Model Reduction of Hybrid Systems

High-Technological solutions of today are characterized by complex dynamical models. A
lot of these models have inherent hybrid/switching structure. Hybrid/switched systems are
powerful models for distributed embedded systems design where discrete controls are applied
to continuous processes. Hybrid systems are also an important modeling class for nonlinear
systems because a wide variety of nonlinearities are either piecewise-affine (e.g., a saturated
linear actuator characteristic) or can be approximated as hybrid systems. The complexity of
verifying and assessing general properties of hybrid systems, designing controllers and
implementations is very high so that the use of these models is limited in applications where
the size of the state space is large. To cope with complexity, model reduction is a powerful
technique.
This thesis presents methods for model reduction and stability analysis of hybrid/switched
systems. Methods are designed to approximate hybrid/switched systems to low order models
which adequately describe the behavior of the switched systems. Three frameworks for
model reduction of switched systems are proposed which are based on the notion of the
generalized gramians. Generalized gramians are the solutions to the observability and
controllability Lyapunov inequalities. In the first framework the projection matrices are
found based on the common generalized gramians. This framework preserves the stability of
the original switched system for all switching signals while reducing the subsystems of the
switched systems. The first framework is computationally efficient due to the construction of
a single projection for all subsystems. This framework is used for switched controller
reduction and it is shown that the stability of the closed loop system is guaranteed to be
preserved for arbitrary switching signal. To compute the common generalized gramians
linear matrix inequalities (LMI’s) need to be solved. These LMI’s are not always feasible. In
order to solve the problem of conservatism, the second framework is presented. In this
method the projection matrices are constructed based on the convex combinations of the
generalized gramians. However this framework is less conservative than the first one, it does
not guarantee the stability for all switching signals. The stability preservation is studied for
this reduction technique. The third framework for model reduction of switched systems is
based on the switching generalized gramians. The reduced order switched system is
guaranteed to be stable for all switching signal in this method. This framework uses stability
conditions which are based on switching quadratic Lyapunov functions which are less
conservative than the stability conditions based on common quadratic Lyapunov functions.
The stability conditions which are used for this method are very useful in model reduction
and design problems because they have slack variables in the conditions. Similar conditions
for a class of switched nonlinear systems are derived in this thesis. The results are used for
output feedback control of switched nonlinear systems. Model reduction of piecewise affine
systems is also studied in this thesis. The proposed method is based on the reduction of linear
subsystems inside the polytopes. The methods which are proposed in this thesis are applied to
several numerical examples.