## 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 [22] and in [48].

#### Boundary Control for Output Regulation in Scale-FreePositive Networks

Participants : D. Nikitin, C. Canudas-de-Wit [Contact person] , P. Frasca.

This work addresses the problem of controlling aggregate quantities in large networks. More precisely, we deal with the problem of controlling a scalar output of a large-scale positive scale-free network to a constant reference value. We design an output-feedback controller such that no information about state vector or system matrices is needed. This controller can have arbitrary positivegains, and only one sufficient sign condition on system matricesshould be satisfied. This controller can be used to regulate the average state in a large-scale network with control applied to boundary nodes of the domain [51].

#### A functional approach to target controllability of networks

Participants : C. Commault, J. Van Der Woude [TU Delft] , P. Frasca [Contact person] .

In the control of networks, it is natural to consider the problem of controlling a limited number of *target nodes* of a network. Equivalently, we can see this problem as controlling the target variables of a structured system, where the state variables of the system are associated to the nodes of the network. We deal with this problem from a different point of view as compared to most recent literature. Indeed, instead of considering controllability in the Kalman sense, that is, as the ability to drive the target states to a desired value, we consider the stronger requirement of driving the target variables as time functions. The latter notion is called functional target controllability. We think that restricting the controllability requirement to a limited set of important variables justifies using a more accurate notion of controllability for these variables. Remarkably, the notion of functional controllability allows formulating very simple graphical conditions for target controllability in the spirit of the structural approach to controllability. The functional approach enables us, moreover, to determine the smallest set of steering nodes that need to be actuated to ensure target controllability, where these steering nodes are constrained to belong to a given set. We show that such a smallest set can be found in polynomial time. We are also able to classify the possible actuated variables in terms of their importance with respect to the functional target controllability problem. This research is reported in [16].

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

Participants : F. Garin [Contact person] , A. Kibangou, S. Gracy [KTH Stockholm] , S.m. Fosson [Politecnico di Torino] .

Cyber-physical systems are composed of many simple components (agents) with interconnections giving rise to a global complex behaviour. One line or research on security of cyber-physical systems models an attack as an unknown input being maliciously injected in the system. 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: structural results, true for almost all interaction weights, and strongly structural results, true for all non-zero interaction weights. Our results in 2019 concern structural and strongly structural ISO for time-varying systems [19], strongly structural ISO for time-invariant systems [46]. Moreover in [44] we study delay-L left-invertibility, where the input reconstruction is allowed to take L time steps instead of requiring immediate reconstruction in a single step. We obtain preliminary results for structural delay-L left-invertibility, which include a full characterization for the case where the input is scalar, and for the cases where L is one and two, while the general case remains an open problem. When the conditions for ISO are satisfied, one can run well-known algorithms in the same vein as a Kalman filter, in order to reconstruct the state and the unknown input from noisy measurements. In [43], we consider cases where the system is not ISO, and we exploit compressive sensing techniques in order to obtain nevertheless a unique reconstruction of the input, under the assumption that the input is highly sparse (e.g., when only one or few states are under attack, albeit the attack position is unknown).

#### Collaborative monitoring of network structural robustness

Participants : A. Kibangou [Contact person] , T.m.d. Tran [Univ. of Danang] .

Interacting systems can be naturally viewed as networks modelled by graphs, whose vertices represent the components of the system while edges stand for the interactions between these components. The efficiency of a network of a network can be evaluated through its functional robustness and structural robustness. The former usually stands for robustness against noise while the latter is related to the network performance despite changes in network topology (node or edge failure). Structural robustness has been an important topic in various domains: in distribution networks (e.g. power or water distribution networks), breakdowns can prevent service to customers; in communication networks, equipment failures may disrupt the network and block users from communicating; in contact networks, removing nodes (persons) by means of vaccination can prevent epidemic propagation. In [31] we have considered the critical threshold of a network and the effective graph resistance (Kirchhoff index) of a sub-graph characterizing the interconnection of sub-networks, that are partitioned from the given network as robustness metric. In which, the critical threshold depends only on the two first moments of the degree distribution while the Kirchhoff index can be computed with Laplacian eigenvalues. Therefore, we show how to estimate jointly the Laplacian eigenvalues and the two first moments of the degree distribution in a distributed way.

#### Estimation of the average state in large scale networks

Participants : A. Kibangou [Contact person] , C. Canudas-de-Wit, U. Niazi, D. Deplano [Univ. Cagliari] .

State estimation for monitoring large-scale systems requires tremendous amounts of computational and sensing resources, which is impractical in most applications. However, knowledge of some aggregated quantity of the state suffices in several applications. Processes over physical networks such as traffic, epidemic spread, and thermal control are examples of large-scale systems. Due to the diffusive nature of these systems, the average state is usually sufficient for monitoring purposes. For instance, estimating the average traffic density in some sector of a traffic network helps to monitor the congestion effectively. In the event of an epidemic, estimating the average proportion of infected people over several towns, which are interconnected through people commuting for work or other purposes, helps to devise the preventive measures for controlling the epidemic spread. For the temperature regulation of a building, the thermistors can only be placed either on the walls or the roof, therefore, estimating the average temperature of the interior of a large corridor is crucial. Other examples include the averaging systems such as opinion networks and wireless sensor networks where the average state is of paramount importance. In [40] we address observability and detectability of the average state of a network system when few gateway nodes are available. To reduce the complexity of the problem, the system is transformed to a lower dimensional state space by aggregation. The notions of average observability and average detectability are then defined, and the respective necessary and sufficient conditions are provided. In [25] we provide a computationally tractable necessary and sufficient condition for the existence of an average state observer for large-scale linear time-invariant (LTI) systems. Two design procedures, each with its own significance, are proposed. When the necessary and sufficient condition is not satisfied, a methodology is devised to obtain an optimal asymptotic estimate of the average state. In particular, the estimation problem is addressed by aggregating the unmeasured states of the original system and obtaining a projected system of reduced dimension. This approach reduces the complexity of the estimation task and yields an observer of dimension one. Moreover, it turns out that the dimension of the system also does not affect the upper bound on the estimation error.

#### Structure-based Clustering Algorithm for Model Reduction of Large-scale Network Systems

Participants : C. Canudas-de-Wit [Contact person] , U. Niazi, J. Scherpen [Univ. Groningen] , X. Cheng [Univ. Groningen] .

In [41], A model reduction technique is presented that identifies and aggregates clusters in a large-scale network system and yields a reduced model with tractable dimension. The network clustering problem is translated to a graph reduction problem, which is formulated as a minimization of distance from lumpability. The problem is a non-convex, mixed-integer optimization problem and only depends on the graph structure of the system. We provide a heuristic algorithm to identify clusters that are not only suboptimal but are also connected, that is, each cluster forms a connected induced subgraph in the network system.