ABSTRACT
This chapter introduces a broad and active field, that of linear matrix inequalities (LMI) in automatic control. In broad sense, an LMI is a set of mathematical expressions whose variables are linearly related matrices This ample definition highlights the fact that an LMI can adopt nonobviously linear or matrix expressions. An LMI can appear in seemingly nonlinear or non-matrix expressions. Moreover, some solutions of nonlinear inequalities can be subsumed as the solution of an LMI problem. A number of classical stability issues can be recast as LMI problems and therefore solved, for linear time-invariant (LTI) models as well as for linear parameter-varying (LPV) ones. Some controller design issues for LTI models follow. It is important to keep in mind that LMI applications extend well beyond this survey to LPV and quasi-LPV models, both continuous and discrete-time, uncertain, and delayed. The elementary results in the sequel as well as the proofs behind them give an outline to the aforementioned extensions.
