On ℓ₁-gain control of 2-D positive Roesser systems with directional delays: necessary and sufficient conditions

In this paper, the problem of ℓ1-gain control is addressed for two-dimensional (2-D) positive Roesser systems with directional delays. A complete solution to the design problem is presented for the first time. Specifically, by exploiting a new approach based on 2-D Z-transform, an exact value of ℓ1-induced norm of the input–output operator is obtained. This analytical result is utilized to derive necessary and sufficient conditions under which a 2-D positive system with delays is asymptotically stable with a prescribed ℓ1-gain performance. Then, by utilizing a vertex optimization approach, necessary and sufficient conditions are established for the existence of a state-feedback controller (SFC) that makes the closed-loop system positive and stable subject to ℓ1-induced performance. The proposed synthesis conditions are formulated in terms of linear programming (LP), which can be effectively solved by various convex algorithms to minimize the ℓ1-performance index.