# Orthogonal Representations for Output System Pairs

## TL;DR

This paper introduces a new canonical form for output system pairs using orthogonal representations, enabling efficient state updates and improved system identification conditioning.

## Contribution

It proposes a novel orthogonal canonical form for output system pairs with fast updates and identifiable parameters, enhancing system identification methods.

## Key findings

- State updates require O(nd) operations.
- Derivatives with respect to parameters are efficiently computed.
- System identification is better conditioned due to the identity observability Grammian.

## Abstract

A new class of canonical forms is given proposed in which $(A, C)$ is in Hessenberg observer or Schur form and output normal: $\bf{I} - A^*A =C^*C$. Here, $C$ is the $d \times n$ measurement matrix and $A$ is the advance matrix. The $(C, A)$ stack is expressed as the product of $n$ orthogonal matrices, each of which depends on $d$ parameters. State updates require only ${\cal O}(nd)$ operations and derivatives of the system with respect to the parameters are fast and convenient to compute. Restrictions are given such that these models are generically identifiable. Since the observability Grammian is the identity matrix, system identification is better conditioned than other classes of models with fast updates.