Augmented Lyapunov Function Research Articles

Memory State‐Feedback Control for a Linear System With Time‐Varying Delay Based on the Tuning Matrix Search Algorithm

ABSTRACTA multiple augmented Lyapunov function (MALF) is a function that includes more than one augmented matrix, allowing for a more precise evaluation of the relationships between different states within a system and facilitating the analysis of system stability. However, employing multiple augmented matrices can complicate the process of designing controllers. For this issue, this article proposes a controller synthesis algorithm to address the complexity of controller design resulting from multiple augmentation matrices. First, a memory state‐feedback control criterion with tuning matrices is derived by constructing an MALF. Second, a tuning matrix search algorithm is proposed to provide appropriate tuning matrices for the memory state‐feedback control criterion, leading to the determination of optimal controller gains. Finally, the effectiveness and advantage are verified by simulation of load frequency control for in the power system and control for a linear singular time‐delay system.

Does the application of Jensen's integral inequality and LMIs confirm exponential stability in delayed systems with gapped gamma distribution through augmented Lyapunov function?

This study is dedicated to a comprehensive exploration aimed at advancing our understanding of stability within dynamic systems. The focus is particularly on the intricate domain of delayed systems characterized by gapped gamma distributions. The primary objective of this investigation revolves around evaluating the pragmatic application and efficacy of Jensen's integral inequality in combination with the powerful analytical tools provided by Linear Matrix Inequalities (LMIs). This evaluation is crucial for rigorously assessing exponential stability within these complex systems. Central to our investigative framework is the strategic deployment of augmented Lyapunov functions. These functions play a crucial role in unraveling the intricate stability properties of delayed systems featuring gapped gamma distributions, allowing for a nuanced examination of their inherent stability characteristics under various conditions. The mathematical formulation crafted in this exploration intricately captures the interplay between the distinctive attributes of the gapped gamma distribution and the complex dynamics of the loop traffic flow model within the overarching delayed system. This interconnection serves as the fundamental basis for the stability analysis, providing insights into the interdependence of these key elements. The noteworthy contribution of this study lies in the systematic construction of a robust analytical framework explicitly tailored for stability assessment. A comprehensive investigation is undertaken to elucidate critical aspects, including the convergence rate and the attainment of asymptotic stability within the considered delayed system. Additionally, a dedicated simulation section, focusing on Vehicle Active Suspension Control, has been incorporated to further validate and showcase the applicability of the proposed methodology.

Improved delay-dependent stability analysis of digital filters with generalized overflow arithmetic

This paper introduces a new set of conditions that establish Input-to-State Stability (ISS) for digital filters with external disturbance input and time-varying delays, all within the context of generalized overflow arithmetic. A unique augmented Lyapunov function and a novel summation inequality are applied to formulate a new criterion for ISS. This criterion assesses the ISS of digital filters in the presence of external disturbance, enabling the identification of asymptotic stability even in the absence of such disturbance. The criteria are developed using linear matrix inequalities (LMIs). The efficacy of the results is demonstrated through three numerical examples, highlighting that the proposed criteria are less conservative than some recent results.

Stability Analysis of Linear Time-Varying Delay Systems via a Novel Augmented Variable Approach

This paper investigates the stability issues of time-varying delay systems. Firstly, a novel augmented Lyapunov functional is constructed for a class of bounded time-varying delays by introducing new double integral terms. Subsequently, a time-varying matrix-dependent zero equation is introduced to relax the constraints of traditional constant matrix-dependent zero equations. Secondly, for a class of periodic time-varying delays, considering the monotonicity of the delay and combining it with an augmented variable approach, Lyapunov functionals are constructed for monotonically increasing and monotonically decreasing delay intervals, respectively. Based on the constructed augmented Lyapunov functionals and the employed time-varying zero equation, less conservative stability criteria are obtained separately for bounded and periodic time-varying delays. Lastly, three examples are used to verify the superiority of the stability conditions obtained in this paper.

