A note on equivalence between two integral inequalities for time-delay systems

Jensen's inequality and extended Jensen's inequality are two important integral inequalities when problems of stability analysis and controller synthesis for time-delay systems are considered. The extended Jensen's inequality introduces two additional free matrices and is generally regarded to be less conservative than Jensen's inequality. The equivalence between Jensen's inequality and extended Jensen's inequality in bounding the quadratic term - h ? t - h t x ? T ( s ) Z x ? ( s ) d s in Lyapunov functional of time-delay systems is presented and theoretically proved. It is shown that the extended Jensen's inequality does not decrease the lower bound of this quadratic term obtained using Jensen's inequality, and then it does not reduce the conservativeness though two additional free matrices M 1 and M 2 are involved.

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