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Recently, a polynomials-based integral inequality was proposed by extending the Moon’s inequality into a generic formulation. By imposing certain structures on the slack matrices of this integral inequality, this paper proposes an orthogonal-polynomials-based integral inequality which has lower computational burden than the polynomials-based integral inequality while maintaining the same conservatism. Further, this paper provides notes on relations among recent general integral inequalities constructed with arbitrary degree polynomials. In these notes, it is shown that the proposed integral inequality is superior to the Bessel–Legendre (B–L) inequality and the polynomials-based integral inequality in terms of the conservatism and computational burden, respectively. Moreover, the effectiveness of the proposed method is demonstrated by an illustrative example of stability analysis for systems with additive time-varying delays.

This paper considers a delay-dependent stability criterion for linear systems with multiple time-varying delays. To exploit all possible information for the relationships among the marginally delayed states ( x ( t − τ i M ) , x ( t − τ i + 1 M ) ), the exactly delayed states ( x ( t − τ i ( t ) ) , x ( t − τ i + 1 ( t ) ) ), and the current state x ( t ) for each pair ( i , i + 1 ) of time-varying delays, a delays-dependent region partitioning approach in double integral terms is proposed. By applying the Wirtinger-based integral inequality and the reciprocally convex approach to terms resulted from the region partitioning, a stability criterion is derived in terms of linear matrix inequalities. Numerical examples show that the resulting criterion outperforms the existing one in literature.

This paper employs polynomial functions to extend a free-matrix-based integral inequality into a general integral inequality, say a polynomials-based integral inequality, which also contains well-known integral inequalities as special cases. By specially designing slack matrices and an arbitrary vector containing state terms, it reduces to an extended version of Wirtinger-based integral inequality or the free-matrix-based integral inequality. Numerical examples for stability analysis of linear systems with interval time-varying delays show the improved performance of the proposed integral inequality in terms of maximum delay bounds and numbers of variables.

This paper suggests first-order and second-order generalized zero equalities and constructs a new flexible Lyapunov-Krasovskii functional with more state terms. Also, by applying various zero equalities, improved stability criteria of linear systems with interval time-varying delays are developed. Using Wirtinger-based integral inequality, Jensen inequality and a lower bound lemma, the time derivative of the Lyapunov-Krasovskii functional is bounded by the combinations of various state terms including not only integral terms but also their interval-normalized versions, which contributes to make the stability criteria less conservative. Numerical examples show the improved performance of the criteria in terms of maximum delay bounds.

For the stability analysis of discrete time-delay systems, Jensen inequality has been widely used as the method supporting inequalities for summation quadratic functions. It not only requires a smaller number of decision variables than other approaches but also achieves identical or comparable performance behavior. Based on the analysis for the conservatism of Jensen inequality, however, this paper suggests a new summation inequality say an auxiliary-function-based summation inequality. It is verified that the proposed inequality is a generalized form of the novel summation inequality reported recently. Also, an application to stability analysis for discrete time-delay systems is provided.

Summary This paper suggests a generalized zero equality lemma for summations, which leads to making a new Lyapunov–Krasovskii functional with more state terms in the summands and thus applying various zero equalities for deriving stability criteria of discrete-time systems with interval time-varying delays. Also, using a discrete-time counter part of Wirtinger-based integral inequality, Jensen inequality, and a lower bound lemma for reciprocal convexity, the forward difference of the Lyapunov–Krasovskii functional is bounded by the combinations of various state terms including not only summation terms but also their interval-normalized versions, which contributes to making the criteria less conservative. Numerical examples show the improved performance of the criteria in terms of maximum delay bounds. Copyright © 2016 John Wiley & Sons, Ltd.

For the interval time-varying delay systems, Jensen inequality lemma yields some terms with the inverse of convex and squared convex parameters, which often makes it difficult to find their bounds. Recently, reciprocal and second-order reciprocal convexity approaches have been proposed to handle the difficulties with a set of convex parameters and a set of squared convex parameters, respectively, but these approaches do not investigate the relation between the two sets. This paper offers much tighter bounds of those terms utilizing the relations among the two sets with the lower bound lemma. Based on the new approach and Lyapunov theory, less conservative stability criteria for time delay systems are developed. To show the effectiveness of the new approach, two numerical examples are given.

This paper proposes a new non-parallel distributed compensation (non-PDC) scheme of $H_{\infty}$ control for uncertain discrete-time Takagi-Sugeno (T-S) fuzzy systems. To utilizing the partition to the range of the fuzzy weights, the proposed scheme sets the variables to be second-order parametric in both the current- and past-time fuzzy weights, which results in a third-order parameterized condition. Then, the scheme uses the elimination lemma, where a decision variable is designed to be constant piecewise by partitioning the range of the parameters, so that the resulting condition is first-order parameterized. Consequently, the switching controller is developed by utilizing the extreme points of each partition. Numerical examples are provided to illustrate the effectiveness of the proposed approach.

Finding integral inequalities for quadratic functions plays a key role in the field of stability analysis. In such circumstances, the Jensen inequality has become a powerful mathematical tool for stability analysis of time-delay systems. This paper suggests a new class of integral inequalities for quadratic functions via intermediate terms called auxiliary functions, which produce more tighter bounds than what the Jensen inequality produces. To show the strength of the new inequalities, their applications to stability analysis for time-delay systems are given with numerical examples.

