Optimal controller/observer gains of discounted-cost LQG systems

The linear-quadratic-Gaussian control paradigm is well-known in literature. The strategy of minimizing the cost function is available, both for the case where the state is fully 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 fully known. This paper builds onward from that result, deriving an optimal control strategy when working with an estimated state. Expressions for the resulting optimal expected cost are also given. The result is verified through an experimental validation.