Exponential Stability Analysis Research Articles

An improved exponential stability analysis method for discrete‐time systems with a time‐varying delay

AbstractExponential stability analysis of discrete‐time nonlinear systems with a time‐varying delay is concerned in this article. The main contribution of this article lies in proposing an improved stability analysis method called a delay‐mode‐based functional method. The improved method is presented by weakening a condition in the Lyapunov–Krasovskii functional method. With the help of the new method, less conservative exponential stability criteria of discrete‐time linear systems with a time‐varying delay are obtained and two classical examples are given to show that the delay‐mode‐based functional method is effective.

Exponential stability of integral time-varying delay system

This paper investigates exponential stability problem of integral time-varying delay system. Based on a novel exponential stability theorem, sufficient conditions for exponential stability of integral time-varying delay system are obtained in the form of coupled linear matrix inequalities (LMIs). These sufficient stability conditions cover some previous results as special cases when the integral time-varying delay system reduce to the integral time-invariant delay system. These sufficient conditions cannot be obtained directly from Theorem 1 given in Li, Zheng, and Wang [(2016). Exponential stability analysis of integral delay systems with multiple exponential kernels. Journal of the Frankline Institute, 353, 1639–1653. https://doi.org/10.1016/j.jfranklin.2015.12.016] due to the presence of time-varying delays. This is the main motivation for the research being carried out in this paper. Four example s are provided to show the effectiveness and advantages of the proposed method.

SY2-3. The role of autophagy in pathophysiology of PE

The coupled repressilators, as a special case of cyclic genetic regulatory networks, can be used to adjust oscillations at the cellular level, and hence it have received extensive attention from many scholars. This paper addresses the problem of global exponential stability analysis for nonnegative equilibrium points of a class of coupled repressilator models with multiple time-varying delays. Sufficient conditions are investigated to guarantee that the considered model has a unique nonnegative equilibrium point which is globally exponentially stable. The obtained criteria are composed of several simple linear inequalities that are only related to the model parameters, so they can be easily verified by using the standard software tools. The results of two illustrative examples present the effectiveness of the proposed approach.

On moment exponential stability of Markovian switching integral delay systems

This paper studies the moment exponential stability analysis of a class of Markovian switching integral delay systems (MSIDSs). The existence, uniqueness and stability of the solution are discussed firstly. Secondly, by selecting appropriate Lyapunov-Krasovskii (L-K) functionals, delay-dependent sufficient conditions are given such that the general form of MSIDSs and the special form of MSIDSs having multiple delays are mean square exponentially stable respectively. The results are then generalized to robust stability of MSIDSs having multiple delays with uncertain parameters. Finally, numerical examples are given to illustrate the effectiveness of the proposed theoretical results.

Stability Analysis of Pseudo-Almost Periodic Solution for a Class of Cellular Neural Network with D Operator and Time-Varying Delays

Cellular neural networks with D operator and time-varying delays are found to be effective in demonstrating complex dynamic behaviors. The stability analysis of the pseudo-almost periodic solution for a novel neural network of this kind is considered in this work. A generalized class neural networks model, combining cellular neural networks and the shunting inhibitory neural networks with D operator and time-varying delays is constructed. Based on the fixed-point theory and the exponential dichotomy of linear equations, the existence and uniqueness of pseudo-almost periodic solutions are investigated. Through a suitable variable transformation, the globally exponentially stable sufficient condition of the cellular neural network is examined. Compared with previous studies on the stability of periodic solutions, the global exponential stability analysis for this work avoids constructing the complex Lyapunov functional. Therefore, the stability criteria of the pseudo-almost periodic solution for cellular neural networks in this paper are more precise and less conservative. Finally, an example is presented to illustrate the feasibility and effectiveness of our obtained theoretical results.

