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In this article, an adaptive switching controller is designed for the stabilization control and the tracking control of nonlinear systems suffering from full state constraints, unknown control directions, and system uncertainties. Featured with changing signs and unbounded increasing magnitudes, a switching control structure is proposed to deal with the unknown control direction problem. In order to explore and finally fix the correct control direction, an adaptive switching rule, a switching function, and a reference function are constructed. An integral Lyapunov function is adopted for the design of the control law, which ensures that full state constraints will not be violated. Using this method, the nonlinear system can be properly controlled without extra estimations or approximations. The boundedness of all the states and the stability of the system are analyzed and guaranteed. Finally, simulations are conducted to validate the proposed method, and the results illustrate its effectiveness. [ABSTRACT FROM AUTHOR]

The work addresses the optimized tracking control problem by combining both reinforcement learning (RL) and backstepping technique for the canonical nonlinear unknown dynamic system. Since such dynamic system contains multiple state variables with differential relation, the backstepping technique is considered by making a virtual control sequence in accordance with Lyapunov functions. In the last backstepping step, the optimized actual control is derived by performing the RL under identifier‐critic‐actor structure, where RL is to overcome the difficulty coming from solving Hamilton‐Jacobi‐Bellman (HJB) equation. Different from the traditional RL optimizing methods that find the RL updating laws from the square of the HJB equation's approximation, this optimized control is to find the RL training laws from the negative gradient of a simple positive definite function, which is equivalent to the HJB equation. The result shows that this optimized control can obviously alleviate the algorithm complexity. Meanwhile, it can remove the requirement of known dynamic as well. Finally, theory and simulation indicate the feasibility of this optimized control. [ABSTRACT FROM AUTHOR]

This article is to develop an optimized leader‐follower consensus control for multiple robot manipulator system by combining sliding mode control (SMC) and reinforcement learning (RL). The SMC mechanism aims to steer both position and velocity states of multiple manipulator reaching the predefined trajectory. And the RL is designed and performed under identifier‐critic‐actor architecture for the achievement of optimized control performance. Compared to traditional optimal control, this proposed method has two main advantages: (i) the RL updating laws for training both the actor and critic networks are simpler; (ii) the optimized control can not require the complete dynamic knowledge because the adaptive identifier is designed into the RL learning. Consequently, this optimized control method can smoothly steer the multiple robot manipulator system to achieve the leader‐follower consensus. Finally, feasibility of this control method is verified through both theory and simulation. [ABSTRACT FROM AUTHOR]

This article addresses the backstepping control problem of the nonlinear strict-feedback system with an immeasurable state. To eliminate the effect coming from the immeasurable state variable, a new adaptive state observer method is developed for backstepping design by employing a neural network (NN) approximation strategy. As one of the highlighting contributions, the observer method can be performed with the relaxed condition because it does not require the design parameters to satisfy the Hurwitz equation, therefore it can be more easily applied and extended to serve for nonlinear system control than the existing observer methods. Finally, by integrating the observer dynamic into the backstepping design, the adaptive tracking control is achieved. Lastly, the feasibility of the method is proved by both stability theory and computer simulation. [ABSTRACT FROM AUTHOR]

In this article, a distributed leader‐follower consensus approach is developed for a class of high‐order unknown nonlinear dynamic multi‐agent systems (MASs). Because every agent of the MAS contains multiple state variables, the existing consensus methods are not completely applicable for it. In order to find the qualified consensus protocol for this high‐order MAS, sliding mode mechanism can be naturally considered for designing the consensus control because it can manage multiple state variables with the help of a constructed hyperplane. To this consensus control design, the sliding mode term is composed of all tracking error variables. Since the method does not require the switching control term around sliding surface, it can avoid the chattering phenomenon, which exits in most of the published sliding mode controls (SMCs). Furthermore, to handle the unknown nonlinear dynamic problem, the adaptive approximation strategy is implemented by employing fuzzy logic system (FLS). In the light of Lyapunov stability analysis, it is demonstrated that the proposed control approach can accomplish the consensus tasks. Finally, a numerical example is implemented to further show the desired results. [ABSTRACT FROM AUTHOR]

This paper presents an optimised tracking control scheme based on reinforcement learning (RL) for a class of second-order nonlinear systems with unknown dynamics. Different from the first-order dynamic system control, the second-order case is required to synchronously steer two variables of position and velocity, hence it makes this optimised control more challenging to accomplish. To achieve the optimised control, first, neural network (NN) is employed to approximate the solution of Hamilton–Jacobi–Bellman (HJB), and then an RL is performed by constructing both critic and actor based on the NN approximation. Since the RL training laws are derived from the negative gradient of a simple positive function generated in accordance with the partial derivative of HJB equation, it can make the control algorithm significantly simple to compare with the existing optimal control methods. Meanwhile, it can also release the condition of persistence excitation and compensate for the nonlinear uncertainty. Finally, the proposed adaptive control method can guarantee the desired results that are demonstrated by theorem, proof and simulation. [ABSTRACT FROM AUTHOR]

In this article, a fuzzy logic system (FLS)-based adaptive optimized backstepping control is developed by employing reinforcement learning (RL) strategy for a class of nonlinear strict feedback systems with unmeasured states. For making the virtual and actual controls are optimized solution of the corresponding subsystem, RL of observer-critic-actor architecture based on FLS approximations is constructed in every backstepping step, where the observer aims to estimate the unmeasurable states, and the critic and actor aim to evaluate control performance and perform control behavior, respectively. In the proposed optimized control, on the one hand, the state observer method can avoid to require the design constants making its characteristic polynomial Hurwitz, which is universally demanded in the existing observer methods; on the other hand, the RL is significantly simple in algorithm, because the critic and actor training laws are derived from the negative gradient of a simple positive function, which is produced from the partial derivative of Hamilton–Jacobi–Bellman (HJB) equation, instead of the square of approximated HJB equation. Therefore this optimized scheme can be more easily applied and widely extended. Finally, from two aspects of theory and simulation, it is demonstrated that the desired objective can be fulfilled. [ABSTRACT FROM AUTHOR]

In this article, an optimized leader-follower consensus control is proposed for a class of second-order unknown nonlinear dynamical multiagent system. Different with the first-order multiagent consensus, the second-order case needs to achieve the agreement not only on position but also on velocity, therefore this optimized control is more challenging and interesting. To derive the control, reinforcement learning (RL) can be a natural consideration because it can overcome the difficulty of solving the Hamilton–Jacobi–Bellman (HJB) equation. To implement RL, it needs to iterate both adaptive critic and actor networks each other. However, if this optimized control learns RL from most existing optimal methods that derives the critic and actor adaptive laws from the negative gradient of square of the approximating function of the HJB equation, this control algorithm will be very intricate, because the HJB equation correlated to a second-order nonlinear multiagent system will become very complex due to strong state coupling and nonlinearity. In this work, since the two RL adaptive laws are derived via implementing the gradient descent method to a simple positive function, which is obtained on the basis of a partial derivative of the HJB equation, this optimized control is significantly simple. Meanwhile, it not merely can avoid the requirement of known dynamic acknowledge, but also can release the condition of persistent excitation, which is demanded in most RL optimization methods for training the adaptive parameter more sufficiently. Finally, the proposed control is demonstrated by both theory and computer simulation. [ABSTRACT FROM AUTHOR]

The article develops an optimized control via learning from the design idea of optimized backstepping (OB) technique for the nonlinear strict feedback systems containing the unknown dynamic and control gain functions. OB technique requires to deal with the actual and virtual controls of backstepping as the optimized solutions of corresponding subsystems so that the entire backstepping control is optimized. In the work, for achieving the optimization control, reinforcement learning (RL) of critic‐actor structure is constructed in every backstepping step on the basis of the neural network approximation of the Hamilton–Jacobi–Bellman equation's solution. Since the unknown nonlinear control gain function is considered, the complexity of control algorithm is greatly increased. However, the proposed RL is with the simple training laws, it can greatly alleviate the algorithm complexity for the optimized control. Finally, the feasibility of the method is demonstrated by both theory and simulation. [ABSTRACT FROM AUTHOR]

