Banded Matrix Fraction Representation of Triangular Input Normal Pairs

Abstract

An input pair $(A,B)$ is triangular input normal if and only if $A$ is triangular and $A{A}^{\ast }+B{B}^{\ast }=I_n$, 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.