Systems with uncertainty is a fundamental area of automatic control since the mathematical model of a real plant is almost never exactly known. Generally, the coefficients of such a model depend on uncertain parameters (e.g., linearly, polynomially, etc) confined in a set (e.g., hypercube, polytope, etc). Unfortunately, essential problems such as establishing whether an equilibrium point is robustly stable for all the admissible uncertainties, and determining worst-case system performances, are notoriously very difficult to solve (NP-hard problems), even in the simplest case of linear dependence of the system on the uncertainty. This talk presents our pioneering and recent results for addressing these problems via linear matrix inequalities (LMIs). Various frameworks are considered, such as continuous-time/discrete-time systems, time-invariant/time-varying uncertainties, and linear/nonlinear dependence of the system on the uncertainty. In particular, it is shown that one can
build LMI conditions that are not only sufficient but also necessary for robustness investigation.