Hybrid system is a new emerging area in control society. Since it can intrinsically comprise many different types of systems in its frame, the control technique for this area has been rapidly developed in recent years. A switched system occupies the one important portion of hybrid systems having a decision logic and continuous-time dynamical subsystems. In this dissertation, we study the stability analysis of hybrid switched systems. By introducing a redundancy factor as a basic analysis tool, we present the new methodology for analyzing the switching stability and establishing the switching law.
Because the previous multiple Lyapunov function based stability analyses impose the restrictions not only on the behavior of individual Lyapunov functions being engaged so as to be monotonically non-increasing at all switching times but also on the switching law so as to have multiple appearance of the same engaged subsystems in associated switching sequence, they have some limitations for the cases where the constructed switching sequence has the form of only single engagement of each of subsystems up to infinite time, and the subsystems are locally stable under given attraction regions, and the some of subsystems are unstable.
In this dissertation, we employ locally/globally decrescent functions, in stead of using multiple Lyapunov functions, described by exponential or class KL functions for each of subsystems in order to build a more practical stability criteria. The sufficient condition about switching stability has been built with the concept such that the state trajectory of engaged subsystem at switching instant ends up inside of the attraction region of next subsystem. The minimum (maximum) holding time for each subsystem is defined as the required time interval to preserve the stability at least (at most). The general switching stability is provided by means of introducing the redundancy factors for class KL functions. It extends and relaxes the scope of stability analysis for the case where the switched system consists of locally stable nonlinear subsystems. Moreover, the stability property of switched system with slowly varying exogenous input is proposed to enhance the application area of switching control for linear and nonlinear parameter varying (PV) systems.
For the case of switched system consisted of unstable subsystems, an interesting behavior is found from the review of previous literature. It is shown to be possible to stabilize the such switched system when there exists at least one of quantities of energy-like functions being in decrease for each nonzero state, or open common conic region that all energy-like functions are in decrease. The energy-like functions are employed since no Lyapunov-like function exist for the unstable subsystems.
The system dynamic depended on several parameters (or exogenous inputs) is frequently encountered in real physical systems such as robot, missile, active suspension, car auto-steering system, and electro-magnetic levitation system. One of the most typical approaches for the PV system, whether it is linear or nonlinear,
is gain scheduling that has been successfully proved in a variety of engineering applications. For the PV systems, there are few results for the stability analysis of the switched system obtained by constructing the multi-controller from partitioning a parameter space. Thus, in this dissertation, we provide the design procedures for such type controller and the stability analysis in the context of the multi-controller system. The practical applicability of switching control for nonlinear PV system is shown by the case study for missile autopilot design.
