Problem Of Exponential Stability Research Articles

Robust $\mathcal {H}_{\infty }$-Based Control for Uncertain Stochastic Fuzzy Switched Time-Delay Systems via Integral Sliding Mode Strategy

In this article, we investigate the problems of robust exponential stabilization in mean square and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> -based integral sliding mode controller design for uncertain stochastic Takagi–Sugeno (T-S) fuzzy switched time-delay systems with both matched and unmatched uncertainties under synchronous switching and asynchronous switching, respectively. In these systems, the control input is dependent on a switching signal. Under synchronous switching and asynchronous switching, sufficient conditions are developed to guarantee that the resulting closed-loop systems are robustly exponentially stable in mean square with a prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> -performance, respectively. The integral sliding mode surfaces and the sliding mode controllers are designed for the underlying systems under synchronous switching and asynchronous switching, respectively. By two examples stemming from the mass–spring–damper model and the single-link robot arm model under stochastic perturbation, the effectiveness of the results obtained is verified.

Exponential stabilisation for nonlinear PDE systems via sampled-data static output feedback control

ABSTRACT This paper focuses on the problem of exponential stabilisation for nonlinear partial differential equation (PDE) systems by designing sampled-data static output feedback controller. First, the nonlinear PDE systems are reconstructed by Takagi-Sugeno (T-S) fuzzy model. Subsequently, the closed-loop systems are obtained by introducing sampled-data static output feedback fuzzy controller. Then, stability conditions of the closed-loop system are obtained based on Lyapunov function approach, which can guarantee that the considered systems are exponential stabilisation and satisfy dissipative performance. Finally, the effectiveness of the developed method is demonstrated by a simulation study.

State-estimation-based adaptive sliding mode control of uncertain switched systems: A novel linear sliding manifold approach

A new sliding mode observer design scheme is developed with a novel linear sliding manifold for the exponential stabilization problem of a class of uncertain switched systems in this note. Firstly, the linear sliding manifold is constructed based on a modified observer, through which a new technical approach of the underlying system analysis is exploited to devise a novel exponential stability criteria. Moreover, an associated adaptive switching controller is presented, by which the arrival condition of the designated sliding manifold is fulfilled. Ultimately, illustrative examples are conducted to confirm the feasibility of the proposed scheme.

Global exponential stabilization for a class of nonlinear systems by output feedback control with three gain‐scaling factors

SummaryIn this article, we consider a problem of global exponential stabilization for a class of nonlinear systems with the perturbed nonlinearity by output feedback. We provide a newly designed observer‐based output feedback controller with three gain‐scaling factors to deal with extended structure of nonlinearity and larger growth rate of nonlinearity over the existing results. Utilizing three gain‐scaling factors, we have more flexibility in selecting control parameters. Moreover, we present two procedures for selecting gain‐scaling parameters which are associated with stabilizing the system and improving the convergence rate of the system states and observer errors, respectively. Via numerical and practical examples, we illustrate the improved features of the proposed control scheme over the existing schemes.

On Optimization Techniques for the Construction of an Exponential Estimate for Delayed Recurrent Neural Networks

This work is devoted to the modeling and investigation of the architecture design for the delayed recurrent neural network, based on the delayed differential equations. The usage of discrete and distributed delays makes it possible to model the calculation of the next states using internal memory, which corresponds to the artificial recurrent neural network architecture used in the field of deep learning. The problem of exponential stability of the models of recurrent neural networks with multiple discrete and distributed delays is considered. For this purpose, the direct method of stability research and the gradient descent method is used. The methods are used consequentially. Firstly we use the direct method in order to construct stability conditions (resulting in an exponential estimate), which include the tuple of positive definite matrices. Then we apply the optimization technique for these stability conditions (or of exponential estimate) with the help of a generalized gradient method with respect to this tuple of matrices. The exponential estimates are constructed on the basis of the Lyapunov–Krasovskii functional. An optimization method of improving estimates is offered, which is based on the notion of the generalized gradient of the convex function of the tuple of positive definite matrices. The search for the optimal exponential estimate is reduced to finding the saddle point of the Lagrange function.

