Problem Of Exponential Stability Research Articles

Exponential Stabilization of Takagi–Sugeno Fuzzy Systems With Aperiodic Sampling: An Aperiodic Adaptive Event-Triggered Method

In this paper, we study the exponential stabilization problem for continuous-time Takagi–Sugeno fuzzy systems subject to aperiodic sampling. By aiming to transmission reduction, an appropriate aperiodic event-triggered communication scheme with adaptive mechanism is put forward, which covers the existing periodic mechanisms as special cases. For the sake of reduction in design conservativeness, both the available information of sampling behavior and threshold error are fully acquired by constructing a novel time-dependent Lyapunov functional. Then, a new exponential stability criterion is presented to establish the quantitative relationship among the adaptive adjusted event threshold, the decay rate, the upper bound, and the lower bound of variable sampling period, simultaneously. By resorting to a matrix transformation, the corresponding stabilization criterion is further derived by which the sampled-data controller can be obtained. Finally, two illustrative examples are provided to demonstrate the virtue and applicability of proposed design method.

Sampled-data fuzzy control for a class of nonlinear parabolic distributed parameter systems under spatially point measurements

In this paper, the exponential stabilization problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems via sampled-data fuzzy control approach. Initially, the nonlinear PDE system is accurately represented by the Takagi–Sugeno (T–S) fuzzy PDE model. Then, based on the T–S fuzzy PDE model, a novel time-dependent Lyapunov functional is used to design a sampled-data fuzzy controller under spatially point measurements such that the closed-loop fuzzy PDE system is exponentially stable with a given decay rate. The stabilization conditions are presented in terms of a set of linear matrix inequalities (LMIs). Finally, simulation results on the control of the diffusion equation and the FitzHugh–Nagumo (FHN) equation to illustrate the effectiveness of the proposed design method.

On exponential stabilization of nonholonomic systems with time-varying drift

A class of nonlinear control-affine systems with bounded time-varying drift is considered. It is assumed that the control vector fields together with their iterated Lie brackets satisfy Hörmander’s condition in a neighborhood of the origin. Then the problem of exponential stabilization is treated by exploiting periodic time-varying feedback controls. An explicit parametrization of such controllers is proposed under a suitable non-resonance assumption. It is shown that these controllers ensure the exponential stability of the closed-loop system provided that the period is small enough. The proposed control design methodology is applied for the stabilization of an underwater vehicle model and a front-wheel drive car.

Exponential stability of big data in networked control systems for a class of uncertain time-delay and packet dropout

This paper studies the problem of exponential stability for the networked control system with uncertain time-delay and packet dropout. The controller gain, designed to get better result, is assumed to have additive and multiplicative gain variations. Supposing networked control systems have uncertain time-delay which is not more than one sampling period and has packet dropout. Using the Lyapunov theory and linear matrix inequality formulation, we could obtain the sufficient condition of the asynchronous dynamical system for all admissible uncertainties and packet dropout. Finally, a simulation example illustrates the effectiveness of the approach.

Lagrange Exponential Stability of Complex-Valued BAM Neural Networks With Time-Varying Delays

This paper is concerned with the Lagrange exponential stability problem of complex-valued bidirectional associative memory neural networks with time-varying delays. On the basis of activation functions satisfying different assumption conditions, by combining the Lyapunov function approach with some inequalities techniques, different sufficient criteria including algebraic conditions and the condition in terms of LMI are derived to guarantee Lagrange exponential stability of the addressed system, respectively. Moreover, the estimations of different globally attractive sets named the convergence balls are also provided. In the end, the effectiveness and superiority–inferiority of these different results are verified by illustrative examples.

Mean Square Exponential Stability of Stochastic Complex‐Valued Neural Networks with Mixed Delays

This paper investigates the mean square exponential stability problem of a class of complex‐valued neural networks with stochastic disturbance and mixed delays including both time‐varying delays and continuously distributed delays. Under different assumption conditions concerning stochastic disturbance term from the existing ones, some sufficient conditions are derived for assuring the mean square exponential stability of the equilibrium point of the system based on the vector Lyapunov function method and differential‐integral theorem. The obtained results not only generalize the existing ones, but also reduce the conservatism of the previous stability results about complex‐valued neural networks with stochastic disturbances. Two numerical examples with simulation results are given to verify the feasibility of the proposed results.

