# Output Feedback Control for Irregular LQ Problem

## TL;DR

This paper addresses the irregular output feedback LQ control problem, proposing a modified cost function, a two-layer optimization approach, and deriving an explicit optimal controller that combines Kalman filtering with Riccati equations.

## Contribution

It introduces a novel framework for irregular LQ control with a modified cost function and a two-layer optimization method, extending previous work to include explicit controllers.

## Key findings

- The separation principle still holds under the modified cost function.
- An explicit optimal controller is derived using two Riccati equations.
- The approach ensures the terminal controller is deterministic.

## Abstract

In this paper, we study the irregular output feedback linear quadratic (LQ) control problem, which is a continuous work of previous works for irregular LQ control [33] where the state is assumed to be exactly known priori. Different from the classic output feedback LQ control problem, the cost function must be modified to guarantee the solvability. In the framework of the modified cost function, it is shown that the separation principle holds and the explicitly optimal controller is given in the feedback form of the Kalman filtering. In particular, the feedback gain is calculated by two Riccati equations independently of the Kalman filtering. The key technique is the "two-layer optimization" approach. We also emphasize that the optimal controller at the terminal time is required to be deterministic.