Gronwall-Bellman Lemma Research Articles

Stability Analysis of a Class of Nonlinear Fractional Differential Systems With Riemann-Liouville Derivative

This paper investigates the stability of n-dimensional nonlinear fractional differential systems with Riemann-Liouville derivative. By using the Mittag-Leffler function, Laplace transform and the Gronwall-Bellman lemma, one sufficient condition is attained for the asymptotical stability of a class of nonlinear fractional differential systems whose order lies in (0, 2). According to this theory, if the nonlinear term satisfies some conditions, then the stability condition for nonlinear fractional differential systems is the same as the ones for corresponding linear systems. Several examples are provided to illustrate the applications of our result.

Robustness analysis of exponential stability of fuzzy inertial neural networks through the estimation of upper limits of perturbations

This paper characterizes the robustness of exponential stability of fuzzy inertial neural network which contains time delays or stochastic disturbance through the estimation of upper limits of perturbations. By utilizing Gronwall-Bellman lemma, stochastic analysis, Cauchy inequality, the mean value theorem of integrals, as well as the properties of integrations, the limits of both time delays and stochastic disturbances are derived in this paper which can make the disturbed system keep exponential stability. The constraints between the two types of disturbances are provided in this paper. Examples are offered to validate our results.

Stability of Sampled-Data Systems With Packet Losses: A Nonuniform Sampling Interval Approach.

In this article, inspired by the Halanay inequality, we study stability of sampled-data systems with packet losses by proposing a nonuniform sampling interval approach. First, a sampled-data controller with an exponential gain is put forward to reduce conservatism. We obtain the sufficient condition for linear sampled-data systems to be exponentially stable by extending the famous Halanay inequality to sampled-data systems. The obtained sufficient conditions indicate that the maximal-allowable bound of sampling intervals is determined by the constant terms in the Halanay inequality, and the decay rate is presented in the form of a Lambert function. Compared with some existing results on the stability of sampled-data systems by using the Gronwall-Bellman Lemma, the conservatism induced by the exponential term via the Gronwall-Bellman Lemma can be reduced to some extent. Considering the phenomenon of packet losses, a new lemma is further proposed to generalize the proposed Halanay-like inequality. The results derived by the new lemma permit that there exist some sampling intervals with the upper bound violating the desired condition of the Halanay-like inequality. This permits us to establish exponential stability in significant cases that do not satisfy the Halanay-like inequality needed in the previous results. Finally, the sampled-data local exponential stability is investigated for nonlinear systems with strong nonlinearity.

Robustness Analysis of Exponential Synchronization in Complex Dynamic Networks with Time-Varying Delays and Random Disturbances

This paper aims to investigate the robustness of exponential synchronization in complex dynamic networks (CDNs) with time-varying delays and random disturbances. Via the Gronwall–Bellman lemma and partial inequality methods, by calculating the transcendental equations, the delays limits and maximum disturbance size of the CDNs are estimated. This means that the perturbed system achieves exponential synchronization if the disturbance strength is within our estimation range. The theoretical results are illustrated by several simulations.

Robust H[formula omitted]-PID control Stability of fractional-order linear systems with Polytopic and two-norm bounded uncertainties subject to input saturation

This paper deals a type of H∞ proportional–integral–derivative (PID) control mechanism for a type of structural uncertain fractional order linear systems by convex Polytopic and two-norm bounded uncertainties subject to input saturation which mainly focuses on the case of a fractional order α such that 0<α<1. The Gronwall–Bellman lemma and the sector condition of the saturation function are investigated for system stability analysis and stabilization. The main strategy of the presented strategy is to restore fractional order PID controller design under input saturation problem from static output feedback controller design. Unlike existing strategies, non-iterative strategy is used to get optimal output feedback based on the LMI. On the premise of a linear matrix inequality algorithm, the SOF control laws can be obtained. After that, the fractional-order PID controller is recovered from the SOF controller. A numerical example is provided in order to show the validity and superiority of the proposed method.

