The communication links of multiagent systems (MASs) could be compromised by stealthy cyber attacks. To solve the problem, this paper proposes distributed nodes detection and event-triggered links isolation of stealthy attacks in MASs by topology-oriented watermarking (TOW). Firstly, the limitation of traditional MASs against distributed pole-dynamics attacks (dPDAs) is revealed. Secondly, different from existing water-marking methods for single-agent systems, a novel TOW method with four optional parameters selection schemes is proposed for distributed nodes detection in MASs, where it is proved that only one of optional parameters selection schemes can simultaneously possesses complexity and resistance to collusive dPDAs. Moreover, TOW combines the advantages of multiplicative and additive watermarking to achieve dPDAs detection. Furthermore, the positive relationship between dPDAs detection performance and the coupling factor of attack topology, attack signal and TOW parameters is quantified. Thirdly, a new TOW-based event-triggered links isolation algorithm incorporating multiple link-deletion observers is proposed, where TOW is leveraged to trigger running of link-deletion observers. It is proved that the compromised links can be correctly isolated. Finally, the simulation results verify our proposed scheme.

Multiagent systems (MASs) are increasingly vulnerable to cyber attacks that exploit limited system resource to evade detection while causing disruptions. This paper investigates the resource-sensitive cyber attacks for MASs. Firstly, distributed replay attack (dRAs) and distributed pole-dynamics attack (dPDAs) are proposed, where dRAs utilize available resources of agents historical measurements, and dPDAs utilize available resources of agent parameter. Secondly, the relationship between available resources and stealthiness of cyber attack are revealed, where both dRAs and dPDAs can be with 0-stealthiness under different available resources. Thirdly, the relationship between available resources and destructiveness of cyber attack are revealed, where dRAs is not with ck-s-destructiveness (i.e., cyber attacks can achieve state trajectory destructiveness without disrupting consensus), because the state trajectory can only be driven to the historical one due to requiring historical measurements. Oppositely, dPDAs can be with ck-s-destructiveness, where the state consistency trajectory can be driven to any one designed by the attacker due to requiring agent parameter. Finally, numerical examples are provided to verify the effectiveness of the proposed attacks.

The main problem addressed in this paper is the controller design under composite switching in large-scale interconnected systems. Composite switching systems are widely present in large-scale cyber-physical systems, such as microgrids and industrial process control systems, motivating the need for scalable and efficient control methods. To this end, this paper develops an integrated model for a large-scale system formed by interconnecting individual subsystems, each of which is a composite switched system, and investigates the scalable control problem for such an interconnected composite switched system (ICSS), a unified switching signal governs all subsystems, where the signal is generated through a logical function driven by random variable inputs. First, the semi-tensor product (STP) technique, combined with the dimension expansion method, is used to compress the composite switching signal, treating it as part of the state and cascading it with the states of each subsystem. This results in a new system state and an expanded interconnected system model described in the form of a linear time-invariant (LTI) system. The key contributions of this paper include the establishment of an integrated model that captures the interconnection and composite switching behavior of the large-scale system, as well as the development of a scalable distributed state feedback control algorithm that leverages this unified model. Based on this, the LTI model of the interconnected large system under distributed state feedback control is provided. Next, the necessary and sufficient conditions for the mean-square stability of this large system are given, and the scalability and design method of the distributed state feedback strategy are implemented based on a recursive algorithm. Finally, quantitative simulation results based on a DC microgrid example demonstrate that system state trajectories decay to zero under allowable composite switching, confirming the theoretical mean-square stability and demonstrating the practical feasibility of the control framework.

This paper investigates the voltage regulation problems in DC microgrids (DC-MGs) with nonlinear ZIP loads under malicious cyber-physical attacks, proposing an adaptive decentralized fully-actuated two-step control strategy rooted in the fully actuated system (FAS) approach. At first, a unified fully actuated distributed generation unit (FA-DGU) model is developed, integrating nonlinear load dynamics, renewable energy intermittency, attack-induced disturbances, and power line coupling into a lumped uncertainty framework. Then, a decentralized controller is designed using Radial Basis Function neural networks (RBFNNs) to adaptively approximate unknown ZIP load characteristics and attack-generated perturbations, eliminating dependence on communication networks while enabling plug-and-play (PnP) operations through purely local measurements. Following this, Lyapunov stability analysis rigorously proves the asymptotic convergence of voltage regulation errors at the point-of-common-coupling (PCC) under heterogeneous uncertainties. Finally, various uncertain scenarios in DC-MGs are simulated using MATLAB/Simulink, thereby verifying the feasibility and effectiveness of the proposed control method.

