Relaxed Stability Criteria Research Articles

Relaxed stability criteria of delayed neural networks using delay-parameters-dependent slack matrices

This note aims to reduce the conservatism of stability criteria for neural networks with time-varying delay. To this goal, on the one hand, we construct an augmented Lyapunov–Krasovskii functional (LKF), incorporating some delay-product terms that capture more information about neural states. On the other hand, when dealing with the derivative of the LKF, we introduce several parameter-dependent slack matrices into an affine integral inequality, zero equations, and the S-procedure. As a result, more relaxed stability criteria are obtained by employing the so-called Lyapunov–Krasovskii Theorem. Two numerical examples show that the proposed stability criteria are of less conservatism compared with some existing methods.

A Degree-Dependent Polynomial-Based Reciprocally Convex Matrix Inequality and Its Application to Stability Analysis of Delayed Neural Networks.

In this article, several improved stability criteria for time-varying delayed neural networks (DNNs) are proposed. A degree-dependent polynomial-based reciprocally convex matrix inequality (RCMI) is proposed for obtaining less conservative stability criteria. Unlike previous RCMIs, the matrix inequality in this article produces a polynomial of any degree in the time-varying delay, which helps to reduce conservatism. In addition, to reduce the computational complexity caused by dealing with the negative definite of the high-degree terms, an improved lemma is presented. Applying the above matrix inequalities and improved negative definiteness condition helps to generate a more relaxed stability criterion for analyzing time-varying DNNs. Two examples are provided to illustrate this statement.

Relaxed Stability Criteria for Time-Delay Systems: A Novel Quadratic Function Convex Approximation Approach

Relaxed Stability Criteria for Time-Delay Systems: A Novel Quadratic Function Convex Approximation Approach

Relaxed Stability Criteria for Delayed Memristor-Based Neural Network Systems via a Novel Matrix-Separation Legendre Inequality.

This article studies the issue of stability in memristor-based neural network (MNN) systems with time-varying delays. First, a novel matrix-separation Legendre inequality is proposed to achieve a tight hierarchical bound on augmented-type integral terms. To derive implementable inequality conditions, several delay-dependent matrices are introduced to eliminate the reciprocal terms associated with time-varying delay. Furthermore, a new Lyapunov-Krasovskii (L-K) functional is proposed by incorporating augmented-type double integrals and delay-product terms. A series of free-weighting matrices are introduced into the L-K functional, leveraging the zero-sum equations and the S-procedure pertaining to both the delay and its derivative. Based on the proposed matrix-separation Legendre inequality and L-K functional, the derived stability conditions exhibit reduced conservatism, as validated by three numerical cases and simulation results.

Relaxed Stability Criteria for Delayed Generalized Neural Networks via a Novel Reciprocally Convex Combination

Dear Editor This letter examines the stability issue of generalized neural networks (GNNs) with time-varying delay based on a novel reciprocally convex combination (RCC). By considering a new matrix polynomial, the proposed novel reciprocally convex method leads to a tight bound for integral inequality combination and encompasses several existing approaches as special cases. The relaxed stability conditions with less conservatism are developed by employing the proposed reciprocally convex combination and the Lyapunov-Krasovskii (L-K) functional. Finally, several numerical examples are conducted to show the superiorities of the stability conditions.

Relaxed stability criteria of time-varying delay systems via delay-derivative-dependent slack matrices

This paper is focused on delay-dependent stability problem of time-varying delay systems. By introducing delay-derivative-dependent slack matrices, relaxed stability conditions are derived based on Lyapunov-Krasovskii functional approach. As the delay-derivative-dependent slack matrices provide extra freedom to optimize the Lyapunov matrices, less conservative results are obtained. Two benchmark examples are provided to verify the effectiveness of the proposed approach.