Robustness analysis of exponential synchronization in complex dynamic networks with random perturbations

&lt;abstract&gt;&lt;p&gt;This article discusses the robustness of exponential synchronization (ESy) of complex dynamic networks (CDNs) with random perturbations. Using the Gronwall-Bellman lemma and partial inequality techniques, by solving the transcendental equation, the maximum perturbation intensity of the CDN is estimated. This implies that the disturbed system achieves ESy if the disturbance intensity is within the range of our estimation. We illustrate the theoretical results with two numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Robustness Analysis of Fuzzy Cellular Neural Network With Deviating Argument and Stochastic Disturbances

Robustness analysis of fuzzy cellular neural networks with deviating arguments and stochastic disturbances is the main topic of discussion in this paper. The issue at hand is what the upper bounds of the disturbances and deviating intervals for the fuzzy cellular neural network can withstand before losing its stability. We solve these problems by using Gronwall-Bellman lemma and some inequality techniques. The theoretical results point that for an exponentially stable fuzzy cellular neural network, the perturbed fuzzy cellular neural network still keep its globally exponential stability if the upper bound of the length of deviating intervals or the intensity of stochastic disturbances is less than the upper bound derived in this paper. A number of numerical cases are offered to support the availability of conjectural values.

Robustness analysis of fuzzy BAM cellular neural network with time-varying delays and stochastic disturbances

&lt;abstract&gt;&lt;p&gt;Robustness analysis for the global exponential stability of fuzzy bidirectional associative memory cellular neural network (FBAMCNN) is explored in this paper. By applying Gronwall-Bellman lemma and other inequality techniques, the range limits of both time-varying delays and the intensity of noise that FBAMCNN can withstand to maintain globally exponentially stable is estimated. It means that if the intensities of interference are larger than the bounds we derived, then the perturbed system may lose global exponential stability. Several instances are given to support our main results.&lt;/p&gt;&lt;/abstract&gt;

Robust Stability Analysis and Feedback Control for Uncertain Systems With Time-Delay and External Disturbance

This article addresses the delay-dependent Takagi–Sugeno (T–S) fuzzy state feedback control and exponential admissibility analysis for a class of T–S fuzzy singular uncertain systems. First, the T–S fuzzy model is employed to approximate the singular uncertain system with time-varying delay, saturation input, and unmatched disturbance. Second, the delay-dependent T–S fuzzy state feedback controller is designed by employing the T–S fuzzy model. Third, the free-weighting matrices and delay-dependent Lyapunov–Krasovskii functional with multiple integral terms are employed to derive the delay-dependent exponential admissibility conditions and prescribed H-infinity performance is guaranteed. Compared with previous works, the delay-dependent T–S fuzzy state feedback controller is designed for the T–S fuzzy singular uncertain system to relax system design conditions. The convex hull lemma is employed to convert the closed-loop system with saturation input into the closed-loop system without saturation input to enhance controller design flexibility. The Schur complement lemma and Gronwall Bellman lemma are employed to derive the less conservative delay-dependent stability conditions for determining controller gain matrices. The exact invariant set with less conservativeness is employed to convert the controller design problem into linear matrix inequalities (LMIs) optimization constraints to reduce computation complexity of solving LMIs. Finally, simulation examples are presented to show the effectiveness of the proposed methods.

Robust H∞ output feedback control of uncertain fractional-order systems subject to input saturation

This article considers the problems of robust stability and stabilization for a class of fractional-order linear time-invariant systems via 0&lt;α&lt;1 subject to input saturation with convex polytopic uncertainties, by a kind of [Formula: see text]–proportional–integral–derivative control synthesis. A new saturation-based stability condition is suggested to estimate the stable region ([Formula: see text]) and region of attraction (ε) using the ellipsoid approach. The main concept of the presented strategy is based on transforming the proportional–integral–derivative controller design problem into the static output feedback control problem. Gronwall–Bellman lemma and sector bounded condition of the saturation function is used for stability investigation and stabilization of such systems. The static output feedback control laws are obtained with the aid of a non-iterative linear matrix inequality implication. From the static output feedback controller, the proportional–integral–derivative controller is then restored. Stability problem is addressed as an optimization problem to obtain the controller gains and the stable region, as well as the largest region of attraction. Some theorems and numerical examples are provided to illustrate the results of proposed mechanism.

