## Section: New Results

### Networks: modeling, analysis and estimation

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

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

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, there are two kinds of results: structural results, true for almost all interaction weights, and strongly structural results, true for all non-zero interaction weights.

In [32], we consider linear time-invariant (LTI) systems, for which we provide a full characterization of structural ISO. The characterization of strongly structural ISO is on-going work.

In [33], instead, we consider linear time-varying (LTV) systems, under some assumptions on the input and output matrices, namely that each attack input and each output measurement concerns a single local state, and that there is no direct feedthrough of the input to the output. Under these assumptions, we characterize strongly structural ISO; in [23] we also give the characterization of structural ISO under the same assumptions.

We are currently working on analogous characterizations for the more general case, removing these assumptions.

Observability is also related to privacy issues. In the ProCyPhyS project, started in October 2016, we are studying privacy-preserving properties of cyber-physical systems, by analyzing observability properties of such systems, in order to derive privacy-preserving policies for applications related to smart mobility. Precisely, by assuming scenarios where nodes compute an average of their initial condition in a finite number of steps with have state privacy-preserving conditions and devise a simple policy that guarantee privacy in case of observable networks.

#### Sensor networks: multisensor data fusion for navigation

Participants : H. Fourati [Contact person] , T. Michel.

Attitude estimation consists in the determination of rigid body orientation in 3D space (principally in terms of Euler angles, rotation matrix, or quaternion). In [27], we solved the attitude determination problem based on a single sensor observation. The rotation equation is transformed into a quadratic quaternion form and is then derived to a linear matrix equation with pseudoinverse matrices. The analytic solutions to the equation are computed via elementary row operations. The solutions show that the attitude determination from a single sensor observation has infinite solutions and the general one is governed by two limiting quaternions. Accordingly, the variance analysis is given in view of probabilistic characters. The authors explore the experimental results via the accelerometer attitude determination system. The properties of the two limiting quaternions are investigated in the experiment. The results show that the gravity-determination abilities of the two limiting quaternions are quite different. Using the rotation vector and eigenvalue decomposition of the attitude matrix, the authors prove that one limiting quaternion is better than another one geometrically. The singularity analysis is also performed revealing the non-existence of singularities for limiting quaternions. The above findings are novel, which are quite different from the conclusions made in a previously published study. In [26], we presents a novel linear approach to solve this problem. We name the proposed method the Fast Linear Attitude Estimator (FLAE) because it is faster than known representative algorithms. The original Wahba’s problem is extracted to several 1-dimensional equations based on quaternions. They are then investigated with pseudo-inverse matrices establishing a linear solution to n-dimensional equations, which are equivalent to the conventional Wahba’s problem. To obtain the attitude quaternion in a robust manner, an eigenvalue-based solution is proposed. Symbolic solutions to the corresponding characteristic polynomial is derived showing higher computation speed. Simulations are designed and conducted using test cases evaluated by several classical methods e.g. M. D. Shuster’s QUaternion ESTimator (QUEST), F. L. Markley’s SVD method, D. Mortari’s Second Estimator of the Optimal Quaternion (ESOQ2) and some recent representative methods e.g. Y. Yang’s analytical method and Riemannian manifold method. The results show that FLAE generates attitude estimates as accurate as that of several existing methods but consumes much less computation time (about 50% of the known fastest algorithm). Also, to verify the feasibility in embedded application, an experiment on the accelerometer-magnetometer combination is carried out where the algorithms are compared via C++ programming language. An extreme case is finally studied, revealing a minor improvement that adds robustness to FLAE. We have been interested in other work [28] to some critical issues on Kalman filter observed in navigation solutions of Global Navigation Satellite System (GNSS). The Kalman fltering (KF) is optimal under the assumption that both process and observation noises are independent white Gaussian noise. However, this assumption is not always satisfed in real-world navigation campaigns. In this paper, two types of KF methods are investigated, i.e. augmented KF (AKF) and the second moment information based KF (SMIKF) with colored system noises, including process and observation noises. As a popular noise-whitening method, the principle of AKF is brieﬂy reviewed for dealing with the colored system noises. The SMIKF method is developed for the colored and correlated system noises, which directly compensates for the covariance through stochastic model in the sense of minimum mean square error. To accurately implement the SMIKF, a refned SMIKF is further derived regarding the continuous-time dynamic model rather than the discrete one. The computational burdens of the proposed SMIKF along with representative methods are analyzed and compared. The simulation results demonstrate the performances of proposed methods.

#### 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.

#### The Observability Radius of Networks

Participants : G. Bianchin, P. Frasca [Contact person] , A. Gasparri, F. Pasqualetti.

Our group is undergoing an effort to understand the system-theoretic properties of networks, namely in terms of controllability and observability. In this context, we have studied the observability radius of network systems, which measures the robustness of a network to perturbations of the edges. We consider linear networks, where the dynamics are described by a weighted adjacency matrix and dedicated sensors are positioned at a subset of nodes. We allow for perturbations of certain edge weights with the objective of preventing observability of some modes of the network dynamics. To comply with the network setting, our work considers perturbations with a desired sparsity structure, thus extending the classic literature on the observability radius of linear systems. The paper [14] proposes two sets of results. First, we propose an optimization framework to determine a perturbation with smallest Frobenius norm that renders a desired mode unobservable from the existing sensor nodes. Second, we study the expected observability radius of networks with given structure and random edge weights. We provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks, showing that line networks are inherently more robust than star networks.

#### Distributed Estimation from Relative and Absolute Measurements

Participants : P. Frasca [Contact person] , W.s. Rossi, F. Fagnani.

Important applications in machine learning, in robotic coordination and in sensor networks require distributed algorithms to solve the so-called relative localization problem: a node-indexed vector has to be reconstructed from measurements of differences between neighbor nodes. In [22] we define the problem of least-squares distributed estimation from relative and absolute measurements, by encoding the set of measurements in a weighted undirected graph. The role of its topology is studied by an electrical interpretation, which easily allows distinguishing between topologies that lead to “small” or “large” estimation errors. The least-squares problem is solved by a distributed gradient algorithm, which we have studied in detail. Remarkably, we have observed that the computed solution is approximately optimal after a number of steps that does not depend on the size of the problem or on the graph-theoretic properties of its encoding. This fact indicates that only a limited cooperation between the sensors is necessary to solve this problem.