Relaxed Feasibility and Stability Criteria for Flexible-Step MPC

We provide extensions to the new flexible-step model predictive control (MPC) scheme, which is based on the idea of generalized discrete-time control Lyapunov functions. These facilitate the implementation of a flexible number of control inputs in each iteration of the MPC scheme. We present relaxed recursive feasibility and stability results and provide a converse Lyapunov result. These results combined simplify the design of the flexible-step MPC scheme. We demonstrate the capabilities of the flexible-step MPC algorithm for a nonholonomic system, where the standard one-step implementation may suffer from lack of asymptotic convergence.

Improved Integral Inequality Based on Free Matrices and Its Application to Stability Analysis of Delayed Neural Networks via Matrix-valued Cubic Polynomial Inequality

This paper deals with the stability analysis of delayed neural networks (DNNs). Firstly, an improved integral inequality based on free matrices (I3BFM) is newly proposed. The I3BFM gives the upper bound of the integral quadratic form including the state-related vector stacked by the state, its derivative, and its integration by introducing some free matrices. The upper bound of the I3BFM fully reflects the information not only on each term in the state-related vector but also on the cross-terms between the three terms. Secondly, the double integral Lyapunov-Krasovskii functional (LKF) of the integral quadratic form including the state-related vector is established to utilize the proposed I3BFM in the stability analysis of DNN. From the derivative of the double integral LKF, a matrix-valued cubic polynomial on a time-varying delay is generated in the process of stability analysis. Therefore, a matrix-valued cubic polynomial inequality is applied to make the matrix-valued cubic polynomial numerically tractable. Finally, the relaxed stability criteria utilizing the I3BFM and the double integral inequality are derived, and the superiority of the derived stability criteria is demonstrated by two well-known numerical examples.

Stability of Truncated Sampled-Data Control Systems With Impulsive Effects

This article considers the stability problem of truncated sampled-data control systems with impulsive effects. We propose the concept of average sampling interval (ASI), in which the constraint on the lower/upper bound of sampling intervals is removed to handle the sampling intervals from the holistic perspective. Then, in the framework of the input delay method, we transform the addressed system into an equivalent switched time-delay system subject to the impulsive effects that result from the truncated sampled-data control law. Interestingly, it is shown that the truncation of the sampled-data control input, which is an extension of the classical zero-order hold function, can play a stabilizing role in the presence of impulsive effects. Next, based on the ASI concept, the idea of average truncated length is introduced and the corresponding relaxed stability criteria are derived for the truncated sampled-data control systems. Particularly, the potential destabilizing impact of impulsive effects on the stability is theoretically revealed. It is also shown that under certain conditions, a truncated sampled-data system can remain stable if the magnitude of the impulsive effects is sufficiently small. Finally, two illustrative examples are presented to show the validity and also the advantages of the obtained results.

New Admissibility and Admissibilization Criteria for Nonlinear Discrete-Time Singular Systems by Switched Fuzzy Models

Admissibility analysis and control synthesis for nonlinear discrete-time singular systems are considered in this article. With regard to the type-1 and interval type-2 fuzzy singular systems, the partition of membership functions and scale transform is imposed, and new switched fuzzy systems, which are equivalent to the original systems, are established. A relaxed stability criterion is derived to ensure the admissibility of the system by using the piecewise Lyapunov function and singular value decomposition. Moreover, two classes of switched controllers are designed for the systems. One is for type 1 systems and the membership functions are consistent with those of the systems. The other can be applied to both of the fuzzy systems by introducing linear membership functions in each subregion. Two criteria are obtained to guarantee that the closed-loop systems are admissible. Several illustrative examples are provided to show the effectiveness of the developed methods.

A novel integral inequality and its application to stability analysis of linear system with multiple time delays

The integral inequality plays a key role in developing delay-dependent stability criteria for the linear system with multiple time delays. This paper investigates a novel integral inequality to bound the integral terms in the derivative of the Lyapunov–Krasovskii functional (LKF). Compared with the auxiliary function-based integral inequality, the presented integral inequality is more general. By removing the involved auxiliary function, the integral term in the bound is linearly represented with a sequence of integrals only related to the system state. After a newly augmented LKF is constructed with multiple integral terms concerning the relationships among multiple delays, an adjustable and relaxed stability criterion is established in terms of linear matrix inequalities using the proposed inequality. Two numerical examples are given to show the merits of the proposed method.

Relaxed Stability Criteria for Neural Networks With Time-Varying Delay Using Extended Secondary Delay Partitioning and Equivalent Reciprocal Convex Combination Techniques.

This article investigates global asymptotic stability for neural networks (NNs) with time-varying delay, which is differentiable and uniformly bounded, and the delay derivative exists and is upper-bounded. First, we propose the extended secondary delay partitioning technique to construct the novel Lyapunov-Krasovskii functional, where both single-integral and double-integral state variables are considered, while the single-integral ones are only solved by the traditional secondary delay partitioning. Second, a novel free-weight matrix equality (FWME) is presented to resolve the reciprocal convex combination problem equivalently and directly without Schur complement, which eliminates the need of positive definite matrices, and is less conservative and restrictive compared with various improved reciprocal convex inequalities. Furthermore, by the present extended secondary delay partitioning, equivalent reciprocal convex combination technique, and Bessel-Legendre inequality, two different relaxed sufficient conditions ensuring global asymptotic stability for NNs are obtained, for time-varying delays, respectively, with unknown and known lower bounds of the delay derivative. Finally, two examples are given to illustrate the superiority and effectiveness of the presented method.

Adaptive event-triggered H∞ fuzzy filtering for interval type-2 T–S fuzzy-model-based networked control systems with asynchronously and imperfectly matched membership functions

This paper is concerned with the network-based H∞ fuzzy filtering for non-linear systems with parameter uncertainties under a novel adaptive discrete event-triggered communication scheme (DETCS). Based on interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy model, the non-linear systems with parameter uncertainties are represented as a class of IT2 T–S fuzzy systems. In the design process, a novel adaptive DETCS is proposed to reduce the usage of system resources and adapt the variation of plant output, and a novel networked IT2 T–S fuzzy filter is applied to improve the flexibility of filter design. By employing the time-delay systems modeling method, the filtering-error-system is modeled as a class of interval time-varying delayed IT2 T–S fuzzy systems with asynchronously and imperfectly matched membership functions, and further conditionally expressed as a favorable form. Then, some relaxed stability criteria are established to determine that this class of delayed IT2 T–S fuzzy systems is asymptotically stable with a prescribed H∞ disturbance attenuation performance. Also, the co-design of parameter matrices of adaptive DETCS and filter is implemented. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.

Membership-Function-Dependent Stability and Stabilization Conditions for T–S Fuzzy Time-Delay Systems

ABSTRACTIn this paper, the stability analysis and stabilization of time-delay nonlinear systems represented by a Takagi–Sugeno fuzzy model are investigated. Combining with a recently developed integral inequality, relaxed delay-dependent stability conditions are derived by means of linear matrix inequalities. To obtain the further relaxed stability criteria, the membership-function-dependent method is applied via the piecewise membership functions (PMFs) and some slack variables. The PMFs are selected to approximate the membership functions (MFs), and then the characteristics of the MFs are taken into consideration adequately. Under the imperfect premise matching design scheme, this paper also proposes a flexible fuzzy controller design strategy that the fuzzy controllers do not employ the identical MFs and the number of rules as the fuzzy models. Finally, this proposed approach is tested successfully and its effectiveness is validated by the simulation examples.

