Search

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.

This paper proposes a novel approach called a combined reciprocal convexity approach for the stability analysis of static neural networks with interval time-varying delays. The proposed approach deals with all convex-parameter-dependent terms in the time derivative of the Lyapunov-Krasovskii functional non-conservatively by extending the idea of the conventional reciprocal convexity approach. Based on the proposed technique and a new Lyapunov-Krasovskii functional, two improved delay-dependent stability criteria are derived in terms of linear matrix inequalities(LMIs). Some numerical examples are given to demonstrate the proposed results.

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.

In the field of stability analysis for time-delay systems, finding precise bounds of integral/summation forms of quadratic functions plays a key role in reducing the conservatism. Consequently, there have been many attempts to develop inequalities yielding much tighter bounds. This paper develops novel inequalities using intermediate terms called auxiliary functions, which includes the existing inequalities as special cases. The stability conditions for continuous/discrete time-delay systems are derived in terms of linear matrix inequalities (LMIs) by using the proposed inequalities with simple numerical examples.

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.

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.

This paper proposes a new integral inequality, say combined-slack-matrix-based integral inequality, which extends a free-matrix-based integral inequality. For a single integral quadratic function, it provides a upper bound which depends on not only the concerned integration interval but also another integration interval. Consequently, the proposed integral inequality contains more flexibility, which contributes to developing an improved stability criterion for linear systems with interval-time varying delays.