This brief proposes a novel neuron dynamic-growing (NDG) strategy for radial basis function neural networks (RBF NNs). Only one neuron is selected in advance relying on the system initial states, and other neurons are dynamically generated based on the designed threshold for the distance between the current NN input and the closest neuron. Compared with the RBF NN using neuron fixed evenly spaced strategy (NFES), the improved RBF NN has two major advantages: one is to extremely reduce the number of neurons, especially for the high dimensional NN inputs; and the other is to provide a theoretical criteria for the choice of NN structure parameters including the neuron center and the compact set size. To guarantee the dynamic learning ability of the improved RBF NN, the persistent excitation (PE) is verified strictly by subtly constructing the threshold and the center of newly added neurons. Simulation and experimental results illustrate that the improved RBF NN integrated into the existing dynamic learning control effectively enhances the transient control performance, reduces the computational burden, and saves data storage space.

This article presents a disturbance observer (DO)-based neural network (NN) control for a two-degree-of-freedom (2-DOF) helicopter system with input saturation, external disturbances, and output constraints. First, the uncertainties in the helicopter system are approximated using a radial basis function NN. Subsequently, a DO is used to approximate unknown compound disturbances, involving errors from NN estimation, input saturation, and external disturbances. To address the issue of output constraints imposed at a prescribed time period, a novel time-shift function and an adjusted barrier function are employed. Through the direct Lyapunov method, the boundedness of all control signals in the closed-loop system is verified. Finally, the effectiveness of the proposed control method is validated through numerical simulation results.

This article investigates internal interaction-based dynamic learning control (LC) for uncertain discrete-time strict-feedback systems. On the basis of predict technology, the original system is converted into a common n -step-ahead input-output predict model. The predict model causes every estimated neural weight to converge to n different constants using the existing control framework. To solve such a problem, the predict model is further decomposed into n one-step-ahead subsystems, which can be viewed as n independent agents. Subsequently, the distributed cooperative weight adaptive laws are designed by introducing an undirected and connected interconnection topology among subsystems. By constructing the variable relationship between the subsystems and the n -step-ahead predict model, a new internal weight interaction-based neural dynamic LC framework is proposed for the whole closed-loop system, in which estimated weights at different times share their weight knowledge. The proposed framework ensures the ultimately uniform boundedness of the closed-loop system and achieves the excellent control performance. By combining the consensus theory and a cooperative persistent excitation condition, every estimated weight along the neural input orbit is verified to exponentially converge to a close vicinity of a unique ideal constant, rather than n different constants. Consequently, the developed LC framework facilitates constant weights storage, saves the knowledge storage space, and improves the robustness of knowledge utilization. These characteristics are verified by simulation results.

This article investigates the leader–follower formation learning control (FLC) problem for discrete-time strict-feedback multiagent systems (MASs). The objective is to acquire the experience knowledge from the stable leader–follower adaptive formation control process and improve the control performance by reusing the experiential knowledge. First, a two-layer control scheme is proposed to solve the leader–follower formation control problem. In the first layer, by combining adaptive distributed observers and constructed$i_{n}$-step predictors, the leader’s future state is predicted by the followers in a distributed manner. In the second layer, the adaptive neural network (NN) controllers are constructed for the followers to ensure that all the followers track the predicted output of the leader. In the stable formation control process, the NN weights are verified to exponentially converge to their optimal values by developing an extended stability corollary of linear time-varying (LTV) system. Second, by constructing some specific “learning rules,” the NN weights with convergent sequences are synthetically acquired and stored in the followers as experience knowledge. Then, the stored knowledge is reused to construct the FLC. The proposed FLC method not only solves the leader–follower formation problem but also improves the transient control performance. Finally, the validity of the presented FLC scheme is illustrated by simulations.

This article focuses on the event-triggered (ET) fuzzy tracking control for uncertain state constrained strict-feedback systems with unknown control directions. A novel nonlinear transformed function is developed to transform the constrained system states into the counterpart without any constraints. An ingenious adaptation law is developed to co-design the control law and the ET rule, thereby effectively compensating the sampling error caused by the ET rule under unknown control directions. Based on the presented adaptation law and the Nussbaum gain technique, a novel ET fuzzy tracking control strategy is proposed, which can handle the situations with and without state constraints in a unified way without readjusting the control scheme. Subsequently, the nonlinear transformed function is extended to the time-varying state constraints, and the corresponding ET fuzzy control scheme is also modified to guarantee the closed-loop boundedness. The proposed two control strategies guarantee the satisfactory tracking performance, avoid the violation of the prescribed state constraints, and decrease the communication load effectively. Finally, the usefulness of the developed methods is verified through two simulation examples.

