Necessary and Sufficient Conditions for Matching Conditions in Uncertain Systems: Scalar Input

The uncertain transformation of scalar input uncertain systems to controllable canonical form is used to derive necessary and sufficient conditions for the satisfaction of matching conditions. In particular, it is shown that the matching conditions imply that the uncertain system is completely controllable for all uncertainties. They also imply that the controllable canonical dynamics matrix with zero eigenvalues when transformed back to the original state space by the uncertain transformation is a constant dynamics matrix. Equations are given for computing the matching conditions matrices in terms of the uncertain coefficients of the characteristic polynomial and the uncertain transformation. In addition, the necessary and sufficient conditions provide insight in the development of robust control for uncertain systems in the absence of matching conditions.