[Thesis]. Manchester, UK: The University of Manchester; 2018.
A method is proposed to infer stability properties for non-linear switching under
continuous state feedback. Continuous-time systems which are dissipative in the multiple
storage function sense are considered. A partition of the state space, induced by
cross-supply rates and the feedback function, is used to derive a restriction on switching.
Then, conditions are proposed, under which, systems controlled by the feedback
function and switching according to the rule are stable. In particular, Lyapunov and
asymptotic stability are proved, both in a local and in a global context. Further,
is shown that the approach can be extended when one uses multiple controllers, and,
therefore, is able to construct multiple partitions; conditions for this case are
Finally, it is shown that, for the switching families that satisfy the switching
rule posited by the results, one is able to find elements (that is, stabilising switching
laws for the system) which are non-Zeno. Additional rule-sets that allow this are
It is argued that the conditions proposed here are easier to verify and apply,
and that they offer additional flexibility when compared to those proposed by other
approaches in the literature.
The same infrastructure is used in the study of hybrid systems. For a general class
of non-linear hybrid systems, a new property is proposed, that retains some of the
properties of dissipativity, but it differs from it, crucially in the fact that it
is not purely
input-output. For systems having this property, it is shown that the partition used
the switching case can also be used. This, along with a set of conditions allows for
the characterisation of the system behaviour in two scenaria. First, when the continuous
behaviours and the jumping scheme act co-operatively, leading the system to
lower energy levels (from the dissipativity point of view). Second, when the continuous
behaviours are allowed to increase the stored energy, but the jumping is able to
compensate this increase. In the first case, it is shown that the equilibrium point
study is stable; in the second, it is shown that the system exhibits a type of attractivity,
and, under additional conditions, it is asymptotically stable.
Besides stability, a collection of stabilisation results are given for the case of
switching systems. It is shown that one may design state feedback functions
(controllers) with the objective that they satisfy the conditions of the stability
in this work. Then, systems under the designed controllers are shown to be stable,
provided that the switching adheres to a specific switching rule. This problem is
using a variety of tools taken from analysis, multi-valued functions and the
space of non-switching stabilisation.
In addition to the main results, an extensive overview of the literature in the area
of switching and hybrid systems is offered, with emphasis on the topics of stability
and dissipativity. Finally, a collection of numerical examples are given, validating
results presented here.