TAILORING THE BIFURCATION DIAGRAM OF NONLINEAR DYNAMICAL SYSTEMS: AN OPTIMIZATION BASED APPROACH

2010 ◽  
Vol 20 (04) ◽  
pp. 1027-1040 ◽  
Author(s):  
PIETRO ALTIMARI ◽  
ERASMO MANCUSI ◽  
MARIO DI BERNARDO ◽  
LUCIA RUSSO ◽  
SILVESTRO CRESCITELLI
Keyword(s):  
Bifurcation Diagram ◽  
Nonlinear Dynamical Systems ◽  
Feedforward Control ◽  
Nonlinear Dynamical System ◽  
Chemical Reactor ◽  
Control Law ◽  
Control Laws ◽  
Nonlinear Dynamical ◽  
Output Behavior ◽  
Objective Functional

Bifurcation tailoring is a method developed to design control laws modifying the bifurcation diagram of a nonlinear dynamical system to a desired one. In its original formulation, this method does not account for the possible presence of constraints on state and/or manipulated inputs. In this paper, a novel formulation of the bifurcation tailoring method overcoming this limitation is presented. In accordance with the proposed approach, a feedforward control law generating an optimal bifurcation diagram is computed by constrained minimization of an objective functional. Then, a feedback control system enforcing stability of the computed equilibrium branch is designed. In this context, bifurcation analysis is exploited to select feedback controller parameters ensuring desired output behavior and, at the same time, preventing the occurrence of multistability. The method is numerically validated on the problem of tailoring the bifurcation diagram of an exothermic chemical reactor.

scholarly journals Theorems and Application of Local Activity of CNN with Five State Variables and One Port

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Gang Xiong ◽  
Xisong Dong ◽  
Li Xie ◽  
Thomas Yang
Keyword(s):  
Bifurcation Diagram ◽  
Complex Dynamics ◽  
Nonlinear Dynamical Systems ◽  
Reaction Diffusion ◽  
Cellular Neural Network ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Local Activity ◽  
Homogeneous Lattice ◽  
Coupled Cells

Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems.

scholarly journals Modeling of information channel by using of pseudorandom signals of nonlinear dynamical system

2020 ◽  
Vol 22 (4) ◽  
pp. 79-87
Author(s):  
I. K. Nasyrov ◽  
V. V. Andreev
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Detection Algorithm ◽  
Main Step ◽  
Operating Characteristics ◽  
Nonlinear Dynamical ◽  
Pseudorandom Signals ◽  
Chaotic Signals ◽  
Correlation Properties

Pseudorandom signals of nonlinear dynamical systems are studied and the possibility of their application in information systems analyzed. Continuous and discrete dynamical systems are considered: Lorenz System, Bernoulli and Henon maps. Since the parameters of dynamical systems (DS) are included in the equations linearly, the principal possibility of the state linear control of a nonlinear DS is shown. The correlation properties comparative analysis of these DSs signals is carried out.. Analysis of correlation characteristics has shown that the use of chaotic signals in communication and radar systems can significantly increase their resolution over the range and taking into account the specific properties of chaotic signals, it allows them to be hidden. The representation of nonlinear dynamical systems equations in the form of stochastic differential equations allowed us to obtain an expression for the likelihood functional, with the help of which many problems of optimal signal reception are solved. It is shown that the main step in processing the received message, which provides the maximum likelihood functionals, is to calculate the correlation integrals between the components and the systems under consideration. This made it possible to base the detection algorithm on the correlation reception between signal components. A correlation detection receiver was synthesized and the operating characteristics of the receiver were found.

The Generalized Shooting Arc-Length Method for Periodic Solutions and the Corresponding Periods of Nonlinear Dynamical Systems

2001 ◽  
Author(s):  
Dexin Li ◽  
Jianxue Xu
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Periodic Orbit ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Optimization Method ◽  
Arc Length ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Nonlinear Dynamical

Abstract In this paper, a generalized shooting/arc-length method for determining periodic orbit and its period of nonlinear dynamical system is presented. At first, by changing the time scale the period value of periodic orbit of the nonlinear system is drawn into the governing equation of this system. Then, by using the period value as a parameter, the shooting/arc-length procedure is taken for seeking such a periodic solution and its period simultaneously. The value of increment changed in iteration procedure is selected by using optimization method. The procedure involves the detennining of periodic orbit and its period value of the system. Thereby, the periodic orbit and period value of the system can be sought out rapidly and precisely. At last, the validity of such method is verified by determining the periodic orbit and period value for van der pol equation and nonlinear rotor-bear system.