This article investigates the stealthy distributed pole-dynamics attacks (dPDAs) detection for multiagent systems (MASs) by distributed additive watermarking (DAW). First, the limitation of traditional MASs for dPDAs is revealed, where dPDAs cannot be detected. Second, unlike the well-established single-agent additive watermarking, to eliminate the side effect of watermarking signal on system state and enable dPDAs detection, the proposed DAW adds watermarking to the control signal of any agent for transmission and removes watermarking of the control signal after receiving it. Meanwhile, the covariance of the watermarking signal in DAW for all agents is different from each other to enable compromised links isolation. Furthermore, the relationship between the dPDAs detection performance and DAW is quantified in the sense of expectation, where the dPDAs detection performance is directly proportional to the sum of the watermarking covariance of the compromised links. Third, leveraging the relationship between dPDAs detection performance and DAW, a DAW-based link isolation scheme is proposed to accurately isolate the compromised links by comparing with the detection function and its approximation, where the approximation of the detection function is iteratively calculated on all possible attack links combination for the compromised agent. As a result, the adverse impacts of dPDAs on MASs are mitigated. Finally, simulation results are conducted to validate the theoretical results.

As data sharing is essential in applications such as distributed state estimation of cyber-physical power systems (CPPSs), the issue of data privacy leakage among individual regional system operators is increasingly concerned. To solve the issue, this paper proposes a new distributed security state estimation (DSSE) method based on homomorphic encryption for privacy-preserving consensus. First, considering that regional measurement data managed by individual regional system operators could be attacked by false data injection attacks (FDIAs), a new active attack detection method based on the statistical characteristics of the watermarking signal before and after FDIAs is proposed to improve detection proactivity and accuracy, and it is found that the detection accuracy is positively correlated with the watermarking intensity when the signal to interference plus noise ratio (SINR) is greater than 10db. Second, considering that each regional system operator needs to exchang intermediate data under the premise of protecting data privacy to guarantee the consistency process of distributed state estimation, a homomorphic encryption (HE)-based privacy-preserving consensus method is proposed, where a hash function-based dual verification mechanism is presented to prevent ciphertext data from being tampered by FDIAs. Third, according to the detection results and data compensation mechanism, a local secure state estimation model is proposed, and it is proved that the upper and lower bounds of the reconstructed estimation error covariance are not only related to system parameters and external noise but also negatively related to the compensation error. Furthermore, according to a Lyapunov function including privacy-preserving consensus, sufficient condition for consistency is proven, which depends on Laplacian matrix of the system and the iteration step size. Finally, experimental results demonstrate the feasibility and effectiveness of the proposed dynamic watermarking-based active attack detector and distributed secure state estimation method for CPPSs. Moreover, the computational overhead of incorporating advanced IND-CPA countermeasures (i.e., ciphertext re-randomization and branchless arithmetic) is quantified, which confirms the feasibility of practical deployment.

This article investigates the observer-based adaptive decentralized control problem for a class of uncertain interconnected nonlinear fully actuated systems (FAS), considering nonsmooth actuator dynamics including actuator failures and unknown control gains. Based on the dynamic gain scaling technique, a dynamic state observer is constructed. By utilizing the high-order FAS (HOFAS) approach, an adaptive decentralized output feedback controller is designed and a closed-loop structure of the fully actuated subsystems is derived. This structure takes actuator loss of effectiveness, unknown control gains, and unstructured uncertainties into account in the interconnected time-delay subsystems. By selecting suitable Lyapunov-Krasovskii (L-K) functionals, the time-delay terms can be removed, ensuring that all signals of the overall closed-loop system converge to a bounded region. Finally, two simulation examples validate the efficacy of the proposed strategy.

