A constructive tool of nonlinear control system design, the method of control Lyapunov functions (CLFs), has found numerous applications in stabilization problems for continuous-time, discrete-time, and hybrid systems. In this paper, we address the fundamental question: Given a CLF, corresponding to a continuous-time controller with some predefined (e.g., exponential) convergence rate, can the same convergence rate be provided by an event-triggered controller? Under certain assumptions, we give an affirmative answer to this question and show that the corresponding event-based controllers provide positive dwell times between consecutive events. Furthermore, we prove the existence of self-triggered and periodic event-triggered controllers, providing stabilization with a known convergence rate.

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This paper is concerned with the rate at which a discrete-time, deterministic, and possibly large network of nonlinear systems generates information, and so with the minimum rate of data transfer under which the addressee can maintain the level of awareness about the current state of the network. While being aimed at development of tractable techniques for estimation of this rate, this paper advocates benefits from directly treating the dynamical system as a set of interacting subsystems. To this end, a novel estimation method is elaborated that is alike in flavor to the small gain theorem on input-to-output stability. The utility of this approach is demonstrated by rigorously justifying an experimentally discovered phenomenon. The topological entropy of nonlinear time-delay systems stays bounded as the delay grows without limits. This is extended on the studied observability rates and appended by constructive upper bounds independent of the delay. It is shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics.

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Optimal decisions on the distribution of finite resources are explicitly structured by mathematical models that specify relevant variables, constraints, and objectives. Here we report analysis and evidence that implicit mathematical structures are also involved in group decision-making on resource allocation distributions under conditions of uncertainty that disallow formal optimization. A group’s array of initial distribution preferences automatically sets up a geometric decision space of alternative resource distributions. Weighted averaging mechanisms of interpersonal influence reduce the heterogeneity of the group’s initial preferences on a suitable distribution. A model of opinion formation based on weighted averaging predicts a distribution that is a feasible point in the group’s implicit initial decision space.

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In this paper, systems of nonlinear integro-differential Volterra equations are examined that can be represented as feedback interconnections of linear time-invariant block and periodic nonlinearities. The interest in such systems is motivated by their numerous applications in mechanical, electrical and communication engineering; examples include, but are not limited to, models of phase-locked loops, pendulum-like mechanical systems, coupled vibrational units and electric machines. Systems with periodic nonlinearities are usually featured by multistability and have infinite sequences of (locally) stable and unstable equilibria; their trajectories may exhibit nontrivial (e.g. chaotic) behavior. We offer frequency-domain criteria, ensuring convergence of any solution to one of the equilibria points, and this property is referred to as the gradient-like behavior and corresponds to phase locking in synchronization systems. Although it is hard to find explicitly the equilibrium, attracting a given trajectory, we give a constructive estimate for the distance between this limit equilibrium and the initial condition. The relevant estimates are closely related to the analysis of cycle slipping in synchronization systems. In the case where the criterion of gradient-type behavior fails, a natural question arises-which nonconverging solutions may exist in the system and, in particular, how many periodic solutions it has. We show that a relaxation of the frequency-domain convergence criterion ensures the absence of high-frequency periodic orbits. The results obtained in the paper are based on the method of integral quadratic constraints that has arisen in absolute stability theory and stems from Popov's techniques of "a priori integral indices". We illustrate the analytic results by numerical simulations.

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In this paper, we consider the mean-field model of noisy bounded confidence opinion dynamics under exogenous influence of static radical opinions. The long-term behavior of the model is analyzed by providing a sufficient condition for exponential convergence of the dynamics to stationary state. The stationary state is also characterized by a global estimate for a sufficiently large noise. Furthermore, we consider the order-disorder transition in the model in order to identify the effect of the (relative) mass of the radicals on the critical noise level at which this transition occurs. A numerical scheme for approximating the critical noise level is provided and validated through numerical simulations of the mean-field model and the corresponding agent-based model for a particular distribution of radical opinions.

