Victor N. Timin

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.

Anisotropy-based suboptimal filtering for the linear discrete time-invariant systems

Consideration was given to the problem of robust filtering for the finite-dimensional linear discrete time-invariant system with measured and estimated outputs. The system is exposed to a random disturbance with the imprecisely known probability distribution. In the information-theoretical terms, the stochastic uncertainty of the input disturbance is defined by the functional of mean anisotropy. The error of estimation was quantified by the anisotropy norm. A sufficient condition for an estimator to exist and ensure that the error is less than the given threshold value was derived in the form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities.

Suboptimal anisotropic filtering in a finite horizon

Consideration was given to the problem of robust stochastic filtering in a finite horizon for the linear discrete time-varying system. A random disturbance with inaccurately known probabilistic distribution is fed to the system input. Uncertainty of the input disturbance is defined in the information-theoretical terms by the anisotropy functional of a random vector. The sufficient condition for strict boundedness of the anisotropic norm of linear discrete timevarying system assigned by the threshold value (lemma of real boundedness) was proved in terms of the matrix inequalities. Sufficient conditions for boundedness of the anisotropic norm of two limiting cases of the anisotropy levels of the input disturbance (a = 0 and a → ∞) were established. A sufficient existence condition for the estimator guaranteeing boundedness of the anisotropic norm of the estimation error operator by the given threshold value was formulated and proved. Sufficient existence conditions for the estimators of two limiting cases of the anisotropy levels of input disturbance were obtained. The estimation algorithm relies on the recurrent solution of a system of matrix inequalities.

Design of robust H ∞ -controller for energetic boiler plant

A problem of robust controller design for control of a boiler plant is solved using an H∞ loop-shaping method with constraints imposed on regional pole placement of transfer functions of the closed-loop system, and linear matrix inequalities tools. Requirements to the closed-loop system are formulated both in the form of frequency constraints on singular values of transfer function of the open-loop system, and in the form of pole placement constraints for the transfer function of the closed-loop system as a given region on the complex plane. A reduction of the full-order H∞-controller obtained in the design procedure is performed.

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.

Anisotropy-based analysis for finite horizon time-varying systems with non-centered disturbances

We consider the problem of anisotropy-based analysis of the robust quality linear discrete time-varying systems with finite horizon under random external disturbance. Uncertainty in the probability distributions of disturbance vectors is found with the information-theoretic notion of anisotropy and additional conditions on the first two moments. The quality of operation of the object is defined by the value of the anisotropic norm of the input-output matrix corresponding to the system. We show that computing the anisotropic norm of a time-varying system in the state space with non-centered disturbance is related to solving a system of difference matrix equations and equations of a special form. We show a sample computation of the anisotropic norm for a time-varying system with finite horizon.

Synthesis of anisotropic suboptimal control for linear time-varying systems on finite time horizon

This paper presents the statement and solution to the problem of synthesis of the anisotropic control that guarantees some prescribed level of attenuation of the uncertain stochastic disturbances affecting the linear discrete time-varying system on finite time horizon. The anisotropy of a random vector is considered as a measure of the statistical uncertainty of the disturbance. The closed-loop system capabilities to attenuate the external disturbances are characterized by its anisotropic norm. The synthesis problem solution is formulated in form of sufficient conditions of existence of a controller that guarantees the anisotropic norm of the closed-loop system to be bounded by some given threshold level. The controller synthesis algorithm is based on solving the system of matrix inequalities recursively.

Anisotropic suboptimal control of linear discrete time-varying system on finite time horizon

This paper addresses the problem of synthesis of the anisotropic control that guarantees some prescribed attenuation level for uncertain stochastic disturbances affecting some linear discrete time-varying system on finite time horizon. The anisotropy of a random vector is considered to be a measure for the statistic uncertainty of the disturbance. The ability of the closed-loop system to attenuate the external disturbances is characterized by its anisotropic norm. The solution to the synthesis problem is formulated as sufficient existence conditions for the controller that guarantees the anisotropic norm of the closed-loop to be bounded by given threshold value. The controller synthesis algorithm is based on recurrent solving a system of matrix inequalities.

Suboptimal anisotropy-based/ℋ ∞ filtering for LDTI systems with nonzero-mean input

In this paper, the anisotropy-based/ℋ∞ suboptimal filtering problem for linear discrete time-invariant (LDTI) system with nonzero-mean stochastic input is considered. The original problem is reduced to the optimization problem with convex restrictions. The solution is based on the sufficient condition for boundedness of the anisotropic norm.