Stability Criteria of Delayed Memristor-Based Neural Networks via Continuous-Time Model and Interval Matrix Approach

Modeling and stability analysis of memristor-based neural networks (MNNs) are the premise of designs and applications. Different from most previous research, delayed MNNs are described by continuous differential equations with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{2n^{2}+n}$</tex-math> </inline-formula> variables where the memductances of memristors are continuously dependent on the fluxes. Both system delays and input delays are considered, and the delays and their derivatives may vary in intervals whose lower bounds are not restricted to be zero. The systems are further reduced to continuous-time neural networks (NNs) with interval matrix uncertainties, and a unified method is developed to solve the stability of delayed NNs and MNNs. Stability criteria are obtained for delayed MNNs by augmented Lyapunov functionals, Wirtinger-based integral inequality, reciprocally convex approach, and linear matrix inequalities. In the end, two numerical simulations are used to demonstrate the validity of our theorems.

Delayed stabilization of parabolic PDEs via augmented Lyapunov functionals and Legendre polynomials

We first study stabilization of heat equation with globally Lipschitz nonlinearity. We consider the point measurements with constant delay and use spatial decomposition. Inspired by recent developments in the area of ordinary differential equations (ODEs) with time-delays, for the stability analysis, we suggest an augmented Lyapunov functional depending on the state derivative that is based on Legendre polynomials. Global exponential stability conditions are derived in terms of linear matrix inequalities (LMIs) that depend on the degree N of Legendre polynomials. The stability conditions form a hierarchy of LMIs: if the LMIs hold for N, they hold for N+1. The dual observer design problem with constant delay is also formulated. We further consider stabilization of Korteweg–de Vries–Burgers (KdVB) equation using the point measurements with constant delay. Due to the third-order partial derivative in KdVB equation, the Lyapunov functionals that depend on the state derivative are not applicable here, which is different from the case of heat equation. We suggest a novel augmented Lyapunov functional depending on the state only that leads to improved regional stability conditions in terms of LMIs. Finally, numerical examples illustrate the efficiency of the method.

Regional Stabilization for Discrete Time-Delay Systems With Actuator Saturations via A Delay-Dependent Polytopic Approach

This paper is concerned with the regional stabilization problem for discrete time-delay systems with actuator saturations. Different from some existing delay-independent techniques handling the saturation nonlinearity, a delay-dependent polytopic approach is first proposed in a discrete-time framework. Then, by combining with an augmented Lyapunov function, a discrete Wirtinger-based inequality and some novel analysis techniques, improved delay-dependent regional stabilization conditions are established by means of linear matrix inequalities. Finally, it is shown via two numerical examples that the obtained results are less conservative than some existing ones yet providing a larger estimate of an initial condition set.

Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay

This paper introduces stability conditions in the form of linear matrix inequalities (LMIs) for general linear retarded systems with a delay term described by Stieltjes integral. The derived LMIs provide in the unified form conditions for both discrete and distributed delays. Two Lyapunov-based methods for the asymptotic mean square stability of stochastic linear time-invariant systems are presented. The first one employs neutral type model transformation and augmented Lyapunov functionals. Differently from the existing LMI stability conditions based on neutral type transformation, the proposed conditions do not require the stability of the corresponding integral equations. Moreover, it is shown that in the simplest existing LMIs based on non-augmented Lyapunov functionals, the stability analysis of the integral equation can be omitted. The second method is based on a stochastic extension of simple Lyapunov functionals depending on the state derivative. The same two methods are further applied to delay-induced stability analysis of stochastic vector second-order systems, simplifying the recent results via neutral type transformation and leading to new conditions for stochastic systems via the second method. Numerical examples give comparison of results via different methods.

ISS Criterion for the Realization of Fixed-Point State-Space Digital Filters with Saturation Arithmetic and External Interference

This paper investigates the problem of the input-to-state stability (ISS) of fixed-point state-space digital filters in the presence of saturation overflow arithmetic and external interference. By utilizing an augmented Lyapunov function and the passivity property associated with multiple saturation nonlinearities, a new criterion for the ISS of interfered digital filters with saturation is proposed. When the external interference is present, the criterion can be used to determine ISS of digital filters. Without the external interference, the criterion is capable of detecting the asymptotic stability of digital filters. The proposed ISS criterion is in linear matrix inequality framework and, hence, is computationally tractable. The obtained criterion turns out to be less conservative than a recently reported ISS criterion. The superiority of the presented approach over the existing techniques is confirmed by numerical examples.

