Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations

This paper establishes a framework for robust stability analysis of linear time-invariant uncertain systems, The uncertainty is assumed to belong to an arbitrary subset of complex matrices. The concept used here is well-posedness of feedback systems, leading to necessary and sufficient conditions for robust stability. Based on this concept, some insights into exact robust stability conditions are given, In particular, frequency domain and state-space conditions for well-posedness are provided in terms of Hermitian-form inequalities. It is shown that these inequalities can be interpreted as small-gain conditions with a generalized class of scalings given by linear fractional transformations (LFT). Using the LFT-scaled small-gain condition in the state-space setting, the “duality” is established between the H_∞ norm condition with frequency-dependent scalings and the parameter-dependent Lyapunov condition. Connections to the existing results, including the structured singular value and the integral quadratic constraints, are also discussed. Finally, we show that our well-posedness conditions can be used to give a less conservative, yet computable bound on the real structured singular value. This result is illustrated by numerical examples