A multiresolution approximation method for fast explicit model predictive control

A model predictive control law is given by the solution to a parametric optimization problem that can be pre- computed offline and provides an explicit map from state to control input. In this paper, an algorithm is introduced based on wavelet multiresolution analysis that returns a low complexity explicit model predictive control law built on a hierarchy of second order interpolets. The resulting interpolation is shown to be everywhere feasible and continuous. Further, tests to confirm stability and to compute a bound on the performance loss are introduced. Since the controller approximation is built on a grid hierarchy, convergence to a stabilizing control law is guaranteed and the evaluation of the control law in real-time systems is naturally fast and runs in a bounded logarithmic time. Two examples are provided; A two-dimensional example with an evaluation speed of 31 ns and a four-dimensional example with an evaluation speed of 119 ns

A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations

This paper presents a new resolution strategy for multi-scale streamer discharge simulations based on a second order time adaptive integration and space adaptive multiresolution. A classical fluid model is used to describe plasma discharges, considering drift-diffusion equations and the computation of electric field. The proposed numerical method provides a time-space accuracy control of the solution, and thus, an effective accurate resolution independent of the fastest physical time scale. An important improvement of the computational efficiency is achieved whenever the required time steps go beyond standard stability constraints associated with mesh size or source time scales for the resolution of the drift-diffusion equations, whereas the stability constraint related to the dielectric relaxation time scale is respected but with a second order precision. Numerical illustrations show that the strategy can be efficiently applied to simulate the propagation of highly nonlinear ionizing waves as streamer discharges, as well as highly multi-scale nanosecond repetitively pulsed discharges, describing consistently a broad spectrum of space and time scales as well as different physical scenarios for consecutive discharge/post-discharge phases, out of reach of standard non-adaptive methods.Comment: Support of Ecole Centrale Paris is gratefully acknowledged for several month stay of Z. Bonaventura at Laboratory EM2C as visiting Professor. Authors express special thanks to Christian Tenaud (LIMSI-CNRS) for providing the basis of the multiresolution kernel of MR CHORUS, code developed for compressible Navier-Stokes equations (D\'eclaration d'Invention DI 03760-01). Accepted for publication; Journal of Computational Physics (2011) 1-2

Time-critical multiresolution rendering of large complex models

Very large and geometrically complex scenes, exceeding millions of polygons and hundreds of objects, arise naturally in many areas of interactive computer graphics. Time-critical rendering of such scenes requires the ability to trade visual quality with speed. Previous work has shown that this can be done by representing individual scene components as multiresolution triangle meshes, and performing at each frame a convex constrained optimization to choose the mesh resolutions that maximize image quality while meeting timing constraints. In this paper we demonstrate that the nonlinear optimization problem with linear constraints associated to a large class of quality estimation heuristics is efficiently solved using an active-set strategy. By exploiting the problem structure, Lagrange multipliers estimates and equality constrained problem solutions are computed in linear time. Results show that our algorithms and data structures provide low memory overhead, smooth level-of-detail control, and guarantee, within acceptable limits, a uniform, bounded frame rate even for widely changing viewing conditions. Implementation details are presented along with the results of tests for memory needs, algorithm timing, and efficacy.785-803Pubblicat

Fast Explicit Nonlinear Model Predictive Control Via Multiresolution Function Approximation with Guaranteed Stability Nonlinear Control Systems

In this paper an algorithm for nonlinear explicit model predictive control is introduced based on multiresolution function approximation that returns a low complexity approximate receding horizon control law built on a hierarchy of second order interpolets. Feasibility and stability guarantees for the approximate control law are given using reachability analysis, where interval methods are used to construct a capture basin (feasible region). A constructive algorithm is provided that combines adaptive function approximation with interval methods to build a receding horizon control law that is suboptimal, yet with a region of guaranteed feasibility and stability. The resulting control law is built on a grid hierarchy that is fast to evaluate in real-time systems