This article presents a new version of the Kalman–Yakubovich–Popov lemma for linear systems with their states constrained in proper cones. Based on this lemma, two important applications are introduced. One is the stabilization controller design to satisfy the spectral radius performance while preserving cone invariance. The other is to obtain an $H^\infty$ state-feedback controller such that both $H^\infty$ performance and cone invariance are guaranteed. Moreover, to address these two problems, a practical algorithm based on the linear matrix inequality is provided. Finally, two numerical examples on a linear system defined in the second-order cone are used to illustrate the results.