arXiv subjects

John Lygeros

Publications and source records attributed to John Lygeros.

At least 235 records · Page 13Linked to original sources

arXiv2016

Distributed Learning in the Presence of Disturbances

We consider a problem where multiple agents must learn an action profile that maximises the sum of their utilities in a distributed manner. The agents are assumed to have no knowledge of either the utility functions or the actions and payoffs of other agents. These assumptions arise when modelling the interactions in a complex system and communicating between various components of the system are both difficult. In [1], a distributed algorithm was proposed, which learnt Pareto-efficient solutions in this problem setting. However, the approach assumes that all agents can choose their actions, which precludes disturbances. In this paper, we show that a modified version of this distributed learning algorithm can learn Pareto-efficient solutions, even in the presence of disturbances from a finite set. We apply our approach to the problem of ramp coordination in traffic control for different demand profiles.

eess.SY↗

arXiv2016

The Stochastic Reach-Avoid Problem and Set Characterization for Diffusions

In this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective. Preceding approaches to solving these reachability problems are either confined to the deterministic setting or address almost-sure stochastic requirements. In contrast, we propose a methodology to tackle problems with less stringent requirements than almost sure. To this end, we first establish a connection between two distinct stochastic reach-avoid problems and three classes of stochastic optimal control problems involving discontinuous payoff functions. Subsequently, we focus on solutions of one of the classes of stochastic optimal control problems---the exit-time problem, which solves both the two reach-avoid problems mentioned above. We then derive a weak version of a dynamic programming principle (DPP) for the corresponding value function; in this direction our contribution compared to the existing literature is to develop techniques that admit discontinuous payoff functions. Moreover, based on our DPP, we provide an alternative characterization of the value function as a solution of a partial differential equation in the sense of discontinuous viscosity solutions, along with boundary conditions both in Dirichlet and viscosity senses. Theoretical justifications are also discussed to pave the way for deployment of off-the-shelf PDE solvers for numerical computations. Finally, we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.

math.OC↗

arXiv2016

A Tractable Fault Detection and Isolation Approach for Nonlinear Systems with Probabilistic Performance

This article presents a novel perspective along with a scalable methodology to design a fault detection and isolation (FDI) filter for high dimensional nonlinear systems. Previous approaches on FDI problems are either confined to linear systems or they are only applicable to low dimensional dynamics with specific structures. In contrast, shifting attention from the system dynamics to the disturbance inputs, we propose a relaxed design perspective to train a linear residual generator given some statistical information about the disturbance patterns. That is, we propose an optimization-based approach to robustify the filter with respect to finitely many signatures of the nonlinearity. We then invoke recent results in randomized optimization to provide theoretical guarantees for the performance of the proposed filer. Finally, motivated by a cyber-physical attack emanating from the vulnerabilities introduced by the interaction between IT infrastructure and power system, we deploy the developed theoretical results to detect such an intrusion before the functionality of the power system is disrupted.

math.OC↗

arXiv2016

Motion Planning for Continuous Time Stochastic Processes: A Dynamic Programming Approach

We study stochastic motion planning problems which involve a controlled process, with possibly discontinuous sample paths, visiting certain subsets of the state-space while avoiding others in a sequential fashion. For this purpose, we first introduce two basic notions of motion planning, and then establish a connection to a class of stochastic optimal control problems concerned with sequential stopping times. A weak dynamic programming principle (DPP) is then proposed, which characterizes the set of initial states that admit a control enabling the process to execute the desired maneuver with probability no less than some pre-specified value. The proposed DPP comprises auxiliary value functions defined in terms of discontinuous payoff functions. A concrete instance of the use of this novel DPP in the case of diffusion processes is also presented. In this case, we establish that the aforementioned set of initial states can be characterized as the level set of a discontinuous viscosity solution to a sequence of partial differential equations, for which the first one has a known boundary condition, while the boundary conditions of the subsequent ones are determined by the solutions to the preceding steps. Finally, the generality and flexibility of the theoretical results are illustrated on an example involving biological switches.

math.OC↗

arXiv2015

Calibration algorithm for spatial partial equilibrium models with conjectural variations

When calibrating spatial partial equilibrium models with conjectural variations, some modelers fit the suppliers' sales to the available data in addition to total consumption and price levels. While this certainly enhances the quality of the calibration, it makes it difficult to accommodate user-imposed bounds on the model parameters such as restricting the market power parameters to the interval [0,1], which is a common requirement in conjectural variations approaches. We propose an algorithm to calibrate the suppliers' sales and simultaneously deal with user-defined bounds on parameters. To this end, we fix the suppliers' sales at reference values and obtain the marginal costs for each supplier and market. We then limit the market power parameters to the interval [0,1], and calculate intervals of anchor prices and price elasticities that reproduce the reference supplier sales in the state of equilibrium. If these intervals do not contain the reference price elasticities and prices, we face a mismatch between reality and the model mechanics. We resolve this by altering the reference sales for the critical suppliers, and iterate. Thereby, the user controls whether price elasticities and anchor prices should be close to their reference values, or the suppliers' sales. The algorithm is tested on data from the European gas market, and required less than one minute to identify calibrated parameters. Our algorithm is widely applicable, since it is based on mild (and common) underlying assumptions and can be configured to suit a specific purpose thanks to the inclusion of user-defined bounds on all relevant parameters.

math.OC↗

arXiv2015

European Union gas market development

The recently announced Energy Union by the European Commission is the most recent step in a series of developments aiming at integrating the EU's gas markets to increase social welfare (SW) and security of gas supply. Based on a spatial partial equilibrium model, we analyze the changes in consumption, prices, and SW up to 2022 induced by the infrastructure expansions planned for this period. We find that wholesale prices decrease slightly and converge at Western European levels, the potential of suppliers to exert market power decreases significantly, and consumer surplus increases by 15.9% in the EU. Our results allow us to distinguish three categories of projects: (i) New gas sources developed and brought to the EU markets. These projects decrease prices and increase SW in a large number of countries. The only project in this category is the Trans-Anatolian Gas Pipeline; (ii) Existing gas sources made available to additional countries. This leads to an increase of SW in the newly connected countries, and a decrease everywhere else. These projects mainly involve pipeline and regasification terminal capacity enhancements; (iii) Projects with a marginal effect on the (fully functioning) market. Most storage expansion projects fall into this category, plus the recently announced Turkish Stream. Our results indicate that if all proposed infrastructure projects are realized, the EU's single market will become a reality in 2019. However, we also find that SW can only be increased significantly for the EU as a whole if new gas sources become accessible. Consequently, we suggest that the EU should emphasize on measures to increase the available volumes, in particular once the integration of the market is completed. At the same time, efficiency gains, albeit decreasing SW, help to improve the situation of consumers and decrease the dependency of the EU as a whole on external suppliers.

econ.GN↗

arXiv2015

Multiplicity of equilibria in conjectural variations models of natural gas markets

Spatial partial equilibrium models incorporating conjectural variations are widely used to analyze the development of oligopolistic multi-agent markets, such as international energy and raw material markets. Although this model type can produce multiple equilibria under commonly used assumptions, to the best of our knowledge, the consequences for the interpretation of the model results have not yet been explored in detail. To this end, we derive a linear complementarity model for the gas market and discuss under which assumptions on the model structure a component of the solution is unique. In particular, we find that the gas flow between a trader and a consumer is unique whenever the trader is modeled to exert market power in the consumer's market. We demonstrate our findings by computing the extreme points of the polyhedral solution space and show that erroneous conclusions could be drawn whenever only one (arbitrary) point in the solution space is picked for interpretation. Furthermore, we discuss whether economically meaningful parameter value changes exist which would enforce uniqueness in all components of the solution.

math.OC↗

arXiv2015

On the Sample Size of Random Convex Programs with Structured Dependence on the Uncertainty (Extended Version)