Plug-and-play (PnP) operations of distributed generation units (DGUs) with constant power loads (CPLs) often destabilize dc microgrids (DCmGs). To address this issue, this article proposes a scalable neural network control strategy for nonlinear DCmGs with CPLs, enabling seamless PnP operations of DGUs. A radial basis function neural network is employed to handle the uncertain CPL nonlinearity without requiring any prior knowledge. A structured Lyapunov matrix is utilized to eliminate the coupling effects of power lines by reshaping them into a Laplacian matrix structure. Within this framework, a scalable neural network control approach is proposed, integrating a nominal controller with explicit gain inequalities and an adaptive controller governed by an adaptation law. This approach operates locally, independent of other DGUs and power lines, ensuring PnP operations and maintaining uniformly ultimately bounded stability. The effectiveness of the proposed method is validated through case studies on a modified IEEE 37-bus test system.

Multiplicative watermarking can enhance cyber attacks detection capacity of quantization-involved networked control system (QNCSs). This article presents an iteration-form solution to conventional summation-form multiplicative watermarking (SMW). First, limitation of conventional SMW for QNCSs is revealed, where there exists high design complexity whilst decreasing measurement perturbation from SMW and achieving successful replay attacks (RAs) detection. Second, a new iteration-form multiplicative watermarking (IMW) is designed by leveraging a keys transformation to reconstruct SMW with multiple coupled keys, where design complexity is greatly reduced and there are only dual decoupled keys to be designed. Furthermore, a positive correlation between value of one key and measurement perturbation from IMW is provided. Third, a positive correlation between ratio of the other key at different instants and RAs detection performance of IMW is explored by using residual covariance analysis. Finally, experimental results from networked inverted pendulum systems confirm the feasibility and effectiveness of new IMW.

Coupling, as an important characteristic of cyber-physical systems (CPS), brings opportunities and challenges to the control and application of industrial systems. This article mainly analyses the role of the cyber layer in the process of controlling the physical layer and gives a security state assessment method for the CPS after the Denial-of-Service (DoS) attack. First, we develop an advanced CPS coupling model that intricately captures the interactions between the physical and cyber layers, emphasizing both data and topology coupling. Unlike existing models that often overlook these aspects, our model offers a more comprehensive understanding of the complex interdependencies within CPS. Second, we propose a unique method for partitioning network layer nodes based on the physical layer's control requirements. Unlike traditional approaches that treat network layer nodes as homogeneous, our method categorizes nodes into sensor, controller, and actuator areas, each with distinct roles in the control process. This differentiation ensures that data flow within the network layer follows a fixed and logical direction, enhancing the overall system efficiency and reliability. Third, we introduce a security state evaluation factor that leverages the network layer partitioning to assess the system's resilience under cyber-attacks. This evaluation provides a detailed analysis of the system's security status, ensuring that potential vulnerabilities are identified and mitigated. Together, these innovations contribute to a more robust and practical framework for CPS design and analysis, with significant implications for enhancing system security and performance in real-world applications.

Secure information exchange of the devices among different domains for cyber-physical power systems (CPPSs) is important yet challenging. Conventional blockchain-based authentication schemes generally adopt single blockchain and signature algorithm, only achieving intradomain or interdomain authentication with lower efficiency, and always failing to meet the confidentiality requirement during information interaction in CPPSs. To address these issues, this paper proposes a cross-domain authentication scheme based on distributed two-layer collaborative blockchains for CPPSs. First, a two-layer-blockchain collaborative authentication architecture is designed, deploying edge servers and taking into account the distributed characteristic of CPPSs. Second, a signcryption algorithm is developed by combining elliptic curve cryptography (ECC) with certificateless cryptography (CLC), which guarantees both the confidentiality and non-repudiation of the block information simultaneously. Furthermore, upper-layer alliance blockchain and lower-layer private blockchain are formed and interact collaboratively via index and Merkle proof, achieving intradomain and interdomain authentication with higher efficiency. Finally, a security analysis and experimental results are presented to superiorly demonstrate the security features and performance in comparison to other schemes in literature.

When traditional pole-dynamics attacks (TPDAs) are implemented with nominal models, model mismatch between exact and nominal models often affects their stealthiness, or even makes the stealthiness lost. To solve this problem, this article presents a novel stealthy measurement-aided pole-dynamics attacks (MAPDAs) method with model mismatch. First, the limitations of TPDAs using exact models are revealed. Second, to handle the limitations, the proposed MAPDAs method is designed by using an adaptive control strategy, which can keep the stealthiness. Moreover, it is easier to implement as only the measurements are needed in comparison with the existing methods requiring both measurements and control inputs. Third, the performance of the proposed MAPDAs method is explored using convergence of multivariate measurements, and MAPDAs with model mismatch have the same stealthiness and similar destructiveness as TPDAs. Finally, experimental results from a networked inverted pendulum system confirm the feasibility and effectiveness of the proposed method.

