Igor Vladimirov

Anisotropy-based performance analysis of linear discrete time invariant control systems

The anisotropic norm of a linear discrete-time-invariant system measures system output sensitivity to stationary Gaussian input disturbances of bounded mean anisotropy. Mean anisotropy characterizes the degree of predictability (or colouredness) and spatial non-roundness of the noise. The anisotropic norm falls between the H

On Computing the Anisotropic Norm of Linear Discrete-Time-Invariant Systems

Abstract The anisotropic norm is a quantitative characteristic for sensitivity of a system with respect to a special family of Gaussian signals with upper bounded mean anisotropy. Formulae for the anisotropic norm of linear discrete-time-invariant systems both in frequency domain and in state space are given.

Anisotropy-based robust performance analysis of finite horizon linear discrete time varying systems

We consider a problem of robust performance analysis of linear discrete time varying systems on a bounded time interval. The system is represented in the state-space form. It is driven by a random input disturbance with imprecisely known probability distribution; this distributional uncertainty is described in terms of entropy. The worst-case performance of the system is quantified by its a-anisotropic norm. Computing the anisotropic norm is reduced to solving a set of difference Riccati and Lyapunov equations and a special form equation.

State-Space Solution to Anisotropy-Based Stochastic H∞-Optimization Problem

Abstract Stochastic H ∞ -optimization problem using the anisotropic norm as the closed-loop system performance criterion is stated. A state-space solution to the problem for finite-dimensional linear discrete-time-invariant systems is proposed. The problem includes the standard H 2 - and H ∞ -optimizat.ion ones as two limiting cases.

Stochastic approach to H/sub /spl infin//-optimization

A stochastic H/sub /spl infin//-optimization problem is stated. The links between the problem formulated and the standard H/sub /spl infin//-optimization problem are outlined. By means of the stochastic gain with respect to a Gaussian family, a stochastic interpretation of the entropy integral is given.<<ETX>>

Pseudo-randomness of round-off errors in discretized linear maps on the plane

We analyze the sequences of round-off errors of the orbits of a discretized planar rotation, from a probabilistic angle. It was shown [Bosio & Vivaldi, 2000] that for a dense set of parameters, the discretized map can be embedded into an expanding p-adic dynamical system, which serves as a source of deterministic randomness. For each parameter value, these systems can generate infinitely many distinct pseudo-random sequences over a finite alphabet, whose average period is conjectured to grow exponentially with the bit-length of the initial condition (the seed). We study some properties of these symbolic sequences, deriving a central limit theorem for the deviations between round-off and exact orbits, and obtain bounds concerning repetitions of words. We also explore some asymptotic problems computationally, verifying, among other things, that the occurrence of words of a given length is consistent with that of an abstract Bernoulli sequence.

TRACING DIFFUSION IN POROUS MEDIA WITH FRACTAL PROPERTIES

This work is concerned with conditional averaging methods which can be used for modeling of transport in porous media with volume reactions in the fluid phase and surface reactions at the fluid/solid interface. The model under consideration takes into account convection, diffusion within the pores and on larger scales, and homogeneous and heterogeneous reactions. Near the interface with fractal properties, the fluid flow is slow, and diffusion, as a transport mechanism, dominates over convection. Following the conditional moment closure paradigm, we employ a diffusion tracer as a reference scalar field that makes the conditional averaging sensitive to the proximity of a point to the interface. The resulting conditionally averaged reactive transport equations are governed by the probability density function (PDF) of the diffusion tracer, and this makes the study of its behavior an important problem. We consider a hitting time stochastic interpretation of the diffusion tracer, establish integral equations r...

Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a≥0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.

Frequency measurability, algebras of quasiperiodic sets and spatial discretizations of smooth dynamical systems

The roundoff errors in computer simulations of continuous dynamical systems caused by finiteness of machine arithmetic lead to spatially discrete systems with distinct functional properties. As models for the discretized systems in fixed-point arithmetic, autonomous dynamical systems on multi-dimensional uniform grids can be considered which are generated by composing the transition operator of the original system with a roundoff mapping. To study asymptotic properties of such model systems with increasing refinement of the grid, a probability theoretical approach is developed which is based on equipping the grid with an algebra of frequency-measurable quasiperiodic subsets characterized by frequency functions. The approach is applied to spatial discretizations of smooth and invertible dynamical systems. It is shown that, under some nonresonance condition, events relating to mutual deviations of finite segments of trajectories of the discretized and original systems can be represented by frequency-measurable quasiperiodic sets with frequency functions amenable to explicit computation.

Longitudinal Anisotropy-Based Flight Control in a Wind Shear

Abstract The design of the control of an aircraft encounter wind shear in landing approach is treated as a problem of minimizing of anisotropy gain between the wind shear and two components of the aircraft state space vector: air velocity and the altitude. The feature of application of anisotropy control methods for aircraft control design is pointed. Competition of the discovered anisotropy control to H ∞ control algorithms and LQG control algorithms is given.