Quaternion-valued Neural Networks Research Articles

Settling-Time Estimation for Finite-Time Stabilization of Fractional-Order Quaternion-Valued Fuzzy NNs

This article solves the problems of finite-time control and settling-time estimation for fractional-order quaternion-valued fuzzy neural networks (FQFNNs) with time delays. A novel fractional differential inequality is established to estimate the settling time of the addressed system, which is more general and less conservative than the existing results. In addition, owing to the noncommutativity of multiplication of quaternions, the decomposition method is usually adopted to discuss the finite-time stabilization (FTS) of quaternion-valued neural networks, which inevitably doubles the dimensionality of the system and brings a great computational burden. To avoid the aforementioned issues, some new properties of the quaternion-valued signum function are presented for exploring the FTS of FQFNNs by the direct quaternion method without any decomposition. Then, 1-norm and 2-norm control strategies are designed to stabilize the addressed system in finite time, and several sufficient criteria are derived to ensure the FTS of FQFNNs. Finally, the validity of the obtained theoretical results and the superiority of the proposed estimation method are illustrated by numerical simulations.

Fixed-time passivity of coupled quaternion-valued neural networks with multiple delayed couplings

Fixed-time passivity of coupled quaternion-valued neural networks with multiple delayed couplings

Finite-time stabilization of quaternion-valued neural networks with time delays: An implicit function method

In this study, we propose a non-decomposing method, the implicit Lyapunov function method, for the finite-time stabilization of a class of quaternion-valued neural networks (QVNNs) with time delays. Our approach has certain advantages, including more relaxed constraints on time delays, more flexible controllers without sign functions, and more general activation functions. Specifically, the derivative of an implicit Lyapunov function with quaternions is analyzed. Some properties of the implicit Lyapunov function, such as positive definiteness, monotonicity, and radial unboundedness, are deduced. With regard to two control schemes, two sets of sufficient conditions are provided for the finite-time stabilization of the QVNN, and an improved adaptive control strategy is designed to enhance stabilization efficiency. Two numerical examples illustrate the correctness, applicability and effectiveness of the method.

Fixed-Time Control for Memristor-Based Quaternion-Valued Neural Networks with Discontinuous Activation Functions

Fixed-Time Control for Memristor-Based Quaternion-Valued Neural Networks with Discontinuous Activation Functions

Global Mittag–Leffler stability and synchronization of discrete-time fractional-order delayed quaternion-valued neural networks

This paper is devoted to investigating discrete-time fractional-order delayed quaternion-valued neural networks (DFDQNNs) by utilizing direct quaternion approach. Firstly, a novel lemma and its two corresponding corollaries have been proposed for estimating nabla fractional difference of the quaternion-valued Lyapunov function. Then, the existence and uniqueness of equilibrium point for DFDQNNs is proved by constructing a new quaternion-valued contraction mapping. In addition, by means of our designed Lyapunov functions and the effective feedback controller as well as neoteric nabla difference inequalities, some sufficient criteria have been obtained to ensure the global Mittag–Leffler stability and Mittag–Leffler synchronization of DFDQNNs, respectively. Finally, some numerical examples are provided to verify the yielded results.

New criteria of finite time synchronization of fractional-order quaternion-valued neural networks with time delay

In this paper, to simulate some situations in the real world, a class issue of finite-time synchronization (FTS) about fractional-order quaternion-valued neural networks (FOQVNNs) with time delay is discussed. Firstly, there is no need to divide the QVNNs into several subsystems, some new lemmas related to the sign function of quaternion-valued are established in the quaternion domain. Secondly, based on the proposed sign function framework, a concise and effective controller is directly designed and FTS criteria are derived by constructing the Lyapunov function. Moreover, two settling times of synchronization are effectively estimated. Finally, some examples are shown to testify the correctness of the presented theoretical results.

Master-slave synchronization of a new fractal-fractional order quaternion-valued neural networks with time-varying delays

In this paper, a new fractional differentiation operator is considered the convolution of a power law with a fractal derivative. The novel operator sought to attract more non-local memory effects and self-similarities in chaotic attractors. This paper addresses the problem of fractal-fractional order quaternion-valued neural networks (FFoQVNNs) with time-varying delays. The sufficient conditions for the existence and uniqueness of an equilibrium point are derived for the proposed model by employing contraction mapping. The Lyapunov direct technique and fractal-fractional differential theory achieve the finite-time synchronization criteria by dividing the FFoQVNNs into four real-valued systems. Furthermore, the settling time is determined, which impact by the fractal dimension β, fractional-order α, and control parameters. Finally, a corresponding numerical simulation is demonstrated to show the accuracy of the theoretical results.