Global exponential stability analysis of anti-periodic of discontinuous BAM neural networks with time-varying delays

Discontinuous system is playing an increasingly important role in terms of both theory and applications. In this paper, we are concerned with discontinuous BAM (bidirectional associative memory) neural networks with time-varying delays. Under the basic framework of Filippov solution, by means of differential inclusions theory, inequality technique, fundamental solution matrix of coefficients and the non-smooth analysis theory with Lyapunov-like approach, some new sufficient criteria are given to ascertain the existence and globally exponential stability of the anti-periodic solutions for the considered BAM neural networks. Simulation results of two topical numerical examples are exploited to illustrate the improvement and advantages of the established theoretical results in comparison with some existing results. Some previous known results are extended and complemented.

Aperiodic Sampled-Data Control for Chaotic System Based on Takagi–Sugeno Fuzzy Model

This paper investigates the aperiodic sampled-data control for a chaotic system. Firstly, Takagi–Sugeno (T-S) fuzzy models for the chaotic systems are established. The lower and upper bounds of the sampling period are taken into consideration. Then, the criteria for mean square exponential stability analysis and aperiodic sampled-data controller synthesis are provided by means of linear matrix inequalities. And the real sampling patterns can be fully captured by constructing suitable Lyapunov functions. Finally, an illustrative example shows that the proposed method is effective to guarantee that the system’s states are stable with aperiodic sampled data.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

This paper deals with the exponential stability analysis of decentralized, sampled-data, Linear Time Invariant (LTI) control systems with asynchronous sensors and actuators. We consider the case where each controller in the decentralized setting has its own sampling and actuation frequency, which translates to asynchrony between sensors and actuators. Additionally, asynchrony may be induced by delays between the sampling instants and actuation update instants as relevant in a networked context. The decentralized, asynchronous LTI system is represented as the feedback interconnection of a continuous-time LTI system operator and an operator that captures the effects of asynchrony induced by sampling and delay. By characterizing the properties of the operators using small-gain type Integral Quadratic Constraints (IQC), we provide criteria for exponential stability of the asynchronous, decentralized LTI state-space models. The approach provided in this paper considers two scenarios, namely the ‘large-delay’ case and the ‘small-delay’ case where the delays are larger and smaller than the sampling interval, respectively. The effectiveness of the proposed results is corroborated by a numerical example.

Stability and [formula omitted]-gain analysis of nonlinear positive Markov jump systems based on a switching transition probability

This paper focuses on mean exponential stability and L1-gain analysis for nonlinear positive Markov jump systems (NPMJSs) based on a switching transition probability (STP), where sector nonlinear functions and delays are adopted. By developing a nonlinear stochastic copositive Lyapunov–Krasovskii functional (NSCLKF) approach, a sufficient condition for mean exponential stability of nonlinear positive time-delay Markov jump system is first presented under mode-dependent average dwell time (MDADT) switching by using linear programming (LP) approach. Further, L1-gain performance analysis is obtained. Moreover, the corresponding results for NPMJSs are given under average dwell time (ADT) switching. Illustrative results are provided to verify the validity of the theoretical results.

Exponential stability and numerical analysis of a thermoelastic diffusion beam with rotational inertia and second sound

We study the dynamic behavior of a thermoelastic diffusion beam with rotational inertia and second sound, clamped at one end and free to move between two stops at the other. The contact with the stops is modeled with the normal compliance condition. The system, recently derived by Aouadi (2015), describes the behavior of thermoelastic diffusion thin plates under Cattaneo’s law for heat and mass diffusion transmission to remove the physical paradox of infinite propagation speeds of the classical Fourier’s and Fick’s laws. The system of equations is a coupling of a hyperbolic equation with four parabolic equations. It poses some new mathematical and numerical difficulties due to the lack of regularity and the nonlinear boundary conditions. The exponential stability of the solutions to the contact problem is obtained in the presence of rotational inertia thanks to a structural damping term. We propose a finite element approximation and we prove that the associated discrete energy decays to zero. Finally, we give an error estimate assuming extra regularity on the solution and we present some results of our numerical experiments.

