Performance optimization over positive l∞ cones

In this paper, we first consider the case where the input to a system is restricted to be in the positive cone of l
∞
, denoted by l
∞
⁺
, and seek to characterize the system's induced norm from l
∞
⁺
to l
∞
. We obtain an exact characterization of this norm which is particularly easy to calculate in the case of LTI systems. Furthermore, we consider the model matching problem to show that time-varying linear or nonlinear control or filtering does not improve the performance with respect to this norm for LTI systems. In the second part of the paper, we consider the case when the output is forced to be in the positive l
∞
cone when the input is in this cone. We show if internal positivity is sought, a dynamic optimal controller offers no advantage over a static one.