A New Method for Equilibria and Stability Analysis of Nonlinear Dynamical Systems

2014 ◽  
Vol 534 ◽  
pp. 131-136
Author(s):  
Long Cao ◽  
Yi Hua Cao
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
Vehicle Model ◽  
Nonlinear Dynamical ◽  
Novel Method ◽  
Linearized System ◽  
Continuation Algorithm

A novel method based on numerical continuation algorithm for equilibria and stability analysis of nonlinear dynamical system is introduced and applied to an aircraft vehicle model. Dynamical systems are usually modeled with differential equations, while their equilibria and stability analysis are pure algebraic problems. The newly-proposed method in this paper provides a way to solve the equilibrium equation and the eigenvalues of the locally linearized system simultaneously, which avoids QR iterations and can save much time.

Bifurcation Control of Ro¨ssler System

2003 ◽  
Author(s):  
Z. Q. Wu ◽  
P. Yu
Keyword(s):  
Feedback Control ◽  
Nonlinear Dynamical Systems ◽  
Equilibrium Points ◽  
New Method ◽  
Control Law ◽  
Equilibrium Solutions ◽  
Nonlinear Dynamical ◽  
The Stability ◽  
Parameter Values ◽  
Feasible System

In this paper, a new method is proposed for controlling bifurcations of nonlinear dynamical systems. This approach employs the idea used in deriving the transition variety sets of bifurcations with constraints to find the stability region of equilibrium points in parameter space. With this method, one can design, via a feedback control, appropriate parameter values to delay either static, or dynamic or both bifurcations as one wishes. The approach is applied to consider controlling bifurcations of the Ro¨ssler system. The uncontrolled Ro¨ssler has two equilibrium solutions, one of which exhibits static bifurcation while the other has Hopf bifurcation. When a feedback control based on the new method is applied, one can delay the bifurcations and even change the type of bifurcations. An optimal control law is obtained to stabilize the Ro¨ssler system using all feasible system parameter values. Numerical simulations are used to verify the analytical results.

scholarly journals Design of Composite Feedback and Feedforward Control Law for Aircraft Inertially Stabilized Platforms

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Valerii Azarskov ◽  
Anatoly Tunik ◽  
Olha Sushchenko
Keyword(s):  
Control Systems ◽  
Feedforward Control ◽  
Design Method ◽  
Synergetic Effect ◽  
Measurement Unit ◽  
Control Law ◽  
Control Laws ◽  
Robustness Criterion ◽  
Two Stages ◽  
Invariance Theory

The design of the control systems of the inertially stabilized platforms (ISPs) as part of airborne equipment for the majority of aircraft has its peculiarity. The presence of rate gyros in the inertial measurement unit gives the possibility to measure the rotation rate of the ISP base, which is the main disturbance interfering with the ISP accuracy. Inclusion of the feedforward disturbance gain in the control law with the simplest PI feedback significantly improves the accuracy of stabilization by the invariance theory. A combination of feedback and feedforward controllers produces a synergetic effect, thus, improving ISP accuracy. This article deals with the design of the airborne ISP control systems consisting of two stages: the parametric optimization of the PI feedback control based on composite “performance-robustness” criterion and the augmentation of the obtained system with feedforward gain. To prove the efficiency of the proposed control laws, the simulation of the ISP was undertaken. We have used a simulation of the heading-hold system of the commuter aircraft Beaver and the yaw rate output of this closed-loop system we have used as a source of the disturbance. The results of modeling proved the efficiency of the proposed design method.

COMPETITIVE MODES AND THEIR APPLICATION

2006 ◽  
Vol 16 (03) ◽  
pp. 497-522 ◽  
Author(s):  
WEIGUANG YAO ◽  
PEI YU ◽  
CHRISTOPHER ESSEX ◽  
MATT DAVISON
Keyword(s):  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Point Of View ◽  
Frequency Function ◽  
Mode Competition ◽  
Integration Scheme ◽  
Nonlinear Dynamical ◽  
Competitive Modes ◽  
Definition Of

We investigate nonlinear dynamical systems from the mode competition point of view, and propose the necessary conditions for a system to be chaotic. We conjecture that a chaotic system has at least two competitive modes (CM's). For a general nonlinear dynamical system, we give a simple, dynamically motivated definition of mode suitable for this concept. Since for most chaotic systems it is difficult to obtain the form of a CM, we focus on the competition between the corresponding modulated frequency components of the CM's. Some direct applications result from the explicit form of the frequency functions. One application is to estimate parameter regimes which may lead to chaos. It is shown that chaos may be found by analyzing the frequency function of the CM's without applying a numerical integration scheme. Another application is to create new chaotic systems using custom-designed CM's. Several new chaotic systems are reported.

scholarly journals Lagrange and practical stability criteria for dynamical systems with nonlinear perturbations

2012 ◽  
Vol 22 (1) ◽  
pp. 43-58
Author(s):  
Assen Krumov
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Computer Control ◽  
Nonlinear Dynamical System ◽  
Sufficient Conditions ◽  
Practical Stability ◽  
Nonlinear Perturbations ◽  
Stability Criteria ◽  
Nonlinear Operators ◽  
Nonlinear Dynamical

Lagrange and practical stability criteria for dynamical systems with nonlinear perturbationsIn the paper two classes of nonlinear dynamical system with perturbations are considered. The sufficient conditions for robust Lagrange and practical stability are proven with theorems, applying the theory of nonlinear operators of the functional analysis. The presented criteria give also the bounds of the analyzed dynamical processes. Three examples comparing the numerical computer solutions and the analytical investigation of the stability of the systems are given. The method can be applied to analytical and computer modeling of nonlinear dynamical systems, synthesis of computer control and optimization.

scholarly journals Constrained attractor selection using deep reinforcement learning

2020 ◽  
pp. 107754632093014
Author(s):  
Xue-She Wang ◽  
James D Turner ◽  
Brian P Mann
Keyword(s):  
Reinforcement Learning ◽  
Gradient Method ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Learning Approaches ◽  
Multiple Attractors ◽  
Nonlinear Dynamical ◽  
Cross Entropy Method ◽  
Policy Gradient ◽  
Attractor Selection

This study describes an approach for attractor selection (or multistability control) in nonlinear dynamical systems with constrained actuation. Attractor selection is obtained using two different deep reinforcement learning methods: (1) the cross-entropy method and (2) the deep deterministic policy gradient method. The framework and algorithms for applying these control methods are presented. Experiments were performed on a Duffing oscillator, as it is a classic nonlinear dynamical system with multiple attractors. Both methods achieve attractor selection under various control constraints. Although these methods have nearly identical success rates, the deep deterministic policy gradient method has the advantages of a high learning rate, low performance variance, and a smooth control approach. This study demonstrates the ability of two reinforcement learning approaches to achieve constrained attractor selection.

A Terminal Sliding Mode Control of Disturbed Nonlinear Second-Order Dynamical Systems

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Pawel Skruch
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Sliding Mode ◽  
Second Order ◽  
Control Law ◽  
Sliding Surface ◽  
Terminal Sliding Mode ◽  
Nonlinear Dynamical ◽  
Second Order Differential Equations ◽  
Bounded Disturbance

The paper presents a terminal sliding mode controller for a certain class of disturbed nonlinear dynamical systems. The class of such systems is described by nonlinear second-order differential equations with an unknown and bounded disturbance. A sliding surface is defined by the system state and the desired trajectory. The control law is designed to force the trajectory of the system from any initial condition to the sliding surface within a finite time. The trajectory of the system after reaching the sliding surface remains on it. A computer simulation is included as an example to verify the approach and to demonstrate its effectiveness.

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