Global Exponential Stability of a Delayed HIV Infection Model with a Nonlinear Incidence Rate

Under the assumption that there is a time delay between the time target cells are contacted by the virus particles and the time the contacted cells become actively infected, we investigate the exponential stability of the noninfected equilibrium for a delayed HIV infection model with a nonlinear incidence rate. Compared with the global asymptotic stability analysis based on basic reproduction number, exponential stability analysis reveals the change range of various cells in different time periods.

Exponential stability analysis for discrete-time quaternion-valued neural networks with leakage delay and discrete time-varying delays

In this paper, we consider the issue of exponential stability analysis for discrete-time quaternion-valued neural networks(DTQVNNs) with leakage delay and discrete time-varying delays. The DTQVNNs are investigated directly rather than through real decomposition method or plural decomposition method. Firstly, the existence and uniqueness of the equilibrium point for DTQVNNs are proposed by using homeomorphic mapping theorem and Cauchy-Schwarz inequality. Then, based on Lyapuinov-Krasovskii functional and matrix inequality, a delay-dependent sufficient condition is provided to guarantee the exponential stability of the equilibrium point for DTQVNNs. Finally, a numerical example is utilized to illustrate the effectiveness of the obtained results.

Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay

<p style='text-indent:20px;'>This paper is concerned with the pullback dynamics and asymptotic stability for a 3D Brinkman-Forchheimer equation with infinite delay. The well-posedness of weak solution to the 3D Brinkman-Forchheimer flow with infinite delay is investigated in the weighted space <inline-formula><tex-math id="M1">\begin{document}$ C_\kappa(H) $\end{document}</tex-math></inline-formula> firstly, then the pullback attractors are presented for the process of weak solution. Moreover, the existence of global attractor and the exponential stability analysis of stationary solutions are shown, which is based on the estimate of corresponding steady state equation.

Global Exponential Stability Analysis for A Class of Coupled Cyclic Genetic Regulatory Networks with Multiple Time-Varying Delays

The coupled repressilators, as a special case of cyclic genetic regulatory networks, can be used to adjust oscillations at the cellular level, and hence it have received extensive attention from many scholars. This paper addresses the problem of global exponential stability analysis for nonnegative equilibrium points of a class of coupled repressilator models with multiple time-varying delays. Sufficient conditions are investigated to guarantee that the considered model has a unique nonnegative equilibrium point which is globally exponentially stable. The obtained criteria are composed of several simple linear inequalities that are only related to the model parameters, so they can be easily verified by using the standard software tools. The results of two illustrative examples present the effectiveness of the proposed approach.

Stability analysis for time-varying nonlinear systems

This paper is concerned with practical asymptotic and exponential stability analysis of time-varying nonlinear systems. With the help of the new notion of practical stable functions, some differential Lyapunov inequalities based necessary and sufficient conditions are derived for testing global uniform practical asymptotic stability and practical exponential stability of general time-varying nonlinear systems. Therefore, we generalised some works which are already made in the literature. Furthermore, some illustrative examples are presented.

Exponential stability analysis of neural networks with a time‐varying delay via a generalized Lyapunov‐Krasovskii functional method

SummaryAs is known to all that the Lyapunov‐Krasovskii functional (LKF) method plays a significant role in deriving exponential stability criteria of neural networks with a time‐varying delay. However, when the LKF method is adopted, the condition that a functional is required for a neural network with a delay varying in a delay interval is so strong that it may be hard to be satisfied and lead to a conservative criterion. Therefore, a generalized LKF method is proposed by weakening the strong condition in this paper. Then, new exponential stability criteria are derived via applying the proposed method. Finally, the effectiveness of the derived criteria is verified by two numerical examples.

