Uncertain Systems: Time-Varying Versus Time-Invariant Uncertainties

In this chapter, we survey a few decades of robustness investigation for uncertain systems. We aim at embracing most of the robustness literature, starting from the Lyapunov approach of the ’70s, which involved both quadratic and non-quadratic Lyapunov functions, until recent developments on polynomial techniques for robustness. We consider both time-varying and time-invariant uncertainties, in an inclusive way: important techniques are presented, such as the Lur’e systems framework, qualitative feedback theory, parametric robustness analysis, linear matrix inequalities, parameter-dependent Lyapunov functions, H-infinity, small-gain theorems, non-quadratic Lyapunov functions and Lyapunov–Metzler inequalities. The chapter proposes a critical view on all these techniques, highlighting both advantages and limitations. Illustrative examples and applications are proposed. Technicalities are kept to the least possible level to render the chapter accessible to a broad, possibly interdisciplinary, audience. The chapter is written with a historic view. Nonetheless, future perspectives are emphasized, and several open problems and future research directions are pointed out. The chapter is inspired by the spirit, attitude and fairness of our great friend Roberto Tempo and is written following his invaluable teaching.