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Whereas development of mathematical models describing the endocrine system as a whole remains a challenging problem, visible progress has been demonstrated in modeling its subsystems, or axes. Models of hormonal axes portray only the most essential interactions between the hormones and can be described by low-order systems of differential equations. This paper analyzes the properties of a novel model of a hypothalamic-pituitary axis, portraying the interactions in a chain of a release hormone (secreted by the hypothalamus), a tropic hormone (produced by the pituitary gland) and an effector hormone (secreted by a target gland). This model, unlike previously published ones, captures two prominent features of neurohormonal regulation systems, namely, the pulsatile (episodic) production of the release hormone and a complex non-cyclic feedback mechanism that maintains the involved hormone concentrations within certain biological limits. At the same time, the discussed model is analytically tractable; in particular, the existence of a so-called 1-cycle featured by a single pulse over one period is proven mathematically.

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In this paper, we consider synchronization of dynamical systems interconnected via nonlinear integral coupling. Integral coupling allows one to achieve synchronization with lower interaction levels (coupling gains) than with linear coupling. Previous results on this topic were obtained for synchronization of several systems with all-to-all interconnections. In this paper, we relax the requirement of all-to-all interconnections and provide two results on exponential synchronization under nonlinear integral coupling for networks with topologies different from all-to-all interconnections. In particular, we provide a high-gain result for an arbitrary interconnection topology and a non-high-gain method for analysis of synchronization for specific topologies. The results are illustrated by simulations of Hindmarsh-Rose neuron oscillators.

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The 2017 Nobel Prize in Physiology or Medicine awarded for discoveries of molecular mechanisms controlling the circadian rhythm has called attention to the challenging area of nonlinear dynamics that deals with synchronization and entrainment of oscillations. Biological circadian clocks keep time in living organisms, orchestrating hormonal cycles and other periodic rhythms. The periodic oscillations of circadian pacemakers are self-sustained; at the same time, they are entrainable by external periodic signals that adjust characteristics of autonomous oscillations. Whereas modeling of biological oscillators is a well-established research topic, mathematical analysis of entrainment, i.e. the nonlinear phenomena imposed by periodic exogenous signals, remains an open problem. Along with sustained periodic rhythms, periodically forced oscillators can exhibit various “irregular” behaviors, such as quasiperiodic or chaotic trajectories. This paper presents an overview of the mathematical models of circadian rhythm with respect to endocrine regulation, as well as biological background. Dynamics of the human endocrine system, comprising numerous glands and hormones operating under neural control, are highly complex. Therefore, only endocrine subsystems (or axes) supporting certain biological functions are usually studied. Low-order dynamical models that capture the essential characteristics and interactions between a few hormones can than be derived. Goodwin's oscillator often serves as such a model and is widely regarded as a prototypical biological oscillator. A comparative analysis of forced dynamics arising in two versions of Goodwin's oscillator is provided in the present paper: the classical continuous oscillator and a more recent impulsive one, capturing e.g. pulsatile secretion of hormones due to neural regulation. The main finding of this study is that, while the continuous oscillator is always forced to a periodic solution by a sufficiently large exogenous signal amplitude, the impulsive one commonly exhibits a quasiperiodic or chaotic behavior due to non-smooth dynamics in entrainment.

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To understand the sophisticated control mechanisms of the human's endocrine system is a challenging task that is a crucial step towards precise medical treatment of many dysfunctions and diseases. Although mathematical models describing the endocrine system as a whole are still elusive, recently some substantial progress has been made in analyzing theoretically its subsystems (or axes) that regulate the production of specific hormones. Secretion of many vital hormones, responsible for growth, reproduction and metabolism, is orchestrated by feedback mechanisms that are similar in structure to the model of simple genetic oscillators, proposed first by B.C. Goodwin. Unlike the celebrated Goodwin's model, the endocrine regulation mechanisms are in fact known to have non-cyclic structures and involve multiple feedbacks; a Goodwin-type model thus represents only a part of such a complicated mechanism. In this paper, we examine a non-cyclic feedback system of hormonal regulation, obtained from the classical Goodwin's oscillator by introducing an additional negative feedback. We establish global properties of this model and show, in particular, that the local instability of its unique equilibrium implies that almost all system's solutions oscillate; furthermore, under additional restrictions these solutions converge to periodic or homoclinic orbits.@en

In this paper, we provide an overview of recent works on dynamics of social networks and distributed algorithms for their exploration, contributed by Dr. Roberto Tempo (1956–2017) and his colleagues. These works, based on the recent achievements in multi-agent systems theory and distributed randomized algorithms, contribute in bridging the gap between the two classical sciences of Social Network Analysis and Systems and Control. The topics covered by this survey include distributed algorithms for analysis of complex social networks and novel dynamic models, describing opinion formation processes in such networks.

