I am familiar with the LQR formulation:
$$\dot{x} = A x + B u$$
My question is: why use this formulation, and never the one with an offset:
$$\dot{x} = A x + B u + C$$
where $C$ is a constant "offset" matrix?
While the question is purely the "why" above, here are a few of the hypothesis of answers I could think about:
- why does it make sense to ask for LQR with a $C$ offset?
I suppose being able to have an "offset" would be useful, for example if there is a non-linear system with no obvious stationary points that has to be linearized around the current state to perform a "local" LQR, for example in a MPC context? In this case, I guess we should expect to have an offset $C$ matrix to appear in general?
- if it does is interesting to look at formulations with a $C$ offset, why is this never used in LQR formulation?
My "speculation" is that the offset is considered to be "factored in" the $Bu$ term (or the $Ax$ term), by decomposing $u = u_{stationary} + u_{control}$, where $B u_{stationary} + C = 0$ (or similarly by change of coordinates of $x = x_{stationary} + \tilde{x}$ so that $A x_{stationary} + C = 0$), and that, if such a decomposition / "projection" of $C$ onto a control (state) offset does not exist, the system is not fully controllable anyways. Is that idea correct \ wrong? How is linearizing around a local state that is not a stationary point usually handled in control theory? Is there some other explanation why we do not see an offset matrix in the fomulation of the LQR problem?