Asymptotic root loci of multivariable linear optimal regulators

The asymptotic loci of the closed-loop poles of the multivariable time-invariant linear regulator are considered as the weight on the input in the criterion approaches zero. It is proved that those poles that go to infinity group into several Butterworth configurations of different orders and with different radii. It is furthermore shown that the first-order patterns are easily explicitly determined. Some examples illustrate the results.