In this paper, the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization. Stable feedback schemes obtainable from a given plant are parameterized. Salient properties of sensitivity reducing schemes are derived, and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity. The theory is developed for input-output systems in a general setting of Banach algebras, and then specialized to a class of multivariable, time-invariant systems characterized by n × n matrices of H? frequency response functions, either with or without zeros in the right half-plane. The approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity, and on the concept of an approximate inverse. Approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction. An indicator of approximate invertibility, called a measure of singularity, is introduced. The measure of singularity of a linear time-invariant plant is shown to be determined by the location of its right half-plane zeros. In the absence of plant uncertainty, the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure. In particular, if there are no right half-plane zeros, sensitivity can be made arbitrarily small. The feedback schemes used in the optimization of sensitiviiy resemble the lead-lag networks of classical control design. Some of their properties, and methods of constructing them in special cases are presented.

Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses

Finite-horizon Anisotropy-based Estimation with Packet Dropouts

In this paper, an estimation problem for the sensor networks is studied in the frameworks of anisotropy-based theory. The considered system is described by linear discrete time-varying model with several sensors on finite horizon. Each sensor operates its own measurements and measurements given from some others. Any sensor could fail in random time instant. The external disturbance is assumed to be coloured sequence with bounded anisotropy. The dynamics of the combined error and extended state system is governed by multiplicative noise system. Anisotropic norm boundedness for input-to-error system is guaranteed by applying of anisotropy-based bounded real lemma for multiplicative noise system. The solution of considered problem is reduced to convex optimization problem.

Finite-horizon Anisotropic Estimator Design in Sensor Networks *

In this paper, the anisotropy-based filtering problem is studied for a class of linear discrete time varying finite horizon systems with measurement data packet dropout. The measurement packet dropout phenomenon is considered to be random, and is modeled as binary switching sequence of independent identically distributed scalar random variables. The estimator is designed to satisfy the upper bound constraint on the anisotropic norm chosen to quantitatively describe the performance index of the input-to-error system over the finite horizon. Given the measurements and estimates of the system outputs, the filter parameters are parameterized by means of the solution to a set of discrete linear (with a minor exception of one) matrix inequalities.

Anisotropic Estimator Design for Time Varying System with Measurement Dropouts

We consider a linear discrete time-varying system with multiplicative noise acted upon by
colored exogenous disturbance with nonzero first moment. Multiplicative noises are modeled in
the form of linear combinations of deterministic matrices with pairwise independent random
coefficients. For this system, we describe a method, based on a state-space realization, for
calculating the anisotropic norm in terms of Riccati equations.

Lemma on Boundedness of Anisotropic Norm for Systems with Multiplicative Noises under a Noncentered Disturbance

In this paper, the advanced analysis of linear discrete time varying multiplicative noise system in terms of anisotropy-based theory is considered. Accurate formulae for anisotropic norm of such system with mutually independent multiplicative noises and additive input disturbances are given. There two types of anisotropy-based bounded real lemmas were derived, the first one in terms of backward-time difference Riccati equations, the another one in terms of inequalities.

Anisotropy-Based Bounded Real Lemma for Multiplicative Noise Systems: the Finite Horizon Case

We consider the anisotropy-based filtering problem for a linear discrete time-varying
multiplicative noise system on a finite time horizon. A random disturbance with an inaccurately
known distribution is fed to the system input. The uncertainty in the description of this
distribution is characterized by a constraint on the anisotropic functional. The output of the
original system is estimated using the selected filter of a certain type according to the principle of
minimizing the performance indicator in the form of the anisotropic norm of the filtering error
system. Based on the lemma on the boundedness of the anisotropic norm of the filtering error
system, relations for the unknown filter matrices are derived. As special cases, we consider
anisotropy-based filtering problems for a system with measurement dropouts as well as a similar
problem for a system with a filter of a special type. A numerical example where the method is
used to synthesize an anisotropy-based filter is given.

Anisotropy-Based Filtering for Linear Discrete Time-Varying Systems with Multiplicative Noises on a Finite Horizon

In this paper, the anisotropy-based estimating problem for linear discrete time-varying multiplicative noises finite horizon systems is considered. We examine the linear discrete time-varying system with uncertainty in the measurement output. The anisotropy-based bounded real lemma is obtained in terms of studied system matrices and desired estimator matrices. As result, we have the system of linear matrix inequalities in terms of estimator matrices. The numerical example of anisotropy-based estimating problem solution for the LDTVFH system is illustrated by the graphs.

Ivan R. Belov

Papers

Anisotropy-Based Bounded Real Lemma for Multiplicative Noise Systems: the Finite Horizon Case

Anisotropy-Based Filtering for Linear Discrete Time-Varying Systems with Multiplicative Noises on a Finite Horizon

Anisotropy-Based Estimation for Linear Discrete Time Varying Finite Horizon Systems with Missing Measurements