Pixel matrices: An elementary technique for solving nonlinear systems

High efficiency readout circuits for large matrices of pixels

The factorisation method commonly used in linear supersymmetric quantum mechanics is extended, such that it can be applied to nonlinear quantum mechanical systems. The new method is distinguishable from the linear formalism, as the superpotential is forced to become eigenfunction-dependent. Example solutions are given for the nonlinear Schrödinger equation and its supersymmetric partner equation. This method allows new nonlinear evolution equations to be constructed from the solutions of known nonlinear equations, and has the potential to be a useful tool for mathematicians and physicists working in the field of nonlinear systems, allowing the discovery of previously unknown ‘dualities’ amongst nonlinear evolution equations.

The construction of highresolution differenceschemes for hyperbolic equations has been addressedin numerous works (see [1, 2] and the referencestherein). Several methods are available for improvingthe accuracy of schemes in space. They can be basedon multipoint stencils; differential consequences ofthe original equations; compact approximations ofderivatives [3]; and combinations of grid functionsobtained on different grids, for example, the Richardson method [4, 5]. There are also approaches thatcombine these methods. For example, in the wellknown work [6] concerning gasdynamic computations, a fourthorder compact approximation and differential consequences of the original equation wereused to construct a fourthorder accurate scheme on atwopoint stencil for a thirdorder scalar nonlineardifferential equation. An analysis of the literature shows that [6] is apparently the first to construct a fourthorder accuratescheme on a twopoint stencil in space. Importantly,in the computation of highgradient flows on the basisof hyperbolic conservation laws [1, 2], twopoint compact schemes seem more promising than threepointones [3] because the former have the following important properties. First, twopoint stencil schemes preserve the order of accuracy in the transition from auniform to a nonuniform grid. Second, interpolationacross a discontinuity in compact twopoint schemescan be avoided by placing a grid node at the discontinuity point. The major elements of the technique of [6] for constructing difference schemes include the introductionof derivatives as additional unknown functions inorder to reduce highorder equations to systems offirstorder equations, the use of the integrointerpolation method, and the Simpson and Maclaurin quadrature rules. Note that in this technique replacing theSimpson and Maclaurin formulas by the trapezoidalrule yields secondorder accurate twopoint schemes.In [6, 7], a secondorder scheme on a twopointminimum stencil for hyperbolic equations wasobtained via the introduction of the derivative of theunknown function as an additional independent variable. In [8] the same technique was used to construct afourthorder scheme on a twopoint stencil for theheat equation. In the recent paper [9], higher orderaccurate schemes on a twopoint stencil were called,for brevity, bicompact. Below, we will follow this (inour view apt) terminology.In [10] the bicompact scheme of [6] for a thirdorder scalar differential equation was modified andextended to systems of secondorder nonlinear differential equations. This modified scheme was used in[11, 12] to solve inner and outer problems in viscousgas dynamics.In contrast to [6–12], for deriving bicompactschemes, we use, as auxiliary functions, primitives ofthe original unknown functions rather than their spatial derivatives. Importantly, the auxiliary functionsthus introduced are once more differentiable than theoriginal unknown functions. For example, if a gasdynamic variable is a piecewise continuous function,then its primitive is a piecewise smooth function. In this work, we construct unconditionally stablefourthorder accurate implicit bicompact schemes forlinear and nonlinear equations and systems of equations of the hyperbolic type written in conservativeform. Based on these schemes, the solution to a mixedCauchy problem [13] is determined by the runningcalculation method. The properties of these schemesare examined. Test computations confirm the accuracy and high quality of the resulting solutions.Below, we construct and analyze bicompactschemes for linear and nonlinear hyperbolic equationsand systems of equations wri tten in conservative form.

Identification of continuous-time nonlinear systems: The nonlinear difference equation with moving average noise (NDEMA) framework

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.

Our thermodynamic approach to the study and design of robust optimal control processes in nonlinear (in general global unstable) dynamic systems used soft computing based on genetic algorithms with a fitness function as minimum entropy production. Control objects were nonlinear dynamic systems involving essentially nonlinear stochastic differential equations. An algorithm was developed for calculating entropy production rate in control object motion and in control systems. Part 1 discusses relation of the Lyapunov function (measure of stability) and the entropy production rate (physical measure of controllability). This relation was used to describe the following qualitative properties and important relations: dynamic stability motion (Lyapunov function), Lyapunov exponent and Kolmogorov-Sinai entropy, physical entropy production rates, and symmetries group representation in essentially nonlinear systems as coupled oscillator models. Results of computer simulation are presented for entropy-like dynamic behavior for typical benchmarks of dynamic systems such as Van der Pol, Duffing, and Holmes-Rand, and coupled oscillators. Parts 2 and 3 discuss the application of this approach to simulation of dynamic entropy-like behavior and optimal benchmark control as a 2-link manipulator in a robot for service use and nonlinear systems under stochastic excitation.

This paper describes a design scheme for terminal sliding mode controllers of certain types of non-linear dynamical systems. Two classes of such systems are considered: the dynamic behavior of the first class of systems is described by non-linear second-order matrix differential equations, and the other class is described by non-linear first-order matrix differential equations. These two classes of non-linear systems are not completely disjointed, and are, therefore, investigated together; however, they are certainly not equivalent. In both cases, the systems experience unknown disturbances which are considered bounded. Sliding surfaces are defined by equations combining the state of the system and the expected trajectory. The control laws are drawn to force the system trajectory from an initial condition to the defined sliding surface in finite time. After reaching the sliding surface, the system trajectory remains on it. The effectiveness of the approaches proposed is verified by a few computer simulation examples.

