Asymptotic Stability of LTV Systems With Applications to Distributed Dynamic Fusion

Abstract

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

Dynamic leader-follower algorithms in mobile multi-agent networks

“…Thus, the convergence of the product of all slices (to zero) depends on the size of the slices. The following theorem summarizes the results in [15]:…”

“…The complete proof is beyond the scope of this paper and can be found in [15]. We now explain the intuition behind Theorem 1:…”

“…In [15], we provide an alternative approach, which links the norm properties of subsets of system matrices to the convergence of the infinite product in Eq. (10).…”

“…1, where the th slice is denoted by . Each slice is initiated by a sub-stochastic matrix [15], and the length of the th slice is defined as Using the slice notation, we can study the convergence of…”

“…10, note that > . In [15], we provide the largest upper bound on the infinity norm of a slice, which is strictly less than one and approaches one as the length of a slice increases. Thus, the convergence of the product of all slices (to zero) depends on the size of the slices.…”