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Most control
algorithms used in embedded controller applications are heuristic because of
both theoretical limitations and the limitations of the computing platform. The
necessity of predictable performance is acute in real-time applications, so that
computations whose running time can not be reliably estimated should be avoided.
Existing control theories do not take into account physical constraints, such as
timing, posed by embedded control applications. Therefore the gap between the
world of well-founded algorithms and implementations is large. The models used
to describe the system do not reflect accurately the plant behaviour, or the
interaction between the plant and the controller. Hybrid systems have been
proposed as a source of new models for capturing the mixed nature of real-word
behaviours. The strong expressive power of hybrid systems makes them promising
for their use in the challenging area of embedded control.
In the COLUMBUS project, hybrid systems play the role of an interchange format, which
allows the integration of tools and methods available for hybrid controller
design. In practice, the modelling of many phenomena requires the integration of
both probabilistic and hybrid (mixed discrete – continuous) aspects. Even though
deterministic hybrid models can capture a wide range of behaviours encountered
in practice, stochastic features are very important, because of the uncertainty
inherent in most real world applications. This implies the necessity to
introduce the stochastic hybrid system concept. Roughly speaking,
stochastic hybrid systems are hybrid systems (which are interacting networks of
digital and continuous systems) with some stochastic flavour.
These systems
typically contain variables, or signals that take values from a continuous set
and also variables that take values from a discrete (finite or countable) set.
Differential equations or stochastic differential equations generally give the
continuous dynamics of such systems. A Markov chain generally governs the
discrete-variable dynamics of stochastic hybrid systems. The stochastic features
might be present in the continuous dynamics, or in the discrete dynamics, or in
both. The continuous and discrete dynamics coexist and interact with each other
and because of this it is important to use models that accurately describe the
dynamic behaviour of such hybrid systems. Stochastic
hybrid systems are used, in the COLUMBUS project, as a paradigm for modelling
embedded systems with safety critical performance requirements. Embedded systems
of this type have to operate in an uncertain and often adversarial environment.
Stochastic analysis and control of hybrid systems is therefore essential to
study and improve the performance of embedded systems in the presence of
uncertainty.
In the context of stochastic
hybrid systems we focused on the following main issues:
1. Modelling of general
stochastic hybrid systems;
2. Theoretical foundation of
reachability analysis of stochastic hybrid systems;
3.Stability and stabilisation
of stochastic hybrid systems.
The study of a
number of safety critical situations involving power train, aircraft and air
traffic control motivated the investigation of the first issue. The conclusion
of this study was that different types of models seem to be needed to capture
the variety of different situations that can arise in practice. This implies that a number
of different techniques and tools must be mastered to deal with all the cases of
interest. If a general stochastic hybrid system framework were available then a
single set of results, simulation procedures, etc. could be used in all cases.
In the setting of a general model for stochastic hybrid systems the main problem
is how to integrate the two central modelling paradigms: hybrid systems and
stochastic aspects.
The second
issue, that of reachability, is motivated by the fact that in practice safety
constraints can be naturally formulated as questions of reachability of certain
sets in the state space. In the context of stochastic hybrid systems such
questions have to be addressed in a probabilistic setting.
Roughly
speaking, stability means insensitivity of the state of the system to small
changes in the initial state or the parameters of the systems. For a stable
system, the trajectories, which are close to each other at a specific instant,
should remain close to each other at all subsequent instants. Some property such
as this is required for useful modelling of the real world, since model
parameters and initial conditions are never known perfectly. Several notions of
stability are possible. When stochastic systems are involved, even more
possibilities arise. It often happens that a system is observable only when it
operates in certain modes. Accordingly, in these modes one may be able to design
a feedback controller, based on the observations, which stabilises the given
system. This leads to important questions of stabilisability.
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