Synchronization Analysis of Discrete-Time Fractional-Order Quaternion-Valued Uncertain Neural Networks

This article studies synchronization issues for a class of discrete-time fractional-order quaternion-valued uncertain neural networks (DFQUNNs) using nonseparation method. First, based on the theory of discrete-time fractional calculus and quaternion properties, two equalities on the nabla Laplace transform and nabla sum are strictly proved, whereafter three Caputo difference inequalities are rigorously demonstrated. Next, based on our established inequalities and equalities, some simple and verifiable quasi-synchronization criteria are derived under the quaternion-valued nonlinear controller, and complete synchronization is achieved using quaternion-valued adaptive controller. Finally, numerical simulations are presented to substantiate the validity of derived results.

Event-Triggered Guarantee Cost Control for Partially Unknown Stochastic Systems via Explorized Integral Reinforcement Learning Strategy

In this article, an integral reinforcement learning (IRL)-based event-triggered guarantee cost control (GCC) approach is proposed for stochastic systems which are modulated by randomly time-varying parameters. First, with the aid of the RL algorithm, the optimal GCC (OGCC) problem is converted into an optimal zero-sum game by solving a modified Hamilton-Jacobin-Isaac (HJI) equation of the auxiliary system. Moreover, in order to address the stochastic zero-sum game, we propose an on-policy IRL-based control approach involved by the multivariate probabilistic collocation method (MPCM), which can accurately predict the mean value of uncertain functions with randomly time-varying parameters. Furthermore, a novel GCC method, which combines the explorized IRL algorithm and MPCM, is designed to relax the restriction of knowing the system dynamics for the class of stochastic systems. On this foundation, for the purpose of reducing computation cost and avoiding the waste of resources, we propose an event-triggered GCC approach involved with explorized IRL and MPCM by utilizing critic-actor-disturbance neural networks (NNs). Meanwhile, the weight vectors of three NNs are updated simultaneously and aperiodically according to the designed triggering condition. The ultimate boundedness (UB) properties of the controlled systems have been proved by means of the Lyapunov theorem. Finally, the effectiveness of the developed GCC algorithms is illustrated via two simulation examples.

Master-Slave Synchronization of Neural Networks With Unbounded Delays via Adaptive Method

Master-slave synchronization of two delayed neural networks with adaptive controller has been studied in recent years; however, the existing delays in network models are bounded or unbounded with some derivative constraints. For more general delay without these restrictions, how to design proper adaptive controller and prove rigorously the convergence of error system is still a challenging problem. This article gives a positive answer for this problem. By means of the stability result of unbounded delayed system and some analytical techniques, we prove that the traditional centralized adaptive algorithms can achieve global asymptotical synchronization even if the network delays are unbounded without any derivative constraints. To describe the convergence speed of the synchronization error, adaptive designs depending on a flexible ω -type function are also provided to control the synchronization error, which can lead exponential synchronization, polynomial synchronization, and logarithmically synchronization. Numerical examples on delayed neural networks and chaotic Ikeda-like oscillator are presented to verify the adaptive designs, and we find that in the case of unbounded delay, the intervention of ω -type function can promote the realization of synchronization but may destroy the convergence of control gain, and this however will not happen in the case of bounded delay.

Delay-Dependent Switching Approaches for Stability Analysis of Two Additive Time-Varying Delay Neural Networks

This article analyzes the exponentially stable problem of neural networks (NNs) with two additive time-varying delay components. Disparate from the previous solutions on this similar model, switching ideas, that divide the time-varying delay intervals and treat the small intervals as switching signals, are introduced to transfer the studied problem into a switching problem. Besides, delay-dependent switching adjustment indicators are proposed to construct a novel set of augmented multiple Lyapunov-Krasovskii functionals (LKFs) that not only satisfy the switching condition but also make the suitable delay-dependent integral items be in the each corresponding LKF based on each switching mode. Combined with some switching techniques, some less conservativeness stability criteria with different numbers of switching modes are obtained. In the end, two simulation examples are performed to demonstrate the effectiveness and efficiency of the presented methods comparing other available ones.

