Optimal Homogeneous ℒp$$ {\boldsymbol{\mathcal{L}}}_{\boldsymbol{p}} $$‐Gain Controller

ABSTRACT Nonlinear ‐controllers are designed for arbitrarily weighted, continuous homogeneous systems with a focus on systems affine in the control input. Based on the homogeneous ‐norm, the input–output behavior is quantified in terms of the homogeneous ‐gain as a generalization of the classical ‐gain (i.e., ‐norm in the linear case). The resulting memoryless state‐feedback control law minimizes the upper estimate of the homogeneous ‐gain from disturbance to extended output. This estimate is obtained by constructing a solution, that is, storage function, to a Hamilton–Jacobi inequality (the homogeneous differential dissipation inequality) based on a control Lyapunov function. Both the continuous, homogeneously stabilizing control law and closed‐loop homogeneous ‐gain estimate depend on the choice of storage function. The design procedure and resulting performance are illustrated with examples and simulations.