Structured singular value of implicit systems

Abstract

Implicit systems provide a general framework in which many important properties of dynamic systems can be studied. Implicit systems are especially relevant to behavioural systems theory, the analysis and synthesis of complex interconnected systems, systems identification and robust control. By incorporating algebraic constraints, implicit models provide additional versatility relative to the standard input–output framework. Problems of robust stability in implicit systems lead in a natural way to non‐standard structured singular value (μ) formulations. In this note, it is shown that for a class of uncertainty structures involving repeated scalar parameters, these problems reduce to a standard μ problem which is well studied and for the solution of which several numerical algorithms are available. Our results are based on a matrix dilation technique and the redefinition of the uncertainty structure of the transformed problem. The main results of the paper are illustrated with a numerical example.