Comparison Between SOS Approach and Slack Variable Approach for Non-negativity Check of Polynomial Functions: Single Variable Case

This note addresses the comparison between sums-of-squares (SOS) approach and slack variable (SV) approach for the non-negativity check of single variable polynomial functions, i.e., univariate polynomial functions. The result is as follows: For the unbounded case, both approaches are equivalent; for the bounded case, SV approach is less conservative with increased number of decision variables compared to SOS approach. Several numerical examples, including polynomial matrix function case, are shown to illustrate our conclusions. Finally, we apply SV approach to finite frequency peak gain analysis of single input single output systems, and show the effectiveness of SV approach compared to Generalized KYP (GKYP) lemma approach.