Characterizing ℒ 1 $$ {\mathcal{L}}_1 $$ output-feedback controller for nonlinear systems: Existence conditions via output controlled invariance domain

Motivated by existing works on the ${ℒ}_1$ state-feedback controller for nonlinear systems, in which the ${ℒ}_{\infty }$ norm of the output for the worst disturbance with a unit magnitude is required to be bounded by 1, this paper considers an extension of those works to an output-feedback form. More precisely, the existence of an ${ℒ}_1$ output-feedback controller for nonlinear systems is characterized by developing output regulation map and output controlled invariance domain, which are extended versions of the conventional regulation map and controlled invariance domain in the previous works. We first lead to a sufficient condition for the existence of an ${ℒ}_1$ output-feedback controller by ensuring the lower-semicontinuity of the corresponding output regulation map. It is also shown in this paper that there exists an ${ℒ}_1$ output-feedback controller only if there exists an output controlled invariance domain. Based on these conditions, we further introduce algorithmic guidelines for verifying the existence of ${ℒ}_1$ output-feedback controller. Finally, a numerical example is provided to verify the validity of the overall arguments developed in this paper.