29 research outputs found
Primal robustness and semidefinite cones
This paper reformulates and streamlines the core tools of robust stability
and performance for LTI systems using now-standard methods in convex
optimization. In particular, robustness analysis can be formulated directly as
a primal convex (semidefinite program or SDP) optimization problem using sets
of gramians whose closure is a semidefinite cone. This allows various
constraints such as structured uncertainty to be included directly, and
worst-case disturbances and perturbations constructed directly from the primal
variables. Well known results such as the KYP lemma and various scaled small
gain tests can also be obtained directly through standard SDP duality. To
readers familiar with robustness and SDPs, the framework should appear obvious,
if only in retrospect. But this is also part of its appeal and should enhance
pedagogy, and we hope suggest new research. There is a key lemma proving
closure of a grammian that is also obvious but our current proof appears
unnecessarily cumbersome, and a final aim of this paper is to enlist the help
of experts in robust control and convex optimization in finding simpler
alternatives.Comment: A shorter version submitted to CDC 1
A non-convex alternating direction method of multipliers heuristic for optimal power flow
The optimal power flow (OPF) problem is fundamental to power system planing and operation. It is a non-convex optimization problem and the semidefinite programing (SDP) relaxation has been proposed recently. However, the SDP relaxation may give an infeasible solution to the original OPF problem. In this paper, we apply the alternating direction method of multiplier method to recover a feasible solution when the solution of the SDP relaxation is infeasible to the OPF problem. Specifically, the proposed procedure iterates between a convex optimization problem, and a non-convex optimization with the rank constraint. By exploiting the special structure of the rank constraint, we obtain a closed form solution of the non-convex optimization based on the singular value decomposition. As a result, we obtain a computationally tractable heuristic for the OPF problem. Although the convergence of the algorithm is not theoretically guaranteed, our simulations show that a feasible solution can be recovered using our method
A Convex Approach to Sparse Hโ Analysis & Synthesis
In this paper, we propose a new robust analysis tool motivated by large-scale systems. The Hโ norm of a system measures its robustness by quantifying the worst-case behavior of a system perturbed by a unit-energy disturbance. However, the disturbance that induces such worst-case behavior requires perfect coordination among all disturbance channels. Given that many systems of interest, such as the power grid, the internet and automated vehicle platoons, are large-scale and spatially distributed, such coordination may not be possible, and hence the Hโ norm, used as a measure of robustness, may be too conservative. We therefore propose a cardinality constrained variant of the Hโ norm in which an adversarial disturbance can use only a limited number of channels. As this problem is inherently combinatorial, we present a semidefinite programming (SDP) relaxation based on the โ_1 norm that yields an upper bound on the cardinality constrained robustness problem. We further propose a simple rounding heuristic based on the optimal solution of our SDP relaxation, which provides a corresponding lower bound. Motivated by privacy in large-scale systems, we also extend these relaxations to computing the minimum gain of a system subject to a limited number of inputs. Finally, we also present a SDP based optimal controller synthesis method for minimizing the SDP relaxation of our novel robustness measure. The effectiveness of our semidefinite relaxation is demonstrated through numerical examples
Multi-Objective Predictive Taxi Dispatch via Network Flow Optimization
In this paper, we discuss a large-scale fleet management problem in a
multi-objective setting. We aim to seek a receding horizon taxi dispatch
solution that serves as many ride requests as possible while minimizing the
cost of relocating vehicles. To obtain the desired solution, we first convert
the multi-objective taxi dispatch problem into a network flow problem, which
can be solved using the classical minimum cost maximum flow (MCMF) algorithm.
We show that a solution obtained using the MCMF algorithm is integer-valued;
thus, it does not require any additional rounding procedure that may introduce
undesirable numerical errors. Furthermore, we prove the time-greedy property of
the proposed solution, which justifies the use of receding horizon
optimization. For computational efficiency, we propose a linear programming
method to obtain an optimal solution in near real time. The results of our
simulation studies using real-world data for the metropolitan area of Seoul,
South Korea indicate that the performance of the proposed predictive method is
almost as good as that of the oracle that foresees the future.Comment: 28 pages, 12 figures, Published in IEEE Acces