Full state feedback of a closed loop system

Question

Does Full state feedback gains affect the dynamic or steady state gain of a closed loop system?

Answer 1

Let me phrase this a bit more elaborate. Suppose we have a state space system, ie: $$\dot{x} = Ax + Bu$$ Where the dynamics of the system are presented through $A$. For instance, the poles of the system are equal to the eigenvalues of $A$. The steady-state value can be computed by solving $0 = Ax^{ss}$. Now we have a full state feedback controller $u = -K(x-r)$. The closed loop model is now equal to: $$\dot{x} = (A-BK)x + BKr$$ Therefore, the dynamics of the system are now represented through $(A-BK)$. This alters both the steady-state value, as this becomes a function of $r$ and the new state dynamics. The dynamic response (so how the model responds to an input) is also different as the forced response matrix $B$ in the open loop model now becomes $BK$ in the closed loop. Furthermore, how the state evolves in time due to this new input is also different, hence the dynamic response is different.

Alternative state feedback notations such as $u = -Kx + r$ exist, and while those do not change how the new input acts on the model, how the states behave due to this input do, thus it still changes the dynamics.