Augmented Lyapunov-Krasovskii Functionals Research Articles

Stability Analysis of Discrete-Time Neural Networks With a Time-Varying Delay: Extended Free-Weighting Matrices Zero Equation Approach.

This research investigates the stability of discrete-time neural networks (DNNs) with a time-varying delay by using the Lyapunov-Krasovskii functional (LKF) method. Recent researches acquired some less conservatism stability criteria for time-varying delayed systems via some augmented LKFs. However, the forward difference of such LKFs resulted in high-degree time-varying delay-dependent polynomials. This research aims to develop some augmented state-related vectors and the corresponding extended free-weighting matrices zero equations to avoid the appearance of such high-degree polynomials and help to provide more freedom for the estimation results. Besides, an augmented delay-product-type LKF is also established for ameliorating the stability conditions of the time-varying delayed DNNs. Then, based on the above methods and Jensen's summation inequality, the auxiliary-function-based summation inequality, and the reciprocally convex matrix inequality, some less conservatism stability criteria for time-varying delayed DNNs are formulated. The validity of the proposed time-varying delay-dependent stability criteria is illustrated by two numerical examples.

L2$$ {L}_2 $$‐gain analysis and fault‐tolerant control for uncertain nonlinear switched system with actuator saturation and time‐varying delay

AbstractFor the nonlinear continuous‐time switched system with input saturation and time‐varying delay, the problem of actuator failure is investigated in the presence of external disturbances. First, the sufficient conditions for the tolerance disturbances of system are obtained by using the multiple augmented Lyapunov Krasovskii functionals method combined with the Wirtinger integral inequality method and the reciprocally convex combination inequality. Second, the restricted ‐gain is analyzed. Third, the switching law and the disturbance suppression fault‐tolerant controllers are designed to obtain the maximum value of tolerance disturbance and the minimum upper bound on the restricted ‐gain. Finally, the numerical examples are given to verified the effectiveness of the proposed method by comparing the state trajectories under the action of the fault‐tolerant controllers and the standard controllers.

H∞ dynamic observer design for a class of Lipschitz nonlinear discrete‐time systems with time varying delays

AbstractThis study explores the development of dynamic observer (HDO) for discrete‐time nonlinear systems (DTNLS) with time‐varying delay (TVD) and disturbances. The approach is to construct an augmented Lyapunov–Krasovskii function (LKF) with double summation terms, using the generalized reciprocally convex matrix inequality (GRCMI), as well as the Jensen‐based inequality (JBI) and the Wirtinger‐based inequality (WBI). These lead to less conservative time‐dependent conditions, represented as a set of linear matrix inequalities (LMIs) that can be efficiently solved using the LMI or YALMIP toolboxes. In addition, the proposed observer includes the widely used proportional observer (PO) and proportional integral observer (PIO) as specific cases. Two examples are presented to demonstrate the validity and effectiveness of the results.

Observer based stabilization of linear time delay systems using new augmented LKF

This paper addresses the stabilization problem of linear time-varying delay systems with unmeasurable states. A novel augmented Lyapunov–Krasovskii functional (LKF) is proposed that effectively accounts for the impact of time delays, and an observer based stabilization controller is developed employing linear matrix inequality (LMI) based optimization technique. The utilization of extended reciprocally convex matrix inequality (ERCMI) is employed in this work to establish less conservative stabilization conditions within the framework of linear matrix inequalities (LMIs). By formulating a convex optimization problem, the observer gain and controller gains are determined. Simulation results are used to validate the design, and two numerical examples are considered to prove the usefulness of the proposed method over existing methods.

Novel stability criteria of generalized neural networks with time-varying delay based on the same augmented LKF and bounding technique

This paper researches the stability issue of generalized neural networks (GNN) with time-varying delay. For the delay, its derivative has an upper bound or is unknown. Firstly, the augmented Lyapunov-Krasovskii functional (LKF) is constructed based on the state vectors of the third order integral inequalities. Then, by introducing two sets of state vectors, the LKF derivative is presented as the quadratic and quintic polynomials of the delay, respectively. Next, the new quadratic and quintic polynomial negative definite conditions (NDCs) are proposed to set up the linear matrix inequalities (LMIs). In addition, based on the same LKF and third order integral inequalities, this paper proves that the introduction of extra state vectors increases the conservatism of the derived stability conditions. Eventually, the advantages of the provided conditions are illustrated by several numerical examples.

