Abstract

This article studies the robust tracking control problems of Euler–Lagrange (EL) systems with uncertainties. To enhance the robustness of the control systems, an asymmetric tan-type barrier Lyapunov function (ATBLF) is used to dynamic constraint position tracking errors. To deal with the problems of the system uncertainties, the self-structuring neural network (SSNN) is developed to estimate the unknown dynamics model and avoid the calculation burden. The robust compensator is designed to estimate and compensate neural network (NN) approximation errors and unknown disturbances. In addition, a relative threshold event-triggered strategy is introduced, which greatly saves communication resources. Under the proposed robust control scheme, tracking behavior can be implemented with disturbance and unknown dynamics of the EL systems. All signals in the closed-loop system are proved to be bounded by stability analysis, and the tracking error can converge to the neighborhood near the origin. The numerical simulation results show the effectiveness and the validity of the proposed robust control scheme.

1. Introduction

Many practical systems can be represented by the El system, such as robotic manipulator [1], hydraulic system [2], and underwater marine system [3]. Therefore, due to its wide application, nonlinear Euler–Lagrange systems are a significant class of nonlinear systems. However, because of the unknown disturbances, the model uncertainties and the actuator communication limit always exist, and some traditional control methods are difficult to obtain satisfactory control performance. Therefore, innovation and development with high precision and high applicability control methods are urgent.

The research on robust control of the El system has always been a hot topic [4–7]. Generally, when a system works under uncertain disturbances, we need to improve the robustness of the control as much as possible. Some scholars have studied the trajectory tracking method of EL systems; the common methods include the backstepping technique [8], dynamics surface control (DSC) [9], robust control [10, 11], adaptive control [12, 13], sliding mode control [14], and learning control [15]. Among them, the error restriction method can effectively enhance the robustness of the control. In addition, considering the control security issues cannot be ignored, generally in the form of output constraints. Violation of these constraints not only leads to performance degradation but also causes system corruption. In most studies, BLF is an effective solution for the constraint problem [16–18]. In [19], the guaranteed performance control problem for EL systems with actuator faults is investigated, and the BLF is introduced to handle the performance constraints problems. In [20, 21], a log-type BLF is employed to ensure that the full-state constraints for an EL system with uncertain dynamics. In [22], the BLF-based control method is proposed for robotic systems with full-state constraints, which demonstrated that the BLF design method has advantages in dealing with state constraint problems of the El system. For research position constraint problems, a BLF-based controller is proposed for the marine vessel with uncertainty in [23], which also demonstrates the superiority of BLF in the El system design. The BLF technique can dynamically constrain the error within the specified range and guaranteed the performance of tracking control, which enhances the robustness of the control.

In practice, the influence of unknown disturbances on control is a serious problem. Some identification methods have been proposed to estimate the effects of uncertainties, such as the adaptive observers and compensate methods. The disturbance observer [24, 25] is a useful tool for nonlinear systems to identify unknown external disturbances, and some works have been applied to solve the problem of El system resisting disturbances [26, 27]. However, EL system model parameters are typically dynamically changed; they are difficult to obtain dynamics parameters accurately. The uncertainty seriously affects the stability and control accuracy of the EL system; therefore, the problem of identification of uncertain models is needed to be studied urgently. Some learning estimation methods have been proposed to approximate the system uncertainty, such as neural networks, fuzzy logic, and machine learning. The NN is often used to estimate unknown nonlinear dynamics models because of its good approximation ability. In [28–30], the adaptive method is combined with NN to design control strategies for a class of uncertain nonlinear systems. In [31, 32], the adaptive NN is used to estimate the uncertainty of El system in tracking or cooperative control. An adaptive multilayer NN is developed to estimate the uncertainty and a novel saturated prescribed performance controller for EL dynamic systems in [33]. In [34], to reduce the calculation burden, the adaptive NNs with the epsilon-modification updating laws are developed to approximate the compounded uncertain vector for EL systems. In [35], a self-structuring NN is designed to estimate the uncertain dynamics of each node of multiagents. Because NN has good learning performance, it has become the main tool to estimate system uncertainty.

Furthermore, in most cases, the bandwidth of the actuator communication network is limited. In order to use available resources reasonably, it is very important to design save resource controllers. It is worth noting that the event-triggered strategy is an effective way to reