Controllability and observability of uncertain systems: A robust measure

This paper deals with the class of polynomially uncertain continuous-time linear time-invariant (LTI) systems whose uncertainties belong to a semi-algebraic set. The objective is to determine the minimum of the smallest singular value of the controllability or observability Gramian over the uncertainty region. This provides a quantitative measure for the robust controllability or observability degree of the system. To this end, it is shown that the problem can be recast as a sum-of-squares (SOS) problem. In the special case when the uncertainty region is polytopic, the corresponding SOS formulation can be simplified significantly. One can apply the proposed method to any large-scale interconnected system to identify those inputs and outputs that are more effective in controlling the system. This enables the designer to simplify the control structure by ignoring those inputs and outputs whose contribution to the overall control operation is relatively weak. A numerical example is presented to demonstrate the efficacy of the results.