Pieter Collins

Pieter Collins
Maastricht University | UM · Department of Data Science and Knowledge Engineering

About

78
Publications
5,665
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
1,191
Citations
Citations since 2016
8 Research Items
447 Citations
20162017201820192020202120220204060
20162017201820192020202120220204060
20162017201820192020202120220204060
20162017201820192020202120220204060

Publications

Publications (78)
Article
Uncertainty is unavoidable in modeling dynamical systems and it may be represented mathematically by differential inclusions. In the past, we proposed an algorithm to compute validated solutions of differential inclusions; here we provide several theoretical improvements to the algorithm, including its extension to piecewise constant and sinusoidal...
Chapter
When analysing cyber-physical systems for runtime verification purposes, reachability analysis can be used to identify whether the set of reached points stays within given safe bounds. If the system dynamics exhibits nonlinearity, approximate numerical techniques (with rigorous numerics) are often necessary when dealing with system evolution. Since...
Preprint
Full-text available
Uncertainty is unavoidable in modeling dynamical systems and it may be represented mathematically by differential inclusions. In the past, we proposed an algorithm to compute validated solutions of differential inclusions; here we provide several theoretical improvements to the algorithm, including its extension to piecewise constant and sinusoidal...
Chapter
Full-text available
Uncertainty is unavoidable in modeling dynamical systems and it may be represented mathematically by differential inclusions. In the past, we proposed an algorithm to compute validated solutions of differential inclusions; here we provide several theoretical improvements to the algorithm, including its extension to piecewise constant and sinusoidal...
Article
Full-text available
Mutations in SCN5A can alter the cardiac sodium current INa and increase the risk of potentially lethal conditions such as Brugada and long-QT syndromes. The relation between mutations and their clinical phenotypes is complex, and systems to predict clinical severity of unclassified SCN5A variants perform poorly. We investigated if instead we could...
Conference Paper
Full-text available
Cyber-physical systems (CPS) are hybrid systems that commonly consist of a discrete control part that operates in a continuous environment. Hybrid automata are a convenient model for CPS suitable for formal verification. The latter is based on reachability analysis of the system to trace its hybrid evolution and consequently verify its properties....
Article
Full-text available
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 M...
Conference Paper
Full-text available
Novel protocols for whole-cell patch clamp current recordings have the potential to uniquely identify Markov model parameters. For the fast sodium current, which operates at very small time scales, measurement noise and amplifier imperfections can affect the recorded signal in non-trivial ways. We investigated the unique identification of the kinet...
Chapter
Differential inclusions are mathematical models of nondeterministic continuous-time systems for which no stochastic information on the behavior is known. They arise naturally as reduced models of deterministic systems or as models of components of a distributed system with partial knowledge of the inputs. In order to verify that such systems satisf...
Article
Full-text available
We introduce the verification of hybrid systems as offered by the open-source framework called ARIADNE. The ARIADNE C++ library exploits approximation techniques based on the theory of computable analysis for implementing formal verification algorithms based on reachability analysis. We demonstrate the tool using a classical example of a controlled...
Article
The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the t...
Conference Paper
Full-text available
Computational cellular electrophysiology uses ODE models to study the electrical behavior of cardiomyocytes, and by extension of the heart. This paper introduces Myokit, an open source framework for development and analysis of such models. The framework consists of a dedicated modeling language and a set of tools for simulation, analysis and post-p...
Conference Paper
Full-text available
Numerical simulation of muscle cells and tissue is an established tool in cardiac electrophysiology, where the electrical behavior of excitable heart muscle cells is commonly modeled as a stiff, non-linear system of ordinary differential equations. A common feature of this system’s right-hand side is the heavy use of computationally expensive univa...
Conference Paper
The last decade brought us a whole range of over-approximative algorithms for the reachability analysis of hybrid automata, a widely used modeling language for systems with combined discrete-continuous behavior. Besides theoretical results, there are also some tools available for proving safety in the continuous time domain. However, if a given set...
Article
The control-to-facet problem plays an important role in the design of feedback controllers for piecewise-affine hybrid systems on polytopes. In the literature, necessary conditions and sufficient conditions for solvability by static state feedback exist. In this paper, we extend these results to the case of continuous piecewise-affine static output...
Article
Full-text available
In many applicative fields, there is the need to model and design complex systems having a mixed discrete and continuous behavior that cannot be characterized faithfully using either discrete or continuous models only. Such systems consist of a discrete control part that operates in a continuous environment and are named hybrid systems because of t...
Conference Paper
Full-text available
