Three Mathematica supported proposals for the discretization of nonlinear dynamical control systems

Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

Comparison between iteration schemes for three-dimensional coordinate-transformed saturated–unsaturated flow model

Finite element discretization of the pressure head form of the Richards equation leads to a nonlinear model, which yields numerical convergence difficulties. When the numerical solution to this problem has either an extremely sharp moving front, infiltration into dry soil, flow domains containing materials with spatially varying properties, or involves time-dependent boundary conditions, the corrector iteration used in many time integrators can terminate prematurely, which leads to incorrect results. While the Picard and Newton iteration methods can solve this problem through tightening the tolerances provided to the solvers, there is a more efficient approach to overcome the convergence difficulties. Four tests examples are examined, and each test case is solved with five sufficiently small tolerances to demonstrate the effectiveness of convergence. The numerical results illustrate that the methods greatly improve the convergence and stability. Test experiments show that the Newton method is more complex and expensive on a per iteration basis than the Picard method for simulating variably saturated–unsaturated flow in one spatial dimension. Consequently, it is suggested that the resulting local and global mass balance is exact within the minimum specified accuracy.

This paper presents a method for obtaining optimal stable control for general nonlinear nonautonomous dynamical systems. The approach is inspired by recent developments in analytical dynamics and the observation that the Lyapunov criterion for stability of dynamical systems can be recast as a constraint to be imposed on the system. A closed-form expression for control is obtained that minimizes a user-defined control cost at each instant of time and enforces the Lyapunov constraint simultaneously. The derivation of this expression closely mirrors the development of the fundamental equation of motion used in the study of constrained motion. For this control method to work, the positive definite functions used in the Lyapunov constraint should satisfy a consistency condition. A class of positive definite functions has been provided for mechanical systems that meet this criterion. To illustrate the broad scope of the method, for linear systems it is shown that a proper choice of these positive definite functions results in conventional LQR control. Control of the Lorenz system and a multi-degree of freedom nonlinear mechanical system are considered. Numerical examples demonstrating the efficacy and simplicity of the method are provided.

: 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.

A new recurrent neural-network predictive feedback control structure for a class of uncertain nonlinear dynamic time-delay systems in canonical form is developed and analyzed. The dynamic system has constant input and feedback time delays due to a communications channel. The proposed control structure consists of a linearized subsystem local to the controlled plant and a remote predictive controller located at the master command station. In the local linearized subsystem, a recurrent neural network with on-line weight tuning algorithm is employed to approximate the dynamics of the time-delay-free nonlinear plant. No linearity in the unknown parameters is required. No preliminary off-line weight learning is needed. The remote controller is a modified Smith predictor that provides prediction and maintains the desired tracking performance; an extra robustifying term is needed to guarantee stability. Rigorous stability proofs are given using Lyapunov analysis. The result is an adaptive neural net compensation scheme for unknown nonlinear systems with time delays. A simulation example is provided to demonstrate the effectiveness of the proposed control strategy.

We describe a general method for adaptive model-based control of nonlinear dynamic systems using neural networks, fuzzy logic and fractal theory. The new neuro-fuzzy-fractal method combines soft computing techniques with the concept of the fractal dimension for the domain of nonlinear dynamic system control. The new method for adaptive model-based control has been implemented as a computer program to show that our neuro-fuzzy-fractal approach is a good alternative for controlling nonlinear dynamic systems. We illustrate our new methodology with the case of controlling aircraft dynamic systems. For this case, we use mathematical models for the simulation of aircraft dynamics during flight. The goal of constructing these models is to capture the dynamics of the aircraft, so as to have a way of controlling this dynamics to avoid dangerous behavior of the system.

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.

This chapter has proposed a systematic method to design mathematical models. These models have been associated with counting of white blood cells, counting of red blood cells, population of mosquitoes, and counting of foreign bodies like virus, bacteria, and parasite in a human body. Interpretations for critical points or equilibrium points have been given for network systems of differential equations associated with models. A practical method of applying these interpretations in administrating medicines to get control over diseases has been given. Order of priority in three types of critical points, namely, stable critical points, unstable critical points, and asymptotically stable critical points, has been given. Conversions of differential equations of models into integral equations and applying Picard's iteration method to solve integral equations have been explained. A step-by-step approach has been used in designing models, solving models, and interpreting solutions of models for tropical diseases.

Stochastic differential equations in ℝn with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space Wβ2 for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in Wα 2 for 0 < α < 1. Linear equations are considered as a special case. This approach leads to fast computer algorithms basing on Picard's iteration method. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

This chapter introduces the basic concepts of dynamical systems theory and several basic mathematical methods for controlling chaos. The main goal of this chapter is to provide an introduction to and a summary of the theory of dynamical systems, with particular emphasis on fractal theory, chaos theory, and chaos control. We first define what is meant by a dynamical system, then we define an attractor, and then the concept of the fractal dimension of a geometrical object. We also define the Lyapunov exponents as a measure of the chaotic behavior of a dynamical system. On the other hand, the fractal dimension can be used to classify geometrical objects because it measures the complexity of an object. The chapter also describes mathematical methods for controlling chaos in dynamic systems. These methods can be used to control a real dynamic system; however, due to efficiency and accuracy requirements we were forced to use fuzzy logic to model the uncertainty, which is present when numerical simulations are performed. We also describe in this chapter a new theory of chaos using fuzzy logic techniques. Chaotic behavior in nonlinear dynamical systems is very difficult to detect and control. Part of the problem is that mathematical results for chaos are difficult to use in many cases, and even if one could use them there is an underlying uncertainty in the accuracy of the numerical simulations of the dynamical systems. For this reason, we can model the uncertainty of detecting the range of values where chaos occurs, using fuzzy set theory. Using fuzzy sets, we can build a theory of fuzzy chaos, where we can use fuzzy sets to describe the behaviors of a system. We illustrate our approach with two cases: Chua’s circuit and Duffing’s oscillator.

The main issue to be addressed here is whether the ideas embodied in nonlinear dynamical systems and chaos are of any value in aiding our understanding of complicated physical phenomena. The views expressed will necessarily be selective and will reflect those features of modern thinking in dynamical systems theory which the author has found to be most useful in interpreting the outcome of experiments. The field may be viewed in the following way. On the one hand there are many splendid abstract mathematical ideas in discussions of dynamical systems and chaos, and on the other there are so-called ‘real world’ phenomena. As is usually the case in physics, the interpretation of experimental results in terms of a set of theoretical ideas is not always a straightforward matter and things are no different here. This is despite some claims in popular accounts that ‘chaos theory’ is the ‘answer’ to a wide variety of problems. An example of this oversimplification is to be found in the phenomenon of turbulence in fluid flows. Over the past decade the words ‘chaos’ and ‘turbulence’ have become synonymous in many popular accounts and yet the connection between the two is far from obvious even today. In the author’s opinion the ideas of chaos have thus far added very little to the understanding of the phenomenon of turbulence. Yet many research groups, including the author’s, continue to study this topic because a significant breakthrough in this very difficult subject seems tantalizingly close.

Constrained exact controllability of semilinear systems

On the Problem of Package Guidance for Nonlinear Control System via Fuzzy Approach