Complex-valued Linear Matrix Inequalities Research Articles

Impulsive Control of Complex-Valued Neural Networks with Mixed Time Delays and Uncertainties

This paper investigates the global exponential stability of uncertain delayed complex-valued neural networks (CVNNs) under an impulsive controller. Both discrete and distributed time-varying delays are considered, which makes our model more general than previous works. Unlike most existing research methods of decomposing CVNNs into real and imaginary parts, some stability criteria in terms of complex-valued linear matrix inequalities (LMIs) are obtained by employing the complex Lyapunov function method, which is valid regardless of whether the activation functions can be decomposed. Moreover, a new impulsive differential inequality is applied to resolve the difficulties caused by the mixed time delays and delayed impulse effects. Finally, an illustrative example is provided to back up our theoretical results.

Lag synchronization of complex-valued interval neural networks via distributed delayed impulsive control

<abstract><p>This paper investigates the lag synchronization problem of complex-valued interval neural networks with both discrete and distributed time-varying delays under delayed impulsive control. A distributed delayed impulsive controller that depends on the accumulation of the states over a history time period is designed to guarantee the exponential lag synchronization between the drive and the response systems. By employing the complex Lyapunov method and a novel impulsive differential inequality technique, some delay-dependent synchronization criteria are established in terms of complex-valued linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.</p></abstract>

Global asymptotic stability of fractional-order complex-valued neural networks with probabilistic time-varying delays

The stability of fractional-order complex-valued neural networks (FOCVNNs) with probabilistic time-varying delays is investigated in this paper. By constructing suitable Lyapunov–Krasovskii functional and utilizing inequality technique, a complex-valued linear matrix inequality (LMI) criterion guaranteeing the global asymptotic stability of the proposed FOCVNNs is deduced. A numerical example with simulations is provided to demonstrate the feasibility and availability of the obtained theoretical result.

Quantized Sampled-Data Control for Exponential Stabilization of Delayed Complex-Valued Neural Networks

This paper addresses the problem of quantized sampled-data control for CVNNs with time-varying delay under the assumption that only quantized measurements are transmitted to the controller. Based on the discrete-time Lyapunov stability theory, reciprocally convex approach, a sector bound approach, and some estimation techniques, a reduced conservative stabilization criterion is obtained to guarantee the exponential stabilization of the considered CVNNs. The desired quantized sampled-data controller is designed via converting the complex-valued linear matrix inequality into real-valued ones. The effectiveness of the derived criteria are shown via an illustrative simulation example.

State-Feedback Filtering for Delayed Discrete-Time Complex-Valued Neural Networks.

This article explores a new filtering problem for the class of delayed discrete-time complex-valued neural networks (CVNNs) via state-feedback control design. The novelty of this article comes from the consideration of the newly developed complex-valued reciprocal convex matrix inequality as well as the complex-valued Jensen-based summation inequalities (JSIs). By employing an appropriate Lyapunov-Krasovskii functional (LKF) and by using newly proposed complex-valued inequalities, attention is concentrated on the design of a state-feedback filter such that the associated filtering error system is asymptotically stable with prescribed filter and control gain matrices. The proposed theoretical results are presented in terms of complex-valued linear matrix inequalities (LMIs) that can be solved numerically by using the YALMIP toolbox in MATLAB software. Additionally, one numerical example is given to confirm the validity of the resulting sufficient conditions with the availability of the suitable control and filter design.

Global Mittag-Leffler Boundedness of Fractional-Order Fuzzy Quaternion-Valued Neural Networks With Linear Threshold Neurons

This article focuses on the global Mittag-Leffler boundedness for fractional-order fuzzy quaternion-valued neural networks (QVNNs) with linear threshold neurons. In order to avert the nonexchangeability for quaternion multiplication, the considered QVNN is separated into four real-valued or two complex-valued systems. By employing Lyapunov function method and fractional-order differential inequalities, several effective conditions according to algebraic inequality and complex-valued linear matrix inequalities are deduced to guarantee the global Mittag-Leffler boundedness of the addressed network. And, the frame of the global Mittag-Leffler attracting sets is presented as well. Here, the numbers of the equilibrium points need not to be concerned. Finally, a numerical example is provided to demonstrate the correctness of the proposed results.

State Estimation of Quaternion-Valued Neural Networks with Leakage Time Delay and Mixed Two Additive Time-Varying Delays

In this paper, the state estimation of quaternion-valued neural networks (QVNNs) with leakage time delay, both discrete and distributed two additive time-varying delays is studied. By considering the QVNNs as a whole, instead of decomposing it into two complex-valued neural networks or four real-valued neural networks. Via constructing suitable Lyapunov–Krasovskii functionals, combining free weight matrix, reciprocally convex approach, and matrix inequalities, the sufficient criteria for time delays are given in the form of quaternion-valued linear matrix inequalities and complex-valued linear matrix inequalities. Some observable output measurements are used to estimate the state of neurons, which ensures the global asymptotic stability of the error-state system. Finally, the effectiveness of theoretical analysis is illustrated by a numerical simulation.

