Sets Of Nonlinear Dynamical Systems Research Articles

Compact neural network modeling of nonlinear dynamical systems via the standard nonlinear operator form

Dynamic artificial neural networks (DANNs) have become popular for the data-driven modelling of nonlinear dynamical systems. This article elucidates properties of a compact DANN model structure called the standard normal operator form (SNOF). Sets of nonlinear dynamical systems are characterized for which the SNOF can achieve the same model identification error with fewer neurons than needed by the popular DANN model structures. The results are demonstrated in a case study for a multi-stage bioreactor, which is a highly nonlinear dynamical system in which the cells have a sharp change in dynamics during a change in the feed composition as the bioreactor shifts from growth mode to production mode. The ability of the SNOF to model highly nonlinear dynamical systems with a very small number of neurons suggests its potential for serving as a basis for the design of model-based optimal control systems with theoretical guarantees of closed-loop stability and performance.

Estimation problems for uncertain nonlinear dynamical systems with ellipsoidal state constraints

The paper deals with the estimation problems for uncertain nonlinear dynamical systems in the case when a probabilistic description of noise and errors is not available, but only bounds on them known. Such models are found in many applied areas ranged from engineering problems from physics to economics as well as to biological and ecological modeling when it occurs that a stochastic nature of the errors is questionable because of limited data or because of complexity and nonlinearity of the model. As alternative to a stochastic characterization, a so-called set-membership approach has been proposed and intensively developed in the last decades. The solution of many control and estimation problems under uncertainty involves constructing reachable sets and their analogs. For models with linear dynamics under such set-membership uncertainty, there are several constructive approaches, which allow finding effective estimates of reachable sets. Certainly, concrete problems are mostly nonlinear in their parameters and the set of feasible system states is usually non-convex or even non-connected. The key issue in nonlinear set-membership estimation is to find suitable techniques, which produce related bounds for the set of unknown system states without being too computationally demanding. Here the problem of estimating reachable sets of nonlinear dynamical control systems with combined bilinear and quadratic nonlinearity and with uncertainty in initial states is studied. Applying results of the theory of trajectory tubes of control systems and related techniques of differential inclusions theory we present approaches that allow finding the upper ellipsoidal estimates of reachable sets. The main new result consists in obtaining ellipsoidal estimates for reachable sets of nonlinear dynamical system with additional state constraint of ellipsoidal type. Related numerical algorithms, examples and results of computer simulations are included.

The Nonlinear Modeling and Control movement of a Human Forearm for Prosthetic Applications

The model and control of a human forearm is analyzed. In this work, we study the problem of human hand control carrying a mass. The equation of motion and the natural frequency of the forearm for small angular displacement are derived. We develop new methods that use vector fields in the controller construction for a set of nonlinear dynamical systems. The paper deals with compensate of non-linear system which has a similar idea as the method mentioned in linear system. The nonlinear control design procedure, as in the case of linear systems, involves three steps. The first step is the devise of a state-feedback control law, the second step involves the design of a state estimator, and the third step merges the first two steps to obtain a collective controller–estimator compensator. We have managed to design a control law for the non-linear representation of a system, in such a way that the representation of a closed loop system is affine, controllable, and observable and a closed loop system is asymptotically stable. Throughout any motion, the forearm can be considered a one-link robot manipulator which could be exploited to benefit people with disabilities (missing extremities).

Controlling the Movement of a Human Forearm based on Finite Time Optimal Regulator and Nonlinear Control System

human forearm trajectories structure was investigated using Optimal Regulator. First, the continuous dynamics of the human forearm were established on the basis of the relation between muscle torque and neural control signal and employed Riccati differential equation to obtain the optimal movement trajectory of the human forearm. The model and control of a human forearm is analyzed. In the advanced control of robotic manipulators, it is important for manipulators to track trajectories in a wide range of work place. If speed and accuracy is required, the control using conventional methods is difficult to realize because of the high nonlinearity of the robot system. In this work; we study the problem of human hand control carrying a mass. The equation of motion and the natural frequency of the forearm for small angular displacement are derived. We develop new methods that use vector fields in the controller construction for a set of nonlinear dynamical systems. The paper deals with compensate of non-linear system which has a similar idea as the method mentioned in linear system. We utilize an optimal regulator and suggest human arm activities under dynamic environments. Humans must pay compensation for loads arising from interaction with the physical environment. We have managed to design a control law for the non-linear arm-mass-system, in such a way that the representation of a closed loop system is affine, controllable, and a closed loop system is asymptotically stable. Throughout any motion, the forearm can be considered a one-link robot manipulator which could be exploited to benefit people with disabilities (missing extremities). An optimal controller calculates the motor command, which minimize the cost function for nonlinear arm dynamics. Therefore, the purpose of our study is to construct a computational model of arm movement. For this purpose, we use an optimal regulator as the model and compare simulated arm movements with actual nonlinear model human movements. In this paper, we present two approaches to human arm modeling: finite time optimal regulator control approach and modern nonlinear control approach. The latter one is formalized using Lie-Derivative based controllers.