Extended dissipative analysis of affine transformed IT2 fuzzy control systems with time delay and disturbances

This paper focuses on the problem of stability and stabilization for interval type-2 (IT2) fuzzy systems with time-varying delays and external disturbances using the extended dissipative theory. In particular, our approach is to develop a new type of IT2 fuzzy controller that uses affine parameter-based weighting variables. Based on appropriate augmented Lyapunov-Krasovskii functionals and recently developed integral inequalities, some less conservative sufficient conditions are derived in the existence of the zeros equations. Specifically, the required conditions for co-design of affine matched membership functions-based IT2 fuzzy control law are provided such that the generalized performance index, the L2−L∞, H∞, passivity and (Q,S,R)−γ-dissipative problems are investigated. Finally, four examples are represented to verify the advantages of the developed method by comparing it with some existing results.

Stability analysis of load frequency control for power systems with interval time-varying delays

This study investigates the stability problem of load frequency control (LFC) for power systems with interval time-varying delays. The two categories of time delays, the lower bound being zero and non-zero, are considered. The systems can be described as time delay systems of load disturbances. First, an augmented Lyapunov–Krasovskii functional (LKF) is constructed. Some delay-dependent nonintegral terms and single integral terms are additionally introduced to make full use of the information on the system state variables and the time-varying delays. Second, to overcome the problem of nonlinear inequalities caused by the augmented LKF, the nonlinear inequalities are converted into linear matrix inequalities (LMIs) by applying the new negative definite inequality equivalence transformation lemma, which can be solved easily by the MATLAB LMI toolbox. A new stability criterion is presented by applying the Lyapunov stability theory. The stability criterion is less conservative than some existing literature studies, which further improves the stability margin for the power systems based on LFC. Finally, some numerical examples are given to show the effectiveness of the proposed method and the superiority of the results.

Stability Analysis of Linear Neutral Delay Systems With Two Delays via Augmented Lyapunov–Krasovskii Functionals

Stability analysis is considered in this paper for linear neutral delay systems subject to two different delays in both the state variables and the retarded derivatives of state variables. By choosing a suitable state vector indexed by an integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , a new augmented Lyapunov-Krasovskii functional (LKF) is constructed, and a stability criterion based on linear matrix inequalities is developed accordingly. It is shown that the proposed condition is less conservative than the existing methods due to the introduction of the delay-product-type integral terms in the LKF. The resulting stability criterion is then applied to the robust stability analysis of neutral delay systems with norm-bounded uncertainty. Moreover, a delay-independent stability criterion is developed based on the proposed LKF, and its frequency-domain interpretation is also given. These developed stability criteria indexed by an integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> exhibit a hierarchical character: the larger the integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , the less conservatism of the resulting stability criterion. Finally, two numerical examples are carried out to illustrate the effectiveness of the proposed method.

Two relaxed quadratic function negative-determination lemmas: Application to time-delay systems

Complicated construction of Lyapunov–Krasovskii functional (LKF) makes the quadratic correlation terms of time-delay appear in its derivative. This paper proposes two relaxed negative-determination lemmas to deduce the negative definite condition of a quadratic function with respect to a time-varying delay, which contain some popularly lemmas as their special cases. Combining a novel augmented LKF and generalized free-matrix-based integral inequality (GFMBII), the developed lemmas are respectively applied to derive the stability criteria for nominal and uncertain systems. Three numerical examples are given to prove the potential gain of the two lemmas and the superiority of the criteria over the previous work.

Advanced controller design for uncertain linear systems with time‐varying delays via augmented zero equality approach

This paper deals with the stability analysis and controller design for linear systems with time‐varying delays and parameter uncertainties. By choosing appropriate augmented Lyapunov–Krasovskii functionals, a set of linear matrix inequalities is derived to get advanced feasible region of stability and controller gain matrices that guarantee the asymptotic stability of the concerned systems within maximum bound of time delays and its time derivative. To further reduce the conservatism of stabilization criterion, a recently developed mathematical technique that constructed a new augmented zero equality is applied. Finally, two numerical examples are utilized to show the validity and superiority of the proposed methods.

