Lower Bound Analysis of L_p+ Induced Norm for LTI Systems

In this paper, we focus on the lower bounds of the L_p (p ∈ [1, ∞), p = ∞) induced norms of continuous-time LTI systems where input signals are restricted to be nonnegative. This induced norm, called the L_p+ induced norm, is particularly useful for the stability analysis of nonlinear feedback systems constructed from linear systems and static nonlinearities where the nonlinearities provide only nonnegative signals for the case p = 2. To have deeper understanding on the L_p+ induced norm, we analyze its lower bounds with respect to the standard L_p induced norm in this paper. As the main result, we show that the L_p+ induced norm of an LTI system cannot be smaller than the L_p induced norm scaled by 2^(1−p)/p for ∈ [1, ∞) (scaled by 2⁻¹ for p = ∞). On the other hand, in the case where p = 2, we further propose a method to compute better (larger) lower bounds for single-input systems via reduction of the lower bound analysis problem into a semi-infinite programming problem. The effectiveness of the lower bound computation method, together with an upper bound computation method proposed in our preceding paper, is illustrated by numerical examples.