Comments on “New finite-sum inequalities with applications to stability of discrete time-delay systems”

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

“…It is clear that the proposed summation inequalities are more general than the existing ones. Thus, the results of the Jensen inequality, the discrete‐time Wirtinger‐based inequality, and auxiliary‐function‐based summation inequalities can be substituted with those of Theorem for ( m = 0), Theorem for ( m = 1), and Theorem for m ∈ {2,3}, respectively. Theorem already demonstrated that Corollary , which utilizes generalized free‐weighting matrix‐based summation inequalities, has the same feasible region of Theorem .…”

“…Time delay is a common phenomenon in many dynamic systems such as chemical systems, biological systems, mechanical engineering systems, and networked control systems . Since such phenomenon often causes control performance degradation or even system instability, there have been considerable efforts to solve stability analysis problems for time‐delay systems before implementing control strategies . Stability analysis for delayed discrete‐time systems can be classified into two types of problems depending on delay properties: constant time delays and time‐varying delays.…”

“…In the case of constant time delays, it has been noted that necessary and sufficient stability criteria can be obtained using a system augmentation method at a price of large computational burden. However, in the case of time‐varying delays, the stability analysis of delayed discrete‐time systems is still an open problem, and thus, various numerical methods such as reciprocally convex combination lemma, summation inequalities, and free‐weighting matrices approach have been proposed to obtain less conservative stability criteria . In the presence of time‐varying delays in systems, the Lyapunov‐Krasovskii (L‐K) approach is one of major methods to derive numerically tractable optimization problems, which are usually formulated in terms of linear matrix inequalities (LMIs), for stability analysis of time‐delay systems.…”

“…Similarly, for stability analysis of delayed discrete‐time systems, there have been various efforts to reduce a bounding gap between the Jensen summation inequality and a summation quadratic function by using zero‐, first‐, second‐, and third‐order discrete orthogonal polynomials obtained from Gram‐Schmidt orthogonalization process . These efforts have resulted in the discrete‐time Wirtinger‐based inequality, auxiliary‐function‐based summation inequalities, and free‐weighting matrix‐based summation inequalities . Differently from the case of integral inequalities, however, the existing summation inequalities are hardly extensible to high‐order or arbitrary‐order cases since there have still not been general discrete orthogonal polynomials applicable to the deriving summation inequalities.…”

“…The proof of the general summation inequalities is straightforwardly obtained from the Bessel inequality by defining an inner product, and thus, the main difficulty when deriving general inequalities comes from a lack of general form of discrete orthogonal polynomial function given in terms of a binomial coefficient form or a rising‐factorial form. Indeed, while there exist many papers handling summation inequalities of some orders or iterative processes to derive summation inequalities, there have still not been general summation inequalities scalable with respect to the order of polynomials. Many summation inequalities are based on low‐order discrete orthogonal polynomials obtained from Gram Schmidt orthogonalization process.…”