* Received 19 February 1997; Received in final form 17 Febru-
ary 1998. This paper was recommended for publication in re-
vised form by Editor Peter Dorato. Corresponding author K. G.
Arvanitis. Tel. #30 1 7722501; Fax #30 1 7722459; E-mail
karvan@control.ece.ntua.gr.
-National Technical University of Athens, Department of
Electrical and Computer Engineering, Division of Computer
Science, Zographou 15773, Athens, Greece.
PII: S0005 –1098(98)00041–7
Automatica, Vol. 34, No. 8, pp. 1021—1024, 1998
1998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain
0005-1098/98 $19.00#0.00
Technical Communique
On the Localization of Intersample Ripples of Linear
Systems Controlled by Generalized Sampled-Data
Hold Functions*
K. G. ARVANITIS-
Key Words—Sampled-data systems; generalized sampled-data hold functions; exact model matching;
intersample ripples; linear nonself-adjoint operators.
Abstract—In this note, the intersample performance of linear
systems, which are controlled on the basis of generalized
sampled-data hold functions, in order to achieve exact model
matching at the sampling instants, is analyzed. The proposed
technique relies on an appropriate error system and of the ex-
pansion of its output signal in suitable basis functions, which are
selected such that some finite rank linear nonself-adjoint oper-
ators are represented exactly. As it is shown, it is plausible to
localize the intersample ripples, which may cause a degradation
of the control performance, by appropriately selecting the arbit-
rary elements of the general forms of the modulating hold
functions. This guarantees that the performance of the closed-
loop system is the desired one, not only at the sampling instants,
but even between them. 1998, Elsevier Science Ltd. All rights
reserved.
1. Introduction
Generalized sampled-data hold functions (GSHF), first pro-
posed by Kabamba (1987), constitute a powerful tool for the
control of linear multivariable systems, alternative to standard
dynamic compensation and especially to state observers. The
basic idea of the GSHF approach is to periodically sample the
plant output and generate the control by means of a hold
function applied to the resulting sequence. Until now, the GSHF
approach has successfully been applied, in order to solve some
very interesting control problems. This approach provides
a series of remarkable advantages over other well-established
feedback control design techniques (for an overview, see Ka-
bamba, 1987; Arvanitis, 1995 and the references cited therein).
From an overview of the so far reported results on the subject,
it becomes clear that, although the particular control objective
sought can be achieved using the GSHF approach at the samp-
ling instants, little can be said about the performance of the
control system between them. Generally speaking, when design-
ing sampled-data controllers for continuous-time linear systems,
it is of crucial importance to take into account the intersample
behaviour of the system, which may not be the desired one, even
though at the sampling instants the system has the desired
control performance. For example, it is well known that, in
sampled control, a degradation of the control performance may
be caused by intersample oscillations, which are usually called
intersample ripples (Astrom and Wittenmark, 1984). Recently,
much research related to the intersample control performance,
has been reported in the literature and several techniques have
been developed to settle this important concern (see, for
example, Bamieh and Pearson, 1992; Chen and Francis,
1991a, 1995; Chen and Qiu, 1994; Dullerud and Francis,
1991; Kabamba and Hara, 1993; Leung et al., 1991; Tadmor,
1992; Toivonen, 1992; Yamamoto, 1994; Arvanitis and
Paraskevopoulos, 1995 and the references cited therein). Never-
theless, it appears that, a thorough study concerning the inter-
sample performance analysis of sampled-data systems
controlled on the basis of the GSHF approach does not exist
in the relevant literature.
In this note, our aim is to investigate the possibility of ameli-
orating the closed-loop system performance in the intersample
time instants, when GSHF are designed in order to achieve the
control objective at the sampling instants. Our interest is fo-
cused on the case where, the control objective is the exact
matching of the closed-loop system to a prespecified discrete-
time model. In this case, the modulating hold functions are
tailored to a given system, in such a way that, for the closed-loop
system, a desired discrete-time transfer function matrix to be
assigned. In Paraskevopoulos and Arvanitis (1994), a new alge-
braic technique is presented, for the solution of the discrete exact
model matching problem using the GSHF approach. This tech-
nique provides necessary and sufficient solvability conditions
and the general expressions of the modulating GSHF sought, as
functions of arbitrary parameters. In this respect, our purpose
here, is to investigate the possibility to improve the ‘‘robustness’’
properties of the aforementioned GSHF technique, by appro-
priately selecting the values of these arbitrary parameters. The
technique proposed to treat this problem, is mainly based on an
appropriate error system and on the expansion of the output
signal of this system, in suitable basis functions, which are
chosen such that some finite rank linear nonself-adjoint oper-
ators are represented exactly. It is worth noticed, at this point
that, the proposed technique is similar to the well-known lifting
technique for continuous-time signals and sampled data systems
(see Bamieh et al., 1991; Bamieh and Pearson, 1992; Tadmor,
1992; Yamamoto, 1994; Chen and Francis, 1995). As it is shown
in the present note, localization of intersample ripples can be
accomplished by a suitable selection of the degrees of freedom
incorporated in the general forms of the modulating hold func-
tions. This guarantees that the performance of the closed-loop
system is the desired one, even between the sampling instants.
2. Preliminaries
Consider the reachable linear time-invariant system of the
form
x (t)"Ax(t)#Bu(t), y(t)"Cx(t), (1)
where x(t)3 1L, u(t)3 1K, y(t)3 1N and A, B, C are real matrices
of appropriate dimensions.
Consider now applying to system (1) the following control
law:
u(t)"F(t)y(k¹)#G(t)w(k¹), (2)
1021