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

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  • $\begingroup$ 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. $\endgroup$
    – Petrus1904
    Aug 27 at 11:12
  • $\begingroup$ @Petrus1904 So full state feedback gains are there to adjust both the dynamic response and the steady state gain of a closed loop? $\endgroup$
    – Pix
    Aug 27 at 11:20
  • $\begingroup$ @Petrus1904 as I read that state feedback does not adjust the dynamic response but instead it changes the state of it. $\endgroup$
    – Pix
    Aug 27 at 11:25
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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.

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

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