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Showing papers on "Lyapunov function published in 2016"


Book
22 Sep 2016
TL;DR: Sampled-data and networked control systems: a time-delay approach and Lyapunov-based stability analysis.
Abstract: Introduction.- Linear time-delay systems.- Lyapunov-based stability analysis.- Performance analysis of time-delay systems.- Control design for time-delay systems.- Discrete-time delay systems.- Sampled-data and networked control systems: a time-delay approach.

828 citations


Journal ArticleDOI
01 Mar 2016
TL;DR: In this article, an adaptive impedance controller for a robotic manipulator with input saturation was developed by employing neural networks. But the adaptive impedance control was not considered in the tracking control design, and the input saturation is handled by designing an auxiliary system.
Abstract: In this paper, adaptive impedance control is developed for an ${n}$ -link robotic manipulator with input saturation by employing neural networks. Both uncertainties and input saturation are considered in the tracking control design. In order to approximate the system uncertainties, we introduce a radial basis function neural network controller, and the input saturation is handled by designing an auxiliary system. By using Lyapunov’s method, we design adaptive neural impedance controllers. Both state and output feedbacks are constructed. To verify the proposed control, extensive simulations are conducted.

685 citations


Journal ArticleDOI
TL;DR: An adaptive control technique is developed for a class of uncertain nonlinear parametric systems and it is proved that all the signals in the closed-loop system are global uniformly bounded and the tracking error is remained in a bounded compact set.

676 citations


Journal ArticleDOI
TL;DR: Cooperative control laws are proposed and the integral-barrier Lyapunov functions are employed for stability analysis of the closed-loop system and Adaption laws are developed for handling parametric uncertainties.

496 citations


Journal ArticleDOI
Zongyu Zuo1, Lin Tie1
TL;DR: It is shown that the finite settling time of the proposed general framework for robust consensus design is upper bounded for any initial condition, which makes it possible for network consensus problems to design and estimate the convergence time offline for a multi-agent team with a given undirected information flow.
Abstract: This paper investigates the robust finite-time consensus problem of multi-agent systems in networks with undirected topology. Global nonlinear consensus protocols augmented with a variable structure are constructed with the aid of Lyapunov functions for each single-integrator agent dynamics in the presence of external disturbances. In particular, it is shown that the finite settling time of the proposed general framework for robust consensus design is upper bounded for any initial condition. This makes it possible for network consensus problems to design and estimate the convergence time offline for a multi-agent team with a given undirected information flow. Finally, simulation results are presented to demonstrate the performance and effectiveness of our finite-time protocols.

496 citations


Journal ArticleDOI
01 Jan 2016
TL;DR: An adaptive fuzzy decentralized output-feedback tracking control approach is developed for the switched subsystems and the stability of the whole closed-loop system is proved by using the Lyapunov function and the average dwell-time methods.
Abstract: In this paper, the problem of adaptive fuzzy decentralized output-feedback control design is investigated for a class of switched nonlinear large-scale systems in strict-feedback form. The considered nonlinear large-scale systems contain the unknown nonlinearities and dead zones, the switching signals with average dwell time, and without the direct requirement of the states being available for feedback. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, a fuzzy switched decentralized state observer is designed and thus via it the immeasurable states are obtained. By applying the adaptive decentralized backstepping design technique, an adaptive fuzzy decentralized output-feedback tracking control approach is developed for the switched subsystems. The stability of the whole closed-loop system is proved by using the Lyapunov function and the average dwell-time methods. Satisfactory tracking performance is achieved under the switching signals with average dwell time. The simulation example is provided to indicate the effectiveness of the proposed control method.

