A well-known property of linear resistive electrical networks is that the current distribution minimizes the total dissipated power. When the circuit includes resistors with nonlinear monotonic characteristic, the current distribution minimizes in general a different functional. We show that, if the nonlinear characteristic is a threshold-like function and the current generator is concentrated in a single point, as in the case of lightning or dielectric discharge, then the current flow is concentrated along a single path, which is a minimum path to the ground with respect to the threshold. We also propose a dynamic model that explains and qualitatively reproduces the lightning transient behavior: initial generation of several plasma branches and subsequent dismissal of all branches but the one reaching the ground first, which is the optimal one.

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A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition and cooperation is presented and analysed. The state of each agent is represented by its payoff, which the agent aims at maximising. The role of control variables is played by the model parameters, which account for the agents' decisions to either cooperate with or boycott the other agents and can vary in time within assigned intervals. Alliances and enmities can be established at any time, according to either a greedy or a longsighted criterion. The general nonlinear case is first considered. It is proved that, under realistic assumptions, the system evolution is bounded positive (no extinction) and there is a unique globally-stable equilibrium point. As is somehow expected, the optimal decision for all agents corresponds to full cooperation (decision parameters kept at their positive maximum value) in the case of both shortsighted and farsighted criteria. This is not true if some parameters have negative upper bounds (meaning that some agents systematically boycott some others). Then, in the linear case, it is shown that the system is stable for arbitrarily-varying decision parameters, provided that a Metzler matrix associated with full cooperation is Hurwitz. A characterisation of the long-term behaviour of the linear system is also provided. In particular, it is proved that, under stability conditions, a Nash equilibrium exists if a steady strategy is adopted.@en

To control the flow in a dynamical network where the nodes are associated with buffer variables and the arcs with controlled flows, we consider a network-decentralised strategy such that each arc controller makes its decision exclusively based on local information about the levels of the buffers that it connects. We seek a flow control law that asymptotically minimises a cost specified in terms of a weighted L₁-norm. This approach has the advantage of providing a solution that is generally sparse, because it uses a limited number of controlled flows. In particular, in the presence of a resource demand applied on a single node, the asymptotic flow is concentrated along the shortest path.

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This paper proposes a decentralised explicit (closed-form) iterative formula that solves convex programming problems with linear equality constraints and interval bounds on the decision variables. In particular, we consider a team of decision agents, each setting the value of a subset of the variables, and a team of information agents, in charge of ensuring that the equality constraints are fulfilled. The structure of the constraint matrix imposes a communication pattern between decision and information agents, which can be represented as a bipartite graph. We associate each information agent with an integral variable and each decision agent with a saturated function, which takes the interval bounds into account, and we design a decentralised dynamic mechanism that globally converges to the optimal solution. Under mild conditions, the convergence is shown to be exponential. We also provide a discrete-time algorithm, based on the Euler system, and we give an upper bound for the step parameter to ensure convergence. Although the considered optimisation problem is static, we show that the proposed scheme can be successfully applied to find the optimal solution of network-decentralised dynamic control problems.

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To detect faults in a system we can adopt an observer, designed for the healthy system, and monitor the discrepancy between actual and expected behaviour of the residual (difference between the system output and its estimate). To isolate faults, we can compute the invariant sets associated with each fault, and their projection in the residual space (limit set): faults can be isolated if the associated limit sets are separated when a (constant) test input is applied. However, the explicit computation of limit sets can be hard even for low-dimensional systems. As a main contribution, we show that, by adopting an implicit representation of limit sets, very efficient procedures can be used to solve the problem, based on convex quadratic programming or linear programming. Simulations show that the approach is effective in solving even large dimensional problems, which makes it suitable for large- scale networked systems.@en

This paper investigates the topology-independent stability of homogeneous dynamical networks, composed of interconnected equal systems. Precisely, dynamical systems with identical nominal transfer function F(s) are associated with the nodes of a directed graph, whose arcs account for their dynamic interactions, described by a common nominal transfer function G(s). It is shown that topology-independent stability is guaranteed for all possible interconnections with interaction degree (defined as the maximum number of arcs leaving a node) equal at most to N if the ∞-norm of the complementary sensitivity function NF(s)G(s)[1+NF(s)G(s)]^-1 is less than 1. This bound is nonconservative in that there exist graphs with interaction degree N that are unstable for an ∞-norm greater than 1. When nodes and arcs transferences are affected by uncertainties with norm bound K> 0, topology-independent stability is robustly ensured if the ∞-norm is less than 1/(1+2NK). For symmetric systems, stability is guaranteed for all topologies with interaction degree at most N if the Nyquist plot of NF(s)G(s) does not intersect the real axis to the left of-1/2. The proposed results are applied to fluid networks and platoon formation.

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The paper presents conditions for the stability of a dynamical network described by a directed graph, whose nodes represent dynamical systems characterised by the same transfer function F(s) and whose edges account for the interactions between pairs of nodes. In turn, these interactions depend via a transference G(s) on the outputs of the subsystems associated with the connected nodes. The stability conditions are topology-independent, in that they hold for all possible connections of the nodes, and robust, in that they allow for uncertainties in the determination of the transferences. Two types of interactions are considered: bidirectional and unidirectional. In the first case, if nodes i and j are connected, both node i affects node j and node j affects node i, while in the second case only one of the two occurrences is admitted. The robust stability conditions are expressed as constraints for the Nyquist diagram of H = FG.@en

We consider the problem of controlling the transmission rate in a communication network where each node adjusts its own transmission rate exclusively based on the physical medium occupation (the band occupancy of the aggregate complementary nodes). We show how to design a decentralized control for maximizing band occupancy and fairness. If the network is fully connected, the problem admits a simple solution. Difficulties arise in the case of partially connected networks, and in the presence of time-varying network topologies and delays. General conditions are given which, by properly tuning control parameters, ensure stability. These conditions are conservative and affect system performance. However, we show that in the case of symmetric connections, stability can be studied based on the system eigenvalues even in the presence of topology switchings. Less conservative bounds can be inferred by exploiting known properties of the eigenvalues of the adjacency matrix of a graph. We finally consider the multichannel case, in which nodes may jump among channels: The previous scheme can be extended to this case and asymptotically ensures uniform channel exploitation.

