  We present a novel method to compute componentwise transient bounds, ultimate
bounds, and invariant regions for a class of switching continuous-time linear
systems with perturbation bounds that may depend nonlinearly on a delayed
state. The main advantage of the method is its componentwise nature, i.e. the
fact that it allows each component of the perturbation vector to have an
independent bound and that the bounds and sets obtained are also given
componentwise. This componentwise method does not employ a norm for bounding
either the perturbation or state vectors, avoids the need for scaling the
different state vector components in order to obtain useful results, and may
also reduce conservativeness in some cases. We give conditions for the derived
bounds to be of local or semi-global nature. In addition, we deal with the case
of perturbation bounds whose dependence on a delayed state is of affine form as
a particular case of nonlinear dependence for which the bounds derived are
shown to be globally valid. A sufficient condition for practical stability is
also provided. The present paper builds upon and extends to switching systems
with delayed-state-dependent perturbations previous results by the authors. In
this sense, the contribution is three-fold: the derivation of the
aforementioned extension; the elucidation of the precise relationship between
the class of switching linear systems to which the proposed method can be
applied and those that admit a common quadratic Lyapunov function (a question
that was left open in our previous work); and the derivation of a technique to
compute a common quadratic Lyapunov function for switching linear systems with
perturbations bounded componentwise by affine functions of the absolute value
of the state vector components.