473 citations


Journal ArticleDOI
TL;DR: In order to stabilize a class of uncertain nonlinear strict-feedback systems with full-state constraints, an adaptive neural network control method is investigated and it is proved that all the signals in the closed-loop system are semiglobal uniformly ultimately bounded and the output is well driven to follow the desired output.
Abstract: In order to stabilize a class of uncertain nonlinear strict-feedback systems with full-state constraints, an adaptive neural network control method is investigated in this paper. The state constraints are frequently emerged in the real-life plants and how to avoid the violation of state constraints is an important task. By introducing a barrier Lyapunov function (BLF) to every step in a backstepping procedure, a novel adaptive backstepping design is well developed to ensure that the full-state constraints are not violated. At the same time, one remarkable feature is that the minimal learning parameters are employed in BLF backstepping design. By making use of Lyapunov analysis, we can prove that all the signals in the closed-loop system are semiglobal uniformly ultimately bounded and the output is well driven to follow the desired output. Finally, a simulation is given to verify the effectiveness of the method.

404 citations


Book
09 Aug 2016
TL;DR: In this paper, the authors explore the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations and demonstrate the use of Lyapunov functions in this type of analysis.
Abstract: This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control. Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

393 citations


Journal ArticleDOI
TL;DR: This paper investigates an adaptive fuzzy tracking control design problem for single-input and single-output uncertain nonstrict feedback nonlinear systems and proposes both adaptive fuzzy state feedback and observer-based output feedback control designs.
Abstract: This paper investigates an adaptive fuzzy tracking control design problem for single-input and single-output uncertain nonstrict feedback nonlinear systems. For the cases of the states measurable and the states immeasurable, fuzzy logic systems are separately adopted to approximate the unknown nonlinear functions or model the uncertain nonlinear systems. In the unified framework of adaptive backstepping control design, both adaptive fuzzy state feedback and observer-based output feedback control design schemes are proposed. The stability of the closed-loop systems is proved by using Lyapunov function theory. The simulation examples are provided to confirm the effectiveness of the proposed control methods.

382 citations


Journal ArticleDOI
TL;DR: An adaptive fuzzy optimal control design is addressed for a class of unknown nonlinear discrete-time systems that contain unknown functions and nonsymmetric dead-zone and can be proved based on the difference Lyapunov function method.
Abstract: In this paper, an adaptive fuzzy optimal control design is addressed for a class of unknown nonlinear discrete-time systems. The controlled systems are in a strict-feedback frame and contain unknown functions and nonsymmetric dead-zone. For this class of systems, the control objective is to design a controller, which not only guarantees the stability of the systems, but achieves the optimal control performance as well. This immediately brings about the difficulties in the controller design. To this end, the fuzzy logic systems are employed to approximate the unknown functions in the systems. Based on the utility functions and the critic designs, and by applying the backsteppping design technique, a reinforcement learning algorithm is used to develop an optimal control signal. The adaptation auxiliary signal for unknown dead-zone parameters is established to compensate for the effect of nonsymmetric dead-zone on the control performance, and the updating laws are obtained based on the gradient descent rule. The stability of the control systems can be proved based on the difference Lyapunov function method. The feasibility of the proposed control approach is further demonstrated via two simulation examples.

366 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present a methodology that allows safety conditions ( expressed as control barrier functions) to be unified with performance objectives (represented as control Lyapunov functions) in the context of real-time optimization-based controllers.
Abstract: Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions -- expressed as control barrier functions -- to be unified with performance objectives -- expressed as control Lyapunov functions -- in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds.

Journal ArticleDOI
TL;DR: It is shown that under the proposed control method, despite the presence of actuator faults and system uncertainties, the formation tracking errors can converge into arbitrarily small neighborhoods around zero in finite time, while the constraint requirements on the LOS range and angle will not be violated.