In this article, an optimized tracking control scheme is studied for the quadrotor unmanned aerial vehicle (QUAV) system by combining both reinforcement learning (RL) and the backstepping technique. The RL aims to overcome the difficulty coming from solving the Hamilton–Jacobi–Bellman (HJB) equation, and it is performed via iterating both critic and actor each other, where the critic is for improving the control performance and the actor is for executing the control behavior. In mathematics, a QUAV system is composed of two connected subsystems that are, respectively, modeled by the translational and rotational dynamic equations, which are coupled via a rotation matrix; hence, the optimized tracking scheme is composed of two interconnected individual controls corresponding to the position and attitude, respectively. To achieve the two optimized position and attitude controls, the RL is constructed on the basis of the neural network (NN) approximation of the HJB equation’s solution by utilizing NN’s outstanding function approximation ability. Particularly, the position control is accomplished by introducing an intermediate control because the translational dynamic is an underactuated system. Since the proposed RL algorithm is significantly simple in comparison with the published methods, the optimized QUAV control can be easily executed in practical applications. Finally, the results are demonstrated by a Lyapunov stability analysis and a numerical simulation. [ABSTRACT FROM AUTHOR]

This article investigates a decentralized adaptive sliding mode formation control (SMFC) based on the leader–follower formation strategy for a class of second‐order nonlinear unknown dynamic multi‐agent systems. Unlike the first‐order multi‐agent formation, the second‐order formation needs to control not only the position but also the velocity, therefore it is more challenging and interesting. In order to handle the unknown dynamic problem, the adaptive neural network (NN) is employed to approximate the unknown dynamic function. Since every agent is required to directly communicate with leader agent, the adaptive NN law can be trained more sufficient. Moreover, in the control design, because a continuous proportional function is introduced to replace the discontinuous switching signal of the traditional sliding mode method, the chattering phenomenon that is a stubborn shortcoming of sliding strategy is effectively suppressed. According to Lyapunov stability analysis, it is proven that the proposed method can fulfill the control task. Finally, a numerical simulation example is executed to demonstrate the performance of the proposed controller. [ABSTRACT FROM AUTHOR]

In this article, a sliding mode control (SMC) is proposed on the basis of an adaptive neural network (NN) for a class of Single-Input–Single-Output (SISO) nonlinear systems containing unknown dynamic functions. Since the control objective is to steer the system states to track the given reference signals, the SMC method is considered by employing the adaptive neural network (NN) strategy for dealing with the unknown dynamic problem. In order to compress the impaction coming from chattering phenomenon (which inherently exists in most SMC methods because of the discontinuous switching term), the boundary layer technique is considered. The basic design idea is to introduce a continuous proportional function to replace the discontinuous switching control term inside the boundary layer so that the chattering can be effectively alleviated. Finally, both Lyapunov theoretical analysis and computer numerical simulation are used to verify the effectiveness of the proposed SMC method. [ABSTRACT FROM AUTHOR]

Over the past few decades, the study of epidemic propagation has caught widespread attention from many areas. The field of graphs contains a wide body of research, yet only a few studies explore epidemic propagation's dynamics in "signed" networks. Motivated by this problem, in this paper we propose a new epidemic propagation model for signed networks, denoted as S-SIS. To explain our analysis, we utilized the mean field theory to demonstrate the theoretical results. When we compare epidemic propagation through negative links to those only having positive links, we find that a higher proportion of infected nodes actually spreads at a relatively small infection rate. It is also found that when the infection rate is higher than a certain value, the overall spreading in a signed network begins showing signs of suppression. Finally, in order to verify our findings, we apply the S-SIS model on Erdös–Rényi random network and scale-free network, and the simulation results is well consist with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this article, a control scheme based on optimized backstepping (OB) technique is developed for a class of nonlinear strict-feedback systems with unknown dynamic functions. Reinforcement learning (RL) is employed for achieving the optimized control, and it is designed on the basis of the neural-network (NN) approximations under identifier–critic–actor architecture, where the identifier, critic, and actor are utilized for estimating the unknown dynamic, evaluating the system performance, and implementing the control action, respectively. OB control is to design all virtual controls and the actual control of backstepping to be the optimized solutions of corresponding subsystems. If the control is developed by employing the existing RL-based optimal control methods, it will become very intricate because their critic and actor updating laws are derived by carrying out gradient descent algorithm to the square of Bellman residual error, which is equal to the approximation of the Hamilton–Jacobi–Bellman (HJB) equation that contains multiple nonlinear terms. In order to effectively accomplish the optimized control, a simplified RL algorithm is designed by deriving the updating laws from the negative gradient of a simple positive function, which is generated from the partial derivative of the HJB equation. Meanwhile, the design can also release the condition of persistence excitation, which is required in most existing optimal controls. Finally, effectiveness is demonstrated by both theory and simulation. [ABSTRACT FROM AUTHOR]

