Bessel summation inequalities for stability analysis of discrete-time systems with time-varying delays

This paper is concerned with the stability analysis problems of discrete-time systems with time-varying delays using summation inequalities. In the literature focusing on the Lyapunov-Krasovskii approach, the Jensen integral/summation inequalities have played important roles to develop less conservative stability criteria and thus have been widely studied. Recently, the Jensen integral inequality was successfully generalized to the Bessel-Legendre inequalities constructed with arbitrary-order Legendre polynomials. It was also shown that general inequality contributes to the less conservatism of stability criteria. In the case of discrete-time systems, however, the Jensen summation inequality are hardly extensible to general ones since there have still not been general discrete orthogonal polynomials applicable to the developments of summation inequalities. Motivated by such observations, this paper proposes novel discrete orthogonal polynomials and then successfully derives general summation inequalities. The resulting summation inequalities are discrete-time counterparts of the Bessel-Legendre inequalities but are not based on the discrete Legendre polynomials. By developing hierarchical stability criteria based on the proposed summation inequalities, the effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of discrete-time systems with time-varying delays.