Frequency domain analysis of nonlinear systems: general theory

Applications of the shifted Legendre polynomials expansion to the analysis and identification of the nonlinear time-delayed system, described by a memoryless nonlinear element followed by a linear plant with time delay, are studied. The system described here is assumed both controllable and observable. For analysis, by using the shifted Legendre polynomials expansion, the solution of a nonlinear state equation is reduced to the solution of a linear algebraic matrix equation. For identification, through the shifted Legendre expansions of the measured input/output data, the unknown parameters of both the linear delayed plant and the characterisation of the nonlinear element are estimated by using the least-squares method. Algorithms are presented. Numerical examples are given to illustrate the use of this approach.

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.

Nonlinear vibrations for an isotropic cracked plate with different possible boundary conditions subjected to transverse harmonic excitation are evaluated. The first mode is examined in detail around the resonant region. A crack consisting of a continuous line is arbitrarily located at the middle and along the x-axis of the plate. The nonlinear dynamical systems analysis of this cracked plate model begins with the stability of the phase states, and the Poincare map followed by a study of the bifurcations that are observed from the analysis of saddle trajectories, and the estimation of the Lyapunov exponent. This leads to the emergence of strange attractors of fractal dimension, the evolution of saddle orbits into chaos, and to the observation that in this system seemingly chaotic behaviour can emerge from perfectly deterministic origins. In this study, the computational methods required are implementations of the Dynamics 2 software package, and specialized code written in Mathematical. Results shows that the system response could be extremely susceptible to changes in the control parameters, and the variation in the half crack length at the plate centre is one influence on the system s bifurcation and chaos.

Publication in the conference proceedings of EUSIPCO, Trieste, Italy, 1996

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot.

Decision letter: Neural markers of predictive coding under perceptual uncertainty revealed with Hierarchical Frequency Tagging

A NEW SOLUTION TO VOLTERRA SERIES ESTIMATION

This is a short tutorial on Volterra and Wiener series applications to modelling of nonlinear systems and phenomena, and also a survey of the recent achievements in this area. In particular, we show here how the philosophies standing behind each of the above theories differ from each other. On the other hand, we discuss also mathematical relationships between Volterra and Wiener kernels and operators. Also, the problem of a best approximation of large-scale nonlinear systems using Volterra operators in weighted Fock spaces is described. Examples of applications considered are the following: Volterra series use in description of nonlinear distortions in satellite systems and their equalization or compensation, exploiting Wiener kernels to modelling of biological systems, the use of both Volterra and Wiener theories in description of ocean waves and in magnetic resonance spectroscopy. Moreover, connections between Volterra series and neural network models, and also input-output descriptions of quantum systems by Volterra series are discussed. Finally, we consider application of Volterra series to solving some nonlinear problems occurring in hydrology, navigation, and transportation.

The [formula omitted]-compound of a difference–algebraic system

In this paper a clutch torsional model having five degree of freedom and including several kinematic non linearities is studied. Three types of nonlinearity are considered. Smoothened Coulomb friction formulations and smoothened clearance non-linear functions are applied to the non linear system. The differential governing equations are numerically solved by Runge Kutta method, given harmonic torque excitation and a mean load. First, dynamic behaviour analysis shows that the conditioning factors σ2 associated with the smoothened friction model and clearance non linear functions must be carefully selected. Then, effects of the saturation friction torque Tsf, the clearance of the splines (b2) and the stiffness K1 are numerically investigated

Analysis of fast-sampled non-linear systems: generalised frequency response functions for δ-operator models