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This paper presents the mathematical conditions and the associated design methodology of an active fault diagnosis technique for continuous-Time linear systems. Given a set of faults known a priori, the system is modeled by a finite family of linear time-invariant systems, accounting for one healthy and several faulty configurations. By assuming bounded disturbances and using a residual generator, an invariant set and its projection in the residual space (i.e., its limit set) are computed for each system configuration. Each limit set, related to a single system configuration, is parameterized with respect to the system input. Thanks to this design, active fault isolation can be guaranteed by the computation of a test input, either constant or periodic, such that the limit sets associated with different system configurations are separated, and the residual converges toward one limit set only. In order to alleviate the complexity of the explicit computation of the limit set, an implicit dual representation is adopted, leading to efficient procedures, based on quadratic programming, for computing the test input. The developed methodology offers a competent continuous-Time solution to the optimization-based computation of the test input via Hahn-Banach duality. Simulation examples illustrate the application of the proposed active fault diagnosis methods and its efficiency in providing a solution, even in relatively large state-dimensional problems.

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The paper presents and analyses some nonlinear continuous-time dynamic models of social systems whose members, groups or individuals, may change partners and/or opponents at any time, according to a greedy criterion. The main structural properties of these models, which belong to the class of positive switching systems, are investigated with particular regard to the existence of solutions, their positivity, boundedness and asymptotic behaviour. Simulations show how the cooperative or hostile attitudes of the participants affect their yield.@en

A major drawback hinders the application of Model Predictive Control (MPC) to the regulation of electromechanical systems or, more generally, systems with fast dynamics: the time needed for the online computation of the control is often too long with respect to the sampling time. This paper shows how this problem can be overcome by suitably implementing the MPC technique. The main idea is to compute the control law using the discrete-time Euler Auxiliary System (EAS) associated with the continuous-time plant, and apply the control obtained for the discrete-time system to the continuous-time system. In this way the implementation sampling time can be much smaller than the EAS time parameter, which leads to significant savings in computation time. Theoretical results guarantee stabilisation, constraint satisfaction and robustness of such a control strategy, which is applied to the control of an electric drive and a cart-pendulum system.@en

This paper addresses set invariance properties for linear time-delay systems. More precisely, the first goal of the article is to review known necessary and/or sufficient conditions for the existence of invariant sets with respect to dynamical systems described by linear discrete time-delay difference equations (dDDEs). Secondly, we address the construction of invariant sets in the original state space (also called ^D-invariant sets) by exploiting the forward mappings. The notion of D-invariance is appealing since it provides a region of attraction, which is difficult to obtain for delay systems without taking into account the delayed states in some appropriate extended state space model. The present paper contains a sufficient condition for the existence of ellipsoidal D-contractive sets for dDDEs, and a necessary and sufficient condition for the existence of D-invariant sets in relation to linear time-varying dDDE stability. Another contribution is the clarification of the relationship between convexity (convex hull operation) and D-invariance of linear dDDEs. In short, it is shown that the convex hull of the union of two or more D-invariant sets is not necessarily D-invariant, while the convex hull of a non-convex D-invariant set is D-invariant.

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The paper analyses some continuous-time dynamic models that describe the evolution of social systems characterized by the possibility of changing the alliances among the parties involved or damaging one's competitors. At any time each participant, either an individual or a coalition, can decide to form or terminate a bond, or to start or stop damaging an opponent (i.e., to switch from a network configuration to another), based on a greedy, or shortsighted, criterion that does not consider the long-term effects of the decision. The proposed models fall within the domain of positive switching systems. Although they can obviously simulate the behaviour of many real-life situations, in which contenders aim at prevailing over one another to achieve supremacy, the paper does not refer to a specific context and concentrates on the main structural properties of the mathematical models, such as positivity and boundedness of the solutions, existence of coalitions, steady-state behaviour. Simulations show how the different cooperative or hostile attitudes of the participants affect their yield.

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In this chapter, a control design approach based on linear parametervarying (LPV) systems, which can be exploited to solve several problems typically encountered in control engineering, is presented. By means of recent techniques based on Youla–Kucera parametrization, it is shown how it is possible not only to design and optimize stabilizing controllers, but also to exploit the structure of the Youla–Kucera parametrized controller to face and solve side problems, including: (a) dealing with nonlinearities; (b) taking into account control input constraints; (c) performing controller commutation or online adaptation, e.g., in the presence of faults; and (d) dealing with delays in the system. The control scheme is observerbased, namely a prestabilizing observer-based precompensator is applied. Consequently, a Youla–Kucera parameter is applied to produce a supplementary input ignition, which is a function of the residual value (the difference between the output and the estimated output). Based on the fact that any stable operator which maps the residual to the supplementary input preserves stability, several additional features can be added to the compensator, without compromising the loop stability.

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