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In this paper, we investigate asymptotic stability of linear time-varying systems with (sub-) stochastic system matrices. Motivated by distributed dynamic fusion over networks of mobile agents, we impose some mild regularity conditions on the elements of time-varying system matrices. We provide sufficient conditions under which the asymptotic stability of the LTV system can be guaranteed. By introducing the notion of slices, as non-overlapping partitions of the sequence of systems matrices, we obtain stability conditions in terms of the slice lengths and some network parameters. In addition, we apply the LTV stability results to the distributed leader-follower algorithm, and show the corresponding convergence and steady-state. An illustrative example is also included to validate the effectiveness of our approach. where x k ∈ R n is the state vector, P k 's are the system matrices, B k 's are the input matrices, and u k ∈ R s is the input vector. This model is particularly relevant to design and analysis of distributed fusion algorithms when the system matrices, P k 's, are (sub-) stochastic, i.e. they are non-negative and each row sums to at most 1. Examples include leader-follower algorithms, [8],[9], consensus-based control algorithms, [10]- [12], and sensor localization, [13], [14]. † The authors are with the of the classical notion of spectral radius, with the following definition:in which M k is the set of all possible products of the length k ≥ 1, i.e.Joint spectral radius (JSR) is independent of the choice of norm, and represents the maximum growth rate that can be achieved by forming arbitrary long products of the matrices taken from the set M. It turns out that the asymptotic stability of the LTV systems, with system matrices taken from the set M, is guaranteed, [18], if and only if ρ(M) < 1.Although the JSR characterizes the stability of LTV systems, its computation is NP-hard, [19], and the determination of a strict bound is undecidable, [20]. Naturally, much of the existing

In this paper, we develop a distributed algorithm to localize a network of robots moving arbitrarily in a bounded region. In the case of such mobile networks, the main challenge is that the robots may not be able to find nearby robots to implement a distributed algorithm. We address this issue by providing an opportunistic algorithm that only implements a location update when there are nearby robots and does not update otherwise. We assume that each robot measures a noisy version of its motion and the distances to the nearby robots.To localize a network of mobile robots in R m , we provide a simple linear update, which is based on barycentric coordinates and is linear-convex. We abstract the corresponding localization algorithm as a Linear Time-Varying (LTV) system and show that it asymptotically converges to the true locations of the robots.We first focus on the noiseless case, where the distance and motion vectors are known (measured) perfectly, and provide sufficient conditions on the convergence of the algorithm. We then evaluate the performance of the algorithm in the presence of noise and provide modifications to counter the undesirable effects of noise. We further show that our algorithm precisely tracks a mobile network as long as there is at least one known beacon (a node whose location is perfectly known). † The authors are with the

The framework of Integral Quadratic Constraints (IQC) introduced by Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov's accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions.