A New Approach to Stable Optimal Control of Complex Nonlinear Dynamical Systems

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
Firdaus E. Udwadia
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Nonlinear Dynamical System ◽  
Rate Of Change ◽  
Definite Function ◽  
Constrained Motion ◽  
Nonlinear Controller ◽  
Control Cost ◽  
Nonlinear Dynamical ◽  
Negative Definite Function

This paper gives a simple approach to designing a controller that minimizes a user-specified control cost for a mechanical system while ensuring that the control is stable. For a user-given Lyapunov function, the method ensures that its time rate of change is negative and equals a user specified negative definite function. Thus a closed-form, optimal, nonlinear controller is obtained that minimizes a desired control cost at each instant of time and is guaranteed to be Lyapunov stable. The complete nonlinear dynamical system is handled with no approximations/linearizations, and no a priori structure is imposed on the nature of the controller. The methodology is developed here for systems modeled by second-order, nonautonomous, nonlinear, differential equations. The approach relies on some recent fundamental results in analytical dynamics and uses ideas from the theory of constrained motion.

Optimal tracking control of nonlinear dynamical systems

2008 ◽  
Vol 464 (2097) ◽  
pp. 2341-2363 ◽  
Author(s):  
Firdaus E Udwadia
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Weighted Norm ◽  
Analytical Dynamics ◽  
Control Cost ◽  
Nonlinear Dynamical ◽  
First Order ◽  
Time Optimal

This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems. A numerical example illustrates the methodology.

A Topological Study of the Genesis of a Horseshoe Vortex in the Symmetry Plane Due to an Adverse Pressure Gradient

2001 ◽  
Author(s):  
Martijn A. van den Berg ◽  
Michael M. J. Proot ◽  
Peter G. Bakker
Keyword(s):  
Pressure Gradient ◽  
Bifurcation Theory ◽  
Nonlinear Differential Equations ◽  
Nonlinear Dynamical System ◽  
Symmetry Plane ◽  
Adverse Pressure Gradient ◽  
Horseshoe Vortex ◽  
Nonlinear Dynamical ◽  
Separating Flow ◽  
Flow Through

Abstract The present paper describes the genesis of a horseshoe vortex in the symmetry plane in front of a juncture. In contrast to a previous topological investigation, the presence of the obstacle is no longer physically modelled. Instead, the pressure gradient, induced by the obstacle, has been used to represent its influence. Consequently, the results of this investigation can be applied to any symmetrical flow above a flat plate. The genesis of the vortical structure is analysed by using the theory of nonlinear differential equations and the bifurcation theory. In particular, the genesis of a horseshoe vortex can be described by the unfolding of the degenerate singularity resulting from a Jordan Normal Form with three vanishing eigenvalues and one linear term which is related to the adverse pressure gradient. The examination of this nonlinear dynamical system reveals that a horseshoe vortex emanates from a non-separating flow through two subsequent saddle-node bifurcations in different directions and the transition of a node into a focus located in the flow field.

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.

Determining Upconversion Parameters and Invertibility of Nonlinear Dynamical Systems

2015 ◽  
Vol 1753 ◽  
Author(s):  
Makhin Thitsa ◽  
Thanh Q. Ta
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Energy Levels ◽  
Classical System ◽  
Rate Equations ◽  
Unknown Parameters ◽  
Emission Data ◽  
Nonlinear Dynamical ◽  
Material Development ◽  
System Inversion

ABSTRACTDetermining upconversion parameters is of high interest in laser material development. For many materials these parameters cannot be directly measured by experimental methods. These upconversion coefficients appear as unknown parameters in the laser rate equations, which are a system of coupled nonlinear differential equations that are used to model the dynamics of population densities in different energy levels. In this paper we propose the well-established system theoretic tools pertaining to the system inversion to be applied in this case. The unknown parameters can be considered as the inputs and the fluorescence signals can be considered as the outputs of the dynamical system. Therefore the determination of the unknown upconversion rates in the system equations from the output data is a classical system inversion problem. In this paper we demonstrate how to compute the unknown coefficients in the rate equations from the experimental emission data utilizing this 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.

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