When designing embedded systems, often the need arises to model systems having a mixed discrete and continuous behavior. Such hybrid systems commonly consist of a discrete control part that operates in a continuous environment and may be represented by hybrid automata. We recently proposed an open-source framework for hybrid automata analysis, call...
Conference Paper
Full-text available
This work presents the recent development in the definition, analysis and implementation of a new approach to parallel, real-time simulation of dynamic nonlinear power systems. This method combines the advantages of the Resistive Companion Method (RCM) with those of the state equation approach, in order to overcome their respective weaknesses when...
Article
Full-text available
We present a numerical method for rigorous over-approximation of a reachable set of differential inclusions. The method gives high-order error bounds for single step approximations and a uniform bound on the error over the finite time interval. We provide formulas for the local error based on Lipschitz constants and bounds on higher-order derivativ...
Conference Paper
Full-text available
Hybrid systems exhibit all the complexities of finite automata, nonlinear dynamic systems and differential equations, and are extremely difficult to analyze. A rigorous mathematical approach is needed to achieve provable approximation bounds along the computations. In this paper we describe a rigorous numerical calculus for working with functions t...
Article
We present a framework for validated numerical computations with real functions. The framework is based on a formalisation of abstract data types for basic floating-point arithmetic, interval arithmetic and function models based on Banach algebra. As a concrete instantiation, we develop an elementary smooth function calculus approximated by sparse...
Article
In this paper we design and implement an algorithm for computing symbolic dynamics for two dimensional piecewise-affine maps. The algorithm is based on detection of periodic orbits using the Conley index and Szymczak decomposition of Conley index pair. The algorithm is also extended to deal with discontinuous maps. We compare the algorithm with the...
Article
In this paper we consider the semantics for the evolution of hybrid systems, and the computability of the evolution with respect to these semantics. We show that with respect to lower semantics, the finite-time reachable sets are lower-semico mputable, and with respect to upper semantics, the finite-time reachable sets are upper- semicomputable. We...
Article
Full-text available
Shadowing is a method of backward error analysis that plays a important role in hyperbolic dynamics. In this paper, the shadowing by containment framework is revisited, including a new shadowing theorem. This new theorem has several advantages with respect to existing shadowing theorems: It does not require injectivity or differentiability, and its...
Article
Full-text available
Analysis of the dynamic behavior of large-scale biochemical reaction systems can be facili-tated by abstraction followed by model checking. A biochemical reaction system can be approximated by a multi-affine system or an affine system on a rectangle. Either of these systems can be abstracted to an automaton. Model checking can then be employed to d...
Conference Paper
A numerical method for rigorous over-approximation of a solution set of an input-affine system whose inputs represent some bounded noise is presented. The method gives high order error for a single time step and a uniform bound on the error over the finite time interval. The approach is based on the approximations of inputs by linear functions at e...
Article
We present a framework for the verification of the numerical algorithms used in Ariadne, a tool for analysis of nonlinear hybrid system. In particular, in Ariadne, smooth functions are approximated by Taylor models based on sparse polynomials. We use the Coq theorem prover for developing Taylor models as sparse polynomials with floating-point coeff...
Article
In this paper we design and implement rigorous algorithms for computing symbolic dynamics for piecewise-monotone-continuous maps of the interval. The algorithms are based on computing forwards and backwards approximations of the boundary, discontinuity and critical points. We explain how to handle the discontinuities in the symbolic dynamics which...
Conference Paper
In this paper, we develop a theory of computable types suitable for the study of control systems. The theory uses type-two effectivity as the underlying computational model, but we quickly develop a type system which can be manipulated abstractly, but for which all allowable operations are guaranteed to be computable. We apply the theory to the stu...
Conference Paper
The CTL * model-checking problem is thoroughly studied and is fully understood for finite and countable state spaces. Yet, in most models arising in the sciences and engineering the system’s sate space is uncountable. Then, the standard computability and complexity theory is inapplicable but the semantics of CTL * has to be in some sense computable...
Article
Full-text available
In this paper we consider the computability of the solution of the initial- value problem for dierential inclusions with semicontinuous right-hand side. We present algorithms for the computation of the solution using the \ten thousand mon- keys" approach, in which we generate all possible solution tubes, and then check which are valid. In this way,...
Conference Paper
In this note we discuss the information needed to compute the homology groups of a topological space. We argue that the natural class of spaces to consider are the compact absolute neighbourhood retracts, since for these spaces the homology groups are finite. We show that we need to specify both a function which defines a retraction from a neighbou...