Journal ArticleDOI
01 Feb 2016
TL;DR: A disturbance observer-based adaptive neural fault-tolerant control scheme is developed to track the desired system output in the presence of system uncertainty, external disturbance, and actuator faults for the three degrees of freedom model helicopter.
Abstract: In this paper, an adaptive neural fault-tolerant control scheme is proposed for the three degrees of freedom model helicopter, subject to system uncertainties, unknown external disturbances, and actuator faults. To tackle system uncertainty and nonlinear actuator faults, a neural network disturbance observer is developed based on the radial basis function neural network. The unknown external disturbance and the unknown neural network approximation errors are treated as a compound disturbance that is estimated by another nonlinear disturbance observer. A disturbance observer-based adaptive neural fault-tolerant control scheme is then developed to track the desired system output in the presence of system uncertainty, external disturbance, and actuator faults. The stability of the whole closed-loop system is analyzed using the Lyapunov method, which guarantees the convergence of all closed-loop signals. Finally, the simulation results are presented to illustrate the effectiveness of the new control design techniques.

Journal ArticleDOI
TL;DR: This paper proposes fully distributed consensus algorithms over a general directed graph when there exist, respectively, absolute velocity damping and relative velocity damped and shows that one proposed algorithm also works for consensus of agents with intrinsic Lipschitz nonlinear dynamics.
Abstract: In this paper, we study the consensus problem for second-order multi-agent systems with heterogeneous unknown inertias and control gains under a general directed graph. Unlike the existing consensus algorithms for second-order multi-agent systems in which all agents are assumed to have common unit inertias or share common control gains, we allow the inertias and the control gains to be heterogeneous and time-varying for each agent. We propose fully distributed consensus algorithms over a general directed graph when there exist, respectively, absolute velocity damping and relative velocity damping. Novel integral-type Lyapunov functions are proposed to study the consensus convergence. Moreover, the adaptive $\sigma$ - modification schemes for the gain adaptation are proposed, which renders smaller control gains and thus requires smaller amplitude on the control input without sacrificing consensus convergence. Furthermore, we show that one proposed algorithm also works for consensus of agents with intrinsic Lipschitz nonlinear dynamics. The control gains are varying and updated by distributed adaptive laws. As a result, the proposed algorithms require no global information and thus can be implemented in a fully distributed manner.

Journal ArticleDOI
TL;DR: The closed-loop stability of the adaptive control system is rigorously proved via Lyapunov analysis and the satisfactory tracking performance is achieved under the integrated effects of unknown dead zone, actuator fault, and unknown external disturbances.
Abstract: In this paper, an adaptive neural fault-tolerant control scheme is proposed and analyzed for a class of uncertain nonlinear large-scale systems with unknown dead zone and external disturbances. To tackle the unknown nonlinear interaction functions in the large-scale system, the radial basis function neural network (RBFNN) is employed to approximate them. To further handle the unknown approximation errors and the effects of the unknown dead zone and external disturbances, integrated as the compounded disturbances, the corresponding disturbance observers are developed for their estimations. Based on the outputs of the RBFNN and the disturbance observer, the adaptive neural fault-tolerant control scheme is designed for uncertain nonlinear large-scale systems by using a decentralized backstepping technique. The closed-loop stability of the adaptive control system is rigorously proved via Lyapunov analysis and the satisfactory tracking performance is achieved under the integrated effects of unknown dead zone, actuator fault, and unknown external disturbances. Simulation results of a mass–spring–damper system are given to illustrate the effectiveness of the proposed adaptive neural fault-tolerant control scheme for uncertain nonlinear large-scale systems.

Journal ArticleDOI
TL;DR: A novel nonlinear control method for solving the problem of stabilization with guaranteed safety for nonlinear systems based on the merging of the well-known Control Lyapunov Function and the recent concept of Control Barrier Function.

Journal ArticleDOI
TL;DR: It is demonstrated theoretically and numerically that the number of consecutive impulses with minimum impulsive interval of the desynchronizing impulsive sequence should not be too large.
Abstract: This note considers globally finite-time synchronization of coupled networks with Markovian topology and distributed impulsive effects. The impulses can be synchronizing or desynchronizing with certain average impulsive interval. By using $M$ -matrix technique and designing new Lyapunov functions and controllers, sufficient conditions are derived to ensure the synchronization within a setting time, and the conditions do not contain any uncertain parameter. It is demonstrated theoretically and numerically that the number of consecutive impulses with minimum impulsive interval of the desynchronizing impulsive sequence should not be too large. It is interesting to discover that the setting time is related to initial values of both the network and the Markov chain. Numerical simulations are provided to illustrate the effectiveness of the theoretical analysis.

