Input and State Observability of Network Systems with a Single Unknown Input

International audienceThis paper studies network systems affected by a single unknown input, possibly representing an attack or a failure, to be estimated. The main result is a characterization of input and state observability, namely the conditions under which both the whole network state and the unknown input can be reconstructed from some measured local states. This characterization is in terms of observability of a suitably-defined subsystem, which allows the use of known graphical charactizations of observability of network systems, leading to structural results (true for almost all interaction weights) or strong structural results (true for all non-zero interaction weights). We apply our results to an illustrative example, finding a full characterization of input and state observability of a path graph, affected by a single unknown input and with measurement of a small number of local states

Unbiased Filtering for State and Unknown Input with Delay

International audienceIn this paper, we consider linear network systems with unknown inputs. We present an unbiased recursive algorithm that simultaneously estimates states and inputs. We focus on delay-left invertible systems with intrinsic delay l ≥ 1, where the input reconstruction is possible only by using outputs up to l time steps later in the future. By showing an equivalence with a descriptor system, we state conditions under which the time-varying filter converges to a stationary stable filter, involving the solution of a discrete-time algebraic Riccati equation

A Discrete-time Networked Competitive Bivirus SIS Model

The paper deals with the analysis of a discrete-time networked competitive bivirus susceptible-infected-susceptible (SIS) model. More specifically, we suppose that virus 1 and virus 2 are circulating in the population and are in competition with each other. We show that the model is strongly monotone, and that, under certain assumptions, it does not admit any periodic orbit. We identify a sufficient condition for exponential convergence to the disease-free equilibrium (DFE). Assuming only virus 1 (resp. virus 2) is alive, we establish a condition for global asymptotic convergence to the single-virus endemic equilibrium of virus 1 (resp. virus 2) -- our proof does not rely on the construction of a Lyapunov function. Assuming both virus 1 and virus 2 are alive, we establish a condition which ensures local exponential convergence to the single-virus equilibrium of virus 1 (resp. virus 2). Finally, we provide a sufficient (resp. necessary) condition for the existence of a coexistence equilibrium

Unbiased Filtering for State and Unknown Input with Delay

International audienceIn this paper, we consider linear network systems with unknown inputs. We present an unbiased recursive algorithm that simultaneously estimates states and inputs. We focus on delay-left invertible systems with intrinsic delay l ≥ 1, where the input reconstruction is possible only by using outputs up to l time steps later in the future. By showing an equivalence with a descriptor system, we state conditions under which the time-varying filter converges to a stationary stable filter, involving the solution of a discrete-time algebraic Riccati equation

Strongly Structural Input and State Observability for Linear Time Invariant Network Systems

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A State Feedback Controller for Mitigation of Continuous-Time Networked SIS Epidemics

The paper considers continuous-time networked susceptible-infected-susceptible (SIS) diseases spreading over a population. Each agent represents a sub-population and has its own healing rate and infection rate; the state of the agent at a time instant denotes what fraction of the said sub-population is infected with the disease at the said time instant. By taking account of the changes in behaviors of the agents in response to the infection rates in real-time, our goal is to devise a feedback strategy such that the infection level for each agent strictly stays below a pre-specified value. Furthermore, we are also interested in ensuring that the closed-loop system converges either to the disease-free equilibrium or, when it exists, to the endemic equilibrium. The upshot of devising such a strategy is that it allows health administration officials to ensure that there is sufficient capacity in the healthcare system to treat the most severe cases. We demonstrate the effectiveness of our controller via numerical examples

Towards Understanding the Endemic Behavior of a Competitive Tri-Virus SIS Networked Model

This paper studies the endemic behavior of a multi-competitive networked susceptible-infected-susceptible (SIS) model. Specifically, the paper deals with three competing virus systems (i.e., tri-virus systems). First, we show that a tri-virus system, unlike a bi-virus system, is not a monotone dynamical system. Using the Parametric Transversality Theorem, we show that, generically, a tri-virus system has a finite number of equilibria and that the Jacobian matrices associated with each equilibrium are nonsingular. The endemic equilibria of this system can be classified as follows: a) single-virus endemic equilibria (also referred to as the boundary equilibria), where precisely one of the three viruses is alive; b) 2-coexistence equilibria, where exactly two of the three viruses are alive; and c) 3-coexistence equilibria, where all three viruses survive in the network. We provide a necessary and sufficient condition that guarantees local exponential convergence to a boundary equilibrium. Further, we secure conditions for the nonexistence of 3-coexistence equilibria (resp. for various forms of 2-coexistence equilibria). We also identify sufficient conditions for the existence of a 2-coexistence (resp. 3-coexistence) equilibrium. We identify conditions on the model parameters that give rise to a continuum of coexistence equilibria. More specifically, we establish i) a scenario that admits the existence and local exponential attractivity of a line of coexistence equilibria; and ii) scenarios that admit the existence of, and, in the case of one such scenario, global convergence to, a plane of 3-coexistence equilibria.Comment: arXiv admin note: substantial text overlap with arXiv:2209.1182