The article proposes an optimized leader-follower formation control using a simplified reinforcement learning (RL) of identifier–critic–actor architecture for a class of nonlinear multiagent systems. In general, optimal control is expected to be obtained by solving Hamilton–Jacobi–Bellman (HJB) equation, but the equation associated with a nonlinear system is difficult to solve by analytical method. Although the difficulty can be effectively overcome by the RL strategy, the existing RL algorithms are very complex because their updating laws are obtained by carrying out gradient descent algorithm to square of the approximated HJB equation (Bellman residual error). For a multiagent system, due to the state coupling problem, these methods will become difficult implementation. In the proposed optimized scheme, the RL updating laws are derived from negative gradient of a simple positive function, which is the equivalence to HJB equation; therefore, the control algorithm is significantly simple. Furthermore, in order to solve the problem of unknown system dynamics, an adaptive identifier is integrated into the control. Finally, the theory and simulation demonstrate that the optimized formation scheme can guarantee the desired control performance. [ABSTRACT FROM AUTHOR]

In this paper, a tracking control approach for surface vessel is developed based on the new control technique named optimized backstepping (OB), which considers optimization as a backstepping design principle. Since surface vessel systems are modeled by second-order dynamic in strict feedback form, backstepping is an ideal technique for finishing the tracking task. In the backstepping control of surface vessel, the virtual and actual controls are designed to be the optimized solutions of corresponding subsystems, therefore the overall control is optimized. In general, optimization control is designed based on the solution of Hamilton–Jacobi–Bellman equation. However, solving the equation is very difficult or even impossible due to the inherent nonlinearity and complexity. In order to overcome the difficulty, the reinforcement learning (RL) strategy of actor-critic architecture is usually considered, of which the critic and actor are utilized for evaluating the control performance and executing the control behavior, respectively. By employing the actor-critic RL algorithm for both virtual and actual controls of the vessel, it is proven that the desired optimizing and tracking performances can be arrived. Simulation results further demonstrate effectiveness of the proposed surface vessel control. [ABSTRACT FROM AUTHOR]

This paper proposes an optimized tracking control approach using neural network (NN) based reinforcement learning (RL) for a class of nonlinear dynamic systems, which requires both tracking and optimizing to be performed simultaneously. Generally, for obtaining optimal control solution, Hamilton–Jacobi–Bellman equation is expected to be solvable, but, owing to strong nonlinearity, the equation is solved difficultly or even impossibly by analytical methods. Therefore, adaptive NN approximation based RL is usually considered. In the optimized control design, for driving output state following to the desired trajectory, an error term is split from optimal performance index function, and then both actor and critic NNs are built to perform RL algorithm. Actor NN aims to execute control behaviors, and critic NN aims to appraise control performance and make feedback to actor. The proof of stability concludes that the desired control performances are obtained. A numerical simulation is designed and implemented, and the desired results are shown. [ABSTRACT FROM AUTHOR]

The paper proposes an optimized leader–follower formation control for the multi-agent systems with unknown nonlinear dynamics. Usually, optimal control is designed based on the solution of the Hamilton–Jacobi–Bellman equation, but it is very difficult to solve the equation because of the unknown dynamic and inherent nonlinearity. Specifically, to multi-agent systems, it will become more complicated owing to the state coupling problem in control design. In order to achieve the optimized control, the reinforcement learning algorithm of the identifier–actor–critic architecture is implemented based on fuzzy logic system (FLS) approximators. The identifier is designed for estimating the unknown multi-agent dynamics; the actor and critic FLSs are constructed for executing control behavior and evaluating control performance, respectively. According to Lyapunov stability theory, it is proven that the desired optimizing performance can be arrived. Finally, a simulation example is carried out to further demonstrate the effectiveness of the proposed control approach. [ABSTRACT FROM AUTHOR]