Journal ArticleDOI
TL;DR: F fuzzy logic system is introduced to approximate the unknown nonlinear dynamics, and adaptive high-gain observer is designed to estimate the unmeasured states and it is proved that all the signals in the multiagent systems are semiglobally uniformly ultimately bounded.
Abstract: In this paper, the consensus tracking control problem of second-order multiagent systems with unknown nonlinear dynamics, immeasurable states, and disturbances is investigated. The nonlinear dynamics in multiagent systems do not satisfy the matched condition. In this paper, fuzzy logic system is introduced to approximate the unknown nonlinear dynamics, and adaptive high-gain observer is designed to estimate the unmeasured states. Based on backstepping approach and Lyapunov theory, a new adaptive fuzzy distributed controller is proposed for each agent only using the information of itself and its neighbors. Then the consensus tracking is achieved under the designed distributed controller. Moreover, it is proved that all the signals in the multiagent systems are semiglobally uniformly ultimately bounded, and the consensus tracking error converges to a small neighborhood of the origin that can be designed as small as possible. Finally, the simulation result illustrates the effectiveness of the designed controller.

Journal ArticleDOI
TL;DR: By computing the value of the Lyapunov function at the initial point, the finite settling time can be theoretically estimated for second-order multi-agent systems with the proposed control protocols.

Journal ArticleDOI
TL;DR: The main purpose of this brief is to design a filter to guarantee that the augmented Markovian jump fuzzy neural networks are stable in mean-square sense and satisfy a prescribed passivity performance index by employing the Lyapunov method and the stochastic analysis technique.
Abstract: In this brief, the problems of the mixed H-infinity and passivity performance analysis and design are investigated for discrete time-delay neural networks with Markovian jump parameters represented by Takagi–Sugeno fuzzy model. The main purpose of this brief is to design a filter to guarantee that the augmented Markovian jump fuzzy neural networks are stable in mean-square sense and satisfy a prescribed passivity performance index by employing the Lyapunov method and the stochastic analysis technique. Applying the matrix decomposition techniques, sufficient conditions are provided for the solvability of the problems, which can be formulated in terms of linear matrix inequalities. A numerical example is also presented to illustrate the effectiveness of the proposed techniques.

Posted Content
TL;DR: There is an equivalence between the technique of estimate sequences and a family of Lyapunov functions in both continuous and discrete time, which allows for a simple and unified analysis of many existing momentum algorithms.
Abstract: Momentum methods play a significant role in optimization. Examples include Nesterov's accelerated gradient method and the conditional gradient algorithm. Several momentum methods are provably optimal under standard oracle models, and all use a technique called estimate sequences to analyze their convergence properties. The technique of estimate sequences has long been considered difficult to understand, leading many researchers to generate alternative, "more intuitive" methods and analyses. We show there is an equivalence between the technique of estimate sequences and a family of Lyapunov functions in both continuous and discrete time. This connection allows us to develop a simple and unified analysis of many existing momentum algorithms, introduce several new algorithms, and strengthen the connection between algorithms and continuous-time dynamical systems.

