A Small-Gain Theory for Abstract Systems on Topological Spaces

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 underlying 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 design of control systems

Necessary Conditions for Output Regulation with Exosystem Modelled by Differential Inclusions

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

Adaptive Output Regulation For Multivariable Nonlinear Systems Via Hybrid Identification Techniques

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

A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization

In this paper we consider a recently developed distributed optimization algorithm based on gradient tracking. We propose a system theory framework to analyze its structural properties on a preliminary, quadratic optimization set-up. Specifically, we focus on a scenario in which agents in a static network want to cooperatively minimize the sum of quadratic cost functions. We show that the gradient tracking distributed algorithm for the investigated program can be viewed as a sparse closed-loop linear system in which the dynamic state-feedback controller includes consensus matrices and optimization (stepsize) parameters. The closed-loop system turns out to be not completely reachable and asymptotic stability can be shown restricted to a proper invariant set. Convergence to the global minimum, in turn, can be obtained only by means of a proper initialization. The proposed system interpretation of the distributed algorithm provides also additional insights on other structural properties and possible design choices that are discussed in the last part of the paper as a starting point for future developments

A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization

In this paper we consider a recently developed distributed optimization algorithm based on gradient tracking. We propose a system theory framework to analyze its structural properties on a preliminary, quadratic optimization set-up. Specifically, we focus on a scenario in which agents in a static network want to cooperatively minimize the sum of quadratic cost functions. We show that the gradient tracking distributed algorithm for the investigated program can be viewed as a sparse closed-loop linear system in which the dynamic state-feedback controller includes consensus matrices and optimization (stepsize) parameters. The closed-loop system turns out to be not completely reachable and asymptotic stability can be shown restricted to a proper invariant set. Convergence to the global minimum, in turn, can be obtained only by means of a proper initialization. The proposed system interpretation of the distributed algorithm provides also additional insights on other structural properties and possible design choices that are discussed in the last part of the paper as a starting point for future developments

On node ranking in graphs

The ranking of nodes in a network according to their ``importance'' is a classic problem that has attracted the interest of different scientific communities in the last decades. The current COVID-19 pandemic has recently rejuvenated the interest in this problem, as it is related to the selection of which individuals should be tested in a population of asymptomatic individuals, or which individuals should be vaccinated first. Motivated by the COVID-19 spreading dynamics, in this paper we review the most popular methods for node ranking in undirected unweighted graphs, and compare their performance in a benchmark realistic network, that takes into account the community-based structure of society. Also, we generalize a classic benchmark network originally proposed by Newman for ranking nodes in unweighted graphs, to show how ranks change in the weighted case

Post-lockdown abatement of COVID-19 by fast periodic switching

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 at https://github.com/V4p1d/FPSP_Covid19. A software tool has also been developed so that interested parties can explore the proof-of-concept system