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The Goodwin oscillator is a simple yet illustrative model of a biochemical system with a stable limit cycle. Considered as a prototypical biological oscillator, Goodwin's model is broadly used e.g. to describe circadian rhythms, hormonal cycles, self-oscillatory metabolic pathways.These periodic or non-periodic oscillations are self-sustained; at the same time, they are entrainable by external periodic signals, adjusting the characteristics of the autonomous oscillatory behavior. Mathematical analysis of entrainment phenomena, i.e. nonlinear phenomena imposed by periodic exogenous signals, remains an open problem. This paper presents a comparative analysis of forced dynamics arising in two versions of Goodwin's oscillator: the classical continuous oscillator and a more recent impulsive one, e.g. capturing pulsatile secretion of hormones. The main finding of this study is that while the continuous oscillator is always forced to a periodic solution by a sufficiently large exogenous signal amplitude, the impulsive one commonly exhibits a quasiperiodic or chaotic behavior thus highlighting the role of non-smooth dynamics in entrainment.

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Control Lyapunov Functions (CLF) method gives a constructive tool for stabilization of nonlinear systems. To find a CLF, many methods have been proposed in the literature, e.g. backstepping for cascaded systems and sum of squares (SOS) programming for polynomial systems. Dealing with continuous-time systems, the CLF-based controller is also continuous-time, whereas practical implementation on a digital platform requires sampled-time control. In this paper, we show that if the continuous-time controller provides exponential stabilization, then an exponentially stabilizing event-triggered control strategy exists with the convergence rate arbitrarily close to the rate of the continuous-time system.

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Recent years have witnessed a significant trend towards filling the gap between Social Network Analysis (SNA) and control theory. This trend was enabled by the introduction of new mathematical models describing dynamics of social groups, the development of algorithms and software for data analysis and the tremendous progress in understanding complex networks and multi-agent systems (MAS) dynamics. The aim of this tutorial is to highlight a novel chapter of control theory, dealing with dynamic models of social networks and processes over them, to the attention of the broad research community. In its first part (Proskurnikov & Tempo, 2017), we have considered the most classical models of social dynamics, which have anticipated and to a great extent inspired the recent extensive studies on MAS and complex networks. This paper is the second part of the tutorial, and it is focused on more recent models of social processes that have been developed concurrently with MAS theory. Future perspectives of control in social and techno-social systems are also discussed.

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Recent tremendous progress in electronics, complexity theory and network science provides new opportunities for intellectual control of complex large-scale systems operating in turbulent environment via networks of interconnected miniature devices, serving as actuators, sensors and data processors. Actual dynamics of the resulting control systems are too sophisticated to be examined controlled by traditional methods, which primarily deal with ordinary differential equations. However, their complexity can be dramatically reduced by fast processes, organizing the elementary units of the system (called agents) into relatively small number of clusters. The clusters emerge and deteriorate in response to changes in the environment, and the processes of their formation and destruction are very short in time. During the periods of the clusters’ existence, the system’s dynamics is essentially low-dimensional due to synchronization between the agents in each cluster. An enormously complicated system is thus reduced to a finite-dimensional model with time-varying structure of the state vector. The low-dimensionality of the reduced model allows to control it by using classical methods, e.g. model-predictive or adaptive control. This philosophy of complex systems control is illustrated on an experimental setup, called the “airplane with feathers”. The wings of this airplane are equipped with arrays of microsensors, microcomputers, and mi-croactuators (“feathers”). The feathers self-organize into clusters by using a multi-agent consensus protocol; the aim of this coordination is to reduce the perturbing forces, affecting the airplane in a turbulent flow.

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In this paper, we propose an algorithm for path-following control of the nonholonomic mobile robot based on the idea of the guiding vector field (GVF). The desired path may be an arbitrary smooth curve in its implicit form, that is, a level set of a predefined smooth function. Using this function and the robot's kinematic model, we design a GVF, whose integral curves converge to the trajectory. A nonlinear motion controller is then proposed, which steers the robot along such an integral curve, bringing it to the desired path. We establish global convergence conditions for our algorithm and demonstrate its applicability and performance by experiments with wheeled robots.

