Summation Inequalities to Bounded Real Lemmas of Discrete-Time Systems With Time-Varying Delay

Summation inequality is an important technique for analysis of discrete-time systems with a time-varying delay. It seems that from the literature a tighter inequality usually leads to a less conservative criterion. Based on $H_{\infty}$ performance analysis problem, this note presents different findings on the relationship between the conservatism of bounded real lemma (BRL) and the tightness of summation inequality. Firstly, the BRL obtained by the Wirtinger-based inequality (WBI) is not always less conservative than the one by the Jensen-based inequality although the WBI is tighter. Secondly, the WBI is tighter than a general free-matrix-based inequality (GFMBI) developed in this note, while the BRL obtained via the GFMBI is less conservative than the WBI-based BRL. Finally, a numerical example is given to demonstrate those findings.