Secure control for cyber-physical power systems (CPPSs) under cyber attacks is a challenging issue. Existing event-triggered control schemes are generally difficult to mitigate the impact of cyber attacks and improve communication efficiency simultaneously. To solve such two problems, this article studies secure adaptive event-triggered control for the CPPSs under energy-limited denial-of-service (DoS) attacks. A new DoS-dependent secure adaptive event-triggered mechanism (SAETM) is developed, where DoS attacks are taken into account when designing the trigger mechanisms. Sufficient conditions are derived to ensure the CPPSs to be uniformly ultimate boundedness stable, and the entering time when the state trajectories of the CPPSs are guaranteed to stay in the secure region is also given. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed control method.

The plugging-in/-out of renewable distributed generation units (DGUs) often alters the microgrid size and coupling terms, resulting in computational burdens and voltage shocks. This article proposes a novel scalable fuzzy voltage control scheme for nonlinear direct current microgrids (DCmGs) composed of DGUs and constant power loads (CPLs) interconnected via power lines. First, a Takagi–Sugeno fuzzy DCmG model with CPL is formulated to capture nonlinear characteristics and diverse transient behaviors. Then, a scalable fuzzy control approach is developed to mitigate negative coupling effects of power lines. This is achieved by a novel argument that leverages dissipativity theory to transform such effects into linear matrix inequalities and subsequently imposes constraints on their sequential principal minors. Specifically, the proposed control method operates locally and independently of other DGUs and line couplings, enabling seamless plug-and-play operations without updating any controllers. Finally, theoretical results are validated through simulations using the MATLAB/SimPowerSystems toolbox.

Most of the existing signcryption schemes utilize a key generation center to generate pseudonyms without updating, and usually opt for bilinear pairing to design authentication schemes. The disadvantage is that these schemes not only incur heavy computation and communication overheads during information interaction, but also can not eliminate security risks arising from not updating pseudonyms. These limitations render them less effective for smart terminals (STs) with limited computation and communication resources in cyber-physical power systems. The main purpose of this article is to explore a streamlined signcryption scheme tailored for resource-limited STs. To achieve this, a dynamical pseudonym self-generation mechanism (DPSGM) is first introduced to prevent the source from being linked and protect privacy. In addition, a streamlined signcryption scheme is designed based on elliptic curve cryptography and certificateless cryptography, integrating seamlessly with DPSGM. This design significantly reduces computation and communication burdens during information interaction. Finally, a real experimental platform is established to demonstrate the feasibility and effectiveness of the proposed scheme. Visual interfaces show the entire secure interaction process and the resistance to attacks.

When networked control systems (NCSs) suffer from denial-of-service (DoS) attacks, system observability can be destroyed. To analyse this problem, this paper mainly investigates the observability of NCSs under DoS attacks. First, how DoS attacks destroy the observability of NCSs is revealed, where the original observability matrix needs to be reconstructed by considering the characteristics of DoS attacks. Second, the necessary and sufficient conditions for judging the observability of NCSs under arbitrary DoS attacks are proved, where the observability matrix needs to be calculated constantly. Third, the observability index and DoS attacks parameters are analysed, while the observability judging criteria under periodic and aperiodic DoS attacks are provided. Finally, the effectiveness and feasibility of the proposed method are verified by numerical examples.

The existing identity security schemes (e.g., based on bilinear pairing) have high computational complexity and large bytes of variables, which results in high computation and communication costs. It is difficult to apply the schemes to resource-constrained (i.e., computation and communication) devices. Moreover, most of these schemes adopt a fixed cycle key update strategy compromising the security of authentication schemes or dynamic (real-time) key update strategy with high computational cost. To solve these issues, this article explores a novel revocable lightweight authentication scheme for resource-constrained devices in cyber–physical power systems (CPPSs). First, a lightweight authentication scheme combined elliptic-curve cryptography (ECC) and certificateless cryptography (CLC) is proposed to negotiate a secure session key, which can achieve mutual authentication with low computation and communication costs. Second, aiming at security problem caused by key leakage, a real-time key update strategy with low computational cost is designed to improve the security of identity authentication. Third, according to a hardness assumption of the elliptic-curve discrete logarithm problem (ECDLP), theoretical analysis rigorously proves that the proposed authentication scheme ensures the security with respect to existential unforgeability against adaptively chosen message attacks (EUF-CMAs). Finally, experimental results confirm the feasibility and effectiveness of the proposed authentication scheme.