Nonlinear Dynamical Systems as Models of Development of Inorganic Cells (iCHELLs) & their Simple Assemblies

The rapid development of synthetic biology and biomimetics in recent years leads to the creation of prototypes of inorganic artificial cells with membrane-like properties. In this regard there is an obvious need in the development of a universal formalized mathematical description of morphology, dynamics and cooperative behavior, which results in the formation of supracellular assemblies, applicable to both organic and inorganic cells. For this purpose we propose to use invariant sets of nonlinear dynamical systems, which in some variations are isomorphic to biological cells or can be considered as a good morphological metaphor of their morphogenesis and assotiation. This paper compares graphical visualization of invariant sets of nonlinear dynamical systems with the results of experiments on the chemical synthesis of biomimetic structures. A number of maps is considered: Poincare map, Henon map, Chirikov and Gumovski-Mirr maps, systems by Kepler, Volterra, Dyufing and Henon-Heiles. The possibility of approximation of the experimentally observed structures by those mathematical models is shown. The given results of computer simulations are applicable to inorganic cells (iCHELLs), obtained recently in the University of Glasgow, but the proposed method was originally developed for functionally identical inorganic cells, obtained by the author according to the Russian technique, proposed in the early 2000s. The micrographs of such inorganic cells and their assemblies are given below in comparison with the theoretical model.

An Effective Numerical Procedure to Determine Saddle-Type Unstable Invariant Limit Sets in Nonlinear Systems

A new technique that can efficiently approximate the attracting set of a nonlinear dynamical system is proposed under the framework of point mapping with the cell reference method. With the aid of the approximated attracting set, the difficulties encountered by the PIM-triple method and bisection procedure in finding trajectories on the stable manifolds of chaotic saddles in basins of attraction and on basin boundaries can be overcome well. On the basis of this development, an effective method to determine saddle-type invariant limit sets of nonlinear dynamical systems can be devised. Examples are presented for the purposes of illustration and to demonstrate the capabilities of the proposed method.

Stabilization of sets with application to multi-vehicle coordinated motion

In this paper, we develop stability and control design framework for time-varying and time-invariant sets of nonlinear dynamical systems using vector Lyapunov functions. Several Lyapunov functions arise naturally in multi-agent systems, where each agent can be associated with a generalized energy function which further becomes a component of a vector Lyapunov function. We apply the developed control framework to the problem of multi-vehicle coordinated motion to design distributed controllers for individual vehicles moving in a specified formation. The main idea of our approach is that a moving formation of vehicles can be characterized by a time-varying set in the state space, and hence, the problem of distributed control design for multi-vehicle coordinated motion is equivalent to the design of stabilizing controllers for time-varying sets of nonlinear dynamical systems. The control framework is shown to ensure global exponential stabilization of multi-vehicle formations. Finally, we implement the feedback stabilizing controllers for time-invariant sets to achieve global exponential stabilization of static formations of multiple vehicles.

The Structure of Gelfand-Levitan-Marchenko Type Equations for Delsarte Transmutation Operators of Linear Multi-Dimensional Differential Operators and Operator Pencils. Part 2

The differential-geometric and topological structure of Delsarte transmutation operators their associated Gelfand-Levitan-Marchenko type equations are studied making use of the de Rham-Hodge-Skrypnik differential complex. The relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multi-dimensional differential operators are done including the three-dimensional Laplace operator and the two-dimensional classical Dirac operator and its multi-dimensional affine extension, related with self-dual Yang-Mills equations. The soliton like solutions to the related set of nonlinear dynamical systems are discussed.