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The ranking of nodes in a network according to their centrality or ``importance'' is a classic problem that has attracted the interest of different scientific communities in the last decades. The COVID-19 pandemic has recently rejuvenated the interest in this problem, as the ranking may be used to decide who should be tested, or vaccinated, first, in a population of asymptomatic individuals. In this paper, we review classic methods for node ranking and compare their performance in a benchmark network that considers the community-based structure of society. The outcome of the ranking procedure is then used to decide which individuals should be tested, and possibly quarantined, first. Finally, we also review the extension of these ranking methods to weighted graphs and explore the importance of weights in a contact network by providing a toy model and comparing node rankings for this case in the context of disease spread.

We develop a small-gain theory for systems described by set-valued maps between topological spaces. We introduce an abstract notion of stability unifying the continuity properties behind different existing concepts, such as Lyapunov stability of equilibria, sets, or motions, (incremental) input-output stability, asymptotic gain properties, and continuity with respect to fast-switching inputs. Then, we prove that a feedback interconnection enjoying a given abstract small-gain property is stable. While, in general, the proposed small-gain property cannot be decomposed as the union of stability of the subsystems and a contractiveness condition, we show that it is implied by standard assumptions in the context of input-to-state stable systems. Finally, we provide application examples illustrating how the developed theory can be used for the analysis of interconnected systems and synthesis of control systems.

Traffic congestion in urban environment is one of the most critical issue for drivers and city planners for both environment and efficiency reasons. Traffic lights are one of the main tools used to regulate traffic by diverting the drivers between different paths. Rational drivers, in turn, react to the traffic light duration by evaluating their options and, if necessary, by changing direction in order to reach their destination quicker. In this paper, we introduce a macroscopic traffic model for urban intersections that incorporates this rational behavior of the drivers. Then, we exploit it to show that, by providing additional information about the expected red-time duration to the drivers, one can decrease the amount of congestion in the network and the overall length of the queues at the intersections. Additionally, we develop a control policy for the traffic lights that exploits the reaction of the drivers in order to divert them to a different route to further increase the performances. These claims are supported by extensive numerical simulations.

State estimation for traffic networks is a particu-larly challenging problem in view of their large dimensionality, and since models are often inaccurate and the interaction pat-terns unpredictable. In this article, we approach the problem by mixing aggregation-based complexity reduction and nonlinear filtering. We subdivide vehicles into groups and derive a lower-dimensional approximate model where vehicles belonging to the same group are represented by a unique random variable matching their average characteristics. Then, we propose a procedure to estimate the statistical properties of the group variables from partial measurements. Connections to car-following models are discussed, and the developed methodology is illustrated through numerical simulations.

We consider an optimal traffic-light control framework for urban traffic intersections to alleviate congestion phenomena. We analyze a scenario in which we provide drivers with information about the waiting time at the intersections. We model the drivers’ lane-changing information-based behavior as the solution of a convex optimization problem. We compute the optimal traffic-light control mechanism as the solution to a bi-level optimization problem. We provide a complete analysis in terms of (i) the existence of a solution; (ii) an iterative algorithm to compute it; (iii) sufficient conditions for the solution’s uniqueness and the algorithm’s convergence. Early simulation results show the proposed control scheme’s effectiveness compared with an optimal control algorithm in the absence of waiting-time information.

The paper deals with the distributed minimum sharing problem: a set of decision-makers compute the minimum of some local quantities of interest in a distributed and decentralized way by exchanging information through acommunication network. We propose an adjustable approximate solution which enjoys several properties of crucial importance in applications. In particular, the proposed solution has good decentralization properties and it is scalable in that the number of local variables does not grow with the size or topology of the communication network. Moreover, a global and uniform (both in the initial time and in the initial conditions)asymptotic stabilityresult is provided towards a steady state which can be made arbitrarily close to the sought minimum. Exact asymptotic convergence can be recovered at the price of losing uniformity with respect to the initial time.

COVID-19 abatement strategies have risks and uncertainties which could lead to repeating waves of infection. We show—as proof of concept grounded on rigorous mathematical evidence—that periodic, high-frequency alternation of into, and out-of, lockdown effectively mitigates second-wave effects, while allowing continued, albeit reduced, economic activity. Periodicity confers (i) predictability, which is essential for economic sustainability, and (ii) robustness, since lockdown periods are not activated by uncertain measurements over short time scales. In turn—while not eliminating the virus—this fast switching policy is sustainable over time, and it mitigates the infection until a vaccine or treatment becomes available, while alleviating the social costs associated with long lockdowns. Typically, the policy might be in the form of 1-day of work followed by 6-days of lockdown every week (or perhaps 2 days working, 5 days off) and it can be modified at a slow-rate based on measurements filtered over longer time scales. Our results highlight the potential efficacy of high frequency switching interventions in post lockdown mitigation. All code is available on Github athttps://github.com/V4p1d/FPSP_Covid19. A software tool has also been developed so that interested parties can explore the proof-of-concept system.