Exponential stabilization of switched linear systems subject to actuator saturation with stabilizable and unstabilizable subsystems

This paper is concerned with the exponential stabilization of switched linear systems subject to actuator saturation with both stabilizable subsystems and unstabilizable subsystems for continuous-time case and discrete-time case, respectively. Sufficient conditions for the exponential stabilization under dwell time switching under the cases of continuous-time and discrete-time are established by using a novel class of multiple time-varying Lyapunov function. The existence conditions for stabilizing controllers are presented in terms of linear matrix inequalities (LMIs) for the continuous-time case and the discrete-time case, respectively. Two optimization problems are proposed for obtaining the maximal attraction region. The problem of exponential stabilization for switched system subject to actuator saturation with asynchronous switching controller is also studied. Several numerical examples are presented to prove the validity of the obtained results.

Global exponential stabilization of nonlinear systems via width time-dependent periodically intermittent smooth control

In this paper, the global exponential stability and stabilization problems for a class of nonlinear systems are investigated. Some sufficient conditions to guarantee global exponential stable and estimate the minimum admissible value of the control width are presented in virtue of time-dependent width Lyapunov functions. Furthermore, a periodically intermittent smooth controller with variant control width is introduced and theoretical analysis is provided. The smooth index function of periodically intermittent smooth control inputs is defined and the supremum (or least upper bound) of smooth index function set can be solved. On the basis of the analysis, the designed periodically intermittent smooth controller not only can globally exponentially stabilize the nonlinear systems, but also can control the exponential convergence rate of the nonlinear systems. Finally, numerical simulations are given to verify the obtained theoretical results.

Practical Exponential Stability of Impulsive Stochastic Reaction-Diffusion Systems With Delays.

This article studies the practical exponential stability of impulsive stochastic reaction-diffusion systems (ISRDSs) with delays. First, a direct approach and the Lyapunov method are developed to investigate the p th moment practical exponential stability and estimate the convergence rate. Note that these two methods can also be used to discuss the exponential stability of systems in certain conditions. Then, the practical stability results are successfully applied to the impulsive reaction-diffusion stochastic Hopfield neural networks (IRDSHNNs) with delays. By the illustration of four numerical examples and their simulations, our results in this article are proven to be effective in dealing with the problem of practical exponential stability of ISRDSs with delays, and may be regarded as stabilization results.

Almost Sure Stabilization of Continuous-Time Jump Linear Systems via a Stochastic Scheduled Controller.

This article addresses the almost surely exponential (ASE) stabilization problem of continuous-time jump systems realized by a stochastic scheduled controller. In this study, a stochastic scheduled controller based on the anytime algorithm is proposed. It is able to cope with the situation where no controller is added to subsystems during some time slices. Sufficient conditions for the existence of such a controller are established by applying novel techniques to its stochastic transfer matrix, and they are all presented with solvable forms. Particularly, both dwell times of the jump signal and distribution properties of stochastic scheduling are considered and proved to have played positive roles in obtaining better performance and applications. Two special situations about no jump systems with constant and varied dwell times are further studied, respectively. A practical example is offered so as to verify the effectiveness and superiority of the methods proposed in this study.

Exponential stabilization and non-fragile sampled-date dissipative control for uncertain time-varying delay T-S fuzzy systems with state quantization

In this paper, the problem of exponential dissipation stability of T-S fuzzy system with state quantization is studied by using non-fragile sampled-data control. First, A Lyapunov-Krasovskii function containing all sampled-data and quantization information is constructed. Second, the better results can be obtained by using the integral inequality and the Newton Leibniz formula. In addition, a dissipative controller for non-fragile sampled-data is designed. Finally, The reasonable examples illustrate the significant improvement and value of this paper.

Exponential stabilization of an ODE–linear KdV cascaded system with boundary input delay

Abstract This paper considers the well posedness and the exponential stabilization problems of a cascaded ordinary differential equation (ODE)–partial differential equation (PDE) system. The considered system is governed by a linear ODE and the one-dimensional linear Korteweg–de Vries (KdV) equation posed on a bounded interval. For the whole system, a control input delay acts on the left boundary of the KdV domain by Dirichlet condition. Whereas, the KdV acts back on the ODE by Dirichlet interconnection on the right boundary. Firstly, we reformulate the system in question as an undelayed ODE-coupled KdV-transport system. Secondly, we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a well-posed and exponentially stable target system. Finally, by invertibility of such design, we use semigroup theory and Lyapunov analysis to prove the well posedness and the exponential stabilization in a suitable functional space of the original plant, respectively.

