Induced norm analysis of linear systems for nonnegative input signals

This paper is concerned with the analysis of the $L_p(p\in [1,\infty ),p=\infty )$ induced norms of continuous-time linear systems where input signals are restricted to be nonnegative. This norm is referred to as the $L_{p+}$ induced norm in this paper. It has been shown recently that the $L_{2+}$ induced norm is effective for the stability analysis of nonlinear feedback systems where the nonlinearity returns only nonnegative signals. However, the exact computation of the $L_{2+}$ induced norm is essentially difficult. To get around this difficulty, in the first part of this paper, we provide a copositive-programming-based method for the upper bound computation by capturing the nonnegativity of the input signals by copositive multipliers. In the second part, we consider how far the $L_{2+}$ induced norm can be smaller than the standard $L_2$ induced norm, and derive the uniform infimum on the ratio of the $L_{2+}$ induced norm to the $L_2$ induced norm over all linear systems including infinite-dimensional ones. Then, for each linear system, we finally derive a computation method of the lower bounds of the $L_{2+}$ induced norm that are larger than (or equal to) the value determined by the uniform infimum. The effectiveness of the upper/lower bound computation methods is illustrated by numerical examples.