How can I identify eliminated state when using minreal? How can I solve LQR error?

I have an adversity in LQR control. It would be grateful if anyone can suggest good direction.

I had following error:

The "lqrd" command failed to stabilize the discretized plant or find an optimal feedback gain. To remedy this problem: 1. Make sure that all unstable poles of A are controllable through B (use MINREAL to check) 2. Modify the weights Q and R to make [Q N;N' R] positive definite (use EIG to check positivity) 3. Try a different sample time.

I think the reason is first thing because when I use MINREAL, I found that there are state elimination. However, I can't figure out which state was eliminated. How can I do this?

Another question: I'm planning to apply control input to nonlinear plant by using state feedback like u=-Kx. K is LQR gain. If I can get LQR gain from minimal realized state space model by using MINREAL, then I should eliminate such uncontrollable state from state vector x. Here is the question. Is the control input u=-Kx same with those of before MINREAL? Can I expect that this control input using minreal has same effect with before minreal for control system (plant)?

Thank you. Best regards

[Sriram Narayanan]

The easiest way to find the state that was eliminated would be to compute the equivalent transfer function for the state-space model (C(sI - A)-1 B + D) and then observe the zeros/poles of the tranfer function before and after minreal to see which state was cancelled.

LQR controller requires all the states to be controllable, therefore, even though the control input should be identical if there was a cancellation after minreal was applied. In any case, I would suggest you to compute the minimal realization and then design an LQR controller for it rather than designing an LQR controller and then performing a MINREAL.