# Optimal controller/observer gains of discounted-cost LQG systems

## TL;DR

This paper extends the LQG control framework to systems with discounted cost functions, deriving optimal control strategies when the state is estimated rather than known, and providing explicit cost expressions.

## Contribution

It introduces a novel optimal control approach for discounted-cost LQG systems using estimated states, filling a gap in existing control theory.

## Key findings

- Derived explicit optimal control laws for estimated states.
- Provided formulas for the optimal expected cost.
- Extended LQG theory to discounted cost scenarios.

## Abstract

The linear-quadratic-Gaussian (LQG) control paradigm is well-known in literature. The strategy of minimizing the cost function is available, both for the case where the state is known and where it is estimated through an observer. The situation is different when the cost function has an exponential discount factor, also known as a prescribed degree of stability. In this case, the optimal control strategy is only available when the state is known. This paper builds on from that result, deriving an optimal control strategy when working with an estimated state. Expressions for the resulting optimal expected cost are also given.