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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!

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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.

This is useful in the sense that the controller will be applied to actual systems, i.e. involving machines, production line, etc. And you can fit your requirements to your plant/system with the appropriated controller.

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