The "scenario approach" provides an intuitive method to address chance constrained problems arising in control design for uncertain systems. It addresses these problems by replacing the chance constraint with a finite number of sampled constraints (scenarios). The sample size critically depends on Helly's dimension, a quantity always upper bounded by the number of decision variables. However, this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability. We derive improved bounds of Helly's dimension for problems where the chance constraint has certain structural properties. The improved bounds lower the number of scenarios required for these problems, leading both to improved objective value and reduced computational complexity. Our results are generally applicable to Randomized Model Predictive Control of chance constrained linear systems with additive uncertainty and affine disturbance feedback. The efficacy of the proposed bound is demonstrated on an inventory management example.

math.OC↗

arXiv2015

Network Aggregative Games and Distributed Mean Field Control via Consensus Theory

We consider network aggregative games to model and study multi-agent populations in which each rational agent is influenced by the aggregate behavior of its neighbors, as specified by an underlying network. Specifically, we examine systems where each agent minimizes a quadratic cost function, that depends on its own strategy and on a convex combination of the strategies of its neighbors, and is subject to personalized convex constraints. We analyze the best response dynamics and we propose alternative distributed algorithms to steer the strategies of the rational agents to a Nash equilibrium configuration. The convergence of these schemes is guaranteed under different sufficient conditions, depending on the matrices defining the cost and on the network. Additionally, we propose an extension to the network aggregative game setting that allows for multiple rounds of communications among the agents, and we illustrate how it can be combined with consensus theory to recover a solution to the mean field control problem in a distributed fashion, that is, without requiring the presence of a central coordinator. Finally, we apply our theoretical findings to study a novel multi-dimensional, convex-constrained model of opinion dynamics and a hierarchical demand-response scheme for energy management in smart buildings, extending literature results.

eess.SY↗

arXiv2015

Upper bounds for the reach-avoid probability via robust optimization

We consider finite horizon reach-avoid problems for discrete time stochastic systems. Our goal is to construct upper bound functions for the reach-avoid probability by means of tractable convex optimization problems. We achieve this by restricting attention to the span of Gaussian radial basis functions and imposing structural assumptions on the transition kernel of the stochastic processes as well as the target and safe sets of the reach-avoid problem. In particular, we require the kernel to be written as a Gaussian mixture density with each mean of the distribution being affine in the current state and input and the target and safe sets to be written as intersections of quadratic inequalities. Taking advantage of these structural assumptions, we formulate a recursion of semidefinite programs where each step provides an upper bound to the value function of the reach- avoid problem. The upper bounds provide a performance metric to which any suboptimal control policy can be compared, and can themselves be used to construct suboptimal control policies. We illustrate via numerical examples that even if the resulting bounds are conservative, the associated control policies achieve higher reach-avoid probabilities than heuristic controllers for problems of large state-input space dimensions (more than 20). The results presented in this paper, far exceed the limits of current approximation methods for reach-avoid problems in the specific class of stochastic systems considered.

math.OC↗

arXiv2015

Decentralized Convergence to Nash Equilibria in Constrained Deterministic Mean Field Control

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and affected by the aggregate behavior of the overall population. For such large-scale systems, the theory of aggregative and mean field games has been established and successfully applied in various scientific disciplines. While the existing literature addresses the case of unconstrained agents, we formulate deterministic mean field control problems in the presence of heterogeneous convex constraints for the individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several model-free feedback iterations to compute in a decentralized fashion a mean field Nash equilibrium in the limit of infinite population size. We apply our methods to the constrained linear quadratic deterministic mean field control problem and to the constrained mean field charging control problem for large populations of plug-in electric vehicles.

