Projected Spectrahedral Cone-Invariant Realization of an LTI System with Nonnegative Impulse Response

For linear time-invariant systems with nonnegative impulse responses, much research has been devoted to studying their positive realizations. However, the limitations in the eigenvalue positions of positive systems suggest that they are not adequately powerful as a modeling tool. Thus in this paper we propose a more powerful projected spectrahedral cone-invariant (PSCI) realization of a system with nonnegative impulse response. In the study of PSCI realization problem, Lorentz cones play an important role. To be specific, we successfully find minimal Lorentz cone-invariant realizations of a class of systems with nonnegative impulse responses, which may not have positive realizations or have positive realizations with large dimensions. Combining positive realizations and Lorentz cone-invariant realizations, which are still PSCI, we can address a larger class of systems with nonnegative impulse responses. Moreover, a minimal PSCI realization can be obtained whenever a non-minimal PSCI realization exists. These results exhibit the potential power of PSCI systems as a modeling tool.