Alexander P. Kurdyukov

Solution of the stochastic H ∞ -optimization problem for discrete time linear systems under parametric uncertainty

The stochastic H∞-optimization problem for a linear discrete time system with uncertain parameters is formulated and solved. The system operates in the presence of Gaussian random disturbances. The original problem with parametric uncertainty is reduced to the stochastic H∞-optimization problem without uncertainty and having one extra input, which is essentially the mixed H2/H∞-optimization problem. In a sense, the problem considered in this paper incorporates the classical H2/H∞-and H∞-optimization problems as limiting cases.

A solution to anisotropic suboptimal filtering problem by convex optimization

This paper considers a robust filtering problem for a linear discrete time invariant system with measured and estimated outputs. The system is exposed to random disturbances with imprecisely known distributions generated by an unknown stable shaping filter from the Gaussian white noise. The stochastic uncertainty of the input disturbance is measured by the mean anisotropy functional. The estimation error is quantified by the anisotropic norm which is a stochastic analogue of the H∞ norm. A sufficient condition for an estimator to exist and ensure that the error is less than a given threshold value is derived in form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities. The suboptimal problem setting results to a set of the estimators ensuring the anisotropic norm of the error to be strictly bounded thereby providing some additional degree of freedom to impose some additional constraints on the estimator performance specification.

Strict Anisotropic Norm Bounded Real Lemma in Terms of Inequalities

Abstract This paper is aimed at extending the H∞ Bounded Real Lemma to stochastic systems under random disturbances with imprecisely known probability distributions. The statistical uncertainty is measured in entropy theoretic terms using the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is a stochastic counterpart of the H∞ norm. A state-space sufficient criterion for the anisotropic norm of a linear discrete time invariant system to be bounded by a given threshold value is derived. The resulting Strict Anisotropic Norm Bounded Real Lemma involves an inequality on the determinant of a positive definite matrix and a linear matrix inequality. As is shown, these convex constraints can be approximated by two linear matrix inequalities.

Calculation of the anisotropic norm of the descriptor system

A method was proposed to calculate the anisotropic norm characterizing robustness of the linear discrete-time descriptor systems to random perturbations with uncertain statistical properties. A numerical example was presented.

The concept of mean anisotropy of signals with nonzero mean

The paper presents a novel concept of the anisotropy-based analysis for the stochastic sequences with nonzero mean. The formulas for the anisotropy of random vector and mean anisotropy of sequence are obtained. Basic types of shaping filter connections are presented.

HOMOTOPY METHOD FOR SOLVING ANISOTROPY-BASED STOCHASTIC H∞-OPTIMIZATION PROBLEM WITH UNCERTAINTY

Abstract Robust stochastic H∞-optimization problem has been solved for linear discrete time-invariant system with structured parametric uncertainty. This problem solution reduces to solving the system of four cross-coupled Riccati equations, Lyapunov equation, and special-type nonlinear algebraic equation. This paper presents the homotopy method — a numerical method for solving this problem.

Robust Stability of Linear Discrete Stationary Systems with Uncertainty Bounded in the Anisotropic Norm

The requirement that uncertainty of the model parameters should be bounded in the anisotropic norm is regarded as one of the possible ways to relax the conditions of the small-gain theorem which guarantee internal stability of the linear stationary system. Within the framework of this approach, an algorithm was proposed to determine that minimum level of anisotropy for which boundedness of the anisotropic norm is the sufficient condition for robustness. This suggests that the plants with uncertain parameters are robust stabilizable.

Anisotropic Norm Bounded Real Lemma for Linear Discrete Time Varying Systems

We consider a finite horizon linear discrete time varying system whose input is a random noise with an imprecisely known probability law. The statistical uncertainty is described by a nonnegative parameter a which constrains the anisotropy of the noise as an entropy theoretic measure of deviation of the actual noise distribution from Gaussian white noise laws with scalar covariance matrices. The worst-case disturbance attenuation capabilities of the system with respect to the statistically uncertain random inputs are quantified by the a-anisotropic norm which is an appropriately constrained operator norm of the system. We establish an anisotropic norm bounded real lemma which provides a state-space criterion for the a-anisotropic norm of the system not to exceed a given threshold. The criterion is organized as an inequality on the determinants of matrices associated with a difference Riccati equation and extends the Bounded Real Lemma of the H-infinity-control theory. We also provide a necessary background on the anisotropy-based robust performance analysis.

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.

On Simplifying Solution to Normalized Anisotropy-Based Stochastic H∞ Problem

Abstract This paper introduces a simplified solution to the normalized anisotropy-based stochastic H ∞ problem. The problem includes the normalized LQG and H ∞ problems as two limiting cases. It is shown that an order of a cross-coupled nonlinear algebraic equation system defining state-space realization matrices of optimal controller can be reduced, and this equation system can be partially decoupled.