Anisotropy-based Control Research Articles

Axiomatic Foundations of Anisotropy-Based and Spectral Entropy Analysis: A Comparative Study

An axiomatic development of control systems theory can systematize important concepts. The current research article is dedicated to the investigation and comparison of two axiomatic approaches to the analysis of discrete linear time-invariant systems affected by external random disturbances. The main goal of this paper is to explore axiomatics of an anisotropy-based theory in comparison with axiomatics of a spectral entropy approach in detail. It is demonstrated that the use of the spectral entropy approach is mathematically rigorous, which allows one to prove that the minimal disturbance attenuation level in terms of an anisotropy-based control theory provides the desired performance that is not only for ergodic signals. As a result, axiomatics of the spectral entropy approach allows one to rigorously prove that anisotropy-based controllers can be used to guarantee the desired disturbance attenuation level, not only for stationary random sequences, but also for a wider set of input random signals.

In Between the $$LQG/H_{2}$$- and $$H_{\infty } $$-Control Theories

In this survey, we discuss various approaches to control theory that have arisen in the recent decades and reflect the desire to reach a trade-off between the $$LQG /H_2 $$ -control theory and the $$H_{\infty } $$ -control theory. The theories of the kind include the theory of risk-sensitive controllers, the theory of suboptimal control with a constraint on the $$H_{\infty } $$ -entropy functional, the mixed $$H_2/H_{\infty } $$ -control theory, the minimax $$LQG $$ -control theory, the anisotropy-based theory, and some others. The survey discusses in more detail the anisotropy-based control theory, which includes both the $$ LQG/H_2$$ - and the $$H_{\infty } $$ -theory within a single statement of the problem.

Anisotropy-Based Analysis for the Case of Nonzero Initial Condition

Usually in the context of the anisotropy-based robust performance analysis stochastic systems with zero initial condition are investigated. In this paper we extend this analysis and consider a linear discrete time invariant system under random disturbances and with the nonzero initial condition. In accordance with the basic postulates of the anisotropy-based control theory the disturbance attenuation capabilities of system are quantified by the anisotropic generalized gain which is defined as the largest root mean square gain of the system with respect to a random input and the nonzero initial condition, anisotropy of which is bounded by a given nonnegative parameter a.

Anisotropy-Based Controller Design for Linear Discrete-Time Systems with Multiplicative Noise

This paper extends anisotropy-based control theory methods to the class of linear discrete-time systems with multiplicative noise, which involves an informational criterion to describe the stochastic uncertainty of an external disturbance. An approximate calculation method for the anisotropic norm of such systems using decomposition into linear subsystems without multiplicative noise is suggested. The cases of perfect and imperfect state measurements are considered. The existence criterion for anisotropy-based controllers in form of special matrix inequalities is established.

Robust Anisotropy-Based Control for Uncertain Descriptor Systems with Transient Response Constraints

Abstract In this paper, a problem of robust anisotropy-based control with regional pole assignment for descriptor systems with norm-bounded parametric uncertainties is concerned. The goal is to find a state-feedback control law, which guarantees desirable disturbance attenuation level from stochastic input with unknown covariance to controllable output of the closed-loop system, and ensures, that all finite eigenvalues of the closed-loop system belong to the given region inside the unit disk for all uncertainties from the given set.

Suboptimal Anisotropy-based Control for Linear Discrete Time Varying Systems with Noncentered Disturbances

This paper is concerned with a γ-optimal control for linear discrete time varying systems on bounded time intervals driven by statistically uncertain random disturbances with nonzero mean values. The uncertainty is described in information theoretic terms using a previously introduced anisotropy functional. It is shown that, under additional constraints on the mean value and the covariance matrix of the input disturbance, the γ-optimal anisotropy-based controller design problem can be reduced to a multiobjective control problem. In comparison with the original anisotropy-based control setting, which uses the anisotropic norm of a system as a performance criterion, the suboptimal control approach, considered in the present paper, employs a modified version of the norm. The modified anisotropic norm is related to the original norm at a reduced anisotropy level and the H∞-norm of the system. Using this connection, we obtain sufficient conditions for γ-optimal anisotropy-based control. The resulting controller involves the solution of a certain convex optimization problem. A numerical example is presented in order to demonstrate the method proposed.

Robust anisotropy-based control of linear discrete-time descriptor systems with norm-bounded uncertainties

The paper deals with linear discrete-time time-invariant (LDTI) descriptor systems with norm-bounded parametric uncertainties. The input signal is supposed to be a “colored” noise with bounded mean anisotropy. Sufficient conditions of anisotropic norm boundedness for such class of systems are given. The control design procedure, based on these conditions, is proposed. Numerical example is given.

A New Anisotropy-Based Control Design Approach for Descriptor Systems Using Convex Optimization Techniques

In this paper, suboptimal anisotropy-based control problem for linear discrete-time descriptor systems is solved. The obtained conditions are given in terms of LMIs. Numerical example is given.

On the relationship between the problem of anisotropy-based controller synthesis and classical optimal control problems

Consideration is given to the interrelation between different problems of $$\mathcal{H}_2 $$ -and $$\mathcal{H}_\infty $$ -optimization and their relationship with the theory of anisotropy-based control. It was shown that the anisotropy-based controller of the full order for the completely defined linear system minimizes the mean amount of the transmitted information between the input and output of the closed system.