Research Report :
Outline of Master Thesis
Ying Shang
My master thesis consists of two major chapters: modelling and analysis. The following sections will
briefly introduce the contents of this thesis.
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Modelling
It has already been shown that there are a lot of physical systems consisting of continuous and discrete
event dynamics. There are several literature which defined formal hybrid system models to represent such
physical systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The early hybrid model is proposed by Hans Witsenhausen
in 1996 [1]. He presented a hybrid model with interacting continuous plant and discrete elements. The
continuous plants are described by differential equations. The transition of discrete elements are triggered
by the continuous states and not by inputs. Similar model proposed by Tavernini [2], so called differential
automata. Branicky’s hybrid model [7, 8] consists of a collection of dynamical systems and switching map
to describe the transition between them.
There are some hybrid models consisting of continuous plant with discrete controller such as Stiver,
Antsaklis and Lemmon ’s model [3] and Nerode and Kohn’s model [5]. In [3], the model consists of three
parts:the plant, the controller and the interface. The continuous plant is described by differential equations
and the discrete controller is modelled as an automaton. The interaction of continuous plant and controller
is specified by an interface. Another framework in computer science is hybrid automaton model proposed
by Alur et al. [13]. A hybrid automaton model is an extension of traditional finite automata theory in the
sense of the continuous dynamics are included in each discrete modes. The transition of the automaton is
controlled by logical predicates(called guards) on the automaton’s arcs. Hybrid model used in this thesis
is hybrid automaton model.
Chapter 2 will introduce existing hybrid models in more details. Preliminary definitions, properties,
classification of hybrid automata will also be mentioned in Chapter 2.
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Analysis
Analysis of hybrid systems means that to answer such a question of whether or not a hybrid system
model satisfies certain properties, such as safety and stability properties, etc. Stability is one of the most
important properties of a dynamical system. The previous results of stability analysis of hybrid system
are mostly the generations of Lyapunov theory. However, this thesis will focus on safety property, not on
stability issue. In the recent years, a lot of literature on the analysis of hybrid system has been devoted to
the reachability problem, which is to ask whether a specified set of states is reachable from a set of initial
states. Safety property is relative to reachability problem. There are some techniques to establish whether
the reachable sets of hybrid systems are in a specified set [11, 12, 13, 14, 15, 16, 19, 20, 21]. All existing
methods try to compute the flow of the system’s differential equations. However, The primary problem
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with these approaches is that computation is expensive and infeasible. Regarding to these problems, one
often has to resort to approximations [22, 23, 24, 25]. Based on the theoretical results, the researchers
have developed several computational tools to exactly compute the reachable set of states HYTECH [26],
COSPAN [27], KRONOS [28], etc. However, it is well known [14] that for all but the simplest class of
hybrid automata the reachability problem is not decidable, which mean there exists no such an algorithm
approach that can terminate after a finite number of iterations.
Since 1995, Aubin el.[33, 31, 34] extended the viability theory [32] to address the reachability problem
of hybrid systems and impulse differential inclusions. Viability theory formally discuss the fundamental
viable property of the state sets of a dynamical system. Roughly speaking, a set of states K is called a
viable set if for all initial conditions in K there exists a solution of the dynamical system that remains in
K. If the set K is not viable , then the largest subset of K which is viable is called viability kernel. This
thesis will extend their results to hybrid automata[13], which are commonly used mathematical model for
dynamical systems whose states have discrete and continuous variables. Lemmon [29, 30] modified the
reachability problem as the existence of non-terminating solutions to a hybrid automaton, which means
the system’s ability to reach some fixed points. Analysis of hybrid systems will be discussed in Chapter 3
and Chapter 4. Chapter 3 will review the existing results on the reachability analysis.
Chapter 4 is based on the viability theory by Aubin [32] and previous results on the non-terminating
solutions of hybrid automata [29, 30]. The main results in Chapter 4 are the conditions for the existence
of non-terminating solutions to controlled hybrid automaton, which is a hybrid automaton with control
inputs. In prior work, the existence of global solutions for traditional hybrid automata has been tackled as a
problem in algorithmic verification [13, 21]. In algorithmic verification, we first identify a cycle of arcs in the
automaton’s graph, and then recursively compute the set of all states that can satisfy the guard condition
of each arc in that cycle. This backward recursion is sometimes referred to as a reachability analysis. If
the recursion terminates in a non-empty set (that we call the outer viability kernel [33]) then we know
that there exists a global solution generating the specified cycle. If the recursion terminates in an empty
set, then we know that no such solution exists. The primary problem with the algorithmic verification
approach is that computation of the outer viability kernel is expensive, usually doesn’t terminate after a
finite number of iterations, and cannot be used in a compositional manner. Composition here means that
the outer viability kernel of specified cycles cannot be used to establish the existence of global solutions
generating some composition of these cycles. This means that the primary approach being used by most
researchers will not scale well for large scale problems. The motivation of this research is the problems of
previous approaches of reachability problem mentioned earlier.
We present an alternative approach that uses a recursion to compute the inner viability kernel of a
cycle. The inner viability kernel is the set of all states such that any point in this set can be controlled
to reach any other point in this set. Using inner viability kernel methods, a sufficient condition for the
existence of the global non-blocking solutions are derived. The inner viability kernel computation are
easily preformed and always terminate in a finite number of iteration. Given a hybrid system in which
multiple cycles support non-terminating solutions with known inner viability kernels, the composition of
these cycles will also have non-terminating solutions provided the kernels have a non-empty intersection.
Therefore, we can use it to study large scale hybrid systems.
Chapter 4 will state and prove these new results with illustrating examples, including a biped walking
robot example.
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3
Future Research
The principal result of this paper is a sufficient condition for the existence of global non-blocking solutions
to controlled hybrid automata. This condition is derived from the inner recursion algorithm that terminates
after a finite number of iterations to a limit set called the inner viability kernel. The inner viability approach
presented in this paper has the following advantages. First, the inner preset recursion terminates after
a finite number of iterations. Second, the inner viability kernel is easily computed for controlled hybrid
automata whose underlying continuous dynamics have controllability manifolds of dimension n − 1 or
higher. Third, in our opinion, the major benefit is the composition property of inner viability kernel. This
property is useful to the analysis of large scale hybrid systems.
There exist, however, problems with the inner viability approach. All of preceding benefits of the inner
viability approach rely on the fact that the inner viability kernel must be a non-empty set. If it is empty, we
can say nothing about the existence of non-terminating solutions to controlled hybrid automata. Moreover,
although the outer viability kernel does not necessarily ensure the composition of sequential event traces,
we may find it supports cycle composition in some special cases. We have yet to answer under what
conditions, the outer viability kernel supports cycle composition.
The previous observations suggest future research directions. The first direction is to find the conditions
under which the inner viability kernels of a hybrid automaton are non-empty. The second direction is to
find a class of hybrid systems whose outer viability kernels support cycle composition. It’s easy to show if
the continuous subsystems of a hybrid automaton are controllable, the inner viability kernel and the outer
viability kernel are the same. Therefore the outer recusion terminates after a finite number of iterations
and essures the composability analysis. So establishing the controllability conditions for hybrid automata
is also a promising research direction.
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