How can I tune the parameters of a lqr controller?

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.