# Full state feedback of a closed loop system

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

• Yes. including feedback in a closed-loop system (which is the essence of a closed loop) affects the dynamics and steady state. If this is not exactly what you meant to ask, please elaborate further. Aug 27 at 11:12
• @Petrus1904 So full state feedback gains are there to adjust both the dynamic response and the steady state gain of a closed loop?
– Pix
Aug 27 at 11:20
• @Petrus1904 as I read that state feedback does not adjust the dynamic response but instead it changes the state of it.
– Pix
Aug 27 at 11:25

## 1 Answer

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.

• Thanks @Petrus1904 for your elaboration. This helped to conclude my answer.
– Pix
Aug 28 at 11:25