Discrete-time nonlinear systems modeling

Theory of nonlinear systems

This paper mainly focuses on the attitude control problem of rigid spacecraft, and conducts the design and research of control methods based on the theory of nonlinear negative imaginary systems. A feedback control law is designed for the attitude control model of the rigid spacecraft, ensuring that the closed-loop system satisfies the nonlinear negative imaginary property. Subsequently, according to the stability theory of nonlinear negative imaginary interconnected systems, a strictly negative imaginary controller is designed to achieve the asymptotic stability of the attitude of the rigid spacecraft during its operation. The numerical simulation results verify the effectiveness of the proposed control synthesis method for the rigid spacecraft system.

: The Commanding General, Marine Corps Combat Development Command (MCCDC) asked the Center for Naval Analyses to assess the general applicability of the new science to land warfare. New Sciences is a catch-all phrase that refers to the tools and methodologies used in nonlinear dynamics and complex systems theory to study physical dynamical systems exhibiting a complicated dynamics. This report concludes that the concepts, ideas, theories, tools and general methodologies of nonlinear dynamics and complex systems theory show enormous, almost unlimited, potential for not just providing better solutions for certain existing problems of land combat, but for fundamentally altering our general understanding of the basic processes of war, at all levels. Indeed, the new sciences' greatest legacy may, in the end, prove to be not just a set of creative answers to old questions but and entirely new set of questions to be asked of what really happens on the battlefield The central thesis of this paper is that land combat is a complex adaptive system. That is to say, that land combat is essentially a nonlinear dynamical system composed of many interacting semi-autonomous and hierarchically organized agents continuously adapting to a changing environment.

Presenting an important potential in the representation of nonlinear systems, the multimodel approach remains an attractive axis for research. One of the important problems in the multimodel structure concerns the validity calculation which is a fundamental point especially when the process is corrupted with noise and/or its parameters are of high variations. A new approach based on the use of both two type of validity is proposed. A developed specification of the need of each one is explained by an optimization procedure. The conduct of this approach requires, first, the classification of the numerical data into a set of clusters. The frequency-sensitive competitive learning (FSCL) algorithm is used to select the number of models and the fuzzy k-means algorithm identify the operating clusters. From the satisfactory results in terms of precision and robustness obtained on theoretical examples, we are incited to confirm our contribution to real process reactor. The results obtained are compared to the classical approaches showing its ability to represent adequately the nonlinear process with a superior precision and accuracy and from this the classic strategy of multimodel representation is oriented towards a multifaceted approach.

Special issue on: nonlinear systems theory and design

A new discretization method for the calculation of a sampled-data representation of nonlinear continuous-time system is proposed. The suggested method is based on the well-known Taylor-series expansion and zero-order hold (ZOH) assumption. The mathematical structure of the new discretization method is analyzed. On the basis of this structure the sampled-data representation of nonlinear system with time-delayed multi-input is derived. First the new approach is applied to nonlinear systems with two inputs. And then the delayed multi-input general equation has been derived. In particular, the effect of the time-discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. And ’hybrid’ discretization schemes that result from a combination of the ’scaling and squaring’ technique with the Taylor method are also proposed, especially under conditions of very low sampling rates. Practical issues associated with the selection of the method’s parameters to meet CPU time and accuracy requirements, are examined as well. A performance of the proposed method is evaluated using a nonlinear system with time-delay: maneuvering an automobile.KeywordsNonlinear SystemDiscretization MethodNonlinear Control SystemTaylor MethodFast Time ConstantThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We show how frequency domain design procedures can be effectively used in nonlinear fuzzy control systems. For this purpose, theoretical justifications of the approach discussed here are provided to show that solid notions from nonlinear system and control theory can be successfully applied in the fuzzy control domain. The methodology proposed is very simple, but powerful, effective and practical. In addition, it supports systematic control system design and stability analysis in the frequency domain for systems described by mathematical, linguistic or experimental models. Several case studies are included to illustrate the usefulness of frequency response in conventional fuzzy control systems analysis and design.

Fundamentals of the theory of non-linear pluse control systems

The concept of generalized polynomial operators is introduced and applied to the theory of non-linear systems. Several properties similar to those previously derived for systems described e.g. by Volterra functional series are studied. The main attention is given to inverting of the operators in question. A local inverse is constructed and the region where it is valid is determined. The construction is applied to solving of certain types of non-linear differential equations and to seeking sufficient conditions for BIBO stability of the corresponding systems.

Two notions from the theory of non-linear systems without time delays are generalized to non-linear delay systems: flatness and quasi-static state feedback. Depending on the generality of the relations considered, flatness properties of increasing generality are obtained: flatness, δ-flatness, π-flatness, and difference-differential flatness. It is then shown that difference-differentially flat delay systems are linearizable by a certain class of “predictive” quasi-static state feedback, i.e., that they are equivalent by such feedback to a linear controllable system. The mathematical framework is difference-differential algebra.

The current theory of nonlinear systems is still not perfect. The modeling and control of nonlinear system problem has always been the difficulty. In a variety of methods of its study, fuzzy system theory because of having the language descriptive way similar to the human mind, can obtain and deal with the qualitative information intelligently. The theory itself also has non-linear characteristics. Therefore the use of fuzzy systems theory to establish the fuzzy model of nonlinear system can well describe the nonlinear characteristics. T-S fuzzy systems, due to the combination of the good performance of the fuzzy system to deal with nonlinear problems with the simple linear expressions, are not only suitable for modeling the nonlinear system, but also use T-S fuzzy model and the linear control theory method to design the controller. So it has been widely used in nonlinear system control problems, and has also greatly developed the T-S fuzzy system theory, appearing a lot of methods of structural and parameter identification. However, this study of T-S fuzzy rules makes us have to face the difference of different ways to select the number of rules as well as online self-adaptability of the number of rules which off-line method lacks when using T-S fuzzy model to deal with nonlinear system modeling and control problem. In view of this, this paper researches on modeling and controlling of complex nonlinear systems based on TS model from different perspectives.

Exact linearization is a traditional method for nonlinear system in the nonlinear system theory, the basic idea of which is to linearize the whole or parts of a nonlinear system using nonlinear coordinate transformation and nonlinear state feedback method. The input-output relationship of the nonlinear system can be represented by a linearized model which can be built using linear system design method. In order to enhance the robustness of an established system which has been linearized via feedback method, the robust control strategy should be adopted. As a result, the control law is composed of feedback linearization control and robust control. Sliding mode variable structure control is a robust control method which is proposed in this paper as a robust control strategy for the exact linearization system.

This note presents an explicit solution to the problem of disturbance attenuation with internal stability for discrete-time nonlinear descriptor systems. Both the static-state feedback and dynamic output feedback cases are considered. In particular, we characterize a family of H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> controllers solving the problem locally around a neighborhood of the origin. To do this, we first derive two stability criteria for discrete-time nonlinear descriptor systems, and then, a version of a bounded real lemma is also developed based on the concepts of dissipation inequality and differential game. After that, the results are used to derive the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> control theory for nonlinear discrete-time descriptor systems. The approach taken is mainly algebraic, and hence is simple and clear.

Theory of nonlinear control

This paper presents a method to determine a nonlinear state-space model from a finite number of measurements of the inputs and outputs. The method is based on embedding theory for nonlinear systems, and can be viewed as an extension of the subspace identification method for linear systems. The paper describes the underlying theory and provides some guidelines for using the method in practice. To illustrate the use of the identification method, it was applied to a second-order nonlinear system.