Complexities and Performance Limitations in Growing Time-Delay Noisy Linear Consensus Networks

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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.

Section: Computational Complexity Discussionmentioning

confidence: 99%

Growing Linear Dynamical Networks Endowed by Spectral Systemic Performance Measures

Siami

Motee

2018

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“…subject to: (29), (30), (32), (33) In spite of smoothness of the cost function in (34), the structure of the cost function is not appealing since we cannot cast it as an SDP with linear objective function and constraints or solve it using existing and standard solvers or toolboxes. Moreover, if we want to write a solver for the problem using the conventional methods, e.g., interior-point or subgradient methods, we have to find eigenvalues and eigenvectors of L + L F for each step of minimizing the performance function; which significantly increases complexity in terms of both time and details of solver.…”

Section: A Cost Function Approximation and Sdp Relaxationmentioning

confidence: 99%

“…Condition (30) ensures stability of the closed-loop network (27). Since the problem given by ( 28)-( 31) is combinatorial, the exact solution must be found by an exhaustive search and appraising ρss(L + L F ; τ ) for all possible k i=1 |Ec| i cases.…”

Section: Improving Performance By Adding New Feedback Loopsmentioning

confidence: 99%

“…The current manuscript extends results of [29]- [31] and presents a consistent story on how to analyze and improve the performance of noisy linear consensus networks subject to time-delay. In [29], we studied properties of the first-order consensus network's H2-norm in the presence of time-delay, and in [30], [31], we offered growing and sparsification algorithms to enhance the H2-norm of the network. This manuscript extends results of [29]- [31] to the general output matrix and provides detailed proofs and explanation for all theorems and lemmas.…”

Section: Introductionmentioning

confidence: 99%

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Performance Improvement in Noisy Linear Consensus Networks With Time-Delay

Ghaedsharaf

Siami

Somarakis

et al. 2019

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We analyze performance of a class of time-delay first-order consensus networks from a graph topological perspective and present methods to improve it. The performance is measured by network's square of H 2 -norm and it is shown that it is a convex function of Laplacian eigenvalues and the coupling weights of the underlying graph of the network. First, we propose a tight convex, but simple, approximation of the performance measure in order to achieve lower complexity in our design problems by eliminating the need for eigen-decomposition. The effect of time-delay reincarnates itself in the form of non-monotonicity, which results in nonintuitive behaviors of the performance as a function of graph topology. Next, we present three methods to improve the performance by growing, re-weighting, or sparsifying the underlying graph of the network. It is shown that our suggested algorithms provide near-optimal solutions with lower complexity with respect to existing methods in literature.performance measure for network synthesis, (iii) low time complexity algorithms to design state feedback controllers for performance enhancement of time-delay linear consensus networks. In section III, we express the H2-norm performance of a time-delay linear consensus network in terms of its Laplacian spectrum. Furthermore, we prove that this performance measure is convex with respect to coupling weights and Laplacian spectrum, and in addition, it is an increasing function of time-delay. In section IV, we discuss topologies with optimal performance. Furthermore, we quantify a sharp lower bound on the best achievable performance for a network with a fixed timedelay. In presence of time-delay, the H2-norm performance of firstorder consensus network is not monotone decreasing with respect to connectivity, which impose challenges in design of the optimal network as increasing connectivity may deteriorate the performance. Then, we present methods to improve the performance measure. We categorize these procedures as growing, reweighting, and sparsification. Although the H2-norm performance is a convex function of Laplacian eigenvalues, direct use of this spectral function in our network design problems requires eigen-decomposition, which adds to time complexity of our design procedures. To overcome this, our key idea is to calculate an approximation function of the performance measure that spares us eigen-decomposition of the Laplacian matrix. In section V, we tackle the combinatorial problem of improving the non-monotone performance measure of the time-delay network by adding new interconnection links. Our time-complexity analysis of our proposed algorithm to grow a time-delay network shows that it can be done in O(n 3 + mn 2 + kn 2 ) arithmetic operations, where n is number of nodes, m is number of rows of the output matrix C and k is maximum number of new interconnections. Section VI discusses reweighting of the coupling weights as an approach to improve the performance measure. This design problem can be cast as a semidefinite programming (SD...

“…In this paper, we adopt a rigorous terminology and develop tools based on notions of risk to quantify and measure the tendency of noisy time-delay linear consensus networks towards exhibiting undesirable systemic events (e.g., large fluctuations and violating critical constraints). Our analysis is based on the fact that a time-delay linear consensus network can be decomposed into many one-dimensional systems and then analyzed using tools developed in [20,13,14].…”

Section: Introductionmentioning

confidence: 99%

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

Somarakis

Ghaedsharaf

Motee

2019

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For the class of noisy time-delay linear consensus networks, we obtain explicit formulas for risk of large fluctuations of a scalar observable as a function of Laplacian spectrum and its eigenvectors. It is shown that there is an intrinsic tradeoff between risk and effective resistance of the underlying coupling graph of the network. The main implication is that increasing network connectivity, increases the risk of large fluctuations. For vector-valued observables, we obtain computationally tractable lower and upper bounds for joint risk measures. Then, we study behavior of risk measures for networks with specific graph topologies and show how risk scales with network size. * The authors are with the