Exponential stability analysis for singular switched positive systems under dwell-time constraints

The article addresses the exponential stability analysis for singular switched positive systems(SSPSs) under dwell-time constraints which include mode-dependent minimum dwell time(MDMDT), mode-dependent constant dwell time(MDCDT) and mode-dependent ranged dwell time(MDRDT) constraints. For SSPSs in both delay-free case and time-varying delay case, a sufficient exponential stability condition is proposed with MDMDT, MDCDT and MDRDT constraints, respectively, and the exponential decay rate can be set as a free parameter based on diverse circumstances. To analyze the dwell-time stability, a novel discretized linear copositive Lyapunov function(DLCLF) approach is introduced in the article, and compared with the general copositive and homogeneous Lyapunov function approach, the main advantage of the DLCLF technique is itself affine dependence on conditions of system matrix, which will be extended and utilized to systems with uncertainties or/and time-varying parameters quite easily. Meanwhile, the proposed condition of exponential stability under MDMDT(or MDCDT or MDRDT) constraint will be degenerated into the one under minimum dwell time(or constant dwell time or ranged dwell time) constraint for some specific situations, which implies that the considered MDMDT(or MDCDT or MDRDT) case is more general and practical. Finally, the validity and significance of the results are illustrated by seven examples.

Generalized weak rigidity: Theory, and local and global convergence of formations

This paper proposes a generalized weak rigidity theory, and aims to apply the theory to formation control problems with a gradient descent flow law. The generalized weak rigidity theory is utilized in order to characterize desired rigid formations by a general set of pure inter-agent distances and subtended angles, where the rigid formation shape with distances and subtended angles is determined up to translations and rotations (if the formation shape is composed of only subtended angles, then it is determined up to translations, rotations and, additionally, scaling factors). As the first result of its applications, this paper provides analysis of local exponential stability for a formation control system with pure distance/angle or only angle constraints in 2- and 3-dimensional spaces. Then, as the second result, it is shown that if there are three agents in 2-dimensional space then almost global exponential stability is ensured for a formation control system with pure distance/angle or only angle constraints.

Projection-Free Distributed Optimization With Nonconvex Local Objective Functions and Resource Allocation Constraint

We present a novel generalized constrained convex optimization model for multiagent systems that contains both the local, coupled equality, and inequality constraints, and a global resource allocation constraint. This model unifies the traditional constrained optimization problem, the resource allocation problem, and the economic dispatch problem. Unlike the majority of literature where each local objective function is required to be convex, we only require a milder condition that the global objective function is convex. The gradient of the global Lagrangian is estimated locally by each agent using the dynamic average consensus protocol. Synchronously, modified primal-dual dynamics produce the optimal solution via the estimated gradient. The generalized Lagrange multiplier method is introduced to avoid the usual positive projections in the presence of inequality constraints. This leads to smooth dynamics and a continuous Lyapunov derivative, which enables the exponential stability analysis. Simulation examples support the proposed distributed methods.

Exponential Stability Analysis of Fixed Point Interfered Nonlinear Digital Systems in the Presence of Quantization and Overflow Constraints

Finite word length is a practical limitation when discrete-time systems are implemented by using digital hardware. This restriction degrades the performance of a discrete-time system and may even lead it toward instability. This paper, addresses the stability and disturbance attenuation performance analysis of nonlinear discrete-time systems under the influence of energy-bounded external interferences when such systems are subjected to quantization and overflow effects of fixed point hardware. The proposed methodology, in comparison with previous paper, describes exponential stability for the nonlinear discrete-time systems by considering composite nonlinearities of digital hardware. The proposed criteria that ensure exponential stability and [Formula: see text] performance index for the digital systems under consideration are presented in the form of a set of linear matrix inequalities (LMIs) by exploiting Lyapunov stability theory, Lipschitz condition and sector conditions for different types of commonly used quantization and overflow arithmetic properties, and the results are validated for recurrent neural networks. Furthermore, novel stability analysis results for a nonlinear discrete-time system under hardware constraints can also be observed as a special case of the proposed criteria.