Model Predictive Control

308 14 Models of Hybrid Systems
Stability Analysis
Piecewise quadratic Lyapunov stability is becoming a standard in the stability
analysis of hybrid systems [141, 83, 88, 202, 203]. It is a deductive way to prove
the stability of an equilibrium point of a subclass of hybrid systems (piecewise
affine systems). The computational burden is usually low, at the price of a convex
relaxation of the problem, that leads to possibly conservative results. Such
conservativeness can be reduced by constructing piecewise polynomial Lyapunov
functions via semidefinite programming by means of the sum of squares (SOS) decomposition
of multivariate polynomials [201]. SOS methods for analyzing stability
of continuous-time hybrid and switched systems are described in [207]. For the general
class of switched systems of the form ˙x = fi(x), i = 1, . . .,s, an extension of the
Lyapunov criterion based on multiple Lyapunov functions was introduced in [58].
The reader is also referred to the book of Liberzon [167].
The research on stability criteria for PWA systems has been motivated by the
fact that the stability of each component subsystem is not sufficient to guarantee
stability of a PWA system (and vice versa). Branicky [58], gives an example where
stable subsystems are suitably combined to generate an unstable PWA system.
Stable systems constructed from unstable ones have been reported in [245]. These
examples point out that restrictions on the switching have to be imposed in order
to guarantee that a PWA composition of stable components remains stable.
Passivity analysis of hybrid models has received very little attention, except
for the contributions of [66, 175, 260] and [205], in which notions of passivity for
continuous-time hybrid systems are formulated, and of [27], where passivity and
synthesis of passifying controllers for discrete-time PWA systems are investigated.
Reachability Analysis and Verification of Safety Properties
Although simulation allows to probe a model for a certain initial condition and
input excitation, any analysis based on simulation is likely to miss the subtle phenomena
that a model may generate, especially in the case of hybrid models. Reachability
analysis (also referred to as “safety analysis” or “formal verification”), aims
at detecting if a hybrid model will eventually reach unsafe state configurations
or satisfy a temporal logic formula [7] for all possible initial conditions and input
excitations within a prescribed set. Reachability analysis relies on a reach set
computation algorithm, which is strongly related to the mathematical model of
the system. In the case of MLD systems, for example, the reachability analysis
problem over a finite time horizon (also referred to as bounded model checking) can
be cast as a mixed-integer feasibility problem. Reachability analysis was also investigated
via bisimulation ideas, namely by analyzing the properties of a simpler,
more abstract system instead of those of the original hybrid dynamics [199].
Timed automata and hybrid automata have proved to be a successful modeling
framework for formal verification and have been widely used in the literature. The
starting point for both models is a finite state machine equipped with continuous
dynamics. In the theory of timed automata, the dynamic part is the continuous-time
flow ˙x = 1. Efficient computational tools complete the theory of timed automata