Some augmented approaches to the improved stability criteria for linear systems with time-varying delays

This work investigates the improved stability conditions for linear systems with time-varying delays via various augmented approaches. Some augmented approaches are augmented Lyapunov-Krasovskii functionals, augmented zero equalities, and the augmented zero equality approach. At first, by constructing augmented Lyapunov-Krasovskii functionals including the state vectors which were not considered in the previous works and augmented zero equalities, a stability criterion is proposed in the forms of linear matrix inequalities. Through the proposed Lyapunov-Krasovskii functionals and an additional functional derived from the integral inequality, a slightly improved result is derived. The proposed results do not consider the increase in the computational complexity to achieve a larger delay bound. So, by applying the augmented zero equality approach, which is a method of grafting the proposed augmented zero equality proposed in Finsler Lemma, to the proposed result, an enhanced stability result was derived. Also, the computational complexity is reduced by appropriately adjusting any vector of the integral inequality utilized in the proposed criteria. By applying some numerical examples to the proposed conditions, the effectiveness and superiority of the results are confirmed.

Finite-time mixed [formula omitted] and passivity analysis for Takagi–Sugeno fuzzy system with time-varying delays and actuator faults

This paper comes up with an novel augmented Lyapunov–Krasovskii functional (LKF) approach to process finite-time passivity-based H∞ non-fragile state feedback control issue of Takagi–Sugeno fuzzy system (TSFS) via taking the effects of both time-varying delays and actuator faults. Designing a non-fragile controller to guarantee that TSFS is finite-time bounded (FTB) and meet finite-time mixed H∞ and passive performance is the focus of this paper. On the basis of the new augmented LKF and improved boundary techniques, sufficient delay-dependent conditions are inferred to acquire the desired results of the TSFS. Furthermore, the derived finite-time stability conditions are derived in terms of linear matrix inequalities (LMIs) and the controller gain matrices are confirmed through solving LMIs. Subsequently, three examples are addressed to certify the applicability and advantage of the put forward methodology.

Improved results on stability analysis of time-varying delay systems via delay partitioning method and Finsler’s lemma

This paper proposes novel conditions based on linear matrix inequalities (LMI) for stability analysis of arbitrarily-fast time-varying delays systems. The time-varying delay interval is divided into smaller pieces in order to obtain an equivalent switched model with multiple time-varying delays of smaller interval, which differently from other existing approaches, the maximum switching frequency is not required for stability analysis. Thus, by the use of augmented Lyapunov-Krasovskii functionals and the Finsler’s lemma, together with some relationships among state variables intentionally defined, the inherent conservatism can be progressively reduced by refining more and more the delay partition. The superiority of the proposed method is illustrated through two benchmark examples.

Fuzzy Secure Control for Nonlinear $N$-D Parabolic PDE-ODE Coupled Systems Under Stochastic Deception Attacks

This article focuses on the design of fuzzy secure control for a class of coupled systems, which are modeled by a nonlinear <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -dimensional ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -D) parabolic partial differential equation (PDE) subsystem and an ordinary differential equation (ODE) subsystem. Under stochastic deception attacks, a fuzzy secure control scheme is designed, which is effective to tolerate the attacks and ensure the desired performance for the considered systems. A new fuzzy-dependent Poincare–Wirtinger’s inequality (PWI) is proposed. Compared with the traditional Poincare’s inequality, the fuzzy-dependent PWI is more flexible and less conservative. Meanwhile, an augmented Lyapunov–Krasovskii functional (LKF) is newly constructed, which strengthens the correlations of the PDE subsystem and ODE subsystem. Then, on the ground of the fuzzy-dependent PWI and the augmented LKF, new exponential stabilization criteria are set up for the PDE-ODE coupled systems. Finally, a hypersonic rocket car is presented to verify the effectiveness and less conservatism of the obtained results.

Delay-Dependent Stability Analysis and Design of Input-delayed Systems by Smooth Sliding Mode Control: Lagrange Theorem Approach

A new smooth sliding mode control design methodology based on Lagrange mean value theorem is proposed for stabilization of single input delayed systems. The Lagrange mean value theorem as a basic theorem of calculus is used for the design of linear sliding mode time-delay controller for the first time. This controller satisfies the sliding condition using a Zhou and Fisher type continuous control law eliminating the chattering effect. The constructive delay-dependent asymptotically stable sliding conditions are obtained by using the augmented Lyapunov-Krasovskii functionals and formulated in terms of simple (4x4)-matrix inequality with scalar elements. Developed design approach can be extended to robust stabilization of sliding system with unknown but bounded input delay. The maximum upper bounds of delay size can be found by using simple optimization algorithms. Helicopter hover control is considered as a design example for illustrating the usefulness of smooth sliding mode approach. Unstable helicopter dynamics are successfully stabilized by using linear sliding mode time-delay controller. For example, settling time is about 20 sec. Therefore, simulation results confirmed the effectiveness of proposed design methodology. Apparently, the proposed method has a great potential in design of time-delayed controllers.

