Modelling and Control of Complex Cyber-Physical Ecosystems

In this paper, we set up a mathematical framework for the modelling and control of complex cyber-physical ecosystems. In our setting, cyber-physical ecosystems (CPES) are cyber-physical systems of systems that are highly connected. CPES are understood as open and adaptive cyber-physical infrastructures. These networked systems combine cyber-physical systems with an interaction mechanism with other systems and the environment (ecosystem capability). The main focus will be on modelling cyber and physical interfaces that play an important role on the control of the emergent properties like safety and security

Stochastic safety for Markov chains

In this letter, we study the so-called p-safety of a Markov chain. We say that a state is p-safe in a state space S with respect to an unsafe set U if the process stays in the state space and hits the set U with the probability less than p. We show several ways of computing p-safety: by means the Dirichlet problem, the evolution equation, the barrier certificates, and the Martin kernel. The set of barrier certificates forms a cone. We show how to generate barrier certificates from the set of extreme points of a cone base

p-Safe analysis of stochastic hybrid processes

We develop a method for determining whether a stochastic system is safe, i.e., whether its trajectories reach unsafe states. Specifically, we define and solve a probabilistic safety problem for Markov processes. Based on the knowledge of the extended generator, we are able to develop an evolution equation, as a system of integral equations, describing the connection between unsafe and initial states. Subsequently, using the moment method, we approximate the infinite-dimensional optimization problem searching for the largest set of safe states by a finite-dimensional polynomial optimization problem. In particular, we address the above safety problem to a special class of stochastic hybrid processes, namely piecewise-deterministic Markov processes. These are characterized by deterministic dynamics and stochastic jumps, where both the time and the destination of the jumps are stochastic. In addition, the jumps can be both spontaneous (in the style of a Poisson process) and forced (governed by guards). In this case, the extended generator of this process and its corresponding martingale problem turn out to be defined on a rather restricted domain. To circumvent this difficulty, we bring the generalized differential formula of this process into the evolution equation and, subsequently, formulate a polynomial optimization

Robust Correlated Equilibrium: Definition and Computation

We study N-player finite games with costs perturbed due to time-varying disturbances in the underlying system and to that end we propose the concept of Robust Correlated Equilibrium that generalizes the definition of Correlated Equilibrium. Conditions under which the Robust Correlated Equilibrium exists are specified and a decentralized algorithm for learning strategies that are optimal in the sense of Robust Correlated Equilibrium is proposed. The primary contribution of the paper is the convergence analysis of the algorithm and to that end, we propose an extension of the celebrated Blackwell's Approachability theorem to games with costs that are not just time-average as in the original Blackwell's Approachability Theorem but also include time-average of previous algorithm iterates. The designed algorithm is applied to a practical water distribution network with pumps being the controllers and their costs being perturbed by uncertain consumption by consumers. Simulation results show that each controller achieves no regret and empirical distributions converge to the Robust Correlated Equilibrium.Comment: Preprint submitted to Automatic

Constructive Potential Theory: Foundations and Applications.

Stochastic analysis is now an important common part of computing and mathematics. Its applications are impressive, ranging from stochastic concurrent and hybrid systems to finances and biomedicine. In this work we investigate the logical and algebraic foundations of stochastic analysis and possible applications to computing. We focus more concretely on functional analysis theoretic core of stochastic analysis called potential theory. Classical potential theory originates in Gauss and Poincare's work on partial differential equations. Modern potential theory now study stochastic processes with their adjacent theory, higher order differential operators and their combination like stochastic differential equations. In this work we consider only the axiomatic branches of modern potential theory, like Dirichlet forms and harmonic spaces. Due to the inherently constructive character of axiomatic potential theory, classical logic has no enough ability to offer a proper logical foundation. In this paper we propose the weak commutative linear logics as a logical framework for reasoning about the processes described by potential theory. The logical approach is complemented by an algebraic one. We construct an algebraic theory with models in stochastic analysis, and based on this, and a process algebra in the sense of computer science. Applications of these in area of hybrid systems, concurrency theory and biomedicine are investigated. Parts of this paper have been presented, in shorter form, at diverse conferences and workshops. This work represents a common 'umbrella' for all these presentations and offers an extended version for the (some time) very short published materials

An Integrated Specification Framework for Embedded Systems

In this paper, we address the complex issue of representation of continuous behaviour of the environment of the embedded controllers. In our approach we propose two novel ideas. One is to consider the weak solutions to describe the evolutions of the dynamical systems. The second novelty is to make available, at the design stage, the information about concurrent evolutions of the environment. We propose a new logic called the Hilbertean logic for representing continuous behaviours. Then, we use the causal order relations to integrate this logic with a probabilistic process algebra. For the resulting specification framework, we construct a denotational semantics rich in mathematical properties