2,894 research outputs found
Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem - A Computer Assisted Proof
The restricted circular three-body problem is considered for the following
parameter values , - the values for {\em Oterma} comet
in the Sun-Jupiter system.
We present a computer assisted proof of an existence of homo- and
heteroclinic cycle between two Lyapunov orbits and an existence of symbolic
dynamics on four symbols built on this cycle.Comment: 40 pages, 11 figure
Breaking Dense Structures: Proving Stability of Densely Structured Hybrid Systems
Abstraction and refinement is widely used in software development. Such
techniques are valuable since they allow to handle even more complex systems.
One key point is the ability to decompose a large system into subsystems,
analyze those subsystems and deduce properties of the larger system. As
cyber-physical systems tend to become more and more complex, such techniques
become more appealing.
In 2009, Oehlerking and Theel presented a (de-)composition technique for
hybrid systems. This technique is graph-based and constructs a Lyapunov
function for hybrid systems having a complex discrete state space. The
technique consists of (1) decomposing the underlying graph of the hybrid system
into subgraphs, (2) computing multiple local Lyapunov functions for the
subgraphs, and finally (3) composing the local Lyapunov functions into a
piecewise Lyapunov function. A Lyapunov function can serve multiple purposes,
e.g., it certifies stability or termination of a system or allows to construct
invariant sets, which in turn may be used to certify safety and security.
In this paper, we propose an improvement to the decomposing technique, which
relaxes the graph structure before applying the decomposition technique. Our
relaxation significantly reduces the connectivity of the graph by exploiting
super-dense switching. The relaxation makes the decomposition technique more
efficient on one hand and on the other allows to decompose a wider range of
graph structures.Comment: In Proceedings ESSS 2015, arXiv:1506.0325
Forward Invariant Cuts to Simplify Proofs of Safety
The use of deductive techniques, such as theorem provers, has several
advantages in safety verification of hybrid sys- tems; however,
state-of-the-art theorem provers require ex- tensive manual intervention.
Furthermore, there is often a gap between the type of assistance that a theorem
prover requires to make progress on a proof task and the assis- tance that a
system designer is able to provide. This paper presents an extension to
KeYmaera, a deductive verification tool for differential dynamic logic; the new
technique allows local reasoning using system designer intuition about per-
formance within particular modes as part of a proof task. Our approach allows
the theorem prover to leverage for- ward invariants, discovered using numerical
techniques, as part of a proof of safety. We introduce a new inference rule
into the proof calculus of KeYmaera, the forward invariant cut rule, and we
present a methodology to discover useful forward invariants, which are then
used with the new cut rule to complete verification tasks. We demonstrate how
our new approach can be used to complete verification tasks that lie out of the
reach of existing deductive approaches us- ing several examples, including one
involving an automotive powertrain control system.Comment: Extended version of EMSOFT pape
Control Barrier Function Based Quadratic Programs for Safety Critical Systems
Safety critical systems involve the tight coupling between potentially
conflicting control objectives and safety constraints. As a means of creating a
formal framework for controlling systems of this form, and with a view toward
automotive applications, this paper develops a methodology that allows safety
conditions -- expressed as control barrier functions -- to be unified with
performance objectives -- expressed as control Lyapunov functions -- in the
context of real-time optimization-based controllers. Safety conditions are
specified in terms of forward invariance of a set, and are verified via two
novel generalizations of barrier functions; in each case, the existence of a
barrier function satisfying Lyapunov-like conditions implies forward invariance
of the set, and the relationship between these two classes of barrier functions
is characterized. In addition, each of these formulations yields a notion of
control barrier function (CBF), providing inequality constraints in the control
input that, when satisfied, again imply forward invariance of the set. Through
these constructions, CBFs can naturally be unified with control Lyapunov
functions (CLFs) in the context of a quadratic program (QP); this allows for
the achievement of control objectives (represented by CLFs) subject to
conditions on the admissible states of the system (represented by CBFs). The
mediation of safety and performance through a QP is demonstrated on adaptive
cruise control and lane keeping, two automotive control problems that present
both safety and performance considerations coupled with actuator bounds
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