# An Input-Output Parametrization of Stabilizing Controllers: amidst Youla and System Level Synthesis

## TL;DR

This paper introduces a new input-output parametrization method for stabilizing controllers that simplifies design by directly using transfer matrices, avoiding complex factorizations, and enabling convex optimization for optimal control.

## Contribution

It presents a novel input-output parametrization framework that bypasses the need for doubly-coprime factorizations and initial stabilizing controllers, facilitating easier and convex control design.

## Key findings

- Allows direct computation of stabilizing controllers via linear programming.
- Enables norm-optimal control under quadratically invariant constraints.
- Provides an alternative to Youla and System Level Synthesis methods.

## Abstract

This paper proposes a novel input-output parametrization of the set of internally stabilizing output-feedback controllers for linear time-invariant (LTI) systems. Our underlying idea is to directly treat the closed-loop transfer matrices from disturbances to input and output signals as design parameters and exploit their affine relationships. This input-output perspective is particularly effective when a doubly-coprime factorization is difficult to compute, or an initial stabilizing controller is challenging to find; most previous work requires one of these pre-computation steps. Instead, our approach can bypass such pre-computations, in the sense that a stabilizing controller is computed by directly solving a linear program (LP). Furthermore, we show that the proposed input-output parametrization allows for computing norm-optimal controllers subject to quadratically invariant (QI) constraints using convex programming.