7
Positive RealControl
7.1 Introduction
The notion of positive realness has playedanimportantrole in control and
system theory[5, 70, 167]. Awell-knownfact in robust and nonlinear control
is that thepositive realness of acertain loop transfer function will guarantee
the
ove
rall
stabilit
yo
fafe
edbac
ks
ystem
if
uncertain
ty
or
nonlinearit
yc
an
be
characterized by apositive realsystem [167]. This has motivated the study
of the positive realcontrol problem. In thecontext of state-space systems,
results
on
po
sitive
rea
lc
on
trol
can
be
found
in
[153,1
75],
and
ther
eferences
therein.
In
this
ch
apter,w
ec
onsider
thep
osi
tive
rea
lc
on
trol
problem
for
singular
systems in both the continuousand discrete cases.The aim is to design a
state feedbackcontroller suchthat the closed-loop system is regular, impulse-
free
(for
con
tin
uous
singular
systems)
or
causalit
y(
for
discrete
singular
sys-
tems), and extended strictly positive real. First, versions of the positive real
lemmaare proposedinterms of LMIs, whichcan be viewed as extensions of
po
sitive
rea
ll
emmas
for
state-space
systems
to
singular
systems.
Based
on
the proposed positive reallemmas, necessary and sufficientconditionsfor the
solvabilityofthe positive realcontrol problem are obtained. Forcontinuous
singular
systems,ad
esired
state
feedbac
kc
on
troller
can
be
constructed
by
solving an LMI, while for discretesingular systems, adesired state feedback
controller can be constructed by solving amatrix inequality.
S. Xu and J. Lam: Rob. Contr. and Filt. of Sing. Syst., LNCIS 332, pp. 119–140, 2006.
© Springer-Verlag Berlin Heidelberg 2006