48 research outputs found
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
Maximizing concave piecewise affine functions on the unitary group
International audienceWe show that a convex relaxation, introduced by Sridharan, McEneaney, Gu and James to approximate the value function of an optimal controlproblem arising from quantum gate synthesis, is exact. This relaxation appliesto the maximization of a class of concave piecewise affine functions over theunitary grou
Global asymptotic stability for semilinear equations via Thompson's metric
In ordered Banach spaces we prove the global asymptotic stability of the unique strictly positive equilibrium of the semilinear equation u′ = Au + ęŤ(u), if A is the generator of a positive and exponentially stable Câ‚€-semigroup and ęŤ is a contraction with respect to Thompson's metric. The given estimates show that convergence holds with a uniform exponential rate.peerReviewe
Nonlinear Balanced Truncation: Part 1 -- Computing Energy Functions
Nonlinear balanced truncation is a model order reduction technique that
reduces the dimension of nonlinear systems in a manner that accounts for either
open- or closed-loop observability and controllability aspects of the system.
Two computational challenges have so far prevented its deployment on
large-scale systems: (a) the energy functions required for characterization of
controllability and observability are solutions of high-dimensional
Hamilton-Jacobi-(Bellman) equations, and (b) efficient model reduction and
subsequent reduced-order model (ROM) simulation on the resulting nonlinear
balanced manifolds. This work proposes a unifying and scalable approach to the
challenge (a) by considering a Taylor series-based approach to solve a class of
parametrized Hamilton-Jacobi-Bellman equations that are at the core of the
balancing approach. The value of a formulation parameter provides either
open-loop balancing or a variety of closed-loop balancing options. To solve for
coefficients of the Taylor-series approximation to the energy functions, the
presented method derives a linear tensor structure and heavily utilizes this to
solve structured linear systems with billions of unknowns. The strength and
scalability of the algorithm is demonstrated on two semi-discretized partial
differential equations, namely the Burgers equation and the
Kuramoto-Sivashinsky equation.Comment: 16 pages, 5 figure
Benelux meeting on systems and control, 23rd, March 17-19, 2004, Helvoirt, The Netherlands
Book of abstract
Exponential integrators: tensor structured problems and applications
The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed
Automation and Control Architecture for Hybrid Pipeline Robots
The aim of this research project, towards the automation of the Hybrid Pipeline Robot (HPR), is the development of a control architecture and strategy, based on reconfiguration of the control strategy for speed-controlled pipeline operations and self-recovering action, while performing energy and time management.
The HPR is a turbine powered pipeline device where the flow energy is converted to mechanical energy for traction of the crawler vehicle. Thus, the device is flow dependent, compromising the autonomy, and the range of tasks it can perform.
The control strategy proposes pipeline operations supervised by a speed control, while optimizing the energy, solved as a multi-objective optimization problem. The states of robot cruising and self recovering, are controlled by solving a neuro-dynamic programming algorithm for energy and time optimization, The robust operation of the robot includes a self-recovering state either after completion of the mission, or as a result of failures leading to the loss of the robot inside the pipeline, and to guaranteeing the HPR autonomy and operations even under adverse pipeline conditions
Two of the proposed models, system identification and tracking system, based on Artificial Neural Networks, have been simulated with trial data. Despite the satisfactory results, it is necessary to measure a full set of robot’s parameters for simulating the complete control strategy. To solve the problem, an instrumentation system, consisting on a set of probes and a signal conditioning board, was designed and developed, customized for the HPR’s mechanical and environmental constraints.
As a result, the contribution of this research project to the Hybrid Pipeline Robot is to add the capabilities of energy management, for improving the vehicle autonomy, increasing the distances the device can travel inside the pipelines; the speed control for broadening the range of operations; and the self-recovery capability for improving the reliability of the device in pipeline operations, lowering the risk of potential loss of the robot inside the pipeline, causing the degradation of pipeline performance. All that means the pipeline robot can target new market sectors that before were prohibitive
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal