Is the Algebraic Ricatti Equation the same for LQR (Linear Quadratic Regulator) and LQI (Linear Quadratic Regulator with Integral action) as used in optimal feedback control systems?
I can see the ARE equation clearly for LQR: it takes in the state matrices, eg here. In this case, the solved gains are the vector Kx, ie one gain for each system state x = [x1, x2, ..., xn].
For LQI, which includes the integral term, the gains are Kx and Ki, so an additional gain to correspond with the additional system state xi: x = [x1, x2, ..., xn, xi], yielding correspondingly different A, B, C matrices compared with LQR.
Is the same ARE equation used for LQI as LQR, just with different input matrices corresponding to the system with integral state, and it automatically puts out a gain vector -- and we just assume the last gain is xi? Or, does ARE change form for LQI, where state xi has to be dealt with explicitly?
I ask, because i'm unable to find an equivalent explanation that shows ARE as used with LQI, and people just show the higher-level methods, eg here.
Overall, I'm trying to understand the usage, for manual implementation purposes.
Edit: Since the optimization criteria arise from (Ax+Bu)S(Ax+Bu), i'm guessing the corresponding A, B, etc are used for LQI with the same ARE equation.
Yes, it is the same ARE for both control approaches (with correspondig system matrices). LQI is just setting up an extended system (with integrator states) and doing LQR. If you are using matlab (as the link you provided is from there), you can realize that by trying to build the extende system (with integrator states) "by hand" and using the lqr command then. The result should be the same than if you use LQI with the original system (it works, I just implemented it in 5 min).