Reduced-Order Algorithm for Eigenvalue Assignment of Singularly Perturbed Linear Systems

Heonjong Yoo,Kyeong-Hwan Lee,Zoran Gajic

doi:10.1155/2020/3948564

Abstract

In this paper, we present an algorithm for eigenvalue assignment of linear singularly perturbed systems in terms of reduced-order slow and fast subproblem matrices. No similar algorithm exists in the literature. First, we present an algorithm for the recursive solution of the singularly perturbed algebraic Sylvester equation used for eigenvalue assignment. Due to the presence of a small singular perturbation parameter that indicates separation of the system variables into slow and fast, the corresponding algebraic Sylvester equation is numerically ill-conditioned. The proposed method for the recursive reduced-order solution of the algebraic Sylvester equations removes ill-conditioning and iteratively obtains the solution in terms of four reduced-order numerically well-conditioned algebraic Sylvester equations corresponding to slow and fast variables. The convergence rate of the proposed algorithm is Oε, where ε is a small positive singular perturbation parameter.

Highlights

E aim of our developed algorithm is to solve a largescale Sylvester equation in order to overcome the numerical ill-conditioning problem of singularly perturbed systems presented in [11]. is leads to reduced-order regular algebraic Sylvester equations [12], combined with the techniques presented in [13, 14] which solves the eigenvalue assignment problem for singularly perturbed linear systems
We will consider the Sylvester equation encountered in the control system design of linear systems: dx(t) Ax(t) + Bu(t), dt y(t) Cx(t), Mathematical Problems in Engineering where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input vector, y(t) ∈ Rp is the vector of system measurements, and A, B, and C are constant matrices
Forms of the Sylvester algebraic equations that appear in the observer and controller designs are given by

Summary

Introduction

E aim of our developed algorithm is to solve a largescale Sylvester equation in order to overcome the numerical ill-conditioning problem of singularly perturbed systems presented in [11]. is leads to reduced-order regular algebraic Sylvester equations [12], combined with the techniques presented in [13, 14] which solves the eigenvalue assignment problem for singularly perturbed linear systems. E aim of our developed algorithm is to solve a largescale Sylvester equation in order to overcome the numerical ill-conditioning problem of singularly perturbed systems presented in [11]. Is leads to reduced-order regular algebraic Sylvester equations [12], combined with the techniques presented in [13, 14] which solves the eigenvalue assignment problem for singularly perturbed linear systems. Forms of the Sylvester algebraic equations that appear in the observer and controller designs are given by

Results

Conclusion