Projective quasi-synchronization of coupled memristive neural networks with uncertainties and impulsive effect

The dynamic behavior of memristive neural networks (MNNs), including synchronization, effectively keeps the robotic stability against numerous uncertainties from the mimic of the human brain. However, it is challenging to perform projective quasi-synchronization of coupled MNNs with low-consumer control devices. This is partly because complete synchronization is difficult to realize under various projective factors and parameter mismatch. This article aims to investigate projective quasi-synchronization from the perspective of the controller. Here, two approaches are considered to find the event-triggered scheme for lag synchronization of coupled MNNs. In the first approach, the projective quasi-synchronization issue is formulated for coupled MNNs for the first time, where the networks are combined with time-varying delays and uncertainties under the constraints imposed by the frequency of controller updates within limited system communication resources. It is shown that our methods can avoid the Zeno-behavior under the newly determined triggered functions. In the second approach, following classical methods, a novel projective quasi-synchronization criterion that combines the nonlinear property of the memristor and the framework of Lyapunov-Krasovskii functional (LKF) is proposed. Simulation results indicate that the proposed two approaches are useful for coupled MNNs, and they have less control cost for different types of quasi-synchronization.

Quasisynchronization of Memristive Neural Networks With Communication Delays via Event-Triggered Impulsive Control

This article considers the quasisynchronization of memristive neural networks (MNNs) with communication delays via event-triggered impulsive control (ETIC). In view of the limited communication and bandwidth, we adopt a novel switching event-triggered mechanism (ETM) that not only decreases the times of controller update and the amount of data sent out but also eliminates the Zeno behavior. By using an appropriate Lyapunov function, several algebraic conditions are given for quasisynchronization of MNNs with communication delays. More important, there is no restriction on the derivation of the Lyapunov function, even if it is an increasing function over a period of time. Then, we further propose a switching ETM depending on communication delays and aperiodic sampling, which is more economical and practical and can directly avoid Zeno behavior. Finally, two simulations are presented to validate the effectiveness of the proposed results.

Nonfragile H∞ Synchronization of BAM Inertial Neural Networks Subject to Persistent Dwell-Time Switching Regularity

This article concentrates on the synchronization of discrete-time persistent dwell-time (PDT) switched bidirectional associative memory inertial neural networks with time-varying delays. Through the use of the switched system theory related to the PDT, the convex optimization technique together with some straightforward decoupling methods, an appropriate mode-dependent controller with nonfragility is developed to acclimatize itself to some practical circumstances. Simultaneously, sufficient conditions of ensuring the H_∞ performance and exponential stability for the resulting switched synchronization error system are derived. Finally, a numerical example is utilized to show the validity of the model constructed and the influence of the PDT on the H_∞ performance. In addition, an image encryption example is employed to show the potential application prospect of the investigated system.

A GCM Neural Network with Piecewise Logistic Chaotic Map

In order to explore dynamic mechanisms and chaos control of globally coupled map (GCM) chaotic neural networks, a new GCM model, called the PL-GCM model is proposed, of which a piecewise logistic chaotic map is used instead of a logistic map. As a result of the strong chaotic features of the map, the neurons’ period and chaotic characteristics over a wide range of parameters are discussed, the dynamic mechanism is demonstrated in detail, and the numerical simulations such as state evolution, the largest Lyapunov exponent (LLE), contour map, and so on are exhibited. Furthermore, chaos control of the proposed PL-GCM model is investigated by adopting two chaos control methods. It is shown that the network with conventional coupling or delay coupling can be precisely controlled to any specified periodic orbit by feedback control, and its dynamic associative memory is realized by the variable threshold parameter control method with external information. The results of simulations and experiments suggest that the network is controlled successfully and can output period patterns with a specified period that contains the stored pattern closest to the initial pattern. All features suggest that the network is fit for pattern recognition and information processing.

Saturated impulsive control for synchronization of coupled delayed neural networks

The paper focuses on the synchronization problem for a class of coupled neural networks with impulsive control, where the saturation structure of impulse action is fully considered. The coupled neural networks under consideration are subject to mixed delays including transmission delay and coupled delay. The sector condition in virtue of a new constraint of set inclusion is given for a addressed network, based on which a sufficient condition for exponential synchronization problem is obtained by replacing saturation nonlinearity with a dead-zone function. In the framework of saturated impulses, our results relying on the domain of attraction can still achieve the synchronization of coupled delayed neural networks. In addition, the estimating domain of attraction is proposed as large as possible by solving an optimization problem. Finally, a numerical simulation example is presented to demonstrate the effectiveness of the proposed results.