Nonlinear Parity Equation Residual Generation for Fault Detection and Isolation

A general identification technique for nonlinear differential equations of motion

Nonlinear dynamics and chaotic systems have been of great interest to many scientists and engineers over the past two decades. Many nonlinear systems are a result of mathematical models of physical and biological systems that possess inherent nonlinear properties. In this paper, we focus our attention on a second-degree nonlinear differential equation with a sinusoidal forcing function and constant coefficients; such equations, which are commonly employed for mathematical modeling of biological systems usually possess inherent nonlinear properties. We show that the second-degree nonlinear differential equation system could be aperiodic and therefore the long-term behavior is not predictable. However, in the case of the increased frequency of the sinusoidal input, the system shows somewhat periodic, convergent behavior. The simulations demonstrate how small, seemingly insignificant changes to the parameters (e.g., initial conditions) can dramatically alter the system response.

In this paper, we consider an observer design by using a formal linearization based on Fourier expansion for nonlinear dynamic and measurement systems. A non-linear dynamic system is given by a nonlinear ordinary differential equation, and a measurement sysetm is done by a nonlinear equation. Defining a linearization function which consists of the trigonometric functions considered up to the higher-order, a nonlinear dynamic system is transformed into an augmented linear one with respect to this linearization function by using Fourier expansion. Introducing an augmented measurement vector which consists of polynomials of measurement data, a measurement equation is transformed into an augmented linear one with respect to the linearization function in the same way. To these augmented linearized systems, a linear estimation theory is applied to design a new non-linear observer.

Robotic and mechatronic systems, autonomous vehicles, electric power systems and smart grids, as well as manufacturing and industrial production systems can exhibit complex nonlinear dynamics or spatio-temporal dynamics which need to be controlled to ensure good functioning and performance. In this comprehensive reference, the authors present new and innovative control and estimation methods and techniques based on dynamical nonlinear and partial differential equation systems. Such results can be classified in five main domains for the control of complex nonlinear dynamical systems using respectively methods of approximate (local) linearization, methods of exact (global) linearization, Lyapunov stability approaches, control and estimation of distributed parameter systems and stochastic estimation and fault diagnosis methods. Control and Estimation of Dynamical Nonlinear and Partial Differential Equation Systems: Theory and applications will be of interest to electrical engineering, physics, computer science, robotics and mechatronics researchers and professionals working on control problems, condition monitoring, estimation and fault diagnosis and isolation problems. It will also be useful to skilled technical personnel working on applications in robotics, energy conversion, transportation and manufacturing.

Rotorcraft and their associated flight dynamics are highly non-linear systems, especially near extremities of the flight envelope. A global picture of the flight dynamics can be obtained by incorporating in the equations of motion non-linearities in aerodynamics, control system features and inertial terms. This paper deals with the coupling of these phenomena with the non-linear dynamics of an underslung load system. Dynamical systems theory provides the basis for the continuation and bifurcation method of analysing non-linear systems of ordinary differential equations, and forms the basis of this study. The principal strength of the bifurcation-based approach is to identify and give insight into phenomena that globally affect behaviour of an aircraft model, as the full system of non-linear equations is used in analysis. Implemented alongside appropriate numerical simulation runs, the technique promotes an understanding of the underlying structure governing the behaviour of non-linear dynamical systems. Bifurcation and continuation analysis is carried out here on a 21st order non-linear flight dynamics model representing a utility helicopter. Also, the equations of motion for an underslung load system are formulated and then analyzed in isolation. Thereafter they are appended to the helicopter model and a non-linear analysis of the complete system is performed. The effect of the underslung load dynamics on the nominal helicopter system is revealed by the bifurcation-based studies. The influence of varying parameters of the underslung model on the underlying equilibria of the rotorcraft is also investigated. The results reveal changes in helicopter and underslung load stability once the two systems are coupled. The underslung system reduces the extent of the stable equilibrium solution branches, and causes the stability augmentation system to saturate closer to the straight and level trim initial condition. The parameter dependance of the combined dynamics are presented and highlight areas of instability, which may be missed using conventional methods, and show how the transition between stable and unstable solutions is exactly determined as an underslung parameter is varied. This characterisation of the coupled system in terms of stability and response illustrates the potential for bifurcation-based methods to contribute to such non-linear analysis and synthesis problems.

High-precision tracking requires excellent control of slow motion and positioning. Recent advances have provided dynamic friction models that represent almost all experimentally observed properties of friction. The state space formulation of these new mathematical descriptions has the property that the state derivatives are continuous functions. This enables the application of established theories for nonlinear systems. The existence of locally stable fixed points does not imply for nonlinear systems the absence of limit cycles (periodic orbits) or unstable solutions. Therefore, global properties of PI velocity and PID position control are analyzed using a passivity and Lyapunov based approach. These linear control laws are then extended by nonlinear components based on the friction model considered. The applications presented in this work are in the domains of mechatronics and machine-tools.

Prediction of rock mass rating system based on continuous functions using Chaos–ANFIS model