Diophantine Equation and the Euclidean Algorithm: Methods and Philosophy of the Algebraic Approach to Linear Multivariable Control [Lecture Notes]

The story behind the Euclidean algorithm and its relationship to the solution of the Diophantine equation is examined in this article. The Euclidean algorithm appears in Proposition 2 in Book 7 of Euclid’s The Elements. This article explores the relationship between some of the concepts and procedures that appear in this Proposition and the ideas and mathematical tools used in linear multivariable control. In particular, its relationship to the solution of the so-called Diophantine equation for polynomial matrices and the problem of the existence and computation of stabilizing controllers for linear multivariable systems (LMSs) is revealed. The results and tools related to the algebraic structure of polynomial matrix models of LMSs that are useful for the numerical solution of the Diophantine equation for polynomial matrices are stated and demonstrated by a simple example.