Certification of fixed computation time first-order optimization-based controllers for a class of nonlinear dynamical systems

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.

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On Periodic Motions in the First-Order Nonlinear Systems

Volume 4B: Dynamics, Vibration, and Control ◽

10.1115/imece2016-66219 ◽

2016 ◽

Author(s):

Albert C. J. Luo ◽

Yeyin Xu ◽

Zhaobo Chen

Keyword(s):

Dynamical System ◽

Fourier Series ◽

Dynamical Systems ◽

Analytical Solutions ◽

Nonlinear Dynamical Systems ◽

Initial Conditions ◽

Nonlinear Dynamical System ◽

Periodic Motions ◽

Nonlinear Dynamical ◽

First Order

In this paper, analytical solutions of periodic motions in the first-order nonlinear dynamical system are discussed from the finite Fourier series expression. The first-order nonlinear dynamical system is transformed to the dynamical system of coefficients in the Fourier series. From investigation of such dynamical system of coefficients, the analytical solutions of periodic motions are obtained, and the corresponding stability and bifurcation of periodic motions will be determined. In fact, this method provides a frequency-response analysis of periodic motions in nonlinear dynamical systems, which is alike the Laplace transformation of periodic motions for nonlinear dynamical systems. The harmonic frequency-amplitude curves are obtained for different-order harmonic terms in the Fourier series. Through such frequency-amplitude curves, the nonlinear characteristics of periodic motions in the first-order nonlinear system can be determined. From analytical solutions, the initial conditions are obtained for numerical simulations. From such initial conditions, numerical simulations are completed in comparison of the analytical solutions of periodic motions.

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Cleaning the Periodicity in Discrete- and Continuous-Time Nonlinear Dynamical Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414300201 ◽

2014 ◽

Vol 24 (07) ◽

pp. 1430020 ◽

Cited By ~ 2

Author(s):

Paulo C. Rech

Keyword(s):

Dynamical Systems ◽

Continuous Time ◽

Nonlinear Dynamical Systems ◽

Angular Frequency ◽

Two Dimensional ◽

Periodic Forcing ◽

Dimensional Parameter ◽

Parameter Spaces ◽

Nonlinear Dynamical ◽

First Order

We investigate periodicity suppression in two-dimensional parameter-spaces of discrete- and continuous-time nonlinear dynamical systems, modeled respectively by a two-dimensional map and a set of three first-order ordinary differential equations. We show for both cases that, by varying the amplitude of an external periodic forcing with a fixed angular frequency, windows of periodicity embedded in a chaotic region may be totally suppressed.

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