In this thesis the study of convergence of piecewise smooth systems and discontinuous dynamical networks is considered. The contributions of the thesis are related both in the investigation of synchronization of networks with discontinuities in the agents's dynamics or in the inter-agents communication and in the investigation of incremental stability of nonsmooth systems. In particular, a framework for the study of synchronization when the nodes' dynamics may be both piecewise smooth and/or nonidentical across the network is presented. The analysis is performed using nonsmooth analysis tools, in particular set-valued Lyapunov functions. In this way, sufficient conditions are derived in order to guarantee global bounded synchronization, with analytical expression of the estimates of the synchronization bound and of the minimum coupling strength required to achieve synchronization. Different hypotheses on the heterogeneity of the agents' dynamics is considered, and the analysis is conducted both for linear and nonlinear coupling protocols. Differently from the few results available in the literature, boundedness of trajectories of the coupled systems is not required a priori, as well as any condition of synchrony among switching signals. Referring to the networks where discontinuities are present at the level of the communication among the agents, in the thesis novel event-triggered control strategies are developed. Such distributed algorithms are able to synchronize networks of general Lipschitz nonlinear systems. The results represent an innovative contribution on the existing literature for two main reasons. The existing literature considers only synchronization of integrators or linear systems and the control signals and communication signals are not both simultaneously piecewise constant, which is instead guaranteed in the developed strategies. Furthermore, a strategy able to guarantee a lower bound for the inter-event times between consecutive updates of the control law is proposed. More in detail, the considered strategy is model based and each node uses the dynamical model of its neighbours to predict their evolution and evaluate the triggering events. Both the cases of exact and not exact information of the neighbours' dynamical model are addressed and synchronization is analytically proven in both the cases. Furthermore, Zeno behaviour are excluded. The focus on the convergence properties and on the incremental stability of nonsmooth dynamical systems leads in the thesis to extend contraction theory, which is a tool used in the literature to address incremental exponential stability of smooth systems. In particular, the extension of such tool to the class of time switching non differentiable dynamical systems which satisfy Caratheodory conditions for the existence and uniqueness of a solution is considered. More in detail, it is proved that infinitesimal contraction of each mode of a switched system of interest gives a sufficient condition for global exponential convergence of trajectories towards each other. Such extension can be related with analogous results already present in the literature for convergent systems, which instead make use of Lyapunov conditions. The results are then used to develop smooth and piecewise smooth distributed protocols to coordinate and synchronize networks of piecewise smooth agents. Furthermore, the open problem of proving incremental stability of Filippov systems is introduced and preliminary results are given for the case of planar systems. All the results in the thesis are analytically proven and illustrated via a set of representative numerical examples.

The purpose of this study is to design an anti-sway motion for industrial overhead cranes which transport objects on a horizontal plane by adjusting movements of a trolley motor and a girder motor. The movement of a hoist motor has not been considered at this time since its role was assumed to move objects only vertically, therefore, not to affect the swing motion of objects. The dynamic behavior of the swing motion shows nonlinear characteristics, which makes the design of anti-sway motion controller difficult. First of all, the nonlinear state equation for the motion of industrial overhead cranes has been derived. Then they have been linearized about normal operating states determined by the dynamic characteristics of motor motion-acceleration, constant speed, and deceleration, and deceleration, during transportation. The partial state feedback control algorithm based on this linearized state equation has been developed on order to suppress the swing motion. The simulation results have demonstrated satisfactory performance of the proposed controller.

When two or more signals of different frequencies pass through a nonlinear system, intermodulation distortion (IMD) occurs, resulting in the formation of spurious distortion signals. IMD is most commonly found in active circuits of a radio system, but it can also be found in passive wireless components such as lters, transmission lines, connectors, antennas, attenuators, and so on, especially when transmit power is quite high. Passive intermodulation (PIM) distortion is the name given to the IMD in the latter scenario. With the evolution of radio systems and the scarcity of radio spectrum, PIM interference is being recognized as a potential stumbling block to a radio network's maximum capacity. This article classifies the PIM sources in BS radio systems into two categories, internal and external sources. Internal sources are the radio's passive components such transmission lines, connections, antennas, and so on. External sources, on the other hand, are passive items that are located outside of the BS antenna but inside the RF signal path, such as metallic and rusted objects in the antenna near eld. The high power current flowing through such passive devices can cause nonlinear behavior, resulting in IMD for both types of sources. Also, a review of PIM mitigation techniques is presented in the article.

Chaos theory can be used for quantitative dynamics of uncertainty and finding order in its perturbances. Chaos theory is largely a colloquial notion and is referred to as an analysis of non-linear dynamical systems. The studies devoted to linear dynamical systems and theories of complexity involve studies of turbulence, or, more precisely, the transfer from stability to turbulence. The dynamical system by its nature does not follow long-term forecasts. There are two reasons for the unpredictability. Dynamical systems involve both feedback and critical levels. To paraphrase, one can say that a dynamical system forms a system of non-linear feedback. The most important properties of the system include: sensitivity to the change of the initial conditions, occurrence of critical points and a fractal dimension. Non-linear dynamical systems tend to have more than one solution. The case often is that the number of solutions is huge or even infinite. A visual representation of the data forms a finite space called the phase space of the system. The number of dimensions in the space is determined by the number of variable presents in the system. If there are two or three variables, it is possible to visually examine the data. In case of a larger number of dimensions, the data are examined with the use of mathematical data. Another term used in non-linear dynamical systems is the attractor. This is an area of a equilibrium of non-linear system within a time series. A system which aims to achieve an equilibrium in the form of a single value has a point attractor. Besides, there are phase attractors, which form periodic cycles or orbits in space forming a limit cycle. An attractor, in which none of the points in the space overlaps and whose orbits do not cross but both originate in the same area of phase space, is called strange attractor. Such attractors in contrast to point attractors are non-periodic and predominantly have a fractal dimension [10,11].