Stability, ℒ₁ performance and state feedback design for linear systems in ice-cream cones

Abstract

This paper considers linear systems which are positively invariant in a second-order cone (ice-cream cone). Three problems are addressed: (i) stability; (ii) ${\mathcal{L}}_1$ performance; (iii) state feedback design for stabilisation and optimal ${\mathcal{L}}_1$ 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.

Keywords:

• Monotone systems
• cone invariance
• stability
• performance analysis
• state feedback control

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by Fundamental Research Funds for the Central Universities of Ministry of Education of China [30918011206] and National Nature Science Foundation of China [61104197].