49 research outputs found
On Stability and Stabilization of Perturbed Time Scale Systems with Gronwall Inequalities
In the paper, new nonlinear time scale integral inequalities are given. By means of the explicit integral bounds, we derive suffcient conditions for the uniform asymptotic stability of perturbed systems on time scales. In the sequel, basing on Lyapunov's direct method, we develop the required type of stability.В статье представлены новые нелинейные интегральные неравенства на временной шкале. С помощью явных интегральных оценок получены достаточные условия для равномерной асимптотической устойчивости возмущенных систем на временных шкалах. Далее, основываясь на прямом методе Ляпунова, разработан нужный тип стабильности
Quasilinear Control: Multivariate Nonlinearities, Robustness and Numerical Properties, and Applications
Quasilinear Control (QLC) is a theory with a set of tools used for the analysis and design of controllers for nonlinear feedback systems driven by stochastic inputs. It is based on the concept of Stochastic Linearization (SL), which is a method of linearizing a nonlinear function that, unlike traditional Jacobian linearization, uses statistical properties of the input to the nonlinearity to linearize it. Until now in the literature of QLC, SL was applied only to feedback systems with single-variable nonlinearities that appear only in actuators and/or sensors. In this dissertation, my recent contributions to the literature of QLC are summarized. First, the QLC theory is extended to feedback systems with isolated multivariate nonlinearities that can appear anywhere in the loop and applied to optimal controller design problems, including systems with state-multiplicative noise. Second, the numerical properties of SL, particularly, the accuracy, robustness, and computation of SL, are investigated. Upper bounds are provided for the open-loop relative accuracy and, consequently, the closed-loop accuracy of SL. A comparison of the computational costs of several common numerical algorithms in solving the SL equations is provided, and a coordinate transformation proposed to improve most of their success rates. A numerical investigation is carried out to determine the relative sensitivities of SL coefficients to system parameters. Finally, QLC is applied to the optimal primary frequency control of power systems with generator saturation, and control of virtual batteries in distribution feeders.
The expected impacts of this work are far-reaching. On the technical front, this work provides: i) a new set of theoretical and algorithmic tools that can improve and simplify control of complex systems affected by noise, ii) information to control engineers on accuracy guarantees, choice of solvers, and relative sensitivities of SL coefficients to system parameters to guide the analysis and design of nonlinear stochastic systems in the context of QLC, iii) a new computationally efficient method of addressing saturation in generators or virtual batteries in modern electric power systems, resulting in efficient utilization of resources in providing grid services. On the societal front, this work: i) enables technologies that rely on computationally-efficient algorithms for automation of complex systems, e.g., control of soil temperature for agriculture, which depends on multiple factors like soil moisture and net radiation, ii) allows effective coordination of controllable smart devices in people’s homes, so as not to hamper their quality of service, and iii) provides a stepping stone towards key societal challenges like combating climate change by facilitating reliable operation of the grid with significant renewable penetration
The 3D transient semiconductor equations with gradient-dependent and interfacial recombination
We establish the well-posedness of the transient van Roosbroeck system
in three space dimensions under realistic assumptions on the data: non-smooth
domains, discontinuous coefficient functions and mixed boundary conditions.
Moreover, within this analysis, recombination terms may be concentrated on
surfaces and interfaces and may not only depend on chargecarrier densities,
but also on the electric field and currents. In particular, this includes
Avalanche recombination. The proofs are based on recent abstract results on
maximal parabolic and optimal elliptic regularity of divergence-form
operators
The 3D transient semiconductor equations with gradient-dependent and interfacial recombination
We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators
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Percolation, statistical topography, and transport in random-media
A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential psi(x). For rapidly decaying correlations of psi, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.Physic
Control Problems for Conservation Laws with Traffic Applications
Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks