Pieter Collins

Reachability and control synthesis for piecewise-affine hybrid systems on simplices

In this paper, we consider the synthesis of control laws for piecewise-affine hybrid systems on simplices. The construction is based on the solution to the control-to-facet problem at the continuous level, and on dynamic programming at the discrete level. The construction is given as an explicit algorithm using only linear algebra and reach-set computations for automata; no numerical integration is required. The method is conservative, in that it may fail to find a control law where one exists, but one cannot hope for a sharp algorithm for control synthesis since reachability for piecewise-affine hybrid systems is undecidable.

Observability of Piecewise-Affine Hybrid Systems

We consider observability for a class of piecewise-affine hybrid systems without inputs. The aim is to give verifiable conditions for observability in terms of linear equations and inequalities. We first discuss a number of important concepts, such as discrete-event detectability and trajectory observability. We give sufficient conditions for observability, observability in infinitesimal time, and observability after a single discrete event. The former conditions are used to construct an observer for the system, the latter are applied to deduce observability for an example system.

SYMBOLIC DYNAMICS FROM HOMOCLINIC TANGLES

We present a method for finding symbolic dynamics for a planar diffeomorphism with a homoclinic tangle. The method only requires a finite piece of tangle, which can be computed with available numer...

Optimal Semicomputable Approximations to Reachable and Invariant Sets

In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are upper-semicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optimal.

Reachability computation for hybrid systems with Ariadne

Abstract A riadne is an in-progress open environment to design algorithms for computing with hybrid automata, that relies on a rigorous computable analysis theory to represent geometric objects, in order to achieve provable approximation bounds along the computations. In this paper we discuss the problem of reachability analysis of hybrid automata to decide safety properties. We describe in details the algorithm used in A riadne to compute over-approximations of reachable sets. Then we show how it works on a simple example. Finally, we discuss the lower-approximation approach to the reachability problem and how to extend Ariadne to support it.

Myokit: A simple interface to cardiac cellular electrophysiology

Myokit is a new powerful and versatile software tool for modeling and simulation of cardiac cellular electrophysiology. Myokit consists of an easy-to-read modeling language, a graphical user interface, single and multi-cell simulation engines and a library of advanced analysis tools accessible through a Python interface. Models can be loaded from Myokits native file format or imported from CellML. Model export is provided to C, MATLAB, CellML, CUDA and OpenCL. Patch-clamp data can be imported and used to estimate model parameters. In this paper, we review existing tools to simulate the cardiac cellular action potential to find that current tools do not cater specifically to model development and that there is a gap between easy-to-use but limited software and powerful tools that require strong programming skills from their users. We then describe Myokits capabilities, focusing on its model description language, simulation engines and import/export facilities in detail. Using three examples, we show how Myokit can be used for clinically relevant investigations, multi-model testing and parameter estimation in Markov models, all with minimal programming effort from the user. This way, Myokit bridges a gap between performance, versatility and user-friendliness.

Computability of finite-time reachable sets for hybrid systems

In this paper we consider the computability of the evolution of hybrid systems, or equivalently, the computability of finite-time reachable sets. We use the framework of type-two computability theory and computable analysis, which gives a theory of computation for points, sets and maps by Turing machines, and is related to computable approximation. We show that, under suitable hypotheses, the system evolution may be lower or upper semicomputable, but cannot be both in the presence of grazing contact with the guard sets.

Forcing Relations for Homoclinic Orbits of the Smale Horseshoe Map

An important problem in the dynamics of surface homeomorphisms is determining the forcing relation between orbits. The forcing relation between periodic orbits can be computed using existing algorithms. Here we consider forcing relations between homoclinic orbits. We outline a general procedure for computing the forcing relation and apply this to compute the equivalence and forcing relations for homoclinic orbits of the Smale horseshoe map.

Tangency Bifurcations of Global Poincaré Maps

One tool to analyze the qualitative behavior of a periodic orbit of a vector field in

Tinkerbell Is Chaotic

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