Vadim Kaparin

Extended observer form: simple existence conditions

The paper focuses on the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of certain partial derivatives, related to the input–output equation, corresponding to the state equations. Moreover, a certain algorithm for transforming the state equations into the observer form is proposed.

Extended observer form for discrete-time nonlinear control systems

The paper addresses the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of differential one-forms, associated with the input-output equation, corresponding to the state equations.

Transformation of nonlinear discrete-time system into the extended observer form

ABSTRACT The paper addresses the problem of transforming discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of both the extended coordinate and output transformations, solving the problem, are formulated in terms of differential one-forms, associated with the input–output equation, corresponding to the state equations. An algorithm for transformation of state equations into the extended observer form is proposed and illustrated by an example. Moreover, the considered approach is compared with the method of dynamic observer error linearisation, which likewise is intended to enlarge the class of systems transformable into an observer form.

Feedback linearization of an active magnetic bearing system operated with a zero–bias flux

Abstract Input-output linearization by state feedback is applied to a flux-controlled active magnetic bearing (AMB) system, operated in the zero-bias mode. Two models of the AMB system are employed. The first one is described by the third-order dynamics with a flux-dependent voltage switching scheme, whereas the second one is the fourth-order system, called self-sensing AMB, since it does not require the measurement of the rotor position. In the case of that system we had to find the flat outputs to guarantee its stability. The proposed control schemes are verified by means of numerical simulations performed within the Matlab environment.

Necessary conditions for transformation the nonlinear control system into the observer form via state and output coordinate changes

The paper gives more direct and simple necessary conditions for the existence of state and output coordinate transformations, allowing to transform the nonlinear single-input single-output control system into the observer form. Both the old and new conditions require that the certain n differential one-forms, associated with the nth order differential input-output equation (corresponding to the state equations), and depending on a unknown single-variable output function, are the total differentials of certain functions.

Linearization by generalized input-output injections on homogeneous time scales

The problem of linearization by generalized input-output injections is addressed for nonlinear multi-input singleoutput systems, described by higher order input-output deltadifferential equations, defined on homogeneous time scales. Necessary and sufficient conditions are formulated in terms of differential forms, associated with the input-output equation of the system. The paper provides a step-by-step algorithm for verification of the necessary and sufficient conditions and computation of generalized input-output injections.

Stable cones of polynomials via Routh rays

A problem of inner convex approximation of a stability domain is addressed in the paper for continuous-time linear control systems. A constructive procedure for generating stable cones of polynomials is provided. The main idea is based on the construction of so-called Routh rays (stable half-lines) starting from a given stable point. These lines serve as edges for the corresponding polyhedral Routh cones inside the stability domain. Several numerical examples are presented to illustrate introduced concepts and effectiveness of the proposed approach.

Transformation of nonlinear state equations into the observer form: Necessary and sufficient conditions in terms of one-forms

Necessary and sufficient conditions are given for the existence of state and output transformations, that bring single-input single-output nonlinear state equations into the observer form. The conditions are formulated in terms of differential one-forms, associated with an input-output equation of the system. An algorithm for transformation of the state equations into the observer form is presented and illustrated by an example.

A symbolic software package for nonlinear control systems

The paper describes the key elements of the recently developed symbolic software for nonlinear control systems, based on algebraic lattice theory. The software is incorporated into the package NLControl that assists the solution of various modeling, analysis and synthesis problems and using the tools of webMathematica made available for researches having no access to Mathematica. Besides the smooth systems, the tools are also applicable to certain subclasses of non-smooth systems depending on such phenomena as saturation, friction, hysteresis and backlash.