Journal ArticleDOI
TL;DR: This paper presents a novel approximation-based event-triggered control of multi-input multi-output uncertain nonlinear continuous-time systems in affine form that is approximated using a linearly parameterized neural network in the context of event-based sampling.
Abstract: This paper presents a novel approximation-based event-triggered control of multi-input multi-output uncertain nonlinear continuous-time systems in affine form. The controller is approximated using a linearly parameterized neural network (NN) in the context of event-based sampling. After revisiting the NN approximation property in the context of event-based sampling, an event-triggered condition is proposed using the Lyapunov technique to reduce the network resource utilization and to generate the required number of events for the NN approximation. In addition, a novel weight update law for aperiodic tuning of the NN weights at triggered instants is proposed to relax the knowledge of complete system dynamics and to reduce the computation when compared with the traditional NN-based control. Nonetheless, a nonzero positive lower bound for the inter-event times is guaranteed to avoid the accumulation of events or Zeno behavior. For analyzing the stability, the event-triggered system is modeled as a nonlinear impulsive dynamical system and the Lyapunov technique is used to show local ultimate boundedness of all signals. Furthermore, in order to overcome the unnecessary triggered events when the system states are inside the ultimate bound, a dead-zone operator is used to reset the event-trigger errors to zero. Finally, the analytical design is substantiated with numerical results.

Journal ArticleDOI
TL;DR: The approach generalizes the idea of energy methods, and extends the concept of energy function to a more general Lyapunov functions family (LFF) constructed via semidefinite programming techniques, and shows that they can certify stability of a broader set of initial conditions in comparison to the energy function in the closest-UEP method.
Abstract: Analysis of transient stability of strongly nonlinear post-fault dynamics is one of the most computationally challenging parts of dynamic security assessment. This paper proposes a novel approach for assessment of transient stability of the system. The approach generalizes the idea of energy methods, and extends the concept of energy function to a more general Lyapunov functions family (LFF) constructed via semidefinite programming techniques. Unlike the traditional energy function and its variations, the constructed Lyapunov functions are proven to be decreasing only in a finite neighborhood of the equilibrium point. However, we show that they can still certify stability of a broader set of initial conditions in comparison to the energy function in the closest-UEP method. Moreover, the certificates of stability can be constructed via a sequence of convex optimization problems that are tractable even for large scale systems. We also propose specific algorithms for adaptation of the Lyapunov functions to specific initial conditions and demonstrate the effectiveness of the approach on a number of IEEE test cases.

Journal ArticleDOI
TL;DR: A class of upper-triangular LTV systems is carefully investigated based on the developed stability analysis approaches and a couple of numerical examples with some of them borrowed from the literature is provided to illustrate the effectiveness of the proposed theoretical results.

Journal ArticleDOI
TL;DR: Based on Lyapunov functions, Halanay inequality, and linear matrix inequalities, sufficient conditions that depend on the probability distribution of the delay coupling and the impulsive delay were obtained and numerical simulations are used to show the effectiveness of the theoretical results.
Abstract: This paper deals with the exponential synchronization of coupled stochastic memristor-based neural networks with probabilistic time-varying delay coupling and time-varying impulsive delay. There is one probabilistic transmittal delay in the delayed coupling that is translated by a Bernoulli stochastic variable satisfying a conditional probability distribution. The disturbance is described by a Wiener process. Based on Lyapunov functions, Halanay inequality, and linear matrix inequalities, sufficient conditions that depend on the probability distribution of the delay coupling and the impulsive delay were obtained. Numerical simulations are used to show the effectiveness of the theoretical results.

Journal ArticleDOI
TL;DR: This work proposes two new families of homogeneous HOSM controllers of a very simple form, one of which consists of quasi-continuous controllers, which can be done arbitrarily smooth everywhere outside of the HosM manifold.

Journal ArticleDOI
TL;DR: In this paper, an H-infinity control method for a platoon of heterogeneous vehicles with uncertain vehicle dynamics and uniform communication delay is presented, and the requirements of string stability, robustness and tracking performance are explicitly satisfied by casting into the linear fractional transformation format, and a delay-dependent linear matrix inequality is derived to numerically solve the distributed controllers for each vehicle.
Abstract: Platoon formation of highway vehicles has the potential to significantly enhance road safety, improve highway utility, and increase traffic efficiency. However, various uncertainties and disturbances that are present in real-world driving conditions make the implementation of vehicular platoon a challenging problem. This study presents an H-infinity control method for a platoon of heterogeneous vehicles with uncertain vehicle dynamics and uniform communication delay. The requirements of string stability, robustness and tracking performance are systematically measured by the H-infinity norm, and explicitly satisfied by casting into the linear fractional transformation format. A delay-dependent linear matrix inequality is derived to numerically solve the distributed controllers for each vehicle. The performances of the controlled platoon are theoretically analysed by using a delay-dependent Lyapunov function which includes a linear quadratic function of states during the delay period. Simulations with a platoon of heterogeneous vehicles are conducted to demonstrate the effectiveness of the proposed method under random parameters and external disturbances.