This paper analyzes delay-dependent robust stability and H"~ performance of linear systems with an interval time-varying delay, based on a new Lyapunov-Krasovskii functional containing new triple integral terms. The time derivative of the Lyapunov-Krasovskii functional produces not only the strictly proper rational functions but also the non-strictly proper rational functions of the time-varying delays with first-order denominators. The combinations of the rational functions are directly handled via the Jensen inequality lemma and the lower bound lemma for reciprocal convexity, whereas such combinations were approximated in the literature. The proposed criteria become less conservative with the significantly smaller number of decision variables than the existing criteria, which will be demonstrated by some numerical examples.

This article surveys the control theoretic study on time delay systems. Since time delay systems are infinite dimensional, there are not analytic but numerical solutions on almost analysis and synthesis problems, which implies that there are a tremendous number of approximated solutions. To show how to find such solutions, several results are summarized in terms of two different axes: 1) theoretic tools like integral inequality associated with the derivative of delay terms, Jensen inequality, lower bound lemma for reciprocal convexity, and Wirtinger-based inequality and 2) various candidates for Laypunov-Krasovskii functionals. Keywords: time delay, integral inequality, Jensen inequality, lower bound lemma for reciprocal convexity, Wirtinger-based inequality, Lyapunov-Krasovskii functionals I. 서론 시스템의 전송 속도, 계산 시간 등의 물리적인 제약으로 인해 발생하는 시간지연 현상은 네트워크 시스템, 원자로 시스템, 생물·화학 공정 시스템 등의 다양한 현실시스템에 필연적으로 존재한다. 이는 전체 시스템을 불안정하게 하거나 성능을 저하시키는 주요 요소 중 하나이기 때문에, 이러한 현상을 갖는 시간지연시스템의 안정성 해석 및 안정화에 대한 연구는 현실적으로나 이론적으로나 매우 중요하다. 하지만 시간지연시스템은 무한차원시스템이기 때문에 정확한 해가 존재하지 않으며 따라서 덜 보수적인(less conservative) 근사 해를 찾기 위한 여러 가지 방법들이 많은 연구자들에 의해 연구되고 있다. 이러한 시간지연시스템에 관한 연구의 목적은 시스템의 안정성을 보장하는 최대허용 시간지연을 밝혀내기 위한 조건을 가능한 가장 적은 변수들을 가지고 찾는 것이다. 이러한 조건들은 선형 행렬 부등식(LMI) 문제의 형태로 유도되는데, 그 이유는 LMI 문제를 해결하기 위한 많은 수치해석 알고리즘들이 현재 존재하며 이를 통해 비교적 쉽게 해를 구할 수 있기 때문이다. 시간지연시스템의 안정성 해석 및 안정화 조건은 크게 시간지연 독립조건과 시간지연 종속조건 두 가지로 분류될 수 있다. 시간지연 독립조건은 그 결과가 시간지연과 무관하기 때문에 시간지연 종속조건보다 더 보수적이며 상대적으로 단순하기 때문에 관련 연구 초기 단계에 많이 연구되었다. 하지만 시간지연 종속조건이 시간지연에 관한 정보를 포함하고 있기 때문에 시간지연 독립조건보다 훨씬 덜 보수적이라는 연구 결과가 발표된 이후 [2], 현재는 거의 모든 연구가 시간지연 종속조건을 찾는데 집중되고 있다. 본 논문의 목적은 현재까지 진행되어 온 시간지연시스템에 관한 연구 동향을 파악하고 나아가 관련 연구의 활성화를 위한 발판을 제공하는 것이다. 시간지연시스템의 안정성 해석을 위한 다양한 접근방법과 보다 덜 보수적인 조건을 구하기 위해 연구된 이론적 도구들이 소개 될 것이다. 또한 보수성(conservatism)을 판단하기 위해 공통된 하나의 시스템에 대해 서로 다른 조건들로부터 얻은 시스템의 안정성을 보장하는 최대허용 시간지연을 나열하여 비교할 것이다. II. PRELIMINARY 시간지연은 크게 두 가지로 나눌 수 있으며, 상태변수 시간지연과 입력 시간지연이다. 상태변수 시간지연은 시스템의 상태변수에 시간지연이 포함되는 경우이며, 입력 시간지연은 제어기로부터 시스템에 들어오는 입력신호에 시간지연이 포함되는 경우이다. 또한 각각의 시간지연은 단일 시간지연, 다중 시간지연, 분산 시간지연, 이산 시간지연 등 다양한 형태로 시스템에 존재한다. 본 논문에서는 시간지연시스템 중에서도 현재까지 연구가 가장 활발한 다음의 단일 상태변수 시간지연시스템을 다루고자 한다.

This paper proposes a new slack-matrix-based summation inequality which extends a discrete-time counterpart of a free-matrix-based integral inequality. For a single summation quadratic function, it provides a upper bound which depends on not only the concerned summation interval but also other summation interval. Thus, the proposed summation inequality has more flexibility, which leads to improved stability criteria for discrete-time systems with interval-time-varying delays.