Lyapunov‐based iterative learning feedback control design for parabolic MIMO PDEs with time‐varying delays

This paper presents an iterative learning control approach to achieve precise tracking control for a class of repeatable parabolic multiple-input–multiple-output (MIMO) partial differential equations (PDEs) with time-varying delays over a finite time interval. Feedback control is utilised in the iterative learning control system to improve the convergence speed of the repeatable system. A Lyapunov–Krasovskii functional is constructed to deal with the time-varying delay problem caused by the model uncertainty of MIMO PDEs. Collocated piecewise control and piecewise measurement is considered based on the distributions of actuators and sensors to generate multiple inputs and multiple outputs. By using the mean value theorem for integrals, variants of the Poincaré–Wirtinger inequality in the 1D space, Cauchy–Schwartz inequality and Gronwall–Bellman lemma, sufficient conditions that guarantee the convergence of the iterative learning feedback control system are presented. Numerical simulation experiments and comparisons verify the effectiveness and advantages of the proposed method.

Robust Stability Analysis and Feedback Control for Uncertain Systems with Time-Delay and External Disturbance

This paper addresses the delay-dependent Takagi-Sugeno (T-S) fuzzy state feedback control and exponential admissibility analysis for a class of T-S fuzzy singular uncertain systems. Firstly, the T-S fuzzy model is employed to approximate the singular uncertain system with time-varying delay, saturation input and unmatched disturbance. Secondly, the delay-dependent T-S fuzzy state feedback controller is designed by employing the T-S fuzzy model. Thirdly, the free-weighting matrices and delay-dependent Lyapunov-Krasovskii functional with multiple integral terms are employed to derive the delay-dependent exponential admissibility conditions and prescribed H-infinity performance is guaranteed. Compared with previous works, the delay-dependent T-S fuzzy state feedback controller is designed for the T-S fuzzy singular uncertain system to relax system design conditions. The convex hull lemma is employed to convert the closed-loop system with saturation input into the closed-loop system without saturation input to enhance controller design flexibility. The Schur complement lemma and Gronwall Bellman lemma are employed to derive the less conservative delay-dependent stability conditions for determining controller gain matrices. The exact invariant set with less conservativeness is employed to convert the controller design problem into linear matrix inequalities (LMIs) optimization constraints to reduce computation complexity of solving LMIs. Finally, simulation examples are presented to show the effectiveness of the proposed methods.

$ L_{\infty} $-norm minimum distance estimation for stochastic differential equations driven by small fractional Lévy noise

&lt;abstract&gt;&lt;p&gt;This paper is concerned with $ L_{\infty} $-norm minimum distance estimation for stochastic differential equations driven by small fractional Lévy noise. By applying the Gronwall-Bellman lemma, Chebyshev's inequality and Taylor's formula, the minimum distance estimator is established and the consistency and asymptotic distribution of the estimator are derived when a small dispersion coefficient $ \varepsilon\rightarrow 0 $.&lt;/p&gt;&lt;/abstract&gt;

Convergence analysis of the hp-version spectral collocation method for a class of nonlinear variable-order fractional differential equations

In this paper, a general class of nonlinear initial value problems involving a Riemann-Liouville fractional derivative and a variable-order fractional derivative is investigated. An existence result of the exact solution is established by using Weissinger's fixed point theorem and Gronwall-Bellman lemma. An hp-version spectral collocation method is presented to solve the problem in numerical frames. The collocation method employs the Legendre-Gauss interpolations to conquer the influence of the nonlinear term and variable-order fractional derivative. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The error estimates under the H1-norm for smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived. Numerical results are given to support the theoretical conclusions.