Stability analysis of continuous-time systems with time-varying delay using new Lyapunov–Krasovskii functionals

This paper studies the stability of linear continuous-time systems with time-varying delay by employing new Lyapunov–Krasovskii functionals. Based on the new Lyapunov–Krasovskii functionals, more relaxed stability criteria are obtained. Firstly, in order to coordinate with the use of the third-order Bessel-Legendre inequality, a proper quadratic functional is constructed. Secondly, two couples of integral terms {∫t−htsx(s)ds,∫stx(s)ds} and {∫t−hMsx(s)ds,∫st−htx(s)ds} are involved in the integral functionals ∫t−htt(·)ds and ∫t−hMt−ht(·)ds, respectively, so that the coupling information between them can be fully utilized. Finally, two commonly-used numerical examples are given to demonstrate the effectiveness of the proposed method.

An improved summation inequality to discrete-time systems with time-varying delay

The summation inequality plays an important role in developing delay-dependent criteria for discrete-time systems with time-varying delay. This note proposes an improved summation inequality to estimate the summation terms appearing in the forward difference of Lyapunov–Krasovskii functional. Compared with the inequality recently developed by the Wirtinger-based summation inequality and the reciprocally convex lemma, the proposed one reduces the estimation gap while requires the same number of decision variables. A relaxed stability criterion of a linear discrete-time system with a time-varying delay is established by using such novel inequality. Two numerical examples are given to demonstrate the advantages of the proposed method.

Razumikhin-type theorems for hybrid system with memory

In this paper, we address the stability problem of hybrid systems with memory, which are newly developed to model hybrid dynamic systems affected by delays. For such systems, stability criteria are developed based on Razumikhin-type conditions, which are usually applied in the delay systems. We prove that the hybrid system with memory is uniformly globally pre-asymptotically stable by small gain theorem. Besides, a relaxed Razumikhin-type stability criterion is also proposed which does not require strict decrease of the Lyapunov–Razumikhin function during jumps. Several examples are given to illustrate the effectiveness and generality of the proposed results.

Stable Grip Control on Soft Objects With Time-Varying Stiffness

Humans can hold a live animal like a hamster without overly squeezing despite the fact that its soft body undergoes impedance and size variations due to breathing and wiggling. Although the exact nature of such biological motor controllers is not known, existing literature suggests that they maintain metastable interactions with dynamic objects based on prediction rather than reaction. Most robotic gripper controllers find such tasks very challenging mainly due to hard constraints imposed on the stability of closed-loop control and inadequate rates of convergence of adaptive controller parameters. This paper presents experimental and numerical simulation results of a control law based on a relaxed stability criterion of reducing the probability of failure to maintain a stable grip on a soft object that undergoes temporal variations in its internal impedance. The proposed controller uses only three parameters to interpret the probability of failure estimated using a history of grip forces to adjust the grip on the dynamic object. Here, we demonstrate that the proposed controller can maintain smooth and stable grip tightening and relaxing when the object undergoes random impedance variations, compared with a reactive controller that involves a similar number of controller parameters.

Robust Stability Criteria for T-S Fuzzy Systems with Time-Varying Delays via Nonquadratic Lyapunov-Krasovskii Functional Approach

This paper tackles the issue of stability analysis for uncertain T-S fuzzy systems with interval time-varying delays, especially based on the nonquadratic Lyapunov-Krasovskii functional (NLKF). To this end, this paper first provides a less conservative relaxation technique and then derives a relaxed robust stability criterion that enhances the interactions among delayed fuzzy subsystems. The effectiveness of our method is verified by two examples.

Relaxed stability criteria for discrete-time Takagi–Sugeno fuzzy systems via new augmented nonquadratic Lyapunov functions

The problem of relaxed stability analysis of discrete-time Takagi–Sugeno fuzzy systems is investigated in this paper. Different from those existing results in the literature that are only quadratic parameter dependent on system state variables, a new augmented nonquadratic Lyapunov function, which is formulated in a higher order form of system state variables than those existing results, is developed for achieving the task of stability analysis with less conservatism. More importantly, it can be proved that the approach proposed in this paper contains the existing one as a special case. Finally, the effectiveness of the proposed approach is illustrated by means of numerical experiments.