Article
Full-text available
The compositional interchange format for hybrid systems (CIF) supports inter-operability of a wide range of tools by means of model transformations to and from the CIF. Work on the CIF takes place in the FP7 Multiform project, and in several other European projects. The CIF consists of an abstract and a concrete format, used for defining a formal s...
Article
In this paper, we develop a theory of computable types suitable for the study of dynamic systems in discrete and continuous time. The theory uses type-two effectivity as the underlying computational model, but we quickly develop a type system which can be manip-ulated abstractly, but for which all allowable operations are guaranteed to be computabl...
Article
Full-text available
We address the control synthesis of hybrid systems with discrete inputs, disturbances and outputs. The control objective is to ensure that the events of the closed-loop system belong to the language of the control requirements. The controller is sampling-based and it is representable by a finite-state machine. We formalize the control problem and p...
Article
The character of this paper is an extended abstract of the approach developed by the authors for control of piecewise-affine hybrid systems. Control of hybrid systems is a wide ranging research topic with many theoretical problems. The authors have therefore decided early on to restrict attention to piecewise-affine hybrid systems on polytopes. The...
Article
Full-text available
In this note we consider the computability of the solution of the initial-value problem for ordinary differential equations with continuous right-hand side. We present algorithms for the computation of the solution using the “thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, w...
Article
Full-text available
This paper introduces a new algorithm dedicated to the rigorous reachability analysis of nonlinear dynamical systems. The algorithm is initially presented in the context of discrete time dynamical systems, and then extended to continuous time dynamical systems driven by ODEs. In continuous time, this algorithm is called the Reach and Evolve algorit...
Article
Full-text available
In this note we give a new representation for closed sets under which the robust zero set of a function is computable. We call this representation the component cover representation. The computation of the zero set is based on topological index theory, the most powerful tool for finding robust solutions of equations.
Conference Paper
In this paper, we develop an algorithm to compute under- and over-approximations to the discrete dynamics of a hybrid automaton. We represent the approximations to the dynamics as \emph{sofic shifts}, which can be generated by a discrete automaton. We restrict to two-dimensional systems, since these give rise to one-dimensional return maps, which a...
Article
Ariadne 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...
Article
In this paper we give an overview of some aspects of chaotic dy-namics in hybrid systems, which comprise different types of behaviour. Hybrid systems may exhibit discontinuous dependence on initial conditions leading to new dynamical phenomena. We indicate how methods from topological dy-namics and ergodic theory may be used to study hybrid systems...
Article
The control-to-facet problem plays an important role in the design of feedback controllers for piecewise-affine hybrid systems on polytopes. In the litera-ture, necessary and sufficient conditions for solvability by static state feedback exist. In this paper, we extend these results to the case of continuous piecewise-affine out-put feedback. For t...
Article
In this paper we consider the computation of controllers for noisy nonlinear discrete-time systems described by upper-semicontinuous multivalued functions. We show that for the problem of controlling to a target set, if an open-loop solution exists, then a feedback controller with can be effectively computed in finite time from the problem data, an...
Article
Full-text available
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...
Conference Paper
Nonlinear dynamical and control systems are an important source of applications for theories of computation over the the real num- bers, since these systems are usually to complicated to study analytically, but may be extremely sensitive to numerical error. Further, computer- assisted proofs and verification problems require a rigorous treatment of...
Article
In this paper we develop general techniques to study stability of hybrid systems with linear continuous dy-namics. These techniques are based on matrix analysis and study of differentiable manifolds. These techniques operate on the space of switching times of the hybrid systems. Some special techniques for hybrid systems with three dimensional stat...
Conference Paper
Full-text available
The software package ConPAHS facilitates control design of continuous-time piecewise-affine hybrid systems on polytopes. For the control objective of reaching a particular state from a specified initial state, the output of the package is a piecewise-affine control law. After a short review of the control theory for this problem, the paper presents...
Article
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 com...
Conference Paper
The computation of reachable and invariant sets of nonlinear dynamic and control systems are important problems of systems theory. In this paper we consider the computability of these sets using Turing machines to perform approximate computations. We use Weihrauch’s type-two theory of effectivity for computable analysis and topology, which provides...
Conference Paper
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....
Conference Paper
Full-text available
