Reliable Modeling of Ideal Generic Memristors via State-Space Transformation

Many questions of control theory are well studied for systems which satisfy to the relative degree definition. If this definition is fulfilled then there exists linear state-space transform reducing system to a very convenient canonical form where zero dynamics is a part of system’s equations. Algorithms of such reduction are well-known. However, there exist systems which don’t satisfy this definition. Such systems are the subject of investigation in the presented paper. To investigate their properties here we suggest to consider an analogue of the classical relative degree definition – the so-called column-wise relative degree. It turned out that this definition is satisfied in some cases when classical relative degree doesn’t exist. We introduce this notion here, investigate it properties and suggest algorithm for reducing systems to the column-wise relative degree compliant form if possible. It is possible to show that systems with column-wise relative degree also can be reduced to a convenient canonical form by a linear state-space transformation. Some problems arise from the fact that some systems which do not have relative degree can be reduced to a form with it using linear inputs or outputs transform. Here we show that this is an interesting mathematical problem, which can be solved with the help of properties of relative degree, formulated and proved in this paper.

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Application of local coordinates rectification in linearization of selected parameters of dynamic nonlinear systems

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/compel-11-2013-0358 ◽

2014 ◽

Vol 33 (5) ◽

pp. 1819-1830 ◽

Cited By ~ 3

Author(s):

Andrzej Zawadzki

Keyword(s):

Nonlinear System ◽

Coordinate System ◽

State Space ◽

State Equation ◽

The State ◽

Content Type ◽

Electric Circuits ◽

First Order ◽

State Space Transformation ◽

Nonlinear State

Purpose – The purpose of this paper is to aim to an application of element of the theory of differential geometry for building the state space transformation, linearizing nonlinear dynamic system into a linear form. Design/methodology/approach – It is assumed that the description of nonlinear electric circuits with concentrated parameters or electromechanical systems is given by nonlinear system of differential equations of first order (state equations). The goal is to find transformation which leads nonlinear state equation (written in one coordinate system) to the linear in the other – sought coordinate system. Findings – The necessary conditions fulfilled by nonlinear system undergoing linearization process are presented. Numerical solutions of the nonlinear equations of state together with linearized system obtained from direct transformation of the state space are included (transformation input – the state of the nonlinear system). Originality/value – Application of first order exact differential forms for determining the transformation linearizing the nonlinear state equation. Simple linear models obtained with the use of the linearizing transformation are very useful (mainly because of the known and well-mastered theory of linear systems) in solving of various practical technical problems.

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Modeling and Control of a Variable Length Flexible Cable Overhead Crane Using the Modified Galerkin Method

Volume 9: Mechanical Systems and Control, Parts A, B, and C ◽

10.1115/imece2007-41916 ◽

2007 ◽

Author(s):

Kamal A. F. Moustafa ◽

Mohamed B. Trabia ◽

Mohamed I. S. Ismail

Keyword(s):

Differential Equations ◽

Galerkin Method ◽

Operating Conditions ◽

Overhead Crane ◽

Modeling And Control ◽

Flexible Cable ◽

Space Transformation ◽

Non Linear ◽

Non Linear System ◽

State Space Transformation

A mathematical model that accurately represents an overhead crane with flexible cable and load hoisting/lowering is developed. The analysis includes the transverse vibrations of the flexible cable and the trolley motion as well as the load hoisting/lowering motions. A set of highly non-linear partial differential equations and ordinary differential equations that govern the motion of the crane system within time-varying spatial domain is derived via calculus of variation and Hamilton’s principle. Variable-time modified Galerkin method has been used to discretize the non-linear system. State space transformation is then used to get a set of first order ordinary differential equation. A proportional derivative control scheme is applied to derive the underlying crane so that the cable and payload swing are damped out. Numerical simulations for the control performance of the considered system are presented for various operating conditions.

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State Space and Unobserved Component Models

10.1017/cbo9780511617010 ◽

2004 ◽

Cited By ~ 31

Keyword(s):

State Space ◽

Unobserved Component ◽

Component Models

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