Dynamic watermarking schemes can enhance the cyberattack detection capability of networked control systems (NCSs). This article presents a linear event-triggered solution to conventional dynamic watermarking (CDW) schemes. First, the limitations of CDW schemes for event-triggered state estimation-based NCSs are investigated. Second, a new event-triggered dynamic watermarking (ETDW) scheme is designed by treating watermarking as symmetric key encryption, based on the limit convergence theorem in probability. Its security property against the generalized replay attacks (GRAs) is also discussed in the form of bounded asymptotic attack power. Third, finite sample ETDW tests are designed with matrix concentration inequalities. Finally, experimental results of a networked inverted pendulum system demonstrate the validity of our proposed scheme.

Confronted with the increasing penetration of renewable energy and hydrogen-powered vehicles (HVs), energy and transportation systems need to be optimized coordinately to improve the economy and reduce carbon emissions. However, the effect of dynamic hydrogen pricing on the optimization performance of energy and transportation systems is scarcely investigated. To realize the low-carbon and economic operation of energy and transportation systems, a hierarchical game between the distributed energy station (DES) and HVs under dynamic hydrogen pricing is put forward. First, a novel dynamic hydrogen pricing mechanism related to the proportion of renewable energy in the energy supply is presented and incorporated into the game optimization, which will promote hydrogen production using renewable energy and minimize the DES operation cost. At the game equilibrium, the HVs completing equivalent aggregation realize balanced hydrogen refueling at the guidance of dynamic hydrogen pricing. Meanwhile, the raised routing optimization method can select a minimum cost route for HVs to refuel based on traffic information. Finally, the effectiveness of the proposed hierarchical game optimization strategy and model is verified by simulations.

We here investigate the secure control of networked control systems developing a new dynamic watermarking (DW) scheme. First, the weaknesses of the conventional DW scheme are revealed, and the tradeoff between the effectiveness of false data injection attack (FDIA) detection and system performance loss is analyzed. Second, we propose a new DW scheme, and its attack detection capability is interrogated using the additive distortion power of a closed-loop system. Furthermore, the FDIA detection effectiveness of the closed-loop system is analyzed using auto/cross-covariance of the signals, where the positive correlation between the FDIA detection effectiveness and the watermarking intensity is measured. Third, the tolerance capacity of FDIA against the closed-loop system is investigated, and theoretical analysis shows that the system performance can be recovered from FDIA using our new DW scheme. Finally, the experimental results from a networked inverted pendulum system demonstrate the validity of our proposed scheme.

Aiming at the challenges of networked visual servo control systems, which rarely consider network communication duration and image processing computational cost simultaneously, we here propose a novel platform for networked inverted pendulum visual servo control using H∞ analysis. Unlike most of the existing methods that usually ignore computational costs involved in measuring, actuating, and controlling, we design a novel event-triggered sampling mechanism that applies a new closed-loop strategy to dealing with networked inverted pendulum visual servo systems of multiple time-varying delays and computational errors. Using the Lyapunov stability theory, we prove that the proposed system can achieve stability whilst compromising image-induced computational and network-induced delays and system performance. In the meantime, we use H∞disturbance attenuation level γ for evaluating the computational errors, whereas the corresponding H∞controller is implemented. Finally, simulation analysis and experimental results demonstrate the proposed system performance in reducing computational errors whilst maintaining system efficiency and robustness.

