From my internet research, it seems there's no systematic way of tuning the design parameters for the LQR controller. I have found Bryson’s rule as an initialization. However I don't know how to proceed from this point besides trial and error or grid search. Ideally I want to tune the parameters such that the settling time for specific state variables are below a threshold. Additionally I want to make constraints on the sensitivity function, such that disturbances up to a specific frequency are attenuated. Is there a recommended way to tune my parameters such that requirements can be fulfilled ?
There are many ways. But, all of them are either trivial or subjective. The most fair way I have found so far is to use Interactive Genetic Algorithm (IGA). IGA uses the human subjectivity to lead the GA optimization. Here is a research performing all you need for Model Predictive Control (MPC) [the same as LQR but respecting constraints too]. In your case, the constraints are offline. In this work, a multi-objective GA is used and applied constraints on the results. Then I got a huge pareto-front and applied an IGA to get the best result out of them which looks appealing to a human. You can directly apply IGA without a multi-objective GA too. But, a GA before IGA does a great filtering.
If you don't have access to IEEE, follow the chapter 4 of my thesis (sorry for the goatic font. I ran out of the symbols).
You can apply constraints either during optimization (at a GA before IGA) or in real-time (MPC instead of LQR). In my case, both of the constraints are applied.