Analysis of multiplicity of Hankel singular values of control systems

Hankel singular values are important input-output invariants of linear control systems, and in this paper we are interested in determining the amount of different values. It is shown that for SISO systems the problem reduces to the analysis of controllability of a certain auxiliary system. For MIMO systems, a method is proposed for constructing the minimal order polynomial such that its roots are equal to the squared Hankel singular values. In the case of maximal multiplicity of the Hankel singular values, the controllability and observability Gramians of the system are proved to be mutually inverse up to a certain scalar multiplier.