This paper addresses formation control with obstacle avoidance problem for a class of second-order stochastic nonlinear multiagent systems under directed topology. Different with deterministic multiagent systems, stochastic cases are more practical and challenging because the exogenous disturbances depicted by the Wiener process are considered. In order to achieve control objective, both the leader-follower formation approach and the artificial potential field (APF) method are combined together, where the artificial potential is utilized to solve obstacle avoidance problem. For obtaining good system robustness to the undesired side effects of the artificial potential, $H_\infty$ analysis is implemented. Based on the Lyapunov stability theory, it is proven that control objective can be achieved, of which obstacle avoidance is proven by finding an energy function satisfying that its time derivative is positive. Finally, a numerical simulation is carried out to further demonstrate the effectiveness of the proposed formation schemes. [ABSTRACT FROM PUBLISHER]

Combining with artificial potential field and robust $H_\infty$ methods, the neural network (NN)-based adaptive synchronized tracking control is proposed for accommodation vessel (AV). The control task is to drive AV synchronous tracking floating production storage and offloading (FPSO). For finishing the task, NN is employed to approximate the unknown nonlinear dynamics of AV; $H_\infty$ method is to guarantee the system states of AV robust to exogenous disturbances; artificial potential method aims to produce the attractive and repulsive forces to assist AV maintaining desired distance with FPSO so that the gangway connecting both AV and FPSO is operated smoothly. Finally, it is proven that the proposed control scheme can guarantee that all error signals of the tracking control are Semi-Globally Uniformly Ultimately Bounded (SGUUB) and AV can synchronously track FPSO to desired accuracy. The simulation results further demonstrate the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]

ABSTRACT In this article, an optimized inverse dead‐zone control using reinforcement learning (RL) is developed for a class of nonlinear dynamic systems. The dead‐zone is frequently occurred in the nonlinear control system, and it can affect the control performance and even cause the system instable. Hence, it is very requisite to consider the effect of dead‐zone in the design of control strategy. In this proposed optimized inverse dead‐zone control, the basic idea is to find the optimized control as input and the adaptive algorithm to estimate the unknown parameters for the inverse dead‐zone function, so that the available dead‐zone input for system control can be derived. Comparing with traditional methods, on the one hand, the proposed dead zone inverse method is with fewer adaptive parameters, on the other hand, the RL under identifier‐critic‐actor architecture is with the simplified algorithm. Finally, theoretical and simulation results manifest the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

This article addresses a distributed time-varying optimal formation protocol for a class of second-order uncertain nonlinear dynamic multiagent systems (MASs) based on an adaptive neural network (NN) state observer through the backstepping method and simplified reinforcement learning (RL). Each follower agent is subjected to only local information and measurable partial states due to actual sensor limitations. In view of the distributed optimized formation strategic needs, the uncertain nonlinear dynamics and undetectable states may jointly affect the stability of the time-varying cooperative formation control. Furthermore, focusing on Hamilton-Jacobi-Bellman optimization, it is almost incapable of directly dealing with unknown equations. Above uncertainty and immeasurability processed by adaptive state observer and NN simplified RL are further designed to achieve desired second-order formation configuration at the least cost. The optimization protocol can not only solve the undetectable states and realize the prescribed time-varying formation performance on the premise that all the errors are SGUUB, but also prove the stability and update the critics and actors easily. Through the above-mentioned approaches offer an optimal control scheme to address time-varying formation control. Finally, the validity of the theoretical method is proven by the Lyapunov stability theory and digital simulation.

In this article, an optimized backstepping (OB) control scheme is proposed for a class of stochastic nonlinear strict-feedback systems with unknown dynamics by using reinforcement learning (RL) strategy of identifier-critic-actor architecture, where the identifier aims to compensate the unknown dynamic, the critic aims to evaluate the control performance and to give the feedback to the actor, and the actor aims to perform the control action. The basic control idea is that all virtual controls and the actual control of backstepping are designed as the optimized solution of corresponding subsystems so that the entire backstepping control is optimized. Different from the deterministic system, stochastic system control needs to consider not only the stochastic disturbance depicted by the Wiener process but also the Hessian term in stability analysis. If the backstepping control is developed on the basis of the published RL optimization methods, it will be difficult to be achieved because, on the one hand, RL of these methods are very complex in the algorithm thanks to their critic and actor updating laws deriving from the negative gradient of the square of approximation of Hamilton-Jacobi-Bellman (HJB) equation; on the other hand, these methods require persistence excitation and known dynamic, where persistence excitation is for training adaptive parameters sufficiently. In this research, both critic and actor updating laws are derived from the negative gradient of a simple positive function, which is yielded on the basis of a partial derivative of the HJB equation. As a result, the RL algorithm can be significantly simplified, meanwhile, two requirements of persistence excitation and known dynamic can be released. Therefore, it can be a natural selection for stochastic optimization control. Finally, from two aspects of theory and simulation, it is demonstrated that the proposed control can arrive at the desired system performance.

