Robust Parametric Control of Lorenz System via State Feedback

Da-Ke Gu,Yin-Dong Liu,Da-Wei Zhang,Chun-Biao Li

doi:10.1155/2020/6548142

Da-Ke Gu, Yin-Dong Liu + Show 2 more

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https://doi.org/10.1155/2020/6548142

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Abstract

This paper considers the parametric control to the Lorenz system by state feedback. Based on the solutions of the generalized Sylvester matrix equation (GSE), the unified explicit parametric expression of the state feedback gain matrix is proposed. The closed loop of the Lorenz system can be transformed into an arbitrary constant matrix with the desired eigenstructure (eigenvalues and eigenvectors). The freedom provided by the parametric control can be fully used to find a controller to satisfy the robustness criteria. A numerical simulation is developed to illustrate the effectiveness of the proposed approach.

Highlights

Lorenz system, first studied by Edward Lorenz, is a simplified mathematical model for atmospheric convection [1], which can display a phenomenon called chaos
Kim et al put forward a robust control approach to regulate and synchronize the generalized Lorenz system based on the backstepping method, while the nonlinear and uncertain items can be estimated and canceled [3]
Zhang et al design a hybrid controller for the Lorenz system with a piecewise linear memristor and provide criteria to maintain that the trivial solutions are exponentially stable in the mean square [7]

Summary

Introduction

First studied by Edward Lorenz, is a simplified mathematical model for atmospheric convection [1], which can display a phenomenon called chaos. Parametric approach, creatively proposed by Duan [22, 23], solves the controller design of quasi-linear systems via state feedback and output feedback, which develops novel research areas. In this paper, based on the solutions of a class of GSE [31, 32], a whole set of parametric state feedback controllers is established and the closed-loop system is transformed into the desired eigenstructure. E parametric approach provides a group of arbitrary parameters that represent the degrees of design freedom, to implement state feedback and robust optimization. RankA, detA, and eigA represent the rank, determinant, and all eigenvalues of matrix A, respectively. deg (A(s)) denotes the degree n of polynomial matrix A(s) A0 + sA1 + · · · + snAn; diag􏼈λ1, λ2, . . . , λn􏼉 indicates the diagonal matrix with diagonal elements λi, i 1, 2, . . . , n. max and min represent the maximum and minimum

Problem Statement and Preliminaries

Main Results

An Example

Conclusion