May someone, in simple terms, describe to me the difference between a Kalman filter and a linear quadratic regulator? If possible, please use an analogy or maybe even a visual demonstration of the difference. Also, some basic mathematical descriptions would also be helpful...thank you.
The LQR is a linear quadratic regulator (controller) and can be applied if the system is fully controllable. It takes the full state vector and transforms it into an input signal to achieve desired properties of the dynamical system. In many practical problems we are not able to measure the full state vector (e.g. sensors are too expensive). In order to estimate the full state vector we must check if the system is observable (e.g. by Kalman's observability criterion), if the system is observable then it is possible to estimate the full state vector by a Kalman filter (LQE = linear quadratic estimator) and use this estimate of the full state vector as input for the LQR controller. Note that the combination of LQR and LQE, sometimes also called LQG (Linear Quadratic Gaussian) might be very sensitive to model uncertainties. This means that even very small uncertainties can lead to an unstable LQG.