Analysis of Positive Systems Using Copositive Programming

In the field of control, a wide range of analysis and synthesis problems of linear time-invariant (LTI) systems are reduced to semidefinite programming problems (SDPs). On the other hand, in the field of mathematical programming, a class of conic programming problems, so called the copositive programming problem (COP), is actively studied. COP is a convex optimization problem on the copositive cone, and the completely positive cone, the doubly nonnegative cone, and the Minkowski sum of the positive semidefinite cone and the nonnegative cone are also closely related to COP. These four cones naturally appear when we deal with optimization problems described by nonnegative vectors. In this letter, we show that the stability, the H
₂
and the H
_∞
performances of LTI positive systems are basically characterized by the feasibility/optimization problems over these four cones. These results can be regarded as the generalization of well-known LMI/SDP-based results on the positive semidefinite cone. We also clarify that in some performances such direct generalization is not possible due to inherent properties of the copositive or the completely positive cone. We thus capture almost entire picture about how far we can generalize the SDP-based results for positive systems to those on the four cones related to COP.