5 research outputs found
Topology design for stochastically-forced consensus networks
We study an optimal control problem aimed at achieving a desired tradeoff
between the network coherence and communication requirements in the distributed
controller. Our objective is to add a certain number of edges to an undirected
network, with a known graph Laplacian, in order to optimally enhance
closed-loop performance. To promote controller sparsity, we introduce
-regularization into the optimal formulation and cast the
design problem as a semidefinite program. We derive a Lagrange dual, provide
interpretation of dual variables, and exploit structure of the optimality
conditions for undirected networks to develop customized proximal gradient and
Newton algorithms that are well-suited for large problems. We illustrate that
our algorithms can solve the problems with more than million edges in the
controller graph in a few minutes, on a PC. We also exploit structure of
connected resistive networks to demonstrate how additional edges can be
systematically added in order to minimize the norm of the
closed-loop system.Comment: 11 pages; 5 figure
Koopman Performance Analysis of Nonlinear Consensus Networks
Spectral decomposition of dynamical systems is a popular methodology to
investigate the fundamental qualitative and quantitative properties of these
systems and their solutions. In this chapter, we consider a class of nonlinear
cooperative protocols, which consist of multiple agents that are coupled
together via an undirected state-dependent graph. We develop a representation
of the system solution by decomposing the nonlinear system utilizing ideas from
the Koopman operator theory and its spectral analysis. We use recent results on
the extensions of the well-known Hartman theorem for hyperbolic systems to
establish a connection between the original nonlinear dynamics and the
linearized dynamics in terms of Koopman spectral properties. The expected value
of the output energy of the nonlinear protocol, which is related to the notions
of coherence and robustness in dynamical networks, is evaluated and
characterized in terms of Koopman eigenvalues, eigenfunctions, and modes.
Spectral representation of the performance measure enables us to develop
algorithmic methods to assess the performance of this class of nonlinear
dynamical networks as a function of their graph topology. Finally, we propose a
scalable computational method for approximation of the components of the
Koopman mode decomposition, which is necessary to evaluate the systemic
performance measure of the nonlinear dynamic network.Comment: Submitted as a chapter to the book "Introduction to Koopman Operator
Theory", Revised Versio
Abstraction of Linear Consensus Networks with Guaranteed Systemic Performance Measures
A proper abstraction of a large-scale linear consensus network with a dense
coupling graph is one whose number of coupling links is proportional to its
number of subsystems and its performance is comparable to the original network.
Optimal design problems for an abstracted network are more amenable to
efficient optimization algorithms. From the implementation point of view,
maintaining such networks are usually more favorable and cost effective due to
their reduced communication requirements across a network. Therefore,
approximating a given dense linear consensus network by a suitable abstract
network is an important analysis and synthesis problem. In this paper, we
develop a framework to compute an abstraction of a given large-scale linear
consensus network with guaranteed performance bounds using a nearly-linear time
algorithm. First, the existence of abstractions of a given network is proven.
Then, we present an efficient and fast algorithm for computing a proper
abstraction of a given network. Finally, we illustrate the effectiveness of our
theoretical findings via several numerical simulations
Performance Improvement in Noisy Linear Consensus Networks with Time-Delay
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.Comment: 16 pages, 11 figure
Growing Linear Consensus Networks Endowed by Spectral Systemic Performance Measures
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