Global asymptotic stability analysis for neutral-type complex-valued neural networks with random time-varying delays

ABSTRACTThis paper investigates the global asymptotic stability problem for a class of neutral-type complex-valued neural networks with random time-varying delays. By introducing a stochastic variable with Bernoulli distribution, the information of time-varying delay is assumed to be random time-varying delays. By constructing an appropriate Lyapunov–Krasovskii functional and employing inequality technique, several sufficient conditions are obtained to ensure the global asymptotically stability of equilibrium point for the considered neural networks. The obtained stability criterion is expressed in terms of complex-valued linear matrix inequalities, which can be simply solved by effective YALMIP toolbox in MATLAB. Finally, three numerical examples are given to demonstrate the efficiency of the proposed main results.

Synchronization of two nonidentical complex-valued neural networks with leakage delay and time-varying delays

This paper investigates the global asymptotical synchronization of two nonidentical complex-valued neural networks (CVNNs) with leakage delay and time-varying delays via integral sliding mode control approach. A suitable sliding surface and an appropriate Lyapunov–Krasovskii functional are constructed. By using free-weighting matrix method combined with inequality technique, a sliding mode controller is designed to ensure the synchronization for the considered CVNNs. The obtained criterion is presented in the form of complex-valued linear matrix inequalities (LMIs), which could be calculated through YALMIP with solver of SDPT3 in MATLAB. An example is provided to demonstrate the obtained theoretical result.

Passivity Analysis for Quaternion-Valued Memristor-Based Neural Networks With Time-Varying Delay.

This paper is concerned with the problem of global exponential passivity for quaternion-valued memristor-based neural networks (QVMNNs) with time-varying delay. The QVMNNs can be seen as a switched system due to the memristor parameters are switching according to the states of the network. This is the first time that the global exponential passivity of QVMNNs with time-varying delay is investigated. By means of a nondecomposition method and structuring novel Lyapunov functional in form of quaternion self-conjugate matrices, the delay-dependent passivity criteria are derived in the forms of quaternion-valued linear matrix inequalities (LMIs) as well as complex-valued LMIs. Furthermore, the asymptotical stability criteria can be obtained from the proposed passivity criteria. Finally, a numerical example is presented to illustrate the effectiveness of the theoretical results.

Global dissipativity of a class of quaternion-valued BAM neural networks with time delay

This paper is concerned with the problem of the global dissipativity of a class of quaternion-valued BAM neural networks with time delay. Two different types of activation functions are considered, including general bounded and Lipschitz-type activation functions. Based on the plural decomposition method of quaternion, the quaternion-valued BAM neural network is separated into two complex-valued systems. Using Lyapunov second method and inequality techniques, some sufficient conditions in complex-valued linear matrix inequality form are derived to ensure the global dissipativity of the discussed network. Moreover, the framework of the globally attractive sets are given out as well. Here, the existence and the uniqueness of the equilibrium point needs not to be considered. Finally, two examples are presented and analyzed to illustrate the effectiveness of our theoretical results.

Leakage delay-dependent stability analysis for complex-valued neural networks with discrete and distributed time-varying delays

This paper investigates the leakage delay-dependent global asymptotic stability problem for a class of complex-valued neural networks (CVNNs) with discrete and distributed time-varying delays. In order to handle this issue easily, an appropriate Lyapunov–Krasovskii functional (LKF) is constructed with some augmented delay-dependent terms. By employing integral inequalities, several delay-dependent sufficient conditions are derived that ensure the global asymptotic stability of the considered system model. Moreover, the results obtained in this paper have expressed in terms of complex-valued linear matrix inequalities (LMIs), whose feasible solutions can be easily verified by effective YALMIP control toolbox in MATLAB LMI. Finally, two benchmark illustrative examples are given to show the effectiveness and advantages of the proposed results.

Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays

This paper discusses octonion-valued neural networks (OVNNs) with leakage delay, time-varying delays, and distributed delays, for which the states, weights, and activation functions belong to the normed division algebra of octonions. The octonion algebra is a nonassociative and noncommutative generalization of the complex and quaternion algebras, but does not belong to the category of Clifford algebras, which are associative. In order to avoid the nonassociativity of the octonion algebra and also the noncommutativity of the quaternion algebra, the Cayley–Dickson construction is used to decompose the OVNNs into 4 complex-valued systems. By using appropriate Lyapunov–Krasovskii functionals, with double and triple integral terms, the free weighting matrix method, and simple and double integral Jensen inequalities, delay-dependent criteria are established for the exponential stability of the considered OVNNs. The criteria are given in terms of complex-valued linear matrix inequalities, for two types of Lipschitz conditions which are assumed to be satisfied by the octonion-valued activation functions. Finally, two numerical examples illustrate the feasibility, effectiveness, and correctness of the theoretical results.

Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays

Abstract Octonion-valued neural networks (OVNNs) are a type of neural networks for which the states and weights are octonions. The octonion algebra is the only normed division algebra that can be defined over the reals, besides the complex and quaternion algebras. Being nonassociative, it clearly does not belong to the Clifford algebras category, which are all associative. In this paper, sufficient conditions for the global exponential stability of neutral-type OVNNs with time-varying delays are formulated, by considering two types of Lipschitz conditions that must be satisfied by the octonion-valued activation functions. To avoid the nonassociativity of the octonions and the noncommutativity of the quaternions, the OVNNs model is decomposed into 4 complex-valued systems, using the Cayley–Dickson construction. By using Lyapunov–Krasovskii functionals with double, triple, and quadruple integral terms, the free weighting matrix method, and simple, double, and triple Jensen inequalities, the stability criteria are formulated in terms of complex-valued linear matrix inequalities. Two numerical examples are provided in order to demonstrate the effectiveness and feasibility of the theoretical results.

Global exponential convergence of fuzzy complex-valued neural networks with time-varying delays and impulsive effects

In this paper, the global exponential convergence of T–S fuzzy complex-valued neural networks with time-varying delays and impulsive effects is discussed. By employing Lyapunov functional method and matrix inequality technique, we analyze a type of activation functions with Lipschitz function, and sufficient conditions in terms of complex-valued linear matrix inequality are obtained to ensure the global exponential convergence. Moreover, the framework of the exponential convergence ball in the state space of the considered neural networks and the exponential convergence rate index are also given out. Here, the existence and uniqueness of the equilibrium points need not be considered and the results improve existing results on the Lyapunov exponential stability as special cases. Finally, one numerical example with simulations is given to illustrate the effectiveness of our theoretical results.

Global asymptotic stability of complex-valued neural networks with additive time-varying delays

In this paper, we extensively study the global asymptotic stability problem of complex-valued neural networks with leakage delay and additive time-varying delays. By constructing a suitable Lyapunov-Krasovskii functional and applying newly developed complex valued integral inequalities, sufficient conditions for the global asymptotic stability of proposed neural networks are established in the form of complex-valued linear matrix inequalities. This linear matrix inequalities are efficiently solved by using standard available numerical packages. Finally, three numerical examples are given to demonstrate the effectiveness of the theoretical results.

Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects

In this paper, the global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects is discussed. By employing Lyapunov functional method and using matrix inequality technique, several sufficient conditions in complex-valued linear matrix inequality form are obtained to ensure the existence, uniqueness and global exponential stability of equilibrium point for the considered neural networks. Moreover, the exponential convergence rate index is estimated, which depends on the system parameters. The proposed stability results are less conservative than some recently known ones in the literatures, which is demonstrated via two examples with simulations.

具有泄漏时滞的复值神经网络的全局同步性

The global synchronization of complex-valued neural networks with leakage time delays was investigated. For the considered complex-valued neural networks, the activation functions need not be separated into real parts and imaginary parts. Through the construction of appropriate Lyapunov-Krasovskii functionals, with the drive-response synchronization method, the free-weighting matrix method and the matrix inequality technique, a delay-dependent criterion for checking the global synchronization of complex-valued neural networks with leakage time delays was established in the form of complex-valued linear matrix inequality (LMI), meanwhile the design method for the synchronization controllers was given. Finally, a simulation example shows the effectiveness of the present work.

Stability criterion of complex-valued neural networks with both leakage delay and time-varying delays on time scales

In this paper, the problem on the global exponential stability of complex-valued neural networks with both leakage delay and time-varying delays on time scales is discussed. By constructing appropriate Lyapunov–Krasovskii functionals and using matrix inequality technique, a delay-dependent condition to assure the global exponential stability for the considered neural networks is established. The condition is expressed in complex-valued linear matrix inequality, which can be checked numerically using the effective YALMIP toolbox in MATLAB. An example with simulations is given to show the effectiveness of the obtained result.

Complete Stability Analysis of Complex-Valued Neural Networks with Time Delays and Impulses

In this paper, we extensively study the analysis of complete stability of complex-valued neural networks with time delay and impulsive effects. Using stability theory, impulsive effects and by constructing appropriate Lyapunov---Krasovskii functional, some sufficient conditions for the existence and complete stability of complex-valued neural networks with time delay and impulsive effects are derived in the form of complex-valued linear matrix inequality (LMIs) as well as real-valued LMIs. Finally, four numerical examples are given to establish the effectiveness of our theoretical results via standard numerical software.