In this article, a new output-feedback event-triggered (ET) tracking control strategy is investigated for discrete-time strict-feedback nonlinear systems based on the adaptive critic design (ACD). An effective state observer is designed to obtain current states information, which can remove the restriction that the system can be converted to the input–output model. Subsequently, dissimilar to the n-step forward predictor with multiple neural networks (NNs) and n-step delays, variable substitution technique is used to avoid time delays and calculation burden of multiple NNs. ACD structure is constructed to acquire the optimal control strategy by critic and action NNs. An effective iterative strategy is adopted to deal with the approximation error of action NN. In this way, the update law of the controller is constructed, which is more reasonable and easier to be implemented. Meanwhile, ET mechanism is placed between the senor and the controller. A novel ET condition is constructed to moderate the burden of network communications and realize the ideal control performance. All the state signals are guaranteed to be bounded. Numerical simulations are performed to display the feasibility of the presented approach.

In this article, a novel disturbance observer-based adaptive neural control (ANC) scheme is proposed for full-state-constrained pure-feedback nonlinear systems using a new system transformation method. A nonlinear transformation function in a uniformed design framework is constructed to convert the original states with constrained bounds into the ones without any constraints. By combining an auxiliary first-order filter, an augmented nonlinear system without any state constraint is derived to circumvent the difficulty of the controller design caused by the nonaffine input signal. Based on the augmented nonlinear system, a nonlinear disturbance observer (NDO) is designed to enhance the disturbance rejection ability. Subsequently, the NDO-based ANC scheme is presented by combining the second-order filters with backstepping. The proposed scheme confines all states within the predefined bounds, eliminates the condition on both the known sign and bounds of control gains, improves the robustness of the closed-loop system, and alleviates the computational burden. Two simulation examples are performed to show the validity of the presented scheme.

In this article, an event-triggered output feedback adaptive fuzzy control scheme is developed for a class of uncertain discrete-time multiinput–multioutput (MIMO) nonlinear systems with immeasurable states and unknown control gains. Due to the existence of unmeasured states, a set of fuzzy filters are designed, then a filter-based event-triggered adaptive fuzzy control scheme is developed for discrete-time MIMO nonlinear systems. To solve the problem of state estimation caused by the coupling of control input and unknown control gains, a fuzzy filters-based state observer is developed by combining filter states. And then, a parameterized state observer is constructed to effectively estimate immeasurable state signals by the combination of the fuzzy filter states and the gradient-based fuzzy parameter updating law. Based on filter states and estimation states, an event-based adaptive fuzzy control scheme is proposed by novel intermediate errors and backstepping. To overcome the causal problem caused by the low-triangular structure, the variable substitution and a predictor are, respectively, employed to forecast the future state and reference signals. A series of stability analyses illustrates that the proposed scheme achieves immeasurable state estimations, guarantees the ultimately uniformly boundedness of the closed-loop system, and obtains the good tracking performance, while reducing communication occupancy. Finally, simulation studies on a numerical and a practical example are conducted to demonstrate the effectiveness of the proposed scheme.

This article first investigates the issue on dynamic learning from adaptive neural network (NN) control of discrete-time strict-feedback nonlinear systems. To verify the exponential convergence of estimated NN weights, an extended stability result is presented for a class of discrete-time linear time-varying systems with time delays. Subsequently, by combining the n -step-ahead predictor technology and backstepping, an adaptive NN controller is constructed, which integrates the novel weight updating laws with time delays and without the σ modification. After ensuring the convergence of system output to a recurrent reference signal, the radial basis function (RBF) NN is verified to satisfy the partial persistent excitation condition. By the combination of the extended stability result, the estimated NN weights can be verified to exponentially converge to their ideal values. The convergent weight sequences are comprehensively represented and stored by constructing some elegant learning rules with some novel sequences and the mod function. The stored knowledge is used again to develop a neural learning control scheme. Compared with the traditional adaptive NN control, the proposed scheme can not only accomplish the same or similar tracking tasks but also greatly improve the transient control performance and alleviate the online computation. Finally, the validity of the presented scheme is illustrated by numerical and practical examples.