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Systems that can be decomposed as feedback interconnections of stable linear blocks and periodic nonlinearities arise in many physical and engineering applications. The relevant models e.g. describe oscillations of a viscously damped pendulum, synchronization circuits (phase, frequency and delay locked loops) and networks of coupled power generators. A system with periodic nonlinearities usually has multiple equilibria (some of them being locally unstable). Many tools of classical stability and control theories fail to cope with such systems. One of the efficient methods, elaborated to deal with periodic nonlinearities, stems from the celebrated Popov method of 'integral indices', or integral quadratic constraints; this method leads, in particular, to frequency-domain criteria of the solutions' convergence, or, equivalently, global stability of the equilibria set. In this paper, we further develop Popov's method, addressing the problem of robustness of the convergence property against external disturbances that do not oscillate at infinity (allowing the system to have equilibria points). Will the forced solutions also converge to one of the equilibria points of the disturbed system? In this paper, a criterion for this type of robustness is offered.

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The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe chaotic systems via communication channels with finite data rates, with a focus on minimum data-rates needed for various types of observability. With the objective of developing tractable techniques to estimate these rates, the paper discloses benefits from regard to the operational structure of the system in the case where the system is representable as a feedback interconnection of two subsystems with inputs and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general nonlinear time-delay systems by rigorously justifying an experimentally discovered phenomenon: Their topological entropy stays bounded as the delay grows without limits. This is extended on the studied observability rates and appended by constructive finite upper bounds independent of the delay. It is shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics.  @en

Singularly perturbed integro-differential Volterra equations with MIMO periodic nonlinearities are considered, which describe synchronization circuits (such as phase- and frequency-locked loops) and many other “pendulum-like” systems. Similar to the usual pendulum equation, such systems are typically featured by infinite sequences of equilibria points, and none of which can be globally asymptotically stable. A natural extension of the global asymptotic stability is the gradient-like behavior, that is, convergence of any solution to one of the equilibria. In this paper, we offer an efficient frequency-domain criterion for gradientlike behavior. This criterion is not only applicable to a broad class of infinite-dimensional systems with periodic nonlinearities, but in fact ensures the equilibria set stability under singular perturbation. In particular, the proposed criterion guarantees the absence of periodic solutions that are considered to be undesirable in synchronization systems. In this paper we also discuss a relaxed version of this criterion, which guarantees the absence of “high-frequency” periodic solutions, whose frequencies lie beyond a certain bounded interval.

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Synchronization among autonomous agents via local interactions is one of the benchmark problems in multi-agent control. Whereas synchronization algorithms for identical agents have been thoroughly studied, synchronization of heterogeneous networks still remains a challenging problem. The existing algorithms primarily use the internal model principle, assigning to each agent a local copy of some dynamical system (internal model). Synchronization of heterogeneous agents thus reduces to global synchronization of identical generators and local synchronization between the agents and their internal models. The internal model approach imposes a number of restrictions and leads to sophisticated dynamical (and, in general, nonlinear) controllers. At the same time, passive heterogeneous agents can be synchronized by a very simple linear protocol, which is used for consensus of first-order integrators. A natural question arises whether analogous algorithms are applicable to synchronization of agents that do not satisfy the passivity condition. In this paper, we study the synchronization problem for heterogeneous agents that are not passive but satisfy a weaker input feedforward passivity (IFP) condition. We show that such agents can also be synchronized by a simple linear protocol, provided that the interaction graph is strongly connected and the couplings are sufficiently weak. We demonstrate how stability of cooperative adaptive cruise control algorithms and some microscopic traffic flow models reduce to synchronization of heterogeneous IFP agents.

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In recent years, we have observed a significant trend towards filling the gap between social network analysis and control. This trend was enabled by the introduction of new mathematical models describing dynamics of social groups, the advancement in complex networks theory and multi-agent systems, and the development of modern computational tools for big data analysis. The aim of this tutorial is to highlight a novel chapter of control theory, dealing with applications to social systems, to the attention of the broad research community. This paper is the first part of the tutorial, and it is focused on the most classical models of social dynamics and on their relations to the recent achievements in multi-agent systems.

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