Michael M. Tchaikovsky

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.

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.

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.

Multichannel synthesis problems for anisotropic control

This paper considers a problem of attenuation of uncertain stochastic disturbances exciting a linear discrete time-invariant system. The system’s abilities to attenuate the external disturbances are quantitatively characterized by its anisotropic norm. The anisotropic control problem is solved for a standard plant with several groups of channels from the external disturbance inputs to the controlled outputs. These channels have different levels of statistic uncertainty measured in terms of the mean anisotropy. The considered technique also allows to design the anisotropic controllers that ensure the closed-loop poles to be placed in some given convex region of the complex plain.

Static output feedback anisotropic controller design by LMI-based approach: General and special cases

This paper considers an approach to attenuation of uncertain stochastic disturbances for a linear discrete time invariant system. The statistical uncertainty is measured in terms of the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is applied as a performance criterion. The designed anisotropic suboptimal controller is a static output feedback gain which is required to stabilize the closed-loop system and keep its anisotropic norm below a prescribed threshold value. The general static output feedback synthesis procedure implies solving a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities in reciprocal matrices which make the general optimization problem nonconvex. By applying some known standard convexification procedures it is shown that the resulting optimization problem is convex for some specific classes of plants defined by certain structural properties. In the convex cases, the anisotropic γ-optimal controllers can be obtained by minimizing the squared norm threshold value subject to convex constraints. The proposed approach to the anisotropy-based optimization is novel as the static output feedback anisotropic controllers have not been considered before.

Normalized problem of anisotropy-based stochastic H ∞ optimization for closed-loop system order reduction by balanced truncation

This paper addresses the normalized problem of anisotropy-based stochastic H∞ optimization for a linear discrete time-invariant system. The problem includes the problem of linear-quadratic Gaussian controller design and H∞ optimization problem as limiting particular cases. It is shown that the order of a cross-coupled nonlinear algebraic equation system defining the optimal controller realization matrices can be reduced. This equation system can be partially decoupled that results in its simplification.

Stochastic robust controller reduction by anisotropic balanced truncation

This paper addresses the problem of normalized anisotropic optimal controller order reduction by means of anisotropic balanced truncation. This controller is the solution to the problem of normalized anisotropy-based stochastic ℌ∞ optimization. An example of application to flight control in a windshear is given.

Numerical Procedures for Anisotropic Analysis of Time-Invariant Systems and Synthesis of Suboptimal Anisotropic Controllers and Filters

This paper briefly considers solutions of primary statements of problem of anisotropic analysis of time-invariant systems and problems of synthesis of suboptimal and γ-optimal anisotropic controllers and filters for the time-invariant systems. Numerical procedures for finding the respective solutions are described. To demonstrate the efficiency of the proposed algorithms, illustrative numerical examples are given.

Anisotropic Suboptimal Control for Systems with Linear-Fractional Uncertainty

The problem of synthesis of robust anisotropic suboptimal controllers is stated and solved for systems with uncertain parameters. This paper considers a general case of unstructured parametric linear-fractional uncertainty bounded in spectral norm. The initial synthesis problem for the uncertain system is embedded into the synthesis problem for some auxiliary system with certain parameters, augmented controlled output, and additional input.