On exponential stability of hybrid neutral stochastic differential delay equations with different structures

This article discusses the problem of exponential stability for a class of hybrid neutral stochastic differential delay equations with highly nonlinear coefficients and different structures in different switching modes. In such systems, the coefficients will satisfy the local Lipschitz condition and suitable Khasminskii-types conditions. The set of switching states will be divided into two subsets. In different subsets, the coefficients will be dominated by polynomials with different degrees. By virtue of M-matrices and suitable Lyapunov functions dependent on coefficient structures and switching modes, some results including the existence-and-uniqueness, boundedness and exponential stability of the solution are proposed and proved.

Exponential Stability of Fractional-Order Complex Multi-Links Networks With Aperiodically Intermittent Control.

In this article, the exponential stability problem for fractional-order complex multi-links networks with aperiodically intermittent control is considered. Using the graph theory and Lyapunov method, two theorems, including a Lyapunov-type theorem and a coefficient-type theorem, are given to ensure the exponential stability of the underlying networks. The theoretical results show that the exponential convergence rate is dependent on the control gain and the order of fractional derivative. To be specific, the larger control gain, the higher the exponential convergence rate. Meanwhile, when aperiodically intermittent control degenerates into periodically intermittent control, a corollary is also provided to ensure the exponential stability of the underlying networks. Furthermore, to show the practicality of theoretical results, as an application, the exponential stability of fractional-order multi-links competitive neural networks with aperiodically intermittent control is investigated and a stability criterion is established. Finally, the effectiveness and feasibility of the theoretical results are demonstrated through a numerical example.

Fuzzy-Dependent-Switching Control of Nonlinear Systems With Aperiodic Sampling

This article considers the exponential stability and aperiodic sampled-data control problem for nonlinear systems based on a class of Takagi&#x2013;Sugeno fuzzy models. The fuzzy-dependent-switching control strategy together with a novel time-varying sampled-data controller is proposed to deal with the exponential stabilization problem of such systems. Mixed-fuzzy dependent Lyapunov&#x2013;Krasovskii functionals (MFDLKFs), which fully make use of available characteristics of the sampling patterns, the signs and the upper bounds of the time derivative of fuzzy membership functions, are constructed for the purpose of reducing the design conservatism. Based on the proposed MFDLKFs, a novel exponential stabilization criterion for the fuzzy systems with aperiodic sampling is established in terms of linear matrix inequalities, which is less conservative and obtains a larger sampling interval compared with existing results. Finally, a simulation example is employed to demonstrate the effectiveness and superiority of the proposed fuzzy-dependent-switching control scheme.

Exponential stabilization for 1-D linear Itô-type state-dependent stochastic parabolic PDE systems via static output feedback

This paper deals with the problem of exponential stabilization for 1-D linear stochastic parabolic partial differential equation (PDE) systems with state-multiplicative noise in the form of Itô type. A static output feedback (SOF) control scheme is proposed to stabilize the stochastic PDE system in a stochastic framework via locally collocated piecewise uniform actuators and sensors. The closed-loop well-posedness analysis is provided by virtue of the semi-group theory. Based on the infinite-dimensional infinitesimal operator and Lyapunov technique, the SOF controller design is developed to guarantee the exponential stability of the resulting closed-loop system in the mean square sense. Finally, simulation results on a stochastic heat equation are presented to verify the effectiveness of the proposed design method.