Notice of Violation of IEEE Publication Principles: An Improved Result on Exponential Stabilization of Sampled-Data Fuzzy Systems

This brief paper studies the exponential stabilization problem for the Takagi–Sugeno fuzzy systems with a variable sampling. Different from previous results, the gains of fuzzy state feedback controller adopted in this paper are time-varying during two consecutive sampling instants, which can contribute to the enlargement of the allowable sampling interval by choosing a suitable design parameter. To reduce the design conservativeness, a novel fuzzy time-dependent Lyapunov functional (FTDLF) is put forward to fully exploit the accessible information about the sampling pattern and the fuzzy basis functions. Moreover, a more relaxed constraint condition is presented to ensure the positive definiteness of the FTDLF on sampling intervals. By resorting to the novel FTDLF and the relaxed constraint condition, new exponential stabilization criteria dependent on and independent of upper bounds on time derivatives of fuzzy basis functions are established, by which a larger sampling interval can be achieved. One example is offered to demonstrate the validity and superiorities of the obtained new results.

Dynamics of interval Cohen‐Grossberg neural networks with time‐varying delays based on LMI computation

SummaryThe problem of the global robust exponential stability of delayed interval Cohen‐Grossberg neural networks is considered. By constructing suitable Lyapunov functional and using the linear matrix inequality (LMI) technique, some sufficient conditions are derived to ensure the existence, uniqueness and robust exponential stability of the equilibrium point. Based on LMI computation, numerical examples are employed to show that the new results are less restrictive and less conservative than some existing results in the literature.

A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope

The exponential stability problem of the nonlinear Saint-Venant equations is addressed in this paper. We consider the general case where an arbitrary friction and space-varying slope are both included in the system, which lead to non-uniform steady-states. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the steady-states is constructed. Then we show that by a suitable choice of boundary feedback controls, that we give explicitly, the local exponential stability of the nonlinear Saint-Venant equations for the H2-norm is guaranteed.

Sufficient conditions for uniform exponential stability and [formula omitted]-stability of some classes of dynamic equations on arbitrary time scales

This paper investigates the exponential stability and h-stability problems of some classes of nonlinear systems. First, we analyze the global uniform exponential stability for a class of integro-differential equations on time scales. We also derive some sufficient conditions for stability using an integral inequality approach. Then, we give an analysis of the h-stability of some classes of linear systems under Lipschitz-type disturbances. To this end, some h-stability criteria are established. Finally, numerical examples are proposed to illustrate the stability concepts.

Exponential Lagrange stability for impulses in discrete-time delayed recurrent neural networks

ABSTRACTThis paper focuses on the problem of exponential stability in the sense of Lagrange for impulses in discrete-time delayed recurrent neural networks. By establishing a delayed impulsive discrete inequality and a novel difference inequality, combining with inequality techniques, some novel sufficient conditions are obtained to ensure exponential Lagrange stability for impulses in discrete-time delayed recurrent neural networks. Meanwhile, exponentially convergent scope of neural network is given. Finally, several numerical simulations are given to demonstrate the effectiveness of our results.

Robust exponential stability of switched delay interconnected systems under arbitrary switching

The problem of robust exponential stability for a class of switched nonlinear dynamical systems with uncertainties and unbounded delay is addressed. On the assumption that the interconnected functions of the studied systems satisfy the Lipschitz condition, by resorting to vector Lyapunov approach and M-matrix theory, the sufficient conditions to ensure the robust exponential stability of the switched interconnected systems under arbitrary switching are obtained. The proposed method, which neither require the individual subsystems to share a Common Lyapunov Function (CLF), nor need to involve the values of individual Lyapunov functions at each switching time, provide a new way of thinking to study the stability of arbitrary switching. In addition, the proposed criteria are explicit, and it is convenient for practical applications. Finally, two numerical examples are given to illustrate the correctness and effectiveness of the proposed theories.

Exponential Stabilization of Stochastic Memristive Recurrent Neural Networks Under Periodically Intermittent State Feedback Control

AbstractIn this paper, the exponential stabilization problem is investigated for a class of memristive time‐varying delayed neural networks with stochastic disturbance via periodically intermittent state feedback control. First, a periodically intermittent state feedback control rule is designed for the exponential stabilization of stochastic memristive time‐varying delayed neural networks. Then, by adopting appropriate Lyapunov‐Krasovskii functionals in light of the Lyapunov stability theory, some novel stabilization criteria are obtained to guarantee exponential stabilization of stochastic memristive time‐varying delayed neural networks via periodically intermittent state feedback control. Compared with existing results on stabilization of stochastic memristive time‐varying delayed neural networks, the obtained stabilization criteria in this paper are not difficult to verify. Finally, an illustrative example is given to illustrate the validity of the obtained results.

