Stability, ℒ 1 performance and state feedback design for linear systems in ice-cream cones

This paper considers linear systems which are positively invariant in a second-order cone (ice-cream cone). Three problems are addressed: (i) stability; (ii) L 1 performance; (iii) state feedback design for stabilisation and optimal L 1 performance while preserving cone invariance. We derive necessary and sufficient conditions via Linear Matrix Inequalities (LMI) for the solution of problems (i) and (ii). As for problem (iii), a full parametrization of feasible state feedback gains is provided, along with some LMI relaxations useful to compute a feasible gain. Finally, a numerical example is briefly discussed.