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State-space realization of nonlinear control systems: Unification and extension via pseudo-linear algebra
Abstract
In this paper, the tools of pseudo-linear algebra are applied to the realization problem, allowing to unify the study of the continuous- and discrete-time nonlinear control systems under a single algebraic framework. The realization of nonlinear input-output equation, defined in terms of the pseudo-linear operator, in the classical state-space form is addressed by the polynomial approach in which the system is described by two polynomials from the non-commutative ring of skew polynomials. This allows to simplify the existing step-by-step algorithm-based solution. The paper presents explicit formulas to compute the differentials of the state coordinates directly from the polynomial description of the nonlinear system. The method is straight-forward and better suited for implementation in different computer algebra packages such as Mathematica or Maple.
... Note that in order to make the tools of pseudo-linear algebra applicable for nonlinear systems, one has to find first the globally linearized system description in terms to differential 1-forms by applying the differential operator to system equations. The approach has been used to solve the realization problem [19] and for reduction of nonlinear control systems [73]. ...
The paper gives an overview of an algebraic approach based on differential 1-forms, developed for the study of nonlinear control systems. The purpose of the paper is to describe the approach, comment on the necessary assumptions made, and demonstrate the effectiveness and limitations of the approach. Two very important aspects of the approach are as follows: (1) one works with differentials and not with functions, meaning that computations are, up to integration similar to the linear case and (2) the approach is used to study generic properties of control systems that hold for almost every point of a suitable domain. The first point means that solutions to various problems are found in terms of 1-forms and the integrability properties allow transformation of the solution back to the level of functions. The study of generic properties simplifies the presentation of the solutions, since there is no need to specify the working point and its neighbourhood. Finally, the paper includes an extensive list of publications, where the approach of 1-forms is studied or applied to solve different control problems.
This chapter develops various approximate discrete-time models for general linear deterministic systems. It is always possible to obtain an exact sampled models for linear systems. However, approximate models are treated here to provide insights into the structure of discrete-time models, to obtain simpler models, and to be able to construct similar approximate sampled models for nonlinear systems, later in the book. We present models based on simple Euler integration, on the inclusion of asymptotic sampling zeros, on up-sampling, on normal forms and on truncated Taylor series expansions.
The paper applies the pseudo-linear algebra to unify the results on reducibility, reduction and transfer equivalence for continuous- and discrete-time nonlinear control systems. A necessary and sufficient condition for reducibility of nonlinear input-output equation is presented in terms of the greatest common left factor of two polynomials describing the behaviour of the ‘tangent linearized system’ equation. The procedure is given to find the reduced (irreducible) system equation that is transfer-equivalent to the original system equation. Besides unification, the tools of pseudo-linear algebra allow to extend the results also for systems defined in terms of difference, q-shift and q-difference operators.
The package NLControl, developed in the Institute of Cybernetics at Tallinn University of Technology within the Mathematica environment, has been made partially available over the internet using webMathematica tools. The package consists of functions that assist the solution of different modelling, analysis, and synthesis problems for nonlinear control systems, described either by state or by input-output equations. This paper focuses on describing the webMathematica-based tools for discrete-time nonlinear control systems.
Based on the strong tracking filter (STF), a STF-LQR control approach is presented for the control of a ball and beam system (BBS). The simple linearized mathematical model of the BBS is first obtained according to physics laws. In order to model the BBS accurately, an additional control action is introduced as an augmented state variable to compensate for the unmodeled dynamics, such as friction, nonlinearity, disturbance and time-varying characteristics. All the states are adaptively estimated by using the STF, and the controller is designed by employing the discrete-time LQR technique. A BBS control test rig is built based on the NetCon system. Practical experiments have been carried out to demonstrate the feasibility and effectiveness of the proposed approach.
Different notions of observability are compared for systems defined by polynomial difference equations. The main result states that, for systems having the standard property of (multiple-experiment initial-state) observability, the response to a generic input sequence is sufficient for final-state determination. Some remarks are made on results for nonpolynomial and/or continuous-time systems. An identifiability result is derived from the above.
A self-contained introduction to algebraic control for nonlinear systems suitable for researchers and graduate students.
The most popular treatment of control for nonlinear systems is from the viewpoint of differential geometry yet this approach proves not to be the most natural when considering problems like dynamic feedback and realization. Professors Conte, Moog and Perdon develop an alternative linear-algebraic strategy based on the use of vector spaces over suitable fields of nonlinear functions. This algebraic perspective is complementary to, and parallel in concept with, its more celebrated differential-geometric counterpart.
Algebraic Methods for Nonlinear Control Systems describes a wide range of results, some of which can be derived using differential geometry but many of which cannot. They include:
• classical and generalized realization in the nonlinear context;
• accessibility and observability recast within the linear-algebraic setting;
• discussion and solution of basic feedback problems like input-to-output linearization, input-to-state linearization, non-interacting control and disturbance decoupling;
• results for dynamic and static state and output feedback.
Dynamic feedback and realization are shown to be dealt with and solved much more easily within the algebraic framework.
Originally published as Nonlinear Control Systems, 1-85233-151-8, this second edition has been completely revised with new text – chapters on modeling and systems structure are expanded and that on output feedback added de novo – examples and exercises. The book is divided into two parts: the first being devoted to the necessary methodology and the second to an exposition of applications to control problems.
Modelling of physical systems produces as a result a set of differentialalgebraic equations (generally of higher order). As such, state space models are not a natural starting point for modelling, while they have utmost importance in the simulation and control phase. The paper addresses the problem of computing state representations for systems of linear differential-algebraic equations from the behavioral point of view.
For a nonlinear input-output equation defined on homogeneous time scale, an explicit formula is given allowing to compute recursively the differentials of the state coordinates. The recursive formula is based on the Euclidean division algorithm.
El presente libro es una versión actualizada de un trabajo que fue considerado como el trabajo definitivo en este tema desde su publicación inicial por J. Wiley & Sons en 1987. Presenta los mayores avances en el tema de Anillos Noetherianos No Conmutativos desde los 50´s, incluyendo temas como Operadores Diferenciales, Teor'ia de la Localización, Anillos de Grupos, Álgebras de Lie, entre otros. El libro no se restringe a los Anillos Noetherianos, también trata otras clases de anillos. En la presente edición, algunos errores se han corregido, algunos temas han sido expandidos y se ha actualizado la bibliograf'ia. Este libro está recomendado para quienes disfrutan de la Teor'ia de Anillos y quieren investigar sobre temas relacionados