Journal ArticleDOI
TL;DR: A novel class of Lyapunov-like barrier functions is introduced and used to encode multiple, non-trivial control objectives, such as collision avoidance, proximity maintenance and convergence to desired destinations, based on recentered barrier functions and on maximum approximation functions.
Abstract: This paper addresses the problem of multi-agent coordination and control under multiple objectives, and presents a set-theoretic formulation amenable to Lyapunov-based analysis and control design. A novel class of Lyapunov-like barrier functions is introduced and used to encode multiple, non-trivial control objectives, such as collision avoidance, proximity maintenance and convergence to desired destinations. The construction is based on recentered barrier functions and on maximum approximation functions. Thus, a single Lyapunov-like function is used to encode the constrained set of each agent, yielding simple, gradient-based control solutions. The derived control strategies are distributed, i.e., based on information locally available to each agent, which is dictated by sensing and communication limitations. Furthermore, the proposed coordination protocol dictates semi-cooperative conflict resolution among agents, which can be also thought as prioritization, as well as conflict resolution with respect to an agent (the leader) which is not actively participating in collision avoidance, except when necessary. The considered scenario is pertinent to surveillance tasks and involves nonholonomic vehicles. The efficacy of the approach is demonstrated through simulation results.

Journal ArticleDOI
TL;DR: An optimal control method is developed for unknown continuous-time systems with unknown disturbances in this paper and it is proven that the weight errors are uniformly ultimately bounded based on Lyapunov techniques.
Abstract: An optimal control method is developed for unknown continuous-time systems with unknown disturbances in this paper. The integral reinforcement learning (IRL) algorithm is presented to obtain the iterative control. Off-policy learning is used to allow the dynamics to be completely unknown. Neural networks are used to construct critic and action networks. It is shown that if there are unknown disturbances, off-policy IRL may not converge or may be biased. For reducing the influence of unknown disturbances, a disturbances compensation controller is added. It is proven that the weight errors are uniformly ultimately bounded based on Lyapunov techniques. Convergence of the Hamiltonian function is also proven. The simulation study demonstrates the effectiveness of the proposed optimal control method for unknown systems with disturbances.

Journal ArticleDOI
TL;DR: The developed impulsive synchronization method is applied to build a spatiotemporal chaotic cryptosystem that can transmit an encrypted image and verify that the proposed image-encrypting cryptos system has the advantages of large key space and high security against some traditional attacks.
Abstract: This paper presents a new impulsive synchronization criterion of two identical reaction–diffusion neural networks with discrete and unbounded distributed delays. The new criterion is established by applying an impulse-time-dependent Lyapunov functional combined with the use of a new type of integral inequality for treating the reaction–diffusion terms. The impulse-time-dependent feature of the proposed Lyapunov functional can capture more hybrid dynamical behaviors of the impulsive reaction–diffusion neural networks than the conventional impulse-time-independent Lyapunov functions/functionals, while the new integral inequality, which is derived from Wirtinger’s inequality, overcomes the conservatism introduced by the integral inequality used in the previous results. Numerical examples demonstrate the effectiveness of the proposed method. Later, the developed impulsive synchronization method is applied to build a spatiotemporal chaotic cryptosystem that can transmit an encrypted image. The experimental results verify that the proposed image-encrypting cryptosystem has the advantages of large key space and high security against some traditional attacks.