eess.SY↗

arXiv2015

Efficient Approximation of Channel Capacities

We propose an iterative method for approximately computing the capacity of discrete memoryless channels, possibly under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. The presented method requires $O(M^2 N \sqrt{\log N}/\varepsilon)$ to provide an estimate of the capacity to within $\varepsilon$, where $N$ and $M$ denote the input and output alphabet size; a single iteration has a complexity $O(M N)$. We also show how to approximately compute the capacity of memoryless channels having a bounded continuous input alphabet and a countable output alphabet under some mild assumptions on the decay rate of the channel's tail. It is shown that discrete-time Poisson channels fall into this problem class. As an example, we compute sharp upper and lower bounds for the capacity of a discrete-time Poisson channel with a peak-power input constraint.

cs.IT↗

arXiv2014

On the computational complexity and generalization properties of multi-stage and recursive scenario programs

We discuss the computational complexity and feasibility properties of scenario based techniques for uncertain optimization programs. We consider different solution alternatives ranging from the standard scenario approach to recursive variants, and compare feasibility as a function of the total computation burden. We identify trade-offs between the different methods depending on the problem structure and the desired probability of constraint satisfaction. Our motivation for this work stems from the applicability and complexity reduction when making decisions by means of recursive algorithms. We illustrate our results on an example from the area of approximate dynamic programming

math.OC↗

arXiv2014

Topology Design for Optimal Network Coherence

We consider a network topology design problem in which an initial undirected graph underlying the network is given and the objective is to select a set of edges to add to the graph to optimize the coherence of the resulting network. We show that network coherence is a submodular function of the network topology. As a consequence, a simple greedy algorithm is guaranteed to produce near optimal edge set selections. We also show that fast rank one updates of the Laplacian pseudoinverse using generalizations of the Sherman-Morrison formula and an accelerated variant of the greedy algorithm can speed up the algorithm by several orders of magnitude in practice. These allow our algorithms to scale to network sizes far beyond those that can be handled by convex relaxation heuristics.

math.OC↗

arXiv2014

Optimal Sensor and Actuator Placement 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 an important class of metrics based on the controllability and observability Gramians has a strong structural property that allows efficient global optimization: the mapping from possible placements to the trace of the associated Gramian is a modular set function. We illustrate the results via placement of power electronic actuators in a model of the European power grid.

math.OC↗

arXiv2014

Submodularity of Energy Related Controllability Metrics

The quantification of controllability and observability has recently received new interest in the context of large, complex networks of dynamical systems. A fundamental but computationally difficult problem is the placement or selection of actuators and sensors that optimize real-valued controllability and observability metrics of the network. We show that several classes of energy related metrics associated with the controllability Gramian in linear dynamical systems have a strong structural property, called submodularity. This property allows for an approximation guarantee by using a simple greedy heuristic for their maximization. The results are illustrated for randomly generated systems and for placement of power electronic actuators in a model of the European power grid.

math.OC↗

arXiv2014

On the connection between compression learning and scenario based optimization

We investigate the connections between compression learning and scenario based optimization. We first show how to strengthen, or relax the consistency assumption at the basis of compression learning and study the learning and generalization properties of the algorithm involved. We then consider different constrained optimization problems affected by uncertainty represented by means of scenarios. We show that the issue of providing guarantees on the probability of constraint violation reduces to a learning problem for an appropriately chosen algorithm that enjoys compression learning properties. The compression learning perspective provides a unifying framework for scenario based optimization and allows us to revisit the scenario approach and the probabilistically robust design, a recently developed technique based on a mixture of randomized and robust optimization, and to extend the guarantees on the probability of constraint violation to cascading optimization problems.

eess.SY↗

arXiv2014

Variance reduction for steady-state simulation and sensitivity analysis of stochastic chemical systems

We address the problem of estimating steady-state quantities associated to systems of stochastic chemical kinetics. In most cases of interest these systems are analytically intractable, and one has to resort to computational methods to estimate stationary values of cost functions. In this work we consider a previously introduced variance reduction method and present an algorithm for its application in the context of stochastic chemical kinetics. Using two numerical examples, we test the efficiency of the method for the calculation of steady-state parametric sensitivities and evaluate its performance in comparison to other estimation methods.

q-bio.QM↗