On boundedness and projective synchronization of distributed order neural networks

In this work, we introduced the distributed-order neural networks (DONNs) which are the generalization of integer and fractional orders neural networks. We presented and proved two theorems for bounded solutions and solutions that approach zero of these networks. The Gronwall–Bellman lemma, the asymptotical expansion of the generalized Mittag–Leffler function and Laplace transform are used to prove these theorems. We derived analytically the condition under which the solution of this network (DONN) is bounded. The active control and Lyapunov direct methods are applied to study the projective synchronization between two different chaotic DONNs. The analytical control functions are derived to achieve our synchronization. Two different examples of DONNs are given to test the validity of the analytical results of our theorems. The projective synchronization is investigated. Numerical simulations are implemented to show the agreement between both analytical and numerical results.

A spectral collocation method for nonlinear fractional initial value problems with a variable-order fractional derivative

In this paper, we investigate the initial value problems for a class of nonlinear fractional differential equations involving the variable-order fractional derivative. Our goal is to construct the spectral collocation scheme for the problem and carry out a rigorous error analysis of the proposed method. To reach this target, we first show that the variable-order fractional calculus of non constant functions does not have the properties like the constant order calculus. Second, we study the existence and uniqueness of exact solution for the problem using Banach’s fixed-point theorem and the Gronwall–Bellman lemma. Third, we employ the Legendre–Gauss and Jacobi–Gauss interpolations to conquer the influence of the nonlinear term and the variable-order fractional derivative. Accordingly, we construct the spectral collocation scheme and design the algorithm. We also establish priori error estimates for the proposed scheme in the function spaces $$L^{2}[0,1]$$ and $$L^{\infty }[0,1]$$ . Finally, numerical results are given to support the theoretical conclusions.

Robust stability and stabilization of uncertain fractional order systems subject to input saturation

This paper studies the stability and the stabilization for a class of uncertain fractional order (FO) systems subject to input saturation. The Lipschitz condition and the Gronwall–Bellman lemma are adopted and sufficient conditions are derived to stabilize systems by designing a state feedback controller. Numerical examples demonstrate the effectiveness of the proposed method.

Second-order P-type iterative learning control for fractional order non-linear time-delay systems

This paper presents a second-order P-type iterative learning control (ILC) scheme for a class of fractional-order non-linear time-delay systems with fractional order α (0 &le; α < 1). First, by analysing the control and learning processes, a discrete system for P-type ILC is established and the ILC design problem is then converted to a stabilisation problem for such a discrete system. Next, by introducing the (Λ, ξ)-norm and using a generalised Gronwall-Bellman lemma, the sufficient condition for the convergence of the control input and the tracking errors is obtained. Finally, the validity of the method is verified by a numerical example.

Second-order P-type iterative learning control for fractional order non-linear time-delay systems

This paper presents a second-order P-type iterative learning control ILC scheme for a class of fractional-order non-linear time-delay systems with fractional order α 0 ≤ α < 1. First, by analysing the control and learning processes, a discrete system for P-type ILC is established and the ILC design problem is then converted to a stabilisation problem for such a discrete system. Next, by introducing the Λ, ξ-norm and using a generalised Gronwall-Bellman lemma, the sufficient condition for the convergence of the control input and the tracking errors is obtained. Finally, the validity of the method is verified by a numerical example.

Stability analysis of a class of fractional order nonlinear systems with order lying in (0, 2)

This paper investigates the stability of n-dimensional fractional order nonlinear systems with commensurate order 0 <α<2. By using the Mittag-Leffler function, Laplace transform and the Gronwall–Bellman lemma, one sufficient condition is attained for the local asymptotical stability of a class of fractional order nonlinear systems with order lying in (0, 2). According to this theory, stabilizing a class of fractional order nonlinear systems only need a linear state feedback controller. Simulation results demonstrate the effectiveness of the proposed theory.