Performance optimization over positive l∞ cones

In this paper we study linear systems with positivity type of constraints. First, we 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 and solve the model matching problem, and show that time-varying linear or nonlinear control/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. Also, if the measurement matrix satisfies certain conditions, synthesizing an optimal static output feedback controller reduces to a linear program.