What do kr and r represent?

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In state feedback controller design (\(u=-Kx+k_rr\,\)), properly choosing the feedback gain \(K\,\) can only guarantee the stability of the closed loop system. However, to ensure that the output of the system (for example, the vehicle's speed in the cruise control example) tracks the reference input \(r\,\), we need to introduce an addition term \(k_rr\,\) to offset the steady state output. For linear systems, the control input is proportional to \(r\,\) so that we can write \(u = k_rr\,\), where \(k_r\,\) is the associated gain.

Because our textbook mainly deals with single output systems, \(k_r\,\) and \(r\,\) are both scalars in the book. However, they can become vectors in general if the system has more than one output.

--Shuo