Problem Of Exponential Stability Research Articles

Exponential and weak stabilization for distributed bilinear systems with time delay via bounded feedback control

AbstractIn this paper, we deal with the problem of exponential and weak stabilization for a class of distributed bilinear systems with time delay in a Hilbert space by using bounded feedback control. In the case of exponential stabilization, an explicit decay rate estimate of the stabilized state is given provided that a non‐standard observability inequality condition is satisfied. The compactness hypothesis of the control operator prevents the validity of the non‐standard observability inequality, thus it can be relaxed to a weaker sufficient condition to show the weak stabilization result by employing the same feedback control. Finally, numerical examples and simulations to hyperbolic and parabolic partial functional differential equations are considered to illustrate the effectiveness of the obtained theoretical results.

Stability analysis for discrete-time switched GRNs with persistent dwell-time and time delays

In this paper, the problem of exponential stability for discrete-time switched genetic regulatory networks (GRNs) with persistent dwell-time (PDT) switching and time delays is concerned. The novelty of this paper is derived from introducing an extended property of quadratic convex function, and utilizing an improved summation inequality together with an extended reciprocally convex matrix inequality, which is less conservative than the Jensen inequality and the reciprocally convex combination employed in the discrete-time delay systems. In addition, the considered switching regularity is more general than dwell-time (DT) switching and average dwell-time (ADT) switching. Moreover, by introducing the delay partitioning method and the piecewise Lyapunov–Krasovskii functional, a set of sufficient conditions are established in the form of linear matrix inequalities to guarantee the discrete-time PDT switched GRNs with constant time delays and time-varying delays are exponentially stable, respectively. Finally, some numerical examples are exploited to demonstrate the validity and potential of the proposed method.

Stability and control of uncertain ICPT system considering time‐varying delay and stochastic disturbance

SummaryThis paper studies the problem of robust exponential stability for uncertain inductively coupled power transfer (ICPT) system considering time‐varying delay and stochastic disturbance. Firstly, the model of the system is set up via a time domain method. Secondly, based on the Newton‐Leibnitz formula and stability theory, a new stability analysis of the ICPT system with uncertain parameter and time‐varying delay or stochastic disturbance is presented respectively. The proposed approach can be applied for analyzing other similar systems, and the obtained results can be also used for estimating convergence rate and region of stability. Thirdly, based on the Lyapunov‐Krasovskii functional (LKF) approach and the stochastic stability theory, robust exponential stability criteria in the mean square are derived and the relevant controller is designed. The proposed method can further reduce conservatism. Finally, the correctness and effectiveness of the obtained results are verified by an example and simulations indicate that the larger time delay, the slower attenuation. Simultaneously, the uncertain parameter and the stochastic disturbance have a significant influence on the region of stability. In addition, in respect of reducing conservatism, the effectiveness of the proposed method is demonstrated by comparing with other papers. In addition, the designed controller shows better performance for the ICPT system with the parameter uncertainty, the time‐varying delay, and the stochastic disturbance.

Exponential stability and H∞ control of uncertain singular nonlinear switched systems with impulsive perturbations

ABSTRACTThis paper investigates the problems of exponential stability and robust control for a general class of uncertain singular nonlinear switched systems subject to impulsive perturbations. An effective computational method is introduced for designing switched linear feedback controllers, which is based on the novel conditions on the solvability of the control problem. The idea for decomposing the singular impulsive increments is applied in this development, which establishes some criteria on exponential stability. Comparing with some existing results on singular switched systems, this design is not only less conservative but also with reduced computational complexity. In addition, the effectiveness of the proposed approach is illustrated by numerical examples.

Output Feedback Stabilization of an ODE-Schrödinger Cascade System Subject to Boundary Control Matched Unknown Disturbance

In this paper, we consider the output feedback exponential stabilization problem of ODE-Schrodinger cascade systems with the external disturbance. We propose a new extended state observer (ESO) that estimates both state and disturbance by the three output signals, then design a stabilizing control law by utilizing the backstepping technique. The resulting closed-loop system is shown to be exponentially stable guaranteeing that all internal systems involved are uniformly bounded. Finally, some numerical experiments are carried out to verify the effectiveness of the proposed control law.

Robust Output Feedback Stabilization for Uncertain Discrete-Time Stochastic Neural Networks with Time-Varying Delay

This paper investigates the problem of robust exponential stabilization of uncertain discrete-time stochastic neural networks with time-varying delay based on output feedback control. By choosing an augmented Lyapunov–Krasovskii functional, we established the sufficient conditions of the delay-dependent asymptotical stabilization in the mean square for a class of discrete-time stochastic neural networks with time-varying delay. Furthermore, we obtain the criteria of robust global exponential stabilization in the mean square for uncertain discrete-time stochastic neural networks with time-varying delay. Finally, we give numerical examples to illustrate the effectiveness of the proposed results.