Output regulation refers to the class of control problems in which some outputs of the controlled system must be steered to some desired references, while maintaining closed-loop stability and in spite of the presence of unmeasured disturbances and model uncertainties. While for linear systems the problem has been elegantly solved in the 70s, output regulation for nonlinear systems is still a challenging research field, and 30 years of active research left open many fundamental problems. In particular, all the regulators proposed so far are limited to very specific classes of nonlinear systems and, even in those cases, they fail in extending in their full generality the celebrated properties of the linear regulator. The aim of this thesis is to make a decisive step towards the systematic extension of the output regulation theory to embrace more general multivariable problems. To this end, we touch here three fundamental pillars of regulation theory: the structure of regulators, the robustness issue, and the adaptation of the control system. Regarding the structural aspects, we pursue here a design paradigm that is complementary to canonical nonlinear regulators and that trades a conceptually more suitable structure with a strong internal intertwining between the different parts of the regulator. For what concerns robustness, we introduce a new framework to characterize robustness of regulators relative to steady-state properties more general than the usual requirement asking a zero asymptotic error. We characterize in this unifying terms a large part of the existing approaches, and we end conjecturing that general nonlinear regulation admits no robust solution. Regarding the evolution of regulators, we propose an adaptive regulation framework in which adaptation is used online to tune the internal models embedded in the control system. Adaptation is cast as a general system identification problem, allowing for different well-known algorithms to be used.

International audience; We propose an adaptive observer scheme to estimate boundary parameters in first-order hyperbolic systems of Partial Differential Equations (PDE). The considered systems feature an arbitrary number of states travelling in one direction and one counter-convecting state. Uncertainties in the boundary reflection coefficients and boundary additive errors are estimated relying on a pre-existing observer design and a novel Lyapunov-based adaptation law.

We propose an adaptive observer scheme to estimate boundary parameters in first-order hyperbolic systems of Partial Differential Equations (PDE). The considered systems feature an arbitrary number of states traveling in one direction and one counter-convecting state. Uncertainties in the boundary reflection coefficients and boundary additive errors are estimated relying on a pre-existing observer design and a novel Lyapunov-based adaptation law.

International audience; In this paper we propose a general framework in which the robustness properties and requirements of output regulation schemes can be formally described. We introduce a topological definition of robustness relative to arbitrary steady state properties, extending the usual notion of robustness relative to the existence of a steady state in which the regulation error vanishes. We review some of the main control approaches for linear and nonlinear systems, by re-framing their robustness properties within the proposed setting. We show that the celebrated robustness property of the linear regulator, namely the ``internal model principle'' stated by Francis, Wonham and Davison in the 70's, can be generalized to nonlinear systems in a robustness property relative to the Fourier expansion of the regulation error. We then focus on nonlinear regulation, where we show that only practical regulation can be achieved robustly, while asymptotic regulation is achieved in a quite fragile way. The paper concludes with a conjecture stating that, in a general nonlinear context, asymptotic regulation cannot be achieved in a robust way with a finite dimensional regulator.

The problem of output regulation has always been tackled in frameworks in which the references to be tracked and the disturbances to be rejected are generated by an autonomous differential equation, referred to as the exosystem. This assumption, that is routinely used in the design of asymptotic regulators, plays also a fundamental role in the formulation of the regulation problem and in the definition of the basic concepts such as the steady state and the zero dynamics of nonlinear systems. In this paper we show that the concepts of steady state, zero dynamics and the output regulation problem can be equivalently defined in a framework in which the exosystem is generated by a differential inclusion.

In this paper we propose an adaptive regulator for general multivariable linear systems to deal with references and disturbances generated by an unknown exosystem. The proposed regulator merges a continuous-time internal model and a discrete-time least-squares identifier that adapts the internal model parameters. We show that, under a suitable persistence of excitation condition, asymptotic regulation is achieved whenever an upper bound on the dimension of the exosystem is known.

In this paper we review the problem of output regulation for nonlinear systems. We discuss how the robustness properties and the simple structure of the linear regulator are linear artefacts which do not extend in more general nonlinear cases. We talk about the intertwining that is necessarily present between the internal model and the stabilising parts of a nonlinear regulator, in which the role of the exosystem mixes up with those of the residual plant's dynamics. We discuss a general guideline to deal with such structural challenges, by looking at adaptation and universal approximators as a promising way to provide systematic design procedures for approximate regulators in a general nonlinear setting.

This paper deals with the problem of output regulation for minimum-phase single-input single-output nonlinear systems. We propose a variation of the “regression-like” design introduced by Byrnes and Isidori (2004), showing how it is possible to move the internal model from the control input to the regulation error. In this way, we obtain a post-processing scheme that more likely resembles the general design for linear systems. Although these approaches are equivalent for single-input single-output systems, a post-processing design represents a step forward in the systematic extension of the theory to the general multivariable case.

International audience; In this paper we introduce the low-power high-gain observer, developed in [1], to solve problems of output regulation for nonlinear systems. We show how the new tool makes it possible the implementation of high dimensional controllers, that tipically arise when the ideal steady-state control that must be generated to secure zero regulation error is affected by uncertainties.