The consumption of communication resources is an essential issue when control tasks are implemented in a wireless network environment. In order to lessen the network resources, a novel model-based (MB) adaptive event-triggered (ET) tracking control scheme is put forward in this article for strict-feedback discrete-time nonlinear systems. In this article, an event-based adaptive model is constructed by the combination of an$n$-step-ahead predictor and event-sampled neural networks. Then, the adaptive neural model is used for designing the MB ET controller. Besides, a modified ET condition is constructed without any delay. By combining a decoupled backstepping framework, the reverse Lyapunov stability technology is developed to verify the ultimate boundedness of all closed-loop signals and the convergence of the tracking error. Compared to the zero-order hold method, which keeps transmitted state signals unchanged in the interevent period, the proposed MB ET control scheme can keep the real-time update of state signals transmitted to the controller. It means that the triggering error will be smaller by the MB trigger mechanism, thereby improving the event-based tracking performance and further saving communication resources. Comparisons of simulation results are given to verify the effectiveness of the proposed control scheme.

In this article, a novel neural network (NN)-based adaptive event-triggered control scheme is developed for a class of uncertain discrete-time strict-feedback nonlinear systems with asymmetric actuator saturation. To deal with the asymmetric input saturation, a unified smooth nonlinear function is constructed to effectively characterize the limitations between the control signal and the actuator. Subsequently, the novel backstepping design process, instead of the traditional$n$-step-ahead predictor, is developed to design the stable event-triggered adaptive tracking controller by combining one neural approximator. Especially, a modified event-triggering condition is equipped into the designed controller to increase the number of triggering events at the transient-state stage. The proposed control scheme can not only achieve the good tracking performance with the avoidance of the$n$-step time delays and the improvement of the transient-state performance but also alleviate the transmission burden of the network resource, and eliminate the effect of the asymmetric actuator saturation. Numerical simulation results demonstrate the effectiveness of the control scheme proposed in this article.

This paper proposes a novel event-triggered (ET) adaptive neural control scheme for a class of discrete-time nonlinear systems in a strict-feedback form. In the proposed scheme, the ideal control input is derived in a recursive design process, which relies on system states only and is unrelated to virtual control laws. In this case, the high-order neural networks (NNs) are used to approximate the ideal control input (but not the virtual control laws), and then the corresponding adaptive neural controller is developed under the ET mechanism. A modified NN weight updating law, nonperiodically tuned at triggering instants, is designed to guarantee the uniformly ultimate boundedness (UUB) of NN weight estimates for all sampling times. In virtue of the bounded NN weight estimates and a dead-zone operator, the ET condition together with an adaptive ET threshold coefficient is constructed to guarantee the UUB of the closed-loop networked control system through the Lyapunov stability theory, thereby largely easing the network communication load. The proposed ET condition is easy to implement because of the avoidance of: 1) the use of the intermediate ET conditions in the backstepping procedure; 2) the computation of virtual control laws; and 3) the redundant triggering of events when the system states converge to a desired region. The validity of the presented scheme is demonstrated by simulation results.

This paper focuses on the observer-based output-feedback control (OBOFC) problem for a class of discrete-time strict-feedback nonlinear systems (DTSFNSs) with both multiplicative process noises and additive measurement noises. A state observer is first designed to estimate immeasurable system states, and then a novel observer-based backstepping control framework is proposed for DTSFNSs with known model information. To be specific, virtual control laws and the actual control law are derived using a variable substitution method that gets rid of the repeated accumulation of measurement noises in the recursive process. Furthermore, for technical derivation, the multiplicative noise is successively bounded by state estimation errors and controlled errors. Stability conditions are obtained to guarantee the exponential mean-square boundedness of the closed-loop system. Moreover, the nonlinear modeling uncertainties are taken into account to better reflect engineering practices. In virtue of the universal approximation property of fuzzy-logic systems, a fuzzy observer and the corresponding fuzzy output-feedback controller are simultaneously constructed to derive the stability criteria by using novel weight updated laws. Simulation studies are performed to test the validity of the proposed OBOFC scheme.

