Set invariance under output feedback: a set-dynamics approach

The article offers a framework to define and analyse the notion of invariance with respect to output feedback under non-parametric disturbances. The motivation is that the straightforward generalisation of the definition of invariance under state feedback to the output feedback framework, namely relying only on feedback from the output, does not yield a useful notion. Our model follows standard feedback invariance considerations with, however, a crucial modification that is needed when only an observation of the state, rather than the state itself, is available. The model incorporates information gathered by the controller during the process; this is in similarity with the observer-based dynamics model; however our framework represents the information within a set dynamics. The evolution of the resulting information sets determines invariant sets and attractors of the state dynamics. The framework in this article is discrete-time control systems. We offer an analysis of the notion with results on existence of, and convergence to, output feedback invariant sets; illustrative examples related to potentially practical feedback rules are exhibited.