# Does controller K have to be Hurwitz?

Ask a few simple questions but confused me.

1. "A" (plant) is unstable. From ARE (or DRE), we find "K", and obtain Ac = A - BK, which is Hurwitz(stable)

Must K be PD(>0), PSD, ND, or NSD? or no such constraint? any example?

2. "A" (plant) is "stable". Why do we still want to design a controller? any concrete example and motive?

Thanks!

About the first question, $K$ might not be square, that depends on the dimensions of $B$, so there is not restriction actually on $K$, just only on the state transition matrix of the closed loop $Ac$, if you want the closed loop asymptotically stable, then $Ac$ must be Hurwitz. If you want your system stable, then the eigenvalues of $Ac$ must have non-positive real part.
About the second question. Imagine the simple scalar stable system $\dot x = -x$, which trivial solution is $x(t)=e^{-t}$. If we want to make the transition time faster (typically in engineering once the solution $x(t)$ reaches the 66% of the asymptotic value, in this example it would be $0.66$), we just apply a gain controller $\dot x = -kx$ with $k > 1$. For second order systems, you can improve the transient in the sense of having more/less damping, faster/slower transient response, etc.