I am working on a Kalman Filter (KF) design problem and I am struggling to understand the role of the Riccati equations in the design process of a KF.
Some sources discuss the importance of Riccati equations to the design of KF (e.g., Grewal and Andrews) while others do not mention it at all! Unfortunately, I have not actually found any source that really provides a good reason why I should care about these equations beyond a theoretical perspective (most just dive into the theory and do not explain why it is relevant). Given the depth of the math involved in to solve these equations, I would like to understand how important they are to a KF design process before committing (probably a lot) of time to understand them.
Here is what I think I understand about the role of Riccati equations for KFs:
The KF algorithm is an algorithm (numerical?) solution to the Riccati equation for the given system model. More specifically the state covariance matrix, $P$, is the solution to the equations when the KF is run for infinite time (assuming it converges at all).
We can (in principle, for simple systems) solve the Riccati equations ahead of time to find the $P$ matrix that the KF will converge towards if all operational assumptions of the KF are satisfied (e.g., Gaussian noise, etc.).
What I am trying to understand is the role of working out the solution (point 2 above) to the Riccati equations ahead of time? Is this simply a matter of allowing the designer to understand the idealized $P$ matrix for their filter? Why would I bother doing that over just using a simulation package (e.g., Simulink) to work out the value of $P$? What advantage does the analytical solution provide me?