Sliding Mode Stabilization of Memristive Neural Networks With Leakage Delays and Control Disturbance

In this article, we investigate a class of memristive neural networks (MNNs) with time-varying delays and leakage delays via sliding mode control (SMC) with and without control disturbance. SMC is used to ensure MNNs' stability. According to the characteristics of the MNNs, we consider the following three models: the first is the MNNs with time-varying delays, the second is the MNNs with time-varying delays and the control disturbance, and the third is the MNNs with time-varying delays, leakage delays, and the control disturbance. We quote some assumptions and lemmas to ensure that our main results are true. The sliding surface, the corresponding sliding mode controller, and the Lyapunov functions are constructed in different models to ensure MNNs' stability. Finally, some examples and simulations verify the validity of our main results by solving linear matrix inequality (LMI), and the conclusions and analysis of the results are given.

Intermittent Control for Quasisynchronization of Delayed Discrete-Time Neural Networks

This article visits the intermittent quasisynchronization control of delayed discrete-time neural networks (DNNs). First, an event-dependent intermittent mechanism is originally designed, which is described by the Lyapunov function and three non-negative real regions. The distinctive feature is that the controller starts to work only when the trajectory of the Lyapunov function goes into the presupposed work region. The proposed method fundamentally changes the principle of the existing intermittent control schemes. Under the proposed framework of the intermittent mechanism, the work/rest time of the controller is aperiodic, unpredictable, and initial value dependent. Second, several succinct sufficient conditions in terms of linear matrix inequalities are developed to achieve the quasisynchronization of the considered DNNs. A simple optimization algorithm is established to compute the control gains and the Lyapunov matrices such that synchronization error is stabilized to the smallest convergence region. Finally, two simulation examples are provided to demonstrate the feasibility of the designed intermittent mechanism.

Discrete-Time Impulsive Adaptive Dynamic Programming

In this paper, a new iterative adaptive dynamic programming (ADP) algorithm is developed to solve optimal impulsive control problems for infinite horizon discrete-time nonlinear systems. Considering the constraint of the impulsive interval, in each iteration, the iterative impulsive value function under each possible impulsive interval is obtained, and then the iterative value function and iterative control law are achieved. A new convergence analysis method is developed which proves an iterative value function to converge to the optimum as the iteration index increases to infinity. The properties of the iterative control law are analyzed, and the detailed implementation of the optimal impulsive control law is presented. Finally, two simulation examples with comparisons are given to show the effectiveness of the developed method.

Improved Sliding Mode Control for Finite-Time Synchronization of Nonidentical Delayed Recurrent Neural Networks

This brief further explores the problem of finite-time synchronization of delayed recurrent neural networks with the mismatched parameters and neuron activation functions. An improved sliding mode control approach is presented for addressing the finite-time synchronization problem. First, by employing the drive-response concept and the synchronization error of drive-response systems, a novel integral sliding mode surface is constructed such that the synchronization error can converge to zero in finite time along the constructed integral sliding mode surface. Second, a suitable sliding mode controller is designed by relying on Lyapunov stability theory such that all system state trajectories can be driven onto the predefined sliding mode surface in finite time. Moreover, it is found that the presented control approach can be conveniently verified and does not need to solve any linear matrix inequality (LMI) to guarantee the finite-time synchronization of delayed recurrent neural networks. Finally, three numerical examples are exploited to demonstrate the effectiveness of the presented control approach.

Robust modified function projective lag synchronization between two nonlinear complex networks with different-dimensional nodes and disturbances

This paper is concerned with a theoretical investigation on the robust modified function projective lag synchronization (MFPLS) between two complex networks with nonlinear couplings, different-dimensional nodes, parameter disturbances and external disturbances. Some novel robust hybrid controllers are designed to address such synchronization problems for networks with and without disturbances, in which adaptive updated laws of disturbances are adopted to attenuate the disturbances occurred in networks. By utilizing Lyapunov functional method, synchronization criteria are derived by strict theoretical proof. Compared with existing jobs, the proposed model and synchronization type are more general. Two examples are presented to verify our results.

