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Register a new userAnalysis of Contractions in System Graphs: Application to State Estimation
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Observability and estimation are closely tied to the system structure, which can be visualized as a system graph--a graph that captures the inter-dependencies within the state variables. For example, in social system graphs such inter-dependencies represent the social interactions of different individuals. It was recently shown that contractions, a key concept from graph theory, in the system graph are critical to system observability, as (at least) one state measurement in every contraction is necessary for observability. Thus, the size and number of contractions are critical in recovering for loss of observability. In this paper, the correlation between the average-size/number of contractions and the global clustering coefficient (GCC) of the system graph is studied. Our empirical results show that estimating systems with high GCC requires fewer measurements, and in case of measurement failure, there are fewer possible options to find substitute measurement that recovers the systems observability. This is significant as by tuning the GCC, we can improve the observability properties of large-scale engineered networks, such as social networks and smart grid.
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In this work, we use the spectral properties of graphons to study stability and sensitivity to noise of deterministic SIS epidemics over large networks. We consider the presence of additive noise in a linearized SIS model and we derive a noise index to quantify the deviation from the disease-free state due to noise. For finite networks, we show that the index depends on the adjacency eigenvalues of its graph. We then assume that the graph is a random sample from a piecewise Lipschitz graphon with finite rank and, using the eigenvalues of the associated graphon operator, we find an approximation of the index that is tight when the network size goes to infinity. A numerical example is included to illustrate the results.
In this paper, we first consider a pinning node selection and control gain co-design problem for complex networks. A necessary and sufficient condition for the synchronization of the pinning controlled networks at a homogeneous state is provided. A quantitative model is built to describe the pinning costs and to formulate the pinning node selection and control gain design problem for different scenarios into the corresponding optimization problems. Algorithms to solve these problems efficiently are presented. Based on the developed results, we take the existence of a malicious attacker into consideration and a resource allocation model for the defender and the malicious attacker is described. We set up a leader-follower Stackelberg game framework to study the behaviour of both sides and the equilibrium of this security game is investigated. Numerical examples and simulations are presented to demonstrate the main results.
We consider remote state estimation of multiple discrete-time linear time-invariant (LTI) systems over multiple wireless time-varying communication channels. Each system state is measured by a sensor, and the measurements from sensors are sent to a remote estimator over the shared wireless channels in a scheduled manner. We answer the following open problem: what is the fundamental requirement on the multi-sensor-multi-channel system to guarantee the existence of a sensor scheduling policy that can stabilize the remote estimation system? To tackle the problem, we propose a novel policy construction method, and develop a new analytical approach by applying the asymptotic theory of spectral radii of products of non-negative matrices. A necessary and sufficient stability condition is derived in terms of the LTI system parameters and the channel statistics, which is more effective than existing sufficient conditions available in the literature. Explicit scheduling policies with stability guarantees are presented as well. We further extend the analytical framework to cover remote estimation with four alternative network setups and obtain corresponding necessary and sufficient stability conditions.
Although state estimation in networked control systems is a fundamental problem, few efforts have been made to study distributed state estimation via multiple access channels (MACs). In this article, we give a characterization of the zero-error capacity region of an M-input, single-output MAC at any finite block-length. To this end, nonstochastic information-theoretic tools are used to derive the converse and achievability proofs. Next, a tight condition to be able to achieve uniformly bounded state estimation errors over such a MAC is provided. The obtained condition establishes a connection between the intrinsic topological entropies of the linear systems and the zero-error capacity region of the MAC.
We consider a fundamental remote state estimation problem of discrete-time linear time-invariant (LTI) systems. A smart sensor forwards its local state estimate to a remote estimator over a time-correlated $M$-state Markov fading channel, where the packet drop probability is time-varying and depends on the current fading channel state. We establish a necessary and sufficient condition for mean-square stability of the remote estimation error covariance as $rho^2(mathbf{A})rho(mathbf{DM})<1$, where $rho(cdot)$ denotes the spectral radius, $mathbf{A}$ is the state transition matrix of the LTI system, $mathbf{D}$ is a diagonal matrix containing the packet drop probabilities in different channel states, and $mathbf{M}$ is the transition probability matrix of the Markov channel states. To derive this result, we propose a novel estimation-cycle based approach, and provide new element-wise bounds of matrix powers. The stability condition is verified by numerical results, and is shown more effective than existing sufficient conditions in the literature. We observe that the stability region in terms of the packet drop probabilities in different channel states can either be convex or concave depending on the transition probability matrix $mathbf{M}$. Our numerical results suggest that the stability conditions for remote estimation may coincide for setups with a smart sensor and with a conventional one (which sends raw measurements to the remote estimator), though the smart sensor setup achieves a better estimation performance.
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