In this article, an optimized leader-following consensus control scheme is proposed for the nonlinear strict-feedback-dynamic multi-agent system by learning from the controlling idea of optimized backstepping technique, which designs the virtual and actual controls of backstepping to be the optimized solution of corresponding subsystems so that the entire backstepping control is optimized. Since this control needs to not only ensure the optimizing system performance but also synchronize the multiple system state variables, it is an interesting and challenging topic. In order to achieve this optimized control, the neural network approximation-based reinforcement learning (RL) is performed under critic-actor architecture. In most of the existing RL-based optimal controls, since both the critic and actor RL updating laws are derived from the negative gradient of square of the Hamilton-Jacobi-Bellman (HJB) equation's approximation, which contains multiple nonlinear terms, their algorithm are inevitably intricate. However, the proposed optimized control derives the RL updating laws from the negative gradient of a simple positive function, which is correlated with the HJB equation; hence, it can be significantly simple in the algorithm. Meanwhile, it can also release two general conditions, known dynamic and persistence excitation, which are required in most of the RL-based optimal controls. Therefore, the proposed optimized scheme can be a natural selection for the high-order nonlinear multi-agent control. Finally, the effectiveness is demonstrated by both theory and simulation.

In this article, the game-based backstepping control method is proposed for the high-order nonlinear multi-agent system with unknown dynamic and input saturation. Reinforcement learning (RL) is employed to get the saddle point solution of the tracking game between each agent and the reference signal for achieving robust control. Specifically, the approximate optimal solution of the established Hamilton-Jacobi-Isaacs (HJI) equation is obtained by policy iteration for each subsystem, and the single network adaptive critic (SNAC) architecture is used to reduce the computational burden. In addition, based on the separation operation of the error term from the derivative of the value function, we achieve the different proportions of the two agents in the game to realize the regulation of the final equilibrium point. Different from the general use of the neural network for system identification, the unknown nonlinear dynamic term is approximated based on the state difference obtained by the command filter. Furthermore, a sufficient condition is established to guarantee that the whole system and each subsystem included are uniformly ultimately bounded. Finally, simulation results are given to show the effectiveness of the proposed method.

In this article, an adaptive optimized control scheme based on neural networks (NNs) is developed for a class of perturbed strict-feedback nonlinear systems. An optimized backstepping (OB) technique is employed for breaking through the limitation of the matching condition. The disturbance of existing nonlinear systems may degrade system performance or even lead to instability. In order to improve the system's robustness, a disturbance observer is constructed to compensate for the impact coming from the external disturbance. Because the proposed optimized scheme needs to train the adaptive parameters not only for reinforcement learning (RL) but also for the disturbance observer, it will become more challenging no matter designing the control algorithm or deriving the adaptive updating laws. Finally, by virtue of the Lyapunov stability theory, it is proved that all internal signals of the closed-loop systems are semiglobal uniformly ultimately bounded (SGUUB). Simulation results are provided to illustrate the validity of the devised method.

In this paper, a control technique named optimized backstepping is first proposed by implementing tracking control for a class of strict-feedback systems, which considers optimization as a design philosophy of the high-order system control. The basic idea is that designing the actual and virtual controls of backstepping is the optimized solutions of the corresponding subsystems so that overall control of the high-order system is optimized. In general, optimization control is designed based on the solution of Hamilton-Jacobi-Bellman equation, but solving the equation is very difficult due to the inherent nonlinearity and intractability. In order to overcome the difficulty, the neural network (NN)-based reinforcement learning strategy of actor-critic architecture is used. In every backstepping step, the actor and critic NNs are constructed for executing control behavior and evaluating control performance, respectively. According to the Lyapunov stability theorem, it is proven that the desired control performance can be obtained. Finally, a simulation example is carried out to further demonstrate the effectiveness of the proposed control approach.