Impulsive Control of Nonlinear Systems With Time-Varying Delay and Applications

Impulsive control of nonlinear delay systems is studied in this paper, where the time delays addressed may be the constant delay, bounded time-varying delay, or unbounded time-varying delay. Based on the impulsive control theory and some analysis techniques, a new theoretical result for global exponential stability is derived from the impulsive control point of view. The significance of the presented result is that the stability can be achieved via the impulsive control at certain impulse points despite the existence of impulsive perturbations which causes negative effect to the control. That is, the impulsive control provides a super performance to allow the existence of impulsive perturbations. In addition, we apply the theoretical result to the problem of impulsive control of delayed neural networks. Some results for global exponential stability and synchronization control of neural networks with time delays are derived via impulsive control. Three illustrated examples are given to show the effectiveness and distinctiveness of the proposed impulsive control schemes.

Exponential Synchronization and L2 -Gain Analysis of Delayed Chaotic Neural Networks Via Intermittent Control With Actuator Saturation

By using an intermittent control approach, this paper is concerned with the exponential synchronization and L₂ -gain analysis for a class of delayed master-slave chaotic neural networks subject to actuator saturation. Based on a switching strategy, the synchronization error system is modeled as a switched synchronization error system consisting of two subsystems, and each subsystem of the switched system satisfies a dwell time constraint due to the characteristics of intermittent control. A piecewise Lyapunov-Krasovskii functional depending on the control rate and control period is then introduced, under which sufficient conditions for the exponential stability of the constructed switched synchronization error system are developed. In addition, the influence of the exogenous perturbations on synchronization performance is constrained at a prescribed level. In the meantime, the intermittent linear state feedback controller can be derived by solving a set of linear matrix inequalities. More incisively, the proposed method is also proved to be valid in the case of aperiodically intermittent control. Finally, two simulation examples are employed to demonstrate the effectiveness and potential of the obtained results.

Quasi-Synchronization of Delayed Memristive Neural Networks via Region-Partitioning-Dependent Intermittent Control

This paper aims at investigating the master-slave quasi-synchronization of delayed memristive neural networks (MNNs) by proposing a region-partitioning-dependent intermittent control. The proposed method is described by three partitions of non-negative real region and an auxiliary positive definite function. Whether the control input is imposed on the slave system or not is decided by the dynamical relationships among the three subregions and the auxiliary function. From these ingredients, several succinct criteria with the associated co-design procedure are presented such that the synchronization error converges to a predetermined level. The proposed intermittent control scheme is also applied to the event-triggered control, and an intermittent event-triggered mechanism is devised to investigate the quasi-synchronization of MNNs correspondingly. Such mechanism eliminates the events in rest time, and then it reduces the amount of samplings. Finally, two illustrative examples are presented to verify the effectiveness of our theoretical results.

Effects of Subsystem and Coupling on Synchronization of Multiple Neural Networks With Delays via Impulsive Coupling

This paper from new perspectives discusses the global synchronization of multiple recurrent neural networks (MNNs) with time delays via impulsive coupling. A new concept (coupling strength) is introduced, it is a variable parameter and plays a key role on synchronization. The selection of coupling strength can bring more convenience to the design of the impulsive coupling controller. Four results are presented for the synchronization of MNNs with time delays by using impulsive coupling with the coupling gain and variable topology, where two results are dependent on topology and other two results are independent on topological connectivity. In our results, the effects of each NN, coupling topology, and coupling strength can be positive or negative role on synchronization. In addition, three examples are presented to test our results in the theory analysis.

H∞ Consensus for Linear Heterogeneous Discrete-Time Multiagent Systems With Output Feedback Control

This paper studies the H_∞ consensus problem for linear heterogeneous discrete-time multiagent systems (MASs). For a special kind of nonlinear matrix inequality, a novel method is provided to make these inequalities turn into some equivalent conditions that can be solved by utilizing the LMI toolbox. According to this method, a necessary and sufficient condition of H_∞ consensus for linear heterogeneous discrete-time MAS with output feedback control scheme and a corresponding iterative algorithm are proposed, respectively. Moreover, by employing these results, a necessary and sufficient condition that makes a general linear system with one kind of constrained controller achieve H_∞ control is provided. Finally, two numerical examples are given to illustrate the validity of our schemes.