In this paper we introduce the problem of reach- ability analysis of hybrid automata to decide safety properties. Then we describe ARIADNE, an in-progress open environment to design algorithms for computing with hybrid automata. We show that ARIADNE relies on a rigorous computable analysis theory to represent geometric objects, in order to achieve...
Article
In this paper we consider various frameworks for representing the trajectories of a hybrid system. We compare several existing frameworks, paying particular attention to convergence of trajectories. We then generalise the approach based on hybrid time domains, rstly to deal with Zeno behaviour, and then to a highly general abstract framework. We il...
Article
A flow in three-dimensions is universal if the periodic orbits contains all knots and links. Universal flows were shown to exist by Ghrist, and can be constructed by means of templates. Likewise, a planar diffeomorphism is universal if it has a suspension flow which is a universal flow. In this paper we prove the existence of a homoclinic trellis t...
Conference Paper
Turing machines exposed to a small stochastic noise are considered. An exact characterisation of their (≈\({\it \Pi}\) \(_{\rm 2}^{\rm 0}\)) computational power (as noise level tends to 0) is obtained. From a probabilistic standpoint this is a theory of large deviations for Turing machines.
Conference Paper
In this paper we discuss the observability of hybrid systems and turing machines. We give an elementary example to show that observability is undecidable for turing machines with output. Since many classes of system simulate turing machines, we can then show that observability for these classes is undecidable. We discuss the observability of piecew...
Conference Paper
We consider observability for a class of piecewise-affine hy- brid systems without inputs. The aim is to give verifiable conditions for observability in terms of linear equations and inequalities. We first dis- cuss a number of important concepts, such as discrete-event detectability and trajectory observability. We give sufficient conditions for o...
Article
The computation of reachable sets of nonlinear dynamic and control systems is an important problem of systems theory. In this paper we consider the computability of reachable sets using Turing machines to perform approximate computations. We use Weihrauch's type-two theory of effectivity for computable analysis and topology, which provides a natura...
Article
In the theory of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, it is possible to compute a one-dimensional representative map for any irreducible isotopy class. The topological entropy of this graph representative is equal to the growth rate of the number of essential Nielsen classes of a given period, and hence is a lower...
Article
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 standard algorithms, though this does not give much information on the structure of the forcing relation. Here we consider forcing relations between homoclinic orbits,...
Article
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 standard algorithms, though this does not give much information on the structure of the forcing relation. Here we consider forcing relations between homoclinic orbits,...
Article
Later published in EQUADIFF: Proceedings of the International Conference on Differential Equations, by World Scientific Publishing (2005), pp. 871-876, ISBN 9812561692 An optically-injected semiconductor laser exhibits chaotic behaviour for certain values of the parameters. The underlying model is an example of a general three-dimensional system of...
Article
We compute bounds on the topological entropy associated with a chaotic attractor of a semiconductor laser with optical injection. We consider the Poincaré return map to a fixed plane, and are able to compute the stable and unstable manifolds of periodic points globally, even though it is impossible to find a plane on which the Poincaré map is globa...
Article
The Nielsen-Thurston theory of surface diffeomorphisms shows that useful dynamical information can be obtained about a surface diffeomorphism from a finite collection of periodic orbits.In this paper, we extend these results to homoclinic and heteroclinic orbits of saddle points. These orbits are most readily computed and studied as intersections o...
Article
In this paper we give methods for computing lower bounds on the number of periodic points of a self-map of a topological pair (X,Y). These methods are particularly useful if X⧹Y is disconnected, as we obtain a symbolic description of orbits in terms of components of X⧹Y. The theories are homotopy invariant, and also allow the comparison of maps on...
Article
Given a saddle fixed point of a surface diffeomorphism, its stable and unstable curves $W^S$ and $W^U$ often form a homoclinic tangle. Given such a tangle, we use topological methods to find periodic points of the diffeomorphism, using only a subset of the tangle with finitely many points of intersection, which we call a trellis. We typically obtai...
Article
In this paper we present the computation of symbolic dynamics of a one dimensional return map of a piecewise-affine hybrid system. The system arises as a simple electrical circuit with hysteresis switching, and exhibits chaotic dynamics. Our method allows us to rigorously obtain a qualitative description of the discrete behaviour of the system. We...

Network

Cited By

Projects

Projects (2)
Project
A C++ library for formal verification of nonlinear hybrid systems through reachability analysis.
Project
Myokit (http://myokit.org) is an easy-to-use but powerful toolkit for modeling and simulation of the cardiac action potential. It uses an easy-to-read model description language, but can import and export to CellML, C, Matlab, etc. Engines are provided for fast single-cell and GPU-powered multi-cell simulations. Other analysis tools are provided via a python API, including tools to load patch-clamp data and fit ion channel models.