Neural Network-Based Adaptive Control of Robotic Manipulator: Application to a Three Links Cylindrical Robot

A composite PD and sliding mode neural network (NN)-based adaptive controller, for robotic manipulator trajectory tracking, is presented in this paper. The designed neural networks are exploited to approximate the robotics dynamics nonlinearities, and compensate its effect and this will enhance the performance of the filtered error based PD and sliding mode controller. Lyapunov theorem has been used to prove the stability of the system and the tracking error boundedness. The augmented Lyapunov function is used to derive the NN weights learning law. To reduce the effect of breaching the NN learning law excitation condition due to external disturbances and measurement noise; a modified learning law is suggested based on e-modification algorithm. The controller effectiveness is demonstrated through computer simulation of cylindrical robot manipulator.

Improved results on stability and stabilization criteria for uncertain linear systems with time-varying delays

ABSTRACTIn this paper, the problems of stability and stabilization for linear systems with time-varying delays and norm-bounded parameter uncertainties are considered. By constructing augmented Lyapunov functionals and utilizing auxiliary function-based integral inequalities, improved delay-dependent stability and stabilization criteria for guaranteeing the asymptotic stability of the system are proposed with the framework of linear matrix inequalities. Four numerical examples are included to show that the proposed results can reduce the conservatism of stability and stabilization criteria by comparing maximum delay bounds.

Nonfragile Exponential Synchronization of Delayed Complex Dynamical Networks With Memory Sampled-Data Control

This paper considers nonfragile exponential synchronization for complex dynamical networks (CDNs) with time-varying coupling delay. The sampled-data feedback control, which is assumed to allow norm-bounded uncertainty and involves a constant signal transmission delay, is constructed for the first time in this paper. By constructing a suitable augmented Lyapunov function, and with the help of introduced integral inequalities and employing the convex combination technique, a sufficient condition is developed, such that the nonfragile exponential stability of the error system is guaranteed. As a result, for the case of sampled-data control free of norm-bound uncertainties, some sufficient conditions of sampled-data synchronization criteria for the CDNs with time-varying coupling delay are presented. As the formulations are in the framework of linear matrix inequality, these conditions can be easily solved and implemented. Two illustrative examples are presented to demonstrate the effectiveness and merits of the proposed feedback control.

Neural network augmented backstepping control for uncertain nonlinear systems - application to laboratory antilock braking system

A new control approach is proposed to address the tracking problem of a class of uncertain nonlinear systems. In this approach, one relies first on a partially known model of the system to be controlled using a backstepping control strategy. The obtained controller is then augmented by an online artificial neural network (ANN) that serves as an approximator for the neglected dynamics and modelling errors. Thus, the developed method combines backstepping approach and ANN to address the tracking problem for uncertain systems. The proposed approach is systematic, and exploits the known nonlinear dynamics to derive the stepwise virtual stabilising control laws. At the final step, an augmented Lyapunov function is introduced to derive the adaptation laws of the network weights. The suggested control algorithm is tested experimentally on a Laboratory ABS system showing satisfactory results although the system is highly nonlinear and with unknown physical parameters.

Some new results on the stability of direct-form digital filters with finite wordlength nonlinearities

This paper proposes a new criterion for the input/output-to-state stability (IOSS) of interfered direct-form digital filters with finite wordlength nonlinearities based on an augmented Lyapunov function. Without external interferences, the output-to-state stability (OSS) and asymptotic stability of direct-form digital filters are guaranteed under the proposed criterion. An improved criterion is also proposed for the IOSS and OSS of interfered direct-form digital filters, with the introduction of four slack matrices. A new state-norm estimator for direct-form digital filters is constructed based on an exponential-decay IOSS criterion. The criteria presented in this paper are expressed by linear matrix inequalities (LMIs). Two numerical examples demonstrate the effectiveness of the proposed results.

Robust delay-depent stability criteria for uncertain neural networks with two additive time-varying delay components

This paper considers the problem of robust stability of uncertain neural networks with two additive time varying delay components. The activation functions are monotone nondecreasing with known lower and upper bounds. By constructing of a modified augmented Lyapunov function, some new stability criteria are established in terms of linear matrix inequalities, which is easily solved by various convex optimization techniques. Compared with the existing works, the obtained criteria are less conservative due to reciprocal convex technique and an improved inequality, which provides more accurate upper bound than Jensen inequality for dealing with the cross-term. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality

In this paper, the problem of stability analysis for linear systems with time-varying delays is considered. By the consideration of new augmented Lyapunov functionals, improved delay-dependent stability criteria for asymptotic stability of the system are proposed for two cases of conditions on time-varying delays with the framework of linear matrix inequalities (LMIs), which can be solved easily by various efficient convex optimization algorithms. The enhancement of the feasible region of the proposed criteria is shown via three numerical examples by the comparison of maximum delay bounds.

The Improved Method of Delay-Dependent Stability Criterion for Uncertain Systems with Multiple Time-Varying Delays

This paper discusses problems of linear uncertain system with multiple time-varying delays. Through the application of a novel design by own lemma and augmented Lyapunov function, a less conservative results than the existing essay that have been obtained. The new criterion enlarges the existing maximum bound of the time delay. Numerical examples were given to demonstrate the effective of the result.

New and improved results on stability of static neural networks with interval time-varying delays

In this paper, the problem of stability analysis for static neural networks with interval time-varying delays is considered. By the consideration of new augmented Lyapunov functionals, new and improved delay-dependent stability criteria to guarantee the asymptotic stability of the concerned networks are proposed with the framework of linear matrix inequalities (LMIs), which can be solved easily by standard numerical packages. The enhancement of the feasible region of the proposed criteria is shown via two numerical examples by the comparison of maximum delay bounds.

New stability conditions for systems with distributed delays

In the present paper, sufficient conditions for the exponential stability of linear systems with infinite distributed delays are presented. Such systems arise in population dynamics, in traffic flow models, in networked control systems, in PID controller design and in other engineering problems. In the early Lyapunov-based analysis of systems with distributed delays (Kolmanovskii & Myshkis, 1999), the delayed terms were treated as perturbations, where it was assumed that the system without the delayed term is asymptotically stable. Later, for the case of constant kernels and finite delays, less conservative conditions were derived under the assumption that the corresponding system with the zero-delay is stable (Chen & Zheng, 2007). We will generalize these results to the infinite delay case by extending the corresponding Jensen’s integral inequalities and Lyapunov–Krasovskii constructions. Our main challenge is the stability conditions for systems with gamma-distributed delays, where the delay is stabilizing, i.e. the corresponding system with the zero-delay as well as the system without the delayed term are not asymptotically stable. Here the results are derived by using augmented Lyapunov functionals. Polytopic uncertainties in the system matrices can be easily included in the analysis. Numerical examples illustrate the efficiency of the method. Thus, for the traffic flow model on the ring, where the delay is stabilizing, the resulting stability region is close to the theoretical one found in Michiels, Morarescu, and Niculescu (2009) via the frequency domain analysis.

Finite-time H ∞ control for a class of Markovian jump systems with mode-dependent time-varying delay

This paper is concerned with the problem of finite-time control for a class of Markovian jump system with mode-dependent time-varying delay. By using the new augmented multiple Lyapunov function with more general decomposition approach, a novel sufficient condition for finite-time bounded with an performance index is derived. Based on the derived condition, the reliable control problem is solved, and an explicit expression of the desired controller is also given, the system trajectory stays within a prescribed bound during a specified time interval. Finally, numerical examples are given to demonstrate that the proposed approach is more effective than some existing ones.