This paper investigates the event-triggered (ET) tracking control problem for a class of discrete-time strict-feedback nonlinear systems subject to both stochastic noises and limited controller-to-actuator communication capacities. The ET mechanism with fixed triggering threshold is designed to decide whether the current control signal should be transmitted to the actuator. A systematic framework is developed to construct a novel adaptive neural controller by directly applying the backstepping procedure to the underlying system. The proposed framework overcomes the noncausality problem, avoids the possible controller-related singularity problem, and gets rid of the neural approximation of the virtual control laws. Under the ET mechanism, the corresponding ET-based actuator is put forward by introducing an ET threshold compensation operator. Such a compensation operator (with an adjustable design parameter) is subtly designed based on a hyperbolic tangent function and a sign function. The threshold compensation error is analytically characterized in terms of a time-varying parameter, and the error bound is shown to be relatively small that is dependent on the adjustable design parameter. Compared with the traditional ET-based actuator without the compensation operator, the proposed ET-based actuator exhibits several distinguished features including: 1) improvement of the tracking accuracy (especially at the triggering instants); 2) further mitigation of the communication load; and 3) enlargement of the allowable range of the ET threshold. These features are illustrated by numerical and practical examples.

This paper addresses a decentralized leader-follower formation control problem for a group of fully actuated unmanned surface vehicles with prescribed performance and collision avoidance. The vehicles are subject to time-varying external disturbances, and the vehicle dynamics include both parametric uncertainties and uncertain nonlinear functions. The control objective is to make each vehicle follow its reference trajectory and avoid collision between each vehicle and its leader. We consider prescribed performance constraints, including transient and steady-state performance constraints, on formation tracking errors. In the kinematic design, we introduce the dynamic surface control technique to avoid the use of vehicle's acceleration. To compensate for the uncertainties and disturbances, we apply an adaptive control technique to estimate the uncertain parameters including the upper bounds of the disturbances and present neural network approximators to estimate uncertain nonlinear dynamics. Consequently, we design a decentralized adaptive formation controller that ensures uniformly ultimate boundedness of the closed-loop system with prescribed performance and avoids collision between each vehicle and its leader. Simulation results illustrate the effectiveness of the decentralized formation controller.

Robots are often required to generalize the skills learned from human demonstrations to fulfil new task requirements. However, skill generalization will be difficult to realize when facing with the following situations: the skill for a complex multistep task includes a number of features; some special constraints are imposed on the robots during the process of task reproduction; and a completely new situation quite different with the one in which demonstrations are given to the robot. This work proposes a new framework to facilitate robot skill generalization. The basic idea lies in that the learned skills are first segmented into a sequence of subskills automatically, then each individual subskill is encoded and regulated accordingly. Specifically, we adapt each set of the segmented movement trajectories individually instead of the whole movement profiles, thus, making it more convenient for the realization of skill generalization. In addition, human limb stiffness estimated from surface electromyographic signals is considered in the framework for the realization of human-to-robot variable impedance control skill transfer, as well as the generalization of both movement trajectories and stiffness profiles. Experimental study has been performed to verify the effectiveness of the proposed framework.

This paper presents adaptive neural tracking control of underactuated surface vessels with modeling uncertainties and time-varying external disturbances, where the tracking errors consisting of position and orientation errors are required to keep inside their predefined feasible regions in which the controller singularity problem does not happen. To provide the preselected specifications on the transient and steady-state performances of the tracking errors, the boundary functions of the predefined regions are taken as exponentially decaying functions of time. The unknown external disturbances are estimated by disturbance observers and then are compensated in the feedforward control loop to improve the robustness against the disturbances. Based on the dynamic surface control technique, backstepping procedure, logarithmic barrier functions, and control Lyapunov synthesis, singularity-free controllers are presented to guarantee the satisfaction of predefined performance requirements. In addition to the nominal case when the accurate model of a marine vessel is known a priori, the modeling uncertainties in the form of unknown nonlinear functions are also discussed. Adaptive neural control with the compensations of modeling uncertainties and external disturbances is developed to achieve the boundedness of the signals in the closed-loop system with guaranteed transient and steady-state tracking performances. Simulation results show the performance of the vessel control systems.

