sufficient
conditions for the existence of an 
 H output feedback
controller, for a class of TDS described in (1.5) and to propose
* In the literature there are two main approaches one using
Lyapunov-Krasovskii functionals and the other using Lyapunov-
Razumikhin functions. The last one is considered to be more
conservative since it is using the Khargonekar Lemma, and it is
recommended only when the previous one fails. For more details see
Dugard [8].
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