Robust control of uncertain time-delay Markovian jump quaternion-valued neural networks subject to partially known transition probabilities: direct quaternion method.

This paper addresses the issue of robust stochastic stabilization and control of uncertain time-delay Markovian jump quaternion-valued neural networks (MJQVNNs) subject to partially known transition probabilities. First, the direct quaternion method is proposed to analyse the MJQVNNs, which is different from some conventional methods in that the former is without any decomposition for systems. After that, in order to estimate the upper bound of the derivative of the constructed Lyapunov-Krasovskii functional(LKF) more accurately, the real-valued convex inequality is extended to quaternion domain. Then, by designed the mode-dependent state feedback controllers, the robust stochastic stabilization conditions of MJQVNNs are given for the admissible uncertainties, and reduce the influence of input disturbance on the controlled output to a specified performance level. Lastly, two numerical examples are given to illustrate the effectiveness of the proposed method.

Global finite-time stability of delayed quaternion-valued neural networks based on a class of extended Lyapunov-Razumikhin methods.

In this paper, a class of global finite-time stability problem for quaternion-valued neural networks with time-varying delays are investigated by adopting an extended modification Lyapunov-Razumikhin (L-R) method and a new upper bounds estimation of system solution in terms of convergence rate was obtained. Firstly, a new extended method of L-R is proposed to solve the general difficulty to find a proper Lyapunov functional. Then, a new suitable controller is designed, the new conditions of inequalities global finite-time stability are obtained via combining with the former proposed L-R method in the separated real-valued system. Finally, for purpose of verifying the availability of the theorem presented, two given illustrative examples are shown.

Global Dissipativity Analysis and Stability Analysis for Fractional-Order Quaternion-Valued Neural Networks With Time Delays

This article studies dissipativity analysis of fractional-order quaternion-valued neural networks (FOQVNNs) with time delays. Two specific activation functions are considered along with common bounded and activation functions of Lipschitz-kind. Since quaternion multiplication is not commutative, we must divide the model, which is evaluated by quaternion, into four elements that are real-valued elements. On the basis of the construction of novel Lyapunov functional, and applying fractional-calculus theory, new criteria for the test of the global dissipativity and exponential stability of FOQVNNs model are established. FOQVNNs have also been suggested to provide global dissipativity and exponential stability, whereas nonlinear complex activation functions are constrained by the usage of linear matrix inequality methods, which utilize quaternion matrices and positive quaternion definite matrices. Finally, the effectiveness and superiority of the proposed approach is validated through numerical examples.

Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays

This study on the local stability of quaternion-valued neural networks is of great significance to the application of associative memory and pattern recognition. In the research, we study local Lagrange exponential stability of quaternion-valued neural networks with time delays. By separating the quaternion-valued neural networks into a real part and three imaginary parts, separating the quaternion field into 34n subregions, and using the intermediate value theorem, sufficient conditions are proposed to ensure quaternion-valued neural networks have 34n equilibrium points. According to the Halanay inequality, the conditions for the existence of 24n local Lagrange exponentially stable equilibria of quaternion-valued neural networks are established. The obtained stability results improve and extend the existing ones. Under the same conditions, quaternion-valued neural networks have more stable equilibrium points than complex-valued neural networks and real-valued neural networks. The validity of the theoretical results were verified by an example.

Synchronization in fixed/preassigned-time of delayed fully quaternion-valued memristive neural networks via non-separation method

This paper mainly centers on investigating issues of synchronization of delayed memristive quaternion-valued neural networks (MQVNNs), which are first studied with an unconventional non-separation approach for the fully quaternion-valued (QV) parameters (i.e. state vectors, weights, inputs and outputs). First, QV sign function and norms are introduced and new properties of quaternions are established, which are the basis of the non-separation approach. Next, more relaxed conditions with a new pure lemma are explored to guarantee fixed-time (FXT) or preassigned-time (PST) synchronization and the settling times (STs) are measured. Specially, the ST of PST synchronization is pre-specified and irrelevant to any parameter of the systems or controllers, which is more independent. Subsequently, controller strategies are given to realize FXT and PST synchronization of the discussed systems. Lastly, the effectiveness and superiority of the achieved theoretical results are illustrated by numerical examples.

Exponential synchronization for second-order switched quaternion-valued neural networks with neutral-type and mixed time-varying delays

This article focuses on the global exponential synchronization (GES) for second-order state-dependent switched quaternion-valued neural networks (SOSDSQVNNs) with neutral-type and mixed delays. By proposing some new Lyapunov–Krasovskii functionals (LKFs) and adopting some inequalities, several new criteria in the shape of algebraic inequalities are proposed to ensure the GES for the concerned system by using hybrid switched controllers (HSCs). Different from the common reducing order and separation ways, this article presents some new LKFs to straightway discuss the GES of the concerned system based on non-reduction order and nonseparation strategies. Ultimately, an example is provided to validate the effectiveness of the theoretical outcomes.

