Banded Matrix Fraction Representation of Triangular Input Normal Pairs

TL;DR

This paper introduces a new banded matrix fraction representation for triangular input normal pairs, enabling efficient system updates and identification, especially when the matrix has real eigenvalues.

Contribution

It presents a novel system representation where the matrix is expressed as a low-bandwidth triangular matrix fraction, facilitating faster computations and explicit parameterization.

Findings

Efficient state update requires only 3n multiplications for real eigenvalues.

Provides explicit parameterization in terms of eigenvalues.

Enables fast system identification and updates.

Abstract

An input pair $(A,B)$ is triangular input normal if and only if $A$ is triangular and $AA^* + BB^* = I_n$, where $I_n$ is theidentity matrix. Input normal pairs generate an orthonormal basis for the impulse response. Every input pair may be transformed to a triangular input normal pair. A new system representation is given: $(A,B)$ is triangular normal and $A$ is a matrix fraction, $A=M^{-1}N$, where $M$ and $N$ are triangular matrices of low bandwidth. For single input pairs, $M$ and $N$ are bidiagonal and an explicit parameterization is given in terms of the eigenvalues of $A$. This band fraction structure allows for fast updates of state space systems and fast system identification. When A has only real eigenvalues, one state advance requires $3n$ multiplications for the single input case.