Growing Linear Dynamical Networks Endowed by Spectral Systemic Performance Measures

Abstract

Abstract-We propose an axiomatic approach for design and performance analysis of noisy linear consensus networks by introducing a notion of systemic performance measure. This class of measures are spectral functions of Laplacian eigenvalues of the network that are monotone, convex, and orthogonally invariant with respect to the Laplacian matrix of the network. It is shown that several existing gold-standard and widely used performance measures in the literature belong to this new class of measures. We build upon this new notion and investigate a general form of combinatorial problem of growing a linear consensus network via minimizing a given systemic performance measure. Two efficient polynomial-time approximation algorithms are devised to tackle this network synthesis problem: a linearization-based method and a simple greedy algorithm based on rank-one updates. Several theoretical fundamental limits on the best achievable performance for the combinatorial problem is derived that assist us to evaluate optimality gaps of our proposed algorithms. A detailed complexity analysis confirms the effectiveness and viability of our algorithms to handle large-scale consensus networks.

Risk of Collision and Detachment in Vehicle Platooning: Time-Delay–Induced Limitations and Tradeoffs

“…Although f (s1, s2) has a closed-form, it does not admit an explicit form. Consequently, the formulas in (15) and (17) are expressed in terms of improper integrals. This induces a computational burden that quickly becomes an issue as the number of vehicles in the platoon increases.…”

“…These systemic metrics quantify macroscopic features of networks. For instance, in consensus networks, H2-norm measures coherency [1] and H∞-norm quantifies global connectivity [17]. However, these measures cannot scrutinize microscopic behaviors of networks.…”

Time-Delay Origins of Fundamental Tradeoffs Between Risk of Large Fluctuations and Network Connectivity

“…The blue dots are numerically calculated based on the following procedure: (i) randomly generate a connected graph that satisfies stability condition in Assumption 2 , (ii) randomly generate an output vector c, and (iii) compute the scaled pair ( √ Ξ G , R ε (ȳ))/( √ 2S ε (0)). The orange color curve in Figure 3 illustrates a particular family of complete graphs with parametrized identical coupling strengths that is specifically constructed to highlight sharpness of our bounds in (38) and (40). Theorem 9.…”

Space-Time Sampling for Network Observability

“…The algorithm stops whenever either a desired sparsity level s ∈ (0, 1) or a maximum allowable relative estimation error e > 0 has been achieved. This algorithm resembles the procedure of updating a performance measure of a linear consensus network when a new coupling link is added to the network [32]. The performance guarantees of the greedy methods in this context is a well-studied subject.…”