Hybrid control design for Mittag-Leffler projective synchronization on FOQVNNs with multiple mixed delays and impulsive effects

This article discusses the global Mittag-Leffler projective synchronization (GM-LPS) on fractional-order quaternion-valued neural networks (FOQVNNs) including mixed delays and impulses. The effects of various types of delay factors and impulses on Mittag-Leffler projective synchronization performance are simultaneously explored. Based on Banach contractive mapping principle, the existence result of solution to the considered systems is derived Applying the Lyapunov direct method, inequality analysis technique and generalized fractional Razumikhin theorem, the algebraic criteria for ensuring global Mittag-Leffler projective synchronization by designing the hybrid controllers are proposed. The result of algebraic inequality greatly reduces the computational complexity to some extent. Combining the different orders of Caputo derivative and MATLAB toolbox, the simulations are given to further validate the obtained results.

New inequalities to finite-time synchronization analysis of delayed fractional-order quaternion-valued neural networks

New inequalities to finite-time synchronization analysis of delayed fractional-order quaternion-valued neural networks

Existence and Finite-Time Stability of Besicovitch Almost Periodic Solutions of Fractional-Order Quaternion-Valued Neural Networks with Time-Varying Delays

Existence and Finite-Time Stability of Besicovitch Almost Periodic Solutions of Fractional-Order Quaternion-Valued Neural Networks with Time-Varying Delays

New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays.

During the past decades, many works on Hopf bifurcation of fractional-order neural networks are mainly concerned with real-valued and complex-valued cases. However, few publications involve the quaternion-valued neural networks which is a generalization of real-valued and complex-valued neural networks. In this present study, we explorate the Hopf bifurcation problem for fractional-order quaternion-valued neural networks involving leakage delays. Taking advantage of the Hamilton rule of quaternion algebra, we decompose the addressed fractional-order quaternion-valued delayed neural networks into the equivalent eight real valued networks. Then the delay-inspired bifurcation condition of the eight real valued networks are derived by making use of the stability criterion and bifurcation theory of fractional-order differential dynamical systems. The impact of leakage delay on the bifurcation behavior of the involved fractional-order quaternion-valued delayed neural networks has been revealed. Software simulations are implemented to support the effectiveness of the derived fruits of this study. The research supplements the work of Huang et al. (Neural Netw 117:67-93, 2019).

Global Mittag-Leffler stability for fractional-order quaternion-valued neural networks with piecewise constant arguments and impulses

This paper is devoted to analysing the global Mittag-Leffler stability of fractional-order quaternion-valued neural networks with piecewise constant arguments and impulses. The quaternion direct method is adopted to address the considered model avoiding any decomposition. By utilising the matrix inequality technique and Lyapunov direct method, some sufficient conditions to guarantee the global Mittag-Leffler stability of equilibrium point for the considered model are obtained in the form of quaternion-valued linear matrix inequalities (LMIs). To certify the validity of the derived result, a numerical example is presented.

Stochastic stability criteria and event-triggered control of delayed Markovian jump quaternion-valued neural networks

This paper investigates, through designing an event-triggered sampled-data mechanism, the stochastic stabilization of Markovian jump quaternion-valued neural networks (QVNNs) with partially unknown transition probabilities. First, since the multiplication of quaternions is not commutative, the Markovian jump QVNNs are decomposed into four real-valued neural networks. Then, by constructing a novel augmented Lyapunov–Krasovskii functional (LKF), it is containing more hybrid features, in which concerning the practical sampling pattern, and using the convex combinatorial optimization and inequality technique, the stabilization condition is proposed in the form of linear matrix inequalities (LMIs). Furthermore, the desired event-triggered sampled-data controller gains are obtained for the closed-loop Markovian jump QVNNs with partially unknown transition probability. Finally, the viability and validness of the obtained results are confirmed via three numerical examples.

Finite-time synchronization of quaternion-valued neural networks with delays: A switching control method without decomposition

For a class of quaternion-valued neural networks (QVNNs) with discrete and distributed time delays, its finite-time synchronization (FTSYN) is addressed in this paper. Instead of decomposition, a direct analytical method named two-step analysis is proposed. That method can always be used to study FTSYN, under either 1-norm or 2-norm of quaternion. Compared with the decomposing method, the two-step method is also suitable for models that are not easily decomposed. Furthermore, a switching controller based on the two-step method is proposed. In addition, two criteria are given to realize the FTSYN of QVNNs. At last, three numerical examples illustrate the feasibility, effectiveness and practicability of our method.