Fixed-time synchronization criteria for complex networks via quantized pinning control

In this paper, fixed-time (FDT) synchronization of complex networks (CNs) is considered via quantized pinning controllers (QPCs). New control schemes with logarithmic quantization are designed, which not only can reduce control cost but also can save channel resources. The QPC with sign function can be used more generally than the QPC without sign function, but the QPC without sign function can be utilized to overcome the chattering phenomenon in some existing results. Based on designed Lyapunov function and different control schemes, several FDT synchronization criteria expressed by linear matrix inequalities (LMIs) are presented. Moreover, a numerical example is presented to illustrate the theoretical results.

Effects of State-Dependent Impulses on Robust Exponential Stability of Quaternion-Valued Neural Networks Under Parametric Uncertainty

This paper addresses the state-dependent impulsive effects on robust exponential stability of quaternion-valued neural networks (QVNNs) with parametric uncertainties. In view of the noncommutativity of quaternion multiplication, we have to separate the concerned quaternion-valued models into four real-valued parts. Then, several assumptions ensuring every solution of the separated state-dependent impulsive neural networks intersects each of the discontinuous surface exactly once are proposed. In the meantime, by applying the B -equivalent method, the addressed state-dependent impulsive models are reduced to fixed-time ones, and the latter can be regarded as the comparative systems of the former. For the subsequent analysis, we proposed a novel norm inequality of block matrix, which can be utilized to analyze the same stability properties of the separated state-dependent impulsive models and the reduced ones efficaciously. Afterward, several sufficient conditions are well presented to guarantee the robust exponential stability of the origin of the considered models; it is worth mentioning that two cases of addressed models are analyzed concretely, that is, models with exponential stable continuous subsystems and destabilizing impulses, and models with unstable continuous subsystems and stabilizing impulses. In addition, an application case corresponding to the stability problem of models with unstable continuous subsystems and stabilizing impulses for state-dependent impulse control to robust exponential synchronization of QVNNs is considered summarily. Finally, some numerical examples are proffered to illustrate the effectiveness and correctness of the obtained results.

Multistability of Delayed Hybrid Impulsive Neural Networks With Application to Associative Memories

The important topic of multistability of continuous-and discrete-time neural network (NN) models has been investigated rather extensively. Concerning the design of associative memories, multistability of delayed hybrid NNs is studied in this paper with an emphasis on the impulse effects. Arising from the spiking phenomenon in biological networks, impulsive NNs provide an efficient model for synaptic interconnections among neurons. Using state-space decomposition, the coexistence of multiple equilibria of hybrid impulsive NNs is analyzed. Multistability criteria are then established regrading delayed hybrid impulsive neurodynamics, for which both the impulse effects on the convergence rate and the basins of attraction of the equilibria are discussed. Illustrative examples are given to verify the theoretical results and demonstrate an application to the design of associative memories. It is shown by an experimental example that delayed hybrid impulsive NNs have the advantages of high storage capacity and high fault tolerance when used for associative memories.

Dynamical and Static Multisynchronization of Coupled Multistable Neural Networks via Impulsive Control

This paper investigates the dynamical multisynchronization and static multisynchronization problem for delayed coupled multistable neural networks with fixed and switching topologies. To begin with, a class of activation functions as well as several sufficient conditions are introduced to ensure that every subnetwork has multiple equilibrium states. By constructing an appropriate Lyapunov function and by employing impulsive control theory and the average impulsive interval method, several sufficient conditions for multisynchronization in terms of linear matrix inequalities (LMIs) are obtained. Moreover, a unified impulsive controller is designed by means of the established LMIs. Finally, a numerical example is presented to demonstrate the effectiveness of the presented impulsive control strategy.

Protocol-based state estimation for delayed Markovian jumping neural networks

This paper is concerned with the state estimation problem for a class of Markovian jumping neural networks (MJNNs) with sensor nonlinearities, mode-dependent time delays and stochastic disturbances subject to the Round-Robin (RR) scheduling mechanism. The system parameters experience switches among finite modes according to a Markov chain. As an equal allocation scheme, the RR communication protocol is introduced for efficient usage of limited bandwidth and energy saving. The update matrix method is adopted to deal with the periodic time-delays resulting from the RR protocol. The objective of the addressed problem is to construct a state estimator for the MJNNs such that the dynamics of the estimation error is exponentially ultimately bounded in the mean square with a certain upper bound. Sufficient conditions are established for the existence of the desired state estimator by resorting to a combination of the Lyapunov stability theory and the stochastic analysis technique. Furthermore, the estimator gain matrices are characterized in terms of the solution to a convex optimization problem. Finally, a numerical simulation example is exploited to demonstrate the effectiveness of the proposed estimator design strategy.

