John Lygeros

Impulse differential inclusions: a viability approach to hybrid systems

Impulse differential inclusions are introduced as a framework for modeling hybrid phenomena. Connections to standard problems in the area of hybrid systems are discussed. Conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion. For sets that violate these conditions, methods are developed for approximating their viability and invariance kernels, that is the largest subset that is viable or invariant under the action of the impulse differential inclusion. The results are demonstrated on examples.

A probabilistic approach to aircraft conflict detection

Conflict detection and resolution schemes operating at the mid-range and short-range level of the air traffic management process are discussed. Probabilistic models for predicting the aircraft position in the near-term and mid-term future are developed. Based on the mid-term prediction model, the maximum instantaneous probability of conflict is proposed as a criticality measure for two aircraft encounters. Randomized algorithms are introduced to efficiently estimate this measure of criticality and provide quantitative bounds on the level of approximation introduced. For short-term detection, approximate closed-form analytical expressions for the probability of conflict are obtained, using the short-term prediction model. Based on these expressions, an algorithm for decentralized conflict detection and resolution that generalizes potential fields methods for path planning to a probabilistic dynamic environment is proposed. The algorithms are validated using Monte Carlo simulations.

On reachability and minimum cost optimal control

Questions of reachability for continuous and hybrid systems can be formulated as optimal control or game theory problems, whose solution can be characterized using variants of the Hamilton-Jacobi-Bellman or Isaacs partial differential equations. The formal link between the solution to the partial differential equation and the reachability problem is usually established in the framework of viscosity solutions. This paper establishes such a link between reachability, viability and invariance problems and viscosity solutions of a special form of the Hamilton-Jacobi equation. This equation is developed to address optimal control problems where the cost function is the minimum of a function of the state over a specified horizon. The main advantage of the proposed approach is that the properties of the value function (uniform continuity) and the form of the partial differential equation (standard Hamilton-Jacobi form, continuity of the Hamiltonian and simple boundary conditions) make the numerical solution of the problem much simpler than other approaches proposed in the literature. This fact is demonstrated by applying our approach to a reachability problem that arises in flight control and using numerical tools to compute the solution.

In silico feedback for in vivo regulation of a gene expression circuit

We show that difficulties in regulating cellular behavior with synthetic biological circuits may be circumvented using in silico feedback control. By tracking a circuits output in Saccharomyces cerevisiae in real time, we precisely control its behavior using an in silico feedback algorithm to compute regulatory inputs implemented through a genetically encoded light-responsive module. Moving control functions outside the cell should enable more sophisticated manipulation of cellular processes whenever real-time measurements of cellular variables are possible.

Moment-based inference predicts bimodality in transient gene expression

Recent computational studies indicate that the molecular noise of a cellular process may be a rich source of information about process dynamics and parameters. However, accessing this source requires stochastic models that are usually difficult to analyze. Therefore, parameter estimation for stochastic systems using distribution measurements, as provided for instance by flow cytometry, currently remains limited to very small and simple systems. Here we propose a new method that makes use of low-order moments of the measured distribution and thereby keeps the essential parts of the provided information, while still staying applicable to systems of realistic size. We demonstrate how cell-to-cell variability can be incorporated into the analysis obviating the need for the ubiquitous assumption that the measurements stem from a homogeneous cell population. We demonstrate the method for a simple example of gene expression using synthetic data generated by stochastic simulation. Subsequently, we use time-lapsed flow cytometry data for the osmo-stress induced transcriptional response in budding yeast to calibrate a stochastic model, which is then used as a basis for predictions. Our results show that measurements of the mean and the variance can be enough to determine the model parameters, even if the measured distributions are not well-characterized by low-order moments only—e.g., if they are bimodal.

Stochastic Hybrid Systems

STOCHASTIC HYBRID SYSTEMS: RESEARCH ISSUES AND AREAS Christos G. Cassandras and John Lygeros Introduction Modeling of Nondeterministic Hybrid Systems Modeling of Stochastic Hybrid Systems Overview of This Volume STOCHASTIC DIFFERENTIAL EQUATIONS ON HYBRID STATE SPACES Jaroslav Krystul, Henk A.P. Blom, and Arunabha Bagchi Introduction Semimartingales and Characteristics Semimartingale Strong Solution of SDE Stochastic Hybrid Processes as Solutions of SDE Instantaneous Hybrid Jumps at a Boundary Related SDE models on Hybrid State Spaces Markov and Strong Markov Properties Concluding Remarks COMPOSITIONAL MODELING OF STOCHASTIC HYBRID SYSTEMS Stefan Strubbe and Arjan van der Schaft Introduction Semantical Models Communicating PDPs Conclusions STOCHASTIC MODEL CHECKING Joost-Pieter Katoen Introduction The Discrete-Time Setting The Continuous-Time Setting Bisimulation and Simulation Relations Epilogue STOCHASTIC REACHABILITY: THEORY AND NUMERICAL APPROXIMATION Maria Prandini and Jianghai Hu Introduction Stochastic Hybrid System Model Reachability Problem Formulation Numerical Approximation Scheme Reachability Computations Possible Extensions Some Examples Conclusion STOCHASTIC FLOW SYSTEMS: MODELING AND SENSITIVITY ANALYSIS Christos G. Cassandras Introduction Modeling Stochastic Flow Systems Sample Paths of Stochastic Flow Systems Optimization Problems in Stochastic Flow Systems Infinitesimal Perturbation Analysis (IPA) Conclusions PERTURBATION ANALYSIS FOR STOCHASTIC FLOW SYSTEMS WITH FEEDBACK Yorai Wardi, George Riley, and Richelle Adams Introduction SFM with Flow Control Retransmission-Based Model Simulation Experiments Conclusions STOCHASTIC HYBRID MODELING OF ON-OFF TCP FLOWS Joao Hespanha Related Work A Stochastic Model for TCP Analysis of the TCP SHS Models Reduced-Order Models Conclusions STOCHASTIC HYBRID MODELING OF BIOCHEMICAL PROCESSES Panagiotis Kouretas, Konstantinos Koutroumpas, John Lygeros, and Zoi Lygerou Introduction Overview of PDMP Subtilin Production by B. subtilis DNA Replication in the Cell Cycle Concluding Remarks FREE FLIGHT COLLISION RISK ESTIMATION BY SEQUENTIAL MC SIMULATION Henk A.P. Blom, Jaroslav Krystul, G.J. (Bert) Bakker, Margriet B. Klompstra, and Bart Klein Obbink Introduction Sequential MC Estimation of Collision Risk Development of a Petri Net Model of Free Flight Simulated Scenarios and Collision Risk Estimates Concluding Remarks INDEX

On Submodularity and Controllability in Complex Dynamical Networks

Controllability and observability have long been recognized as fundamental structural properties of dynamical systems, but have recently seen renewed interest in the context of large, complex networks of dynamical systems. A basic problem is sensor and actuator placement: choose a subset from a finite set of possible placements to optimize some real-valued controllability and observability metrics of the network. Surprisingly little is known about the structure of such combinatorial optimization problems. In this paper, we show that several important classes of metrics based on the controllability and observability Gramians have a strong structural property that allows for either efficient global optimization or an approximation guarantee by using a simple greedy heuristic for their maximization. In particular, the mapping from possible placements to several scalar functions of the associated Gramian is either a modular or submodular set function. The results are illustrated on randomly generated systems and on a problem of power-electronic actuator placement in a model of the European power grid.

Approximate Model Checking of Stochastic Hybrid Systems

A method for approximate model checking of stochastic hybrid systems with provable approximation guarantees is proposed. We focus on the probabilistic invariance problem for discrete time stochastic hybrid systems and propose a two-step scheme. The stochastic hybrid system is first approximated by a finite state Markov chain. The approximating chain is then model checked for probabilistic invariance. Under certain regularity conditions on the transition and reset kernels governing the dynamics of the stochastic hybrid system, the invariance probability computed using the approximating Markov chain is shown to converge to the invariance probability of the original stochastic hybrid system, as the grid used in the approximation gets finer. A bound on the convergence rate is also provided. The performance of the two-step approximate model checking procedure is assessed on a case study of a multi-room heating system.

Stabilization of a class of stochastic differential equations with Markovian switching

Stability of stochastic differential equations with Markovian switching has been studied quite extensively for a number of years, for example, by Basak et al. (J. Math. Anal. Appl. 202 (1996) 604–622), Ji and Chizeck (IEEE Trans. Automat. Control 35 (1990) 777–788), Mariton (Jump Linear Systems in Automatic Control, Marcel Dekker, New York, 1990), Mao et al. (Stochastic Process. Appl. 79 (1999) 45–67; Bernoulli 6 (2000) 73–90) and Yuan and Lygeros (in: R. Alur, G. Pappas (Eds.), Hybrid Systems: Computation and Control, Seventh International Workshop, HSCC 2004, Lecture Notes in Computer Science, vol. 2993, Springer, Berlin, 2004, pp. 646–659). By contrast, the problem of designing controllers to stabilize systems of this type has received relatively little attention. In this paper we study the problem of mean square exponential stabilization for a class of stochastic differential equations with Markovian switching.

Monte Carlo Optimization for Conflict Resolution in Air Traffic Control

The safety of flights, and, in particular, separation assurance, is one of the main tasks of air traffic control (ATC). Conflict resolution refers to the process used by ATCs to prevent loss of separation. Conflict resolution involves issuing instructions to aircraft to avoid loss of safe separation between them and, at the same time, direct them to their destinations. Conflict resolution requires decision making in the face of the considerable levels of uncertainty inherent in the motion of aircraft. In this paper, a framework for conflict resolution that allows one to take into account such levels of uncertainty using a stochastic simulator is presented. The conflict resolution task is posed as the problem of optimizing an expected value criterion. It is then shown how the cost criterion can be selected to ensure an upper bound on the probability of conflict for the optimal maneuver. Optimization of the expected value resolution criterion is carried out through an iterative procedure based on Markov chain Monte Carlo. Simulation examples inspired by current ATC practice in terminal maneuvering areas and approach sectors illustrate the proposed conflict resolution strategy