This paper presents dynamic learning from adaptive neural control (ANC) with prescribed tracking error performance for an n-link robot manipulator subjected to unknown system dynamics and external disturbances. To achieve the prescribed performance, a performance function is introduced to describe the performance restrictions on tracking errors, and then specific performance requirements are served as a priori condition of tracking control design. By an error transformation method, the constrained tracking control problem of the original robot manipulator is transformed into the stabilization problem of an unconstrained augmented system. Then, a novel ANC scheme is proposed for the unconstrained system by combining a filter tracking error with radial basis function (RBF) neural network (NN) approximator, and all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. The external disturbances might make it difficult to achieve the accurate convergence of NN weight estimates. To overcome this difficulty, an appropriate state transformation is introduced to transform the closed-loop system into a linear time-varying system with small perturbed terms. Under partial persistent excitation condition of RBF NNs, the convergence of NN weight estimates is guaranteed, and then the experienced knowledge on the unknown robot manipulator dynamics can be stored with NN constant weights. Using the experienced knowledge, a static neural learning control is proposed to improve the system performances without time-consuming online parameter adjustment process, and the proposed learning control can also guarantee the prescribed transient and steady-state tracking control performance. Simulation results demonstrate the effectiveness of the proposed method.

This paper focuses on dynamic learning from neural control for a class of nonlinear strict-feedback systems with predefined tracking performance attributes. To reduce the number of neural network (NN) approximators used and make the convergence of neural weights verified easily, state variables are introduced to transform the state-feedback control of the original strict-feedback systems into the output-feedback control of the system in the normal form. Then, using the output error transformation based on performance functions, the constrained tracking control problem of the normal systems is transformed into the stabilization problem of an equivalent unconstrained one. By combining the backstepping method, a high-gain observer with radial basis function (RBF) NNs, a novel adaptive neural control (ANC) scheme is proposed to guarantee the predefined tracking error performance as well as the ultimate boundedness of all other closed-loop signals. In particular, only one NN is employed to approximate the lumped unknown system dynamics during the controller design. Under the satisfaction of the partial persistent excitation condition for RBF NNs, the proposed stable ANC scheme is shown to be capable of achieving knowledge acquisition, expression, and storage of unknown system dynamics. The stored knowledge is reused to develop a neural learning controller for improving the control performance of the closed-loop system. When the initial condition satisfies the predefined performance, the proposed neural learning control can still guarantee the predefined tracking performance. Simulation results on a third-order one-link robot are given to show the effectiveness of the proposed method.

Learning plays an essential role in autonomous control systems. However, how to achieve learning in the nonstationary environment for nonlinear systems is a challenging problem. In this paper, we present learning method for a class of n th-order strict-feedback systems by adaptive dynamic surface control (DSC) technology, which achieves the human-like ability of learning by doing and doing with learned knowledge. To achieve the learning, this paper first proposes stable adaptive DSC with auxiliary first-order filters, which ensures the boundedness of all the signals in the closed-loop system and the convergence of tracking errors in a finite time. With the help of DSC, the derivative of the filter output variable is used as the neural network (NN) input instead of traditional intermediate variables. As a result, the proposed adaptive DSC method reduces greatly the dimension of NN inputs, especially for high-order systems. After the stable DSC design, we decompose the stable closed-loop system into a series of linear time-varying perturbed subsystems. Using a recursive design, the recurrent property of NN input variables is easily verified since the complexity is overcome using DSC. Subsequently, the partial persistent excitation condition of the radial basis function NN is satisfied. By combining a state transformation, accurate approximations of the closed-loop system dynamics are recursively achieved in a local region along recurrent orbits. Then, the learning control method using the learned knowledge is proposed to achieve the closed-loop stability and the improved control performance. Simulation studies are performed to demonstrate the proposed scheme can not only reuse the learned knowledge to achieve the better control performance with the faster tracking convergence rate and the smaller tracking error but also greatly alleviate the computational burden because of reducing the number and complexity of NN input variables.

