## Section: New Results

### Network systems: modeling, analysis, and estimation

#### Network reduction towards a scale-free structure preserving physical properties

Participants : N. Martin, P. Frasca, C. Canudas de Wit [Contact person] .

In the context of the ERC project, we are addressing a problem of graph reduction, where a given arbitrary weighted graph is reduced to a (smaller) scale-free graph while preserving a consistency with the initial graph and some physical properties. This problem can be formulated as a minimization problem. We give specifications to this general problem to treat a particular case: to this end we define a metric to measure the scale-freeness of a graph and another metric to measure the similarity between two graphs with different dimensions, based on a notion of spectral centrality. Moreover, through the reduction we also preserve a property of mass conservation (essentially, Kirchoff's first law). We study the optimization problem and, based on the gained insights, we derive an algorithm allowing to find an approximate solution. Finally, we have simulated the algorithm both on synthetic networks and on real-world examples of traffic networks that represent the city of Grenoble. These results are presented in [57] and in [31]. We also developed an application to the control of epidemics [58].

#### Cyber-Physical Systems: a control-theoretic approach to privacy and security

Participants : F. Garin [Contact person] , A. Kibangou, S. Gracy.

Cyber-physical systems are composed of many simple components (agents) with interconnections giving rise to a global complex behaviour. Interesting recent research has been exploring how the graph describing interactions affects control-theoretic properties such as controllability or observability, namely answering the question whether a small group of agents would be able to drive the whole system to a desired state, or to retrieve the state of all agents from the observed local states only.

A related problem is observability in the presence of an unknown input, where the input can represent a failure or a malicious attack, aiming at disrupting the normal system functioning while staying undetected. We study linear network systems, and we aim at characterizing input and state observability (ISO), namely the conditions under which both the whole network state and the unknown input can be reconstructed from some measured local states. We complement the classical algebraic characterizations with novel structural results, which depend only on the graph of interactions (equivalently, on the zero pattern of the system matrices). More precisely, we obtain two kinds of results (see [24], [25] and the PhD thesis of S. Gracy): structural results, true for almost all interaction weights, and strongly structural results, true for all non-zero interaction weights. We consider both the case where the system graph is time-invariant, and the case where it varies in time.

When the conditions for ISO are satisfied, one can run algorithms in the same vein as a Kalman filter, in order to reconstruct the state and the unknown input from noisy measurements. These algorithms are known for the case where the input can be reconstructed with only one time-step of delay with respect to the measurements; in [54] we propose a (suboptimal) filter for the case when this is not possible, i.e., more measurements are needed for the input reconstruction.

#### Heterogeneity and uncertainty in distributed estimation from relative measurements

Participants : C. Ravazzi, N. K. Chan, P. Frasca [Contact person] .

This work, presented in [34], has studied the problem of estimation from relative measurements in a graph, in which a vector indexed over the nodes has to be reconstructed from pairwise measurements of differences between its components associated to nodes connected by an edge. In order to model heterogeneity and uncertainty of the measurements, we assume them to be affected by additive noise distributed according to a Gaussian mixture. In this original setup, we formulate the problem of computing the Maximum-Likelihood (ML) estimates and we design two novel algorithms, based on Least Squares regression and Expectation-Maximization (EM). The first algorithm (LSEM) is centralized and performs the estimation from relative measurements, the soft classification of the measurements, and the estimation of the noise parameters. The second algorithm (Distributed LS-EM) is distributed and performs estimation and soft classification of the measurements, but requires the knowledge of the noise parameters. We provide rigorous proofs of convergence for both algorithms and we present numerical experiments to evaluate their performance and compare it with solutions from the literature. The experiments show the robustness of the proposed methods against different kinds of noise and, for the Distributed LS-EM, against errors in the knowledge of noise parameters.

#### Average state estimation in large-scale multi-cluster networks

Participants : U. Niazi, A. Kibangou, C. Canudas de Wit [Contact person] .

In the context of the ERC project, we are addressing the problem of estimation of a functional of non-observed states. Indeed, large-scale network systems can be unobservable from the dedicated state measurements at few nodes. By resorting to an aggregation of multiple clusters of unmeasured nodes, we are investigating the observability and detectability of average states of the clusters. The approach is to obtain a reduced network system whose state vector contains the average states of the clusters. The notion of average observability is defined with respect to the observability of this reduced network system. For average observability, we have stated a necessary condition and a sufficient condition depending solely on the structure of the network. Average detectability, which is a milder notion than average observability, is also studied and a sufficient condition, under which an open-loop average state observer converges, is provided. This condition requires clusters of unmeasured nodes to have negatively balanced local outflow centrality.