Ackermann’s Method: Revisited, Extended, and Generalized to Uncontrollabe Systems

Omar M El-Ghezawi

doi:10.1109/access.2021.3111928

Omar M El-Ghezawi

Open Access

https://doi.org/10.1109/access.2021.3111928

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Abstract

The celebrated method of Ackermann for eigenvalue assignment of single-input controllable systems is revisited in this paper, contributing an elegant proof. The new proof facilitates a compact formula which consequently permits an extension of the method to what we call incomplete assignment of eigenvalues. The inability of Ackermann’s formula to deal with uncontrollable systems is considered a weakness inherent in the method. The notion of incomplete assignment leads to a straightforward generalization of our method to eigenvalue assignment of uncontrollable systems, thus mitigating such a drawback of a popular method. Further results concerning the incomplete assignment are stated, verified, and commented. Such results reveal the trace of the state matrix $A$ a worthy feature pertinent to an open loop system. Finally, four numerical examples are worked out to demonstrate cases of incomplete, and uncontrollable eigenvalues assignment. The examples consider a case where the structure of the feedback matrix can be easily simplified. The paper ends with a commentary brief concerning some commonly used MATLAB commands for eigenvalue assignment.

Highlights

The eigenvalue assignment problem is well established in system theory and continues to attract further research over the years
K, and provide an alternative general proof to the above expression given in (3), ending with a compact form for K. Such depiction facilitates incomplete eigenvalue assignment and enables the method to be extended to uncontrollable systems
An advantage of incomplete assignment is that it can help in simplifying the structure for the feedback matrix in certain cases especially when the system is uncontrollable

Summary

INTRODUCTION

The eigenvalue assignment problem is well established in system theory and continues to attract further research over the years. NATURE OF THE ACKERMANN’S FORMULA The method is well known, simple in concept, of explicit nature, and requires no particular transformation It suffers from the drawback of only applying to controllable single-input systems [1,2], [25,26]. K , and provide an alternative general proof to the above expression given in (3) , ending with a compact form for K Such depiction facilitates incomplete eigenvalue assignment and enables the method to be extended to uncontrollable systems. Stands as a compact alternative to the original classical form presented in (3) Such depiction proves to be convenient in extending the method to incomplete eigenvalue assignment and to the assignment of uncontrollable systems as will be shown in sec.V, and sec.VI

FURTHER INSIGHT

INCOMPLETE EIGENVALUES ASSIGNMENT

UNCONTROLLABLE EIGENVALUE ASSIGNMENT

CONCLUSION