Delay-dependent stability and hybrid [formula omitted]-gain analysis of linear impulsive time-delay systems: A continuous timer-dependent Lyapunov-like functional approach

This paper presents a new approach for analyzing delay-dependent stability and hybrid L2×l2-gain performance of linear impulsive delay systems. The new approach is inspired by the delay-partitioning method, the timer-dependent Lyapunov functional method, and the looped-functional method. In the delay-partitioning framework, a new type of timer-dependent Lyapunov functional is constructed, which depends on the partition on impulse intervals and also on impulse dynamics. Different from the previous discontinuous Lyapunov functionals, the introduced Lyapunov functional is continuous along the trajectories of the considered impulsive delay system. Consequently, two different problems of exponential stability and hybrid L2×l2-gain performance are tackled by using the same class of Lyapunov functionals. It is shown that the positive definiteness of this Lyapunov functional inside impulse intervals is not necessary for proving exponential stability. By use of new integral inequalities based techniques, delay-dependent criteria for exponential stability and finite hybrid L2×l2-gain are established in terms of linear matrix inequalities. Numerical examples are provided to illustrate the efficiency of the new approach.

Stochastic Exponential Stabilization for Markov Jump Neural Networks with Time-varying Delays via Adaptive Event-Triggered Impulsive Control

This paper focuses on the exponential stabilization problem for Markov jump neural networks with Time-varying Delays (TDs). Firstly, we provide a new Free-matrix-based Exponential-type Integral Inequality (FMEII) containing the information of attenuation exponent, which is helpful to reduce the conservativeness of stability criteria. To further save control cost, we introduce a sample-based Adaptive Event-triggered Impulsive Control (AEIC) scheme, in which the trigger threshold is adaptively varied with the sampled state. By fully considering the information about sampled state, TDs, and Markov jump parameters, a suitable Lyapunov–Krasovskii functional is constructed. With the virtue of FMEII and AEIC scheme, some novel stabilization criteria are presented in the form of linear matrix inequalities. At last, two numerical examples are given to show the validity of the obtained results.

Observer-based Adaptive Control for a Class of Uncertain Switched Systems with Time-delay: A Sliding Mode Approach

This article is devoted to the exponential stabilization problem of time-varying delay switched systems with nonlinear uncertainties via sliding mode technique. First of all, anon-fragile observer is established to deal with the case that the partial or total measurement of state variables is unavailable, from which a new sliding manifold is constructed. An exponential stability criteria of the dynamics in sliding phase is then devised in accordance with Lyapunov stability theory, linear matrix inequality and average dwell time method. Furthermore, the arrival condition of the designated sliding manifold is satisfied by utilizing the developed adaptive sliding mode controller. At last, the feasibility of the proposed control scheme is confirmed through two illustrative examples.

Exponential stabilization of inertial quaternion‐valued Cohen‐Grossberg neural networks: Lexicographical order method

SummaryIn this article, a general class of delayed interval inertial Cohen‐Grossberg neural networks described by quaternion‐valued parameters is considered. Under the homeomorphism mapping theory and lexicographical order method, we investigate the exponential stabilization problem for the quaternion‐valued Cohen‐Grossberg neural networks. To do so, we verify the existence and uniqueness of the equilibrium point (EP), and then by designing a sampled‐data feedback controller, several sufficient criteria are derived to ascertain the robust stability of the EP for the given system. What should be mentioned is that the state parameters are taking values in an interval, which implies the states are taking values between two different quaternions, thus, a lexicographical order method is employed, which proposed an effective method to determine the “magnitude” of two different quaternions. Finally, numerical example is provided to demonstrate the effectiveness of the developed theoretical results.

Intermittent Stabilization of Fuzzy Competitive Neural Networks With Reaction Diffusions

This article investigates the global exponential stability and stabilization problems for a class of Takagi-Sugeno (T-S) fuzzy competitive neural networks (NNs). In the considered model, we introduce the T-S fuzzy rule to describe the parametric switching causing by complexity and the vagueness in practical environment. Besides, the effects of reaction diffusions and distributed delays, which inherently exist in circuits of NNs, are also taken into consideration. By using the Lyapunov functional theory and Green formula, several stability criteria in terms of \mathbb p-norm are established for the uncompensated fuzzy competitive NNs. Moreover, by designing a fuzzy intermittent controller, the corresponding stabilizability criteria in terms of \mathbb p-norm are derived. We also carry out some discussions and comparisons to further show the less conservativeness and wide applicability of the main theorems. Finally, several examples are presented to verify the obtained results.