Reachable Set Estimation of Delayed Markovian Jump Neural Networks Based on an Improved Reciprocally Convex Inequality.

This brief investigates the reachable set estimation problem of the delayed Markovian jump neural networks (NNs) with bounded disturbances. First, an improved reciprocally convex inequality is proposed, which contains some existing ones as its special cases. Second, an augmented Lyapunov-Krasovskii functional (LKF) tailored for delayed Markovian jump NNs is proposed. Thirdly, based on the proposed reciprocally convex inequality and the augmented LKF, an accurate ellipsoidal description of the reachable set for delayed Markovian jump NNs is obtained. Finally, simulation results are given to illustrate the effectiveness of the proposed method.

A relaxed binary quadratic function negative-determination lemma and its application to neutral systems with interval time-varying delays and nonlinear disturbances

This paper considers the stability problem of neutral systems with interval time-varying delays and nonlinear disturbances. Firstly, an augmented vector containing two double integral terms is introduced into the Lyapunov-Krasovskii functional (LKF). In this case, a binary quadratic function with discrete and neutral delay arises in the time derivative. To gain the negativity condition of such function, by taking full advantage of the idea of partial differential of the binary quadratic function and Taylor's formula, a relaxed binary quadratic function negative-determination lemma with two adjustable parameters is proposed, which contains the existing lemmas as its special cases and shows the great potential of reducing conservatism for the case where the tangent slope at the endpoint is far from zero. Then, based on the improved lemma, more relaxing stability criteria have been obtained via an augmented LKF. Finally, two classic numerical examples are given to attest the effectiveness and strengths of the obtained stability criteria.

Stability analysis of systems with time‐varying delay via a delay‐product‐type integral inequality

This paper is concerned with the issue of stability analysis of systems with time‐varying delay. First, a delay‐product‐type (DPT) integral inequality is proposed, which combines the third‐order Bessel–Legendre (TOBL) integral inequality and the reciprocally convex (RC) inequality into a unified integral inequality; then, a tight upper bound on the time‐derivative of the Lyapunov–Krasovskii functional (LKF) can be obtained. Second, an augmented LKF is tailored for the use of the DPT integral inequality; then, more information about the instant state and the delayed states is taken into account. Third, by using the DPT integral inequality and the augmented LKF, some less conservative conditions that ensure the stability of systems with time‐varying delay are derived. Finally, simulations are provided to illustrate the advantage of our proposed method.

Less conservative stability criteria for general neural networks through novel delay-dependent functional

This work investigates the improved stability conditions for neural networks with time-varying delay. By the construction of newly augmented Lyapunov–Krasovskii functionals including the integral inequality and the use of the augmented zero equality approach, three improved results are proposed in the form of linear matrix inequalities. Two delay-dependent Lyapunov–Krasovskii functionals based on the integral inequality are proposed for the first time. Also, by utilizing the augmented zero equality approach, a less conservative result is obtained while reducing computation complexity. Through some numerical examples, the effectiveness and superiority of the proposed results are confirmed by comparing the existing works.

An Improved Stability Criterion for Discrete-Time Linear Systems With Two Additive Time-Varying Delays

In this paper, the stability analysis problem for discrete-time linear systems with additive time-varying delays is further investigated. In the first place, an augmented Lyapunov-Krasovskii functional (LKF) based on delay interval decomposition is designed, where some augmented vectors are selected to supplement the coupling relationships between some system state variables and different delay subintervals. In the second place, based on the augmented LKF, a new delay-dependent stability criterion is derived vai a general summation inequality lemma. The stability criterion is derived in the form of linear matrix inequality (LMI), which can be solved quickly by Matlab LMI-Tool. In the end, the effectiveness of the proposed method is illustrated by some common numerical examples.