Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Average predictive control for nonlinear discrete dynamical systems

D. Dmitrishin, E. Iacob, A. Stokolos

TL;DR

This work proposes the generalization of predictive control method for resolving the stabilization problem of unstable periodic orbits in discrete nonlinear dynamical systems and embodies the development of control method proposed by B.T. Polyak.

Abstract

We explore the problem of stabilization of unstable periodic orbits in discrete nonlinear dynamical systems. This work proposes the generalization of predictive control method for resolving the stabilization problem. Our method embodies the development of control method proposed by B.T. Polyak. The control we propose uses a linear (convex) combination of iterated functions. With the proposed method auxiliary, the problem of robust cycle stabilization for various cases of its multipliers localization is solved. An algorithm for finding a given length cycle when its multipliers are known is described as a particular case of our method application. Also, we present numerical simulation results for some well-known mappings and the possibility of further generalization of this method.

Average predictive control for nonlinear discrete dynamical systems

TL;DR

This work proposes the generalization of predictive control method for resolving the stabilization problem of unstable periodic orbits in discrete nonlinear dynamical systems and embodies the development of control method proposed by B.T. Polyak.

Abstract

We explore the problem of stabilization of unstable periodic orbits in discrete nonlinear dynamical systems. This work proposes the generalization of predictive control method for resolving the stabilization problem. Our method embodies the development of control method proposed by B.T. Polyak. The control we propose uses a linear (convex) combination of iterated functions. With the proposed method auxiliary, the problem of robust cycle stabilization for various cases of its multipliers localization is solved. An algorithm for finding a given length cycle when its multipliers are known is described as a particular case of our method application. Also, we present numerical simulation results for some well-known mappings and the possibility of further generalization of this method.

Paper Structure

This paper contains 25 sections, 5 theorems, 46 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Problem statement.
  3. Constructing the Jacobi matrix for a controlled system
  4. Main result
  5. Case $M=\left\{\mu_1,\dots,\mu_m\right\}$
  6. Case $M=\{z:\, {\rm Re}\,z\le0\}\cup {\mathbb D}$
  7. Case $M={\mathbb D}\cup\left\{\mu^*\right\},\ \left|\mu^*\right|>1$ 15
  8. Case $M={\mathbb D}\cup\left\{\mu^*,\overline{\mu}^*\right\},\ \left|\mu^*\right|>1$
  9. Case $T=1,\ M=\lfloor-\mu^*,\mu^*\rfloor,\ M=\lfloor-\mu^*,1\rfloor$
  10. The general case
  11. Examples
  12. Logistic mapping
  13. Triangular mapping
  14. Burgers mapping
  15. Tinkerbell mapping
  16. ...and 10 more sections

Key Result

Lemma 3.1

The Jacobi matrix of the cycle $\left\{\eta_1,\ldots,\eta_T\right\}$ in the system 2 can be represented as where $J$ is the Jacobi matrix of the cycle $\left\{\eta_1,\ldots,\eta_T\right\}$ in the system 1.

Figures (12)

  • Figure 1: A 101-cycle of the logistic mapping \ref{['8']}.
  • Figure 2: A 101-cycle of triangular system \ref{['9']}.
  • Figure 3: A 101-cycle of the Burgers system \ref{['10']}.
  • Figure 4: A 101-cycle of the Tinkerbell mapping \ref{['11']}.
  • Figure 5: A 101-cycle of the Gingerbredman mapping \ref{['12']}.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Lemma 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • Theorem 4.4
  • proof