Polynomial and Matrix Fraction Description

Glossary Canonical realization: A state-space realization that has specific properties that makes it unique. Characteristic polynomial: The scalar denominator of a transfer function. Column-reduced: A polynomial matrix with non-singular column leading coefficient matrix. Coprime: See relatively prime. Denominator polynomial: The polynomial appearing in the denominator of a rational transfer function. Elementary operations: Operations that are performed to reduce a polynomial matrix to some special form. Greatest common divisor: A highest degree common factor that can be extracted from two polyno-mials. Hermite form: A triangular canonical form of a polynomial matrix. Irreducible: A transfer function is irreducible when its numerator and denominator polynomials are relatively prime. Leading coefficient matrix: The constant matrix whose entries are built from coefficients of highest powers of the entries of a polynomial matrix. Left MFD: An MFD where the denominator polynomial matrix enters from the left. Matrix fraction description: Ratio of two polynomial matrices describing a matrix transfer function. MFD: See matrix fraction description. MIMO: Multi-input multi-output system. Minimal: A state-space realization is minimal if it has the lowest possible dimension. Monic: A polynomial is monic when its leading coefficient is equal to one. Numerator polynomial: The polynomial appearing in the numerator of a rational transfer function. Polynomial matrix: A matrix with polynomial entries, or equivalently a polynomial with matrix coefficients. Polynomial echelon form: See Popov form. Polynomial Toolbox: The Matlab Toolbox for polynomials, polynomial matrices and their application in systems, signals and control. Popov form: A canonical form of a polynomial matrix with a special structure. Proper: A matrix transfer function is proper if the degree of the denominator polynomial of each entry is greater than or equal to the degree of the numerator polynomial. State-space realization: The state-space equation describing internally a transfer function. Realization: See state-space realization. Relatively prime: Two polynomials or polynomial matrices are relatively prime if they have no common factor. Right MFD: An MFD where the denominator polynomial matrix enters from the right. Row-reduced: A polynomial matrix with non-singular row leading coefficient matrix. SISO: Single-input single-output system. Transfer function: Rational matrix relating the Laplace or z-transform of the output to the Laplace or z-transform of the input, in the absence of initial conditions. Unimodular: A polynomial matrix with a non-zero constant determinant Summary This article illustrates how polynomials and polynomial matrices can be used to describe linear systems. The focus is put on the transformation to and from the state-space …