Smart grid (SG) represents a large-scale network system with the tight integration of a physical power network and an information network, which makes it more vulnerable to hybrid cyber attacks against different regional subsystems. First, an alternating direction method of multipliers-based distributed state estimation method is developed to overcome the limitation of conventional state estimation and performance analysis of SG against a single type of cyber attacks. Regional subsystems are partitioned via the K-means method. Second, a novel distributed state estimation method integrated with the characteristics of data deception attacks and denial of service (DoS) attacks is proposed to account for the simultaneous presence of different cyber attacks on individual regional subsystems. Third, the convergence of a distributed state estimation algorithm under hybrid cyber attacks is proved theoretically. Furthermore, the relationships between the convergence and algorithm parameters as well as the occurring probability of DoS attacks are established. Finally, the simulations on a modified IEEE 118-bus system are given to demonstrate the feasibility and effectiveness of the proposed method.

As a novel family member of the redox flow batteries (RFBs), the single flow zinc-nickel battery (ZNB) without ion exchange membranes has attracted a lot of interests in recent years due to the high charging and discharging efficiencies. To understand the electrical behaviour is a key for proper battery management system. Unlike the electrochemical mechanism models and equivalent circuit models, the neural network based black-box model does not need knowledge about the electrochemical reactions and is a promising and adaptive approach for the ZNB battery modelling. In this paper, a compact radial basis function neural network is developed using a two-stage layer selection strategy to determine the network structure. While Jaya optimization is utilized to determine the non-linear parameters in the selected hidden nodes of the resultant RBF neural network (RBF-NN) model. The proposed method is implemented to model the ZNB to capture the non-linear electric behaviours through the readily measurable input signals. Experimental results manifest the accurate prediction capability of the resultant neural model and confirm the effectiveness of the proposed approach.

Currently cyber-security has attracted a lot of attention, in particular in wireless industrial control networks (WICNs). In this paper, the stability of wireless networked control systems (WNCSs) under deception attacks is studied with a token-based protocol applied to the data link layer (DLL) of WICNS. Since deception attacks cause the stability problem of WNCSs by changing the data transmitted over wireless network, it is important to detect deception attacks, discard the injected false data and compensate for the missing data (i.e., the discarded original data with the injected false data). The main contributions of this paper are: (1) With respect to the character of the token-based protocol, a switched system model is developed. Different from the traditional switched system where the number of subsystems is fixed, in our new model this number will be changed under deception attacks. (2) For this model, a new Kalman filter (KF) is developed for the purpose of attack detection and the missing data reconstruction. (3) For the given linear feedback WNCSs, when the noise level is below a threshold derived in this paper, the maximum allowable duration of deception attacks is obtained to maintain the exponential stability of the system. Finally, a numerical example based on a linearized model of an inverted pendulum is provided to demonstrate the proposed design.

This paper is concerned with the stability for visual servo inverted pendulum system with variable image processing time delay. Firstly, the image processing time delay is inevitably introduced in visual servo inverted pendulum system due to the visual feedback, where its upper and lower bounds are obtained via the statistics analysis on time delay data. Specially, a Markov chain is then employed to describe the characters of image processing time delay. Furthermore, a discrete model of inverted pendulum system with time-varying image processing time delay is presented and the state-feedback controller is designed. Finally, simulation results confirm the feasibility and effectiveness of the proposed method.

This study focuses on networked H ∞ fuzzy filtering for Markovian jump Takagi–Sugeno (T–S) fuzzy systems. Since the traditional PDC method is ineffective under network environments, a flexible filter design method is provided with imperfect premise matching. First, a unified T–S fuzzy error model is provided by considering the mismatched grades of membership. Second, by use of the constructed model and the Markovian switched Lyapunov–Krasovskii functional, a stability and a stabilisation criteria of the fuzzy filtering error system are derived, in which it is no required that the fuzzy filter shares the same membership functions with the fuzzy systems. Therefore, the designed filter is available under network environments since the mismatched premises induced by networks are well considered. Finally, the effectiveness of the proposed new design method is illustrated by two examples.

Most existing networked filtering methods are proposed to solve the filtering problem under single (wireless or wired) network environment. However, with the development of the networked technology, hybrid networks that integrate wireless and wired networks are gradually employed in industrial fields. The main objectiveness of the paper is to investigate H2/H∞filtering of hybrid networks with random packet losses. Firstly, two different Bernoulli random processes are used to describe the characters of the random packet losses of wireless and wired networks, respectively. A model of filtering error dynamic system is then proposed, and a sufficient condition of stochastic stability is further obtained, where the relationship between the system stability, the filter parameters and the packet losses parameters is established. Finally, simulation results confirm the effectiveness and feasibility of the proposed method.