270 research outputs found
Minimally Constrained Stable Switched Systems and Application to Co-simulation
We propose an algorithm to restrict the switching signals of a constrained
switched system in order to guarantee its stability, while at the same time
attempting to keep the largest possible set of allowed switching signals. Our
work is motivated by applications to (co-)simulation, where numerical stability
is a hard constraint, but should be attained by restricting as little as
possible the allowed behaviours of the simulators. We apply our results to
certify the stability of an adaptive co-simulation orchestration algorithm,
which selects the optimal switching signal at run-time, as a function of
(varying) performance and accuracy requirements.Comment: Technical report complementing the following conference publication:
Gomes, Cl\'audio, Beno\^it Legat, Rapha\"el Jungers, and Hans Vangheluwe.
"Minimally Constrained Stable Switched Systems and Application to
Co-Simulation." In IEEE Conference on Decision and Control. Miami Beach, FL,
USA, 201
Distributed Averaging via Lifted Markov Chains
Motivated by applications of distributed linear estimation, distributed
control and distributed optimization, we consider the question of designing
linear iterative algorithms for computing the average of numbers in a network.
Specifically, our interest is in designing such an algorithm with the fastest
rate of convergence given the topological constraints of the network. As the
main result of this paper, we design an algorithm with the fastest possible
rate of convergence using a non-reversible Markov chain on the given network
graph. We construct such a Markov chain by transforming the standard Markov
chain, which is obtained using the Metropolis-Hastings method. We call this
novel transformation pseudo-lifting. We apply our method to graphs with
geometry, or graphs with doubling dimension. Specifically, the convergence time
of our algorithm (equivalently, the mixing time of our Markov chain) is
proportional to the diameter of the network graph and hence optimal. As a
byproduct, our result provides the fastest mixing Markov chain given the
network topological constraints, and should naturally find their applications
in the context of distributed optimization, estimation and control
Tropical Kraus maps for optimal control of switched systems
Kraus maps (completely positive trace preserving maps) arise classically in
quantum information, as they describe the evolution of noncommutative
probability measures. We introduce tropical analogues of Kraus maps, obtained
by replacing the addition of positive semidefinite matrices by a multivalued
supremum with respect to the L\"owner order. We show that non-linear
eigenvectors of tropical Kraus maps determine piecewise quadratic
approximations of the value functions of switched optimal control problems.
This leads to a new approximation method, which we illustrate by two
applications: 1) approximating the joint spectral radius, 2) computing
approximate solutions of Hamilton-Jacobi PDE arising from a class of switched
linear quadratic problems studied previously by McEneaney. We report numerical
experiments, indicating a major improvement in terms of scalability by
comparison with earlier numerical schemes, owing to the "LMI-free" nature of
our method.Comment: 15 page
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