$L_{2+}$ Induced Norm Analysis of Continuous-Time LTI Systems Using Positive Filters and Copositive Programming

Abstract

This paper is concerned with the analysis of the $L_{2}$ induced norm of continuous-time LTI systems where the input signals are restricted to be nonnegative. This induced norm is referred to as the $L_{2+}$ induced norm in this paper. It has been shown very recently that the $L_{2+}$ induced norm is particularly useful for the stability analysis of nonlinear feedback systems constructed from linear systems and static nonlinearities where the nonlinear elements only provide nonnegative signals. For the upper bound computation of the $L_{2+}$ induced norm, an approach with copositive programming has also been proposed. It is nonetheless true that this approach becomes effective only for multi-input systems, and for single-input systems this approach does not bring any improvement over the trivial upper bound, the standard $L_2$ norm. To overcome this difficulty, we newly introduce positive filters to increase the number of positive signals. This enables us to enlarge the size of the copositive multipliers so that we can obtain better (smaller) upper bounds with copositive programming.

Figures (3)

• Figure 1: Nonlinear Feedback System.
• Figure 2: The Values of $\overline{\overline{\gamma}}_{a,\alpha,N}$: Upper Bounds of $\|G\|_{2+}$.
• Figure 3: The Values of $\overline{\overline{\gamma}}_{a,\alpha,N}$: Upper Bounds of $\|G\|_{2+}$.