3,816 research outputs found
Differential-Algebraic Equations
Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed
A new solution approach to polynomial LPV system analysis and synthesis
Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial LPV system analysis and control synthesis problems. Instead of solving matrix variables over a positive definite cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach
Contingency Model Predictive Control for Automated Vehicles
We present Contingency Model Predictive Control (CMPC), a novel and
implementable control framework which tracks a desired path while
simultaneously maintaining a contingency plan -- an alternate trajectory to
avert an identified potential emergency. In this way, CMPC anticipates events
that might take place, instead of reacting when emergencies occur. We
accomplish this by adding an additional prediction horizon in parallel to the
classical receding MPC horizon. The contingency horizon is constrained to
maintain a feasible avoidance solution; as such, CMPC is selectively robust to
this emergency while tracking the desired path as closely as possible. After
defining the framework mathematically, we demonstrate its effectiveness
experimentally by comparing its performance to a state-of-the-art deterministic
MPC. The controllers drive an automated research platform through a left-hand
turn which may be covered by ice. Contingency MPC prepares for the potential
loss of friction by purposefully and intuitively deviating from the prescribed
path to approach the turn more conservatively; this deviation significantly
mitigates the consequence of encountering ice.Comment: American Control Conference, July 2019; 6 page
A Structural Analysis of Field/Circuit Coupled Problems Based on a Generalised Circuit Element
In some applications there arises the need of a spatially distributed
description of a physical quantity inside a device coupled to a circuit. Then,
the in-space discretised system of partial differential equations is coupled to
the system of equations describing the circuit (Modified Nodal Analysis) which
yields a system of Differential Algebraic Equations (DAEs). This paper deals
with the differential index analysis of such coupled systems. For that, a new
generalised inductance-like element is defined. The index of the DAEs obtained
from a circuit containing such an element is then related to the topological
characteristics of the circuit's underlying graph. Field/circuit coupling is
performed when circuits are simulated containing elements described by
Maxwell's equations. The index of such systems with two different types of
magnetoquasistatic formulations (A* and T-) is then deduced by showing
that the spatial discretisations in both cases lead to an inductance-like
element
Numerical Methods for PDE Constrained Optimization with Uncertain Data
Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization.
The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods
Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations
A recurring theme in attempts to break the curse of dimensionality in the
numerical approximations of solutions to high-dimensional partial differential
equations (PDEs) is to employ some form of sparse tensor approximation.
Unfortunately, there are only a few results that quantify the possible
advantages of such an approach. This paper introduces a class of
functions, which can be written as a sum of rank-one tensors using a total of
at most parameters and then uses this notion of sparsity to prove a
regularity theorem for certain high-dimensional elliptic PDEs. It is shown,
among other results, that whenever the right-hand side of the elliptic PDE
can be approximated with a certain rate in the norm of
by elements of , then the solution can be
approximated in from to accuracy
for any . Since these results require
knowledge of the eigenbasis of the elliptic operator considered, we propose a
second "basis-free" model of tensor sparsity and prove a regularity theorem for
this second sparsity model as well. We then proceed to address the important
question of the extent such regularity theorems translate into results on
computational complexity. It is shown how this second model can be used to
derive computational algorithms with performance that breaks the curse of
dimensionality on certain model high-dimensional elliptic PDEs with
tensor-sparse data.Comment: 41 pages, 1 figur
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