Adaptive Neural Network Control for Robotic Manipulators With Unknown Deadzone

This paper addresses the problem of robotic manipulators with unknown deadzone. In order to tackle the uncertainty and the unknown deadzone effect, we introduce adaptive neural network (NN) control for robotic manipulators. State-feedback control is introduced first and a high-gain observer is then designed to make the proposed control scheme more practical. One radial basis function NN (RBFNN) is used to tackle the deadzone effect, and the other RBFNN is also proposed to estimate the unknown dynamics of robot. The proposed control is then verified on a two-joint rigid manipulator via numerical simulations and experiments.

Stability Analysis of Continuous-Time and Discrete-Time Quaternion-Valued Neural Networks With Linear Threshold Neurons

This paper addresses the problem of stability for continuous-time and discrete-time quaternion-valued neural networks (QVNNs) with linear threshold neurons. Applying the semidiscretization technique to the continuous-time QVNNs, the discrete-time analogs are obtained, which preserve the dynamical characteristics of their continuous-time counterparts. Via the plural decomposition method of quaternion, homeomorphic mapping theorem, as well as Lyapunov theorem, some sufficient conditions on the existence, uniqueness, and global asymptotical stability of the equilibrium point are derived for the continuous-time QVNNs and their discrete-time analogs, respectively. Furthermore, a uniform sufficient condition on the existence, uniqueness, and global asymptotical stability of the equilibrium point is obtained for both continuous-time QVNNs and their discrete-time version. Finally, two numerical examples are provided to substantiate the effectiveness of the proposed results.

Chaos synchronization of uncertain chaotic systems using composite nonlinear feedback based integral sliding mode control

This paper proposes a combination of composite nonlinear feedback and integral sliding mode techniques for fast and accurate chaos synchronization of uncertain chaotic systems with Lipschitz nonlinear functions, time-varying delays and disturbances. The composite nonlinear feedback method allows accurate following of the master chaotic system and the integral sliding mode control provides invariance property which rejects the perturbations and preserves the stability of the closed-loop system. Based on the Lyapunov- Krasovskii stability theory and linear matrix inequalities, a novel sufficient condition is offered for the chaos synchronization of uncertain chaotic systems. This method not only guarantees the robustness against perturbations and time-delays, but also eliminates reaching phase and avoids chattering problem. Simulation results demonstrate that the suggested procedure leads to a great control performance.

Synchronization of General Chaotic Neural Networks With Nonuniform Sampling and Packet Missing: A Switched System Approach

This paper is concerned with the exponential synchronization issue of general chaotic neural networks subject to nonuniform sampling and control packet missing in the frame of the zero-input strategy. Based on this strategy, we make use of the switched system model to describe the synchronization error system. First, when the missing of control packet does not occur, an exponential stability criterion with less conservatism is established for the resultant synchronization error systems via a superior time-dependent Lyapunov functional and the convex optimization approach. The characteristics induced by nonuniform sampling can be used to the full because of the structure and property of the constructed Lyapunov functional, that is not necessary to be positive definite except sampling times. Then, a criterion is obtained to guarantee that the general chaotic neural networks are synchronous exponentially when the missing of control packet occurs by means of the average dwell-time technique. An explicit expression of the sampled-data static output feedback controller is also gained. Finally, the effectiveness of the proposed new design methods is shown via two examples.

Synchronization of Switched Neural Networks With Communication Delays via the Event-Triggered Control

This paper addresses the issue of synchronization of switched delayed neural networks with communication delays via event-triggered control. For synchronizing coupled switched neural networks, we propose a novel event-triggered control law which could greatly reduce the number of control updates for synchronization tasks of coupled switched neural networks involving embedded microprocessors with limited on-board resources. The control signals are driven by properly defined events, which depend on the measurement errors and current-sampled states. By using a delay system method, a novel model of synchronization error system with delays is proposed with the communication delays and event-triggered control in the unified framework for coupled switched neural networks. The criteria are derived for the event-triggered synchronization analysis and control synthesis of switched neural networks via the Lyapunov-Krasovskii functional method and free weighting matrix approach. A numerical example is elaborated on to illustrate the effectiveness of the derived results.