This paper studies the problem of learning from adaptive neural network (NN) control of a class of nonaffine nonlinear systems in uncertain dynamic environments. In the control design process, a stable adaptive NN tracking control design technique is proposed for the nonaffine nonlinear systems with a mild assumption by combining a filtered tracking error with the implicit function theorem, input-to-state stability, and the small-gain theorem. The proposed stable control design technique not only overcomes the difficulty in controlling nonaffine nonlinear systems but also relaxes constraint conditions of the considered systems. In the learning process, the partial persistent excitation (PE) condition of radial basis function NNs is satisfied during tracking control to a recurrent reference trajectory. Under the PE condition and an appropriate state transformation, the proposed adaptive NN control is shown to be capable of acquiring knowledge on the implicit desired control input dynamics in the stable control process and of storing the learned knowledge in memory. Subsequently, an NN learning control design technique that effectively exploits the learned knowledge without re-adapting to the controller parameters is proposed to achieve closed-loop stability and improved control performance. Simulation studies are performed to demonstrate the effectiveness of the proposed design techniques.

This paper studies learning from adaptive neural control (ANC) for a class of pure-feedback nonlinear systems with unknown non-affine terms. The existence of the cascade structure and unknown non-affine terms makes it very difficult to achieve learning using previous methods. To overcome these difficulties, firstly, the implicit function theorem and the mean value theorem are combined to transform the closed-loop system into a semi-affine form during the control design process. Then, we decompose the stable closed-loop system into a series of linear time-varying (LTV) perturbed subsystems with the appropriate state transformation. Using a recursive design, the partial persistent excitation (PE) condition for the radial basis function (RBF) neural network (NN) is satisfied during tracking control to a recurrent reference trajectory. Under the PE condition, accurate approximations of the closed-loop system dynamics are recursively achieved in a local region along recurrent orbits of closed-loop signals. Subsequently, the NN learning control method which effectively utilizes the learned knowledge without re-adapting to the unknown system dynamics is proposed to achieve the closed-loop stability and the improved control performance. Simulation studies are performed to demonstrate the effectiveness of the proposed scheme.

This paper studies learning from adaptive neural control (ANC) for a class of nonlinear strict-feedback systems with unknown affine terms. To achieve the purpose of learning, a simple input-to-state stability (ISS) modular ANC method is first presented to ensure the boundedness of all the signals in the closed-loop system and the convergence of tracking errors in finite time. Subsequently, it is proven that learning with the proposed stable ISS-modular ANC can be achieved. The cascade structure and unknown affine terms of the considered systems make it very difficult to achieve learning using existing methods. To overcome these difficulties, the stable closed-loop system in the control process is decomposed into a series of linear time-varying (LTV) perturbed subsystems with the appropriate state transformation. Using a recursive design, the partial persistent excitation condition for the radial basis function neural network (NN) is established, which guarantees exponential stability of LTV perturbed subsystems. Consequently, accurate approximation of the closed-loop system dynamics is achieved in a local region along recurrent orbits of closed-loop signals, and learning is implemented during a closed-loop feedback control process. The learned knowledge is reused to achieve stability and an improved performance, thereby avoiding the tremendous repeated training process of NNs. Simulation studies are given to demonstrate the effectiveness of the proposed method.

This paper presents adaptive neural tracking control for a class of non-affine pure-feedback systems with multiple unknown state time-varying delays. To overcome the design difficulty from non-affine structure of pure-feedback system, mean value theorem is exploited to deduce affine appearance of state variables x(i) as virtual controls α(i), and of the actual control u. The separation technique is introduced to decompose unknown functions of all time-varying delayed states into a series of continuous functions of each delayed state. The novel Lyapunov-Krasovskii functionals are employed to compensate for the unknown functions of current delayed state, which is effectively free from any restriction on unknown time-delay functions and overcomes the circular construction of controller caused by the neural approximation of a function of u and [Formula: see text] . Novel continuous functions are introduced to overcome the design difficulty deduced from the use of one adaptive parameter. To achieve uniformly ultimate boundedness of all the signals in the closed-loop system and tracking performance, control gains are effectively modified as a dynamic form with a class of even function, which makes stability analysis be carried out at the present of multiple time-varying delays. Simulation studies are provided to demonstrate the effectiveness of the proposed scheme.

This paper proposes a novel adaptive neural control scheme for a class of perturbed strict-feedback nonlinear time-delay systems with unknown virtual control coefficients. Based on the radial basis function neural network online approximation capability, an adaptive neural controller is presented by combining the backstepping approach and Lyapunov-Krasovskii functionals. The proposed controller guarantees the semiglobal boundedness of all the signals in the closed-loop system and contains minimal learning parameters. Finally, three simulation examples are given to demonstrate the effectiveness and applicability of the proposed scheme.