Mean square exponential stabilization of sampled‐data Markovian jump systems

SummaryIn this paper, the problem of mean square exponential stabilization for sampled‐data Markovin jump systems is studied. A time‐scheduled Lyapunov functional consisting of a exponential‐type looped function is constructed using segmentation technology and linear interpolation. Based on this new Lyapunov functional, a less conservative mean square exponential stability criterion is obtained such that a bigger maximum decay rate can be easily calculated. Meanwhile, the quantitative relationship among some system parameters, maximum sampling period, and decay rate is established. Moreover, a time‐dependent state feedback sample‐data controller is designed. Significant improvements of the proposed exponential‐type time‐scheduled Lyapunov functional method over some existing ones are verified by numerical examples.

Robust Exponential Stabilization for Switched Neutral Neural Networks with Mixed Time-Varying Delays

This paper studies the problems of exponential stabilization for a class of uncertain switched neutral neural networks with mixed time-varying delays. Based on the multiple Lyapunov-like functional method and the average dwell time method, the sufficient conditions which guarantee exponential stabilization of the uncertain switched neutral neural networks with mixed time-varying delays are presented. Averaged well time of switching signals is also given. Moreover, a design scheme for the stabilizing feedback controllers is proposed to guarantee exponential stability of corresponding closed-loop systems. Finally, two examples are given to illustrate the applicability and the effectiveness of the proposed method.

Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semilinear Parabolic PDE Systems

This paper employs a Takagi–Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial $L^\infty$ norm $\Vert \cdot \Vert _\infty$ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov–Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincare–Wirtinger inequality in one-dimensional spatial domain, as well as a vector-valued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of $\Vert \cdot \Vert _\infty$ , if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples.

Stabilization of Nonlinear Fornasini–Marchesini Systems

This paper considers the 2D systems described by the Fornasini–Marchesini statespace model. Direct and converse theorems on the exponential stability of such systems are proved in terms of vector Lyapunov functions. The concepts of exponential passivity and a vector storage function are introduced for solving exponential stabilization problems. An example is given to illustrate the efficiency of the new results.

Discrete-time feedback stabilization for hybrid neutral stochastic systems

In this paper, we study the exponential stabilization problem in [Formula: see text]th moment of hybrid neutral stochastic systems by discrete-time feedback controls. By extending Mao’s method, we first derive an upper bound of state observation duration of the corresponding systems with continuous-time state observation feedback controls. We further establish exponential stabilization conditions in [Formula: see text]th moment for the hybrid neutral stochastic system. The obtained results have potential applications to stabilize practical hybrid systems with interconnected stochastic oscillators.

Exponential Stabilization for a Class of Nonlinear Switched Systems with Mixed Delays under Asynchronous Switching

This paper deals with the exponential stabilization problem for a class of nonlinear switched systems with mixed delays under asynchronous switching. The switching signal of the switched controller involves delay, which results in the asynchronous switching between the candidate controllers and subsystems. By constructing the parameter-dependent Lyapunov-Krasovskii functional and the average dwell time approach, some sufficient conditions in forms of linear matrix inequalities are presented to ensure the exponential stability of the switched nonlinear system under arbitrary switching signals. In addition, through the special deformation of the matrix and Schur complement, the controllers with asynchronous switching are designed. Finally, a numerical example and a practical example of river pollution control are provided to show the validity and potential of the developed results.

Robust H∞ control for uncertain switched nonlinear polynomial systems: Parameterization of controller approach

SummaryThis paper considers the global feedback exponential stabilization and L2 gain disturbance attenuation problems of the switched nonlinear polynomial systems with passive and nonpassive subsystems for any given average dwell time. In the existing result, it needs that there exists at least one open loop passive subsystem in the switched nonlinear system, which is unnecessary for the switched nonlinear polynomial system in this paper because the passivity of the subsystem can be obtained according to the passivity‐based feedback dissipative Hamiltonian realization. In addition, a parameterization of controller approach–based feedback exponential stabilization method of nonlinear polynomial system is described and utilized such that the Lyapunov function of the passive subsystem possesses the required decay rate to exponentially stabilize the switched nonlinear polynomial system. Comparing with the controller designed with the existing method, the obtained controller in this paper has much simpler structure and better performance. An example simulation is provided to illustrate the effectiveness of the proposed approach.