Integrated design of switching control and mobile actuator/sensor guidance for a linear diffusion process

This paper is concerned with the exponential stabilization problem for a class of diffusion processes described by a linear parabolic partial differential equation (PDE) using mobile collocated actuators and sensors. Each collocated actuator/sensor pair is capable of moving within the respective spatial domain divided in advance and a mode indicator function is employed to indicate the different modes for all actuator/sensor pairs according to whether each actuator/sensor pair is static or mobile. By utilizing Lyapunov direct method, an integrated design of switching controllers and mobile actuator/sensor guidance laws for the diffusion process is developed such that the resulting closed-loop system is exponentially stable. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method.

Exponential stability of positive neural networks in bidirectional associative memory model with delays

This paper is concerned with the problem of exponential stability of positive neural networks in bidirectional associative memory (BAM) model with multiple time‐varying delays and nonlinear self‐excitation rates. On the basis of a systematic approach involving extended comparison techniques via differential inequalities, we first prove the positivity of state trajectories initializing from a positive cone called the admissible set of initial conditions. In combination with the use of Brouwer's fixed point theorem and M‐matrix theory, we then derive conditions for the existence and global exponential stability of a unique positive equilibrium of the model. An extension to the case of BAM neural networks with proportional delays is also presented. The effectiveness of the obtained results is illustrated by a numerical example with simulations.

Exponential stability analysis for a class of neutral singular Markovian jump systems with time-varying delays

This paper deals with the exponential stability problem for a class of neutral singular systems with Markovian jump parameters. The considered systems involve time-varying delays not only in their state but also in their derivatives of state. By using the Lyapunov–Krasovskii functional method, some sufficient conditions are derived, which ensure that the considered systems are regular, impulse-free and exponentially stable. Finally, some numerical examples are employed to demonstrate the effectiveness of the obtained approaches.

Fixed-time switching control of underactuated surface vessels with dead-zones: Global exponential stabilization

This paper investigates the problem of global exponential stabilization of underactuated surface vessels (USVs) with actuator dead-zones. By utilizing input and state transformations, the dynamic model of USV is converted into an equivalent system consisting of two cascade connected subsystems. For the transformed system, a switching scheme based on fixed-time control theory is presented. For one of the subsystems with input dead-zones, a fixed-time control method is proposed, which addresses the dead-zone problem while guaranteeing global exponential stabilization. Simulations are given to demonstrate the effectiveness of presented method.

Deterministic learning from sampling data

In this paper, based on the deterministic learning mechanism, we present an alternative systematic scheme for dynamics identification from sampling data sequences. The proposed scheme belongs to a dynamical machine learning framework based on Lyapunov stability theory rather than optimization estimation approaches. Given a sampling data sequence collected from an unknown deterministic nonlinear dynamical system, we show that the inherent dynamics of the sampling data sequence can be locally accurately captured and represented as a converged constant neural network. This accurate identification result can be derived from the exponential stability of a class of linear time-varying (LTV) error systems usually appearing in adaptive control and identification fields. However, there are few existing results concerning the exponential stability problem of the discrete-time LTV case. To this end, by using the small gain theorem of input-to-state stability (ISS), we provide a rigorous proof for establishing the exponential stability of the LTV system under the persistent excitation (PE) condition. Based on the exponential stability results, the exponential convergence of neural weights to their optimal values can be analytically guaranteed, which leads to the achievement of the accurate dynamics identification. The significance of the paper is that we preliminarily explore the essential problem of machine learning of dynamical systems by innovatively transforming tools and methodologies from dynamic system stability and control theory. Numerical simulation results are provided to validate the effectiveness of the proposed scheme with comparison to previously mentioned approaches.

Distributed adaptive consensus control for networked robotic manipulators with time-varying delays under directed switching topologies

This work focus on consensus control problem for networked multiple robotic manipulators with and without time-varying delays under directed switching topologies. Two different consensus protocols are proposed with considering dynamic uncertainties in networked robotic manipulators. First, we investigate a distributed adaptive consensus algorithm without communication delays under directed switching topologies in networks. A new analysis method for demonstrating the stability of the designed algorithm is provided by constructing a new matrix and utilizing matrix theory, graph theory, Lyapunov stability and Barbalat’s Lemma. Then a consensus controller considering time-varying delays under directed switching topologies is presented. Furthermore, combining with the method for the case without delays, LMI technology is utilized to demonstrate the exponential stability problem. The distributed adaptive consensus control for multiple robotic manipulators with parametric uncertainties, time-varying delays and the directed switching topology is designed and analyzed in a unified framework. Numerical simulations finally prove the validity of the obtained results.

Exponential stabilization and sampled-date H∞ control for uncertain T–S fuzzy systems with time-varying delay

In this paper, the exponential stabilization problem of uncertain T–S fuzzy systems with time-varying delay is emulated by fuzzy sampled-data H∞ control. Firstly, a novel suitable Lyapunov–Krasovskii function is constructed, which contains all the information about the sampling pattern. Secondly, a less conservative result is achieved by using an extended Jensen inequality, and purposefully using a compact free weighting matrix. In addition, according to the linear matrix inequality (LMI), some sampled-data H∞ exponential stability sufficient conditions and controller design of T–S fuzzy systems are established. Finally, effectiveness gives some illustrative examples may be used to display the value of the current proposed method as well as a significant improvement.

Stability and synchronization for impulsive Markovian switching CVNNs: matrix measure approach

This paper devotes to the global exponential stability and synchronization problem for impulsive Markovian switching complex-valued neural networks (CVNNs) with time-varying delays. Based on the matrix measure approach and the impulsive differential inequality, some sufficient conditions are firstly derived to guarantee the impulsive network to be exponentially stable, where the exponential convergence rate is explicitly estimated. After that, the synchronization problem is investigated for the coupled impulsive complex-valued networks, and the obtained criteria are easy to be verified and implemented in practice. Finally, two examples are presented to illustrate effectiveness of the proposed theoretical results.

Exponential stability of impulsive positive switched systems with discrete and distributed time‐varying delays

SummaryThis paper studies the exponential stability problems of discrete‐time and continuous‐time impulsive positive switched systems with mixed (discrete and distributed) time‐varying delays, respectively. By constructing novel copositive Lyapunov‐Krasovskii functionals and using the average dwell time technique, delay‐dependent sufficient conditions for the solvability of considered problems are given in terms of fairly simple linear matrix inequalities. Compared with the most existing results, by introducing an extra real vector, restrictive conditions on derivative of the time‐varying delays (less than 1) are relaxed, thus the obtained improved stability criteria can deal with a wider class of continuous‐time positive switched systems with time‐varying delays. Finally, two simple examples are provided to verify the validity of theoretical results.

Exponential Stability of Neural Networks with Markovian Switching Parameters and General Noise

This paper investigates the problem of exponential stability of Neural Networks (NNs) with Markovian parameters and general noise. The model in this paper with general noise is more suitable for many real nervous systems than NNs with white noise. Criteria for the exponential stability of the NNs with Markovian switching parameters and general noise in both the mean square and p-th moment are derived by utilizing the random analysis method and Lyapunov functional method techniques. The exponential stability of NNs without Markovian switching is given as a special case. Finally, simulation result in two examples are discussed to illustrate the theoretical results.

Exponential Stabilization of Markov Jump Systems with Mode-Dependent Mixed Time-Varying Delays and Unknown Transition Rates

This paper investigates the problem of exponential stability in mean square sense for stochastic Markov jump systems with mixed time-varying delays and partly unknown transition rates. By employing a class of appropriate stochastic Lyapunov functionals, the analysis process of stability for stochastic Markov jump systems can be effectively carried out. Based on the linear matrix inequalities technique, the mean square exponential stability criteria are presented for stochastic Markov jump systems with partly unknown transition rates. Furthermore, by expanding this case to uncertain Markov jump systems, we derive the sufficient conditions for guaranteeing the stability of uncertain Markov jump systems. A numerical example is presented to illustrate the effectiveness of the proposed results.

Locally Exponential Stability of Discrete-time Complex Networks with Impulsive Input Saturation

In this paper, the problem of exponential stabilization for a class of discrete-time complex network with saturated impulse input is investigated. Based on the inductive method, convex analysis, and auxiliary matrix, several Lyapunov-type stability criteria are derived for exponential stability of discrete-time complex network with impulsive input saturation. Two examples are also presented to illustrate the effectiveness and the feasibility of the obtained results.

Delay-dependent global exponential stability for neural networks with time-varying delay

This paper deals with the global exponential stability problem of neural networks with time-varying delay. A novel Lyapounov–Krasovskii functional (LKF), which contains a common double integral term, an augmented double integral term and two delay-product-type terms, is constructed to analyze the exponential stability. An auxiliary function-based integral inequality (AFBII) and two special forms of it are applied to estimate the upper bounds of single integral terms produced in the time derivation of the LKF candidate. By using the novel LKF and AFBII, some new cross terms of matrix variables are included in linear matrix inequalities (LMIs). As a result, a less conservative delay-dependent global exponential stability criterion is proposed. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed criterion.

Stability analysis of sample data systems with input missing: A hybrid control approach

In this paper, we establish exponential stability criteria for the sampled-data impulsive control of the linear time-invariant system. With average impulse interval (AII), less conservative conditions are obtained on the exponential stability problem for the sampled-data systems. It is proved that when the AII of the impulsive sequences is fixed, the upper bound of the impulsive intervals could be very large, which guarantees the less conservativeness of the obtained result concerning the sampling intervals. The control input missing is also studied and we establish a new stability criterion for the exponential decay rate and the sampling period which is less conservative than the ones obtained for variable sampling intervals. Two examples are given to show the effectiveness of the obtained result.