Lyapunov-like equations and reachability/observabiliy Gramians for descriptor systems

Reachability and observability Gramians are defined for the descriptor systems
Ex_{k-1} = Ax_{k} + Bu_{k}, y_{k} = Cx_{k}
and
E\dot{x}(t) = Ax(t) + Bu(t), y(t) = Cx(t)
. The pencils
(zE - A)
and
(sE - A)
are assumed to be regular. If
E
is singular,these Gramians do not, in general, satisfy the "expected" Lyapunov-like (Sylvester) equations (for example,
EPA^{T] + APE^{T} = - BB^{T}
if
P
is the continuous-time teachability Gramian). This is shown by expressing the solution of the descriptor system in terms of the parameters of the Laurent expansion of
(zE - A)^{-1}
or
(sE - A)^{-1}
. We derive the Lyapunov-like equations which these Gramians do satisfy.