Christos G. Cassandras

Optimal control of a class of hybrid systems

We present a modeling framework for hybrid systems intended to capture the interaction of event-driven and time-driven dynamics. This is motivated by the structure of many manufacturing environments where discrete entities (termed jobs) are processed through a network of workcenters so as to change their physical characteristics. Associated with each job is a temporal state subject to event-driven dynamics and a physical state subject to time-driven dynamics. Based on this framework, we formulate and analyze a class of optimal control problems for single-stage processes. First-order optimality conditions are derived and several properties of optimal state trajectories (sample paths) are identified which significantly simplify the task of obtaining explicit optimal control policies.

Optimal control of hybrid systems in manufacturing

Hybrid systems combine time-driven and event-driven dynamics. This is a natural framework for manufacturing processes: The physical characteristics of production parts undergo changes at various operations described by time-driven models, while the timing control of operations is described by event-driven models. Accordingly, in the framework we propose, manufactured parts are characterized by physical states (e.g. temperature, geometry) subject to time-driven dynamics and by temporal states (e.g., operation start and stop times) subject to event-driven dynamics. We first provide a tutorial introduction to this hybrid system framework and associated optimal control problems through a single-stage manufacturing process model. We then show how the structure of the problem can be exploited to decompose what is a hard nonsmooth, nonconvex optimization problem into a collection of simpler problems. Next, we present extensions to multistage manufacturing processes for which we develop solution algorithms that make use of Bezier approximation techniques. Emphasis is given to the issue of deriving solutions through efficient algorithms, and some explicit numerical results are included.

Perturbation analysis for online control and optimization of stochastic fluid models

Uses stochastic fluid models (SFMs) for control and optimization (rather than performance analysis) of communication networks, focusing on problems of buffer control. We derive gradient estimators for packet loss and workload related performance metrics with respect to threshold parameters. These estimators are shown to be unbiased and directly observable from a sample path without any knowledge of underlying stochastic characteristics, including traffic and processing rates (i.e., they are nonparametric). This renders them computable in online environments and easily implementable for network management and control. We further demonstrate their use in buffer control problems where our SFM-based estimators are evaluated based on data from an actual system.

Distributed Cooperative coverage Control of Sensor Networks

We present a distributed coverage control scheme for cooperating mobile sensor networks. The mission space is modeled using a density function representing the frequency of random events taking place, with mobile sensors operating over a limited range defined by a probabilistic model. A gradient-based algorithm is designed requiring local information at each sensor and maximizing the joint detection probabilities of random events. We also incorporate communication costs into the coverage control problem, viewing the sensor network as a multi-source, single-basestation data collection network. Communication cost is modeled as the power consumption needed to deliver collected data from sensor nodes, thus trading off sensing coverage and communication cost. The control Scheme is tested in a simulation environment to illustrate its adaptive, distributed, and asynchronous properties.

Brief paper: Infinitesimal and finite perturbation analysis for queueing networks

The sample-path pertrubation analysis technique introduced in refs. (1-4) is extended to included finite (and possibly large) pertrubations typically introduced by changes in queue sizes or other parameter. It is shown that there is a natural hierarchy of perturbation analysis which takes care of increasingly large perturbations. Experiments with zeroth (infinitesmal) and first order (finite) pertrubation analysis show that significant accuracy improvement can be obtained with small increase in computational effort.

Discrete-Event Systems

Discrete Event Systems (DES) are characterized by the occurrence of discrete events asynchronously over time which are responsible for driving all dynamics. Such systems are ubiquitous in modern technological environments, ranging from communication networks and manufacturing to transportation and logistics. This class of event-driven dynamic systems is first compared to traditional time-driven systems described by differential (or difference) equations. We subsequently overview some of the major modeling frameworks for DES, based on automata, Petri nets, and dioid algebras. The use of these models is illustrated through queuing systems, a common class of DES.

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

Distributed coverage control and data collection with mobile sensor networks

We study wireless sensor networks whose objective is to control the locations of mobile nodes so as to maximize the probability of detecting randomly occurring events in a mission space and to extract information from data sources, when detected, with maximal effectiveness. The control system for the mission is composed of three components executed in parallel on sensor nodes: coverage control, data source detection, and data collection. In order to maximize the joint detection probability of random events in a given mission space, we build upon a previously developed distributed gradient-based coverage control scheme to include three additional capabilities: allowing polygonal obstacles, including limited sensing field-of-view constraints, and preserving network connectivity through a provably correct algorithm using each nodes routing information. In order to combine coverage and data collection, we formulate and solve a modified optimization problem with these two objectives. The interactions among the three components of the sensor network control system are discussed and simulation examples are presented to illustrate our results.

Optimal dispatching control for elevator systems during uppeak traffic

In this paper we develop optimal dispatching controllers for elevator systems during uppeak traffic. An uppeak traffic period arises when the bulk of the passenger traffic is moving from the first door up into the building (e.g., the start of a business day in an office building). The cars deliver the passengers and then return empty to the first floor to pick up more passengers. We show that the structure of the optimal dispatching policy minimizing the discounted or average passenger waiting time is a threshold-based policy. That is, the optimal policy is to dispatch an available car from the first floor when the number of passengers inside the car reaches or exceeds a threshold that depends on several factors including the passenger arrival rate, elevator performance capabilities, and the number of elevators available at the first floor. Since most elevator systems have sensors to determine the car locations and the number of passengers in each car, such a threshold policy is easily implemented. Our analysis is based on a Markov decision problem formulation with a batch service queueing model consisting of a single queue served by multiple finite-capacity bulk servers. We use dynamic programming techniques to obtain the structure of the optimal control policy and to derive some of its important properties. Several numerical examples are included to illustrate our results and to compare the optimal threshold policy to some known ad hoc approaches. Finally, since many transportation systems can be modeled as multiserver batch service queueing systems, we expect our results to be useful in controlling those systems as well.

Paper: A new approach to the analysis of discrete event dynamic systems

We present a new, time domain approach to the study of discrete event dynamical systems (DEDS), typified by queueing networks and production systems. A general state-space representation is developed and perturbation analysis is carried out. Observation of a single sample realization of such a system can be used to predict behavior over other sample realizations, when some parameter is perturbed, without having to make additional observations. Conditions under which this is always possible are investigated and explicit results for some special cases are included.