Learning Controllers for Performance Through LMI Regions

In an experiment, an input sequence is applied to an unknown linear time-invariant system (in continuous or discrete time) affected also by an unknown-but-bounded disturbance sequence; the corresponding state sequence (and state derivative sequence, in continuous time) is measured. The goal is to design directly from the input and state sequences a controller that enforces a certain performance specification on the transient behaviour of the unknown system. The performance specification is expressed through a subset of the complex plane where closed-loop eigenvalues need to belong, a so called LMI region. For this control design problem, we provide here convex programs to enforce the performance specification from data in the form of linear matrix inequalities (LMI). For generic LMI regions, these are sufficient conditions to assign the eigenvalues within the LMI region for all possible dynamics consistent with data, and become necessary and sufficient conditions for special LMI regions. In this way, we extend classical model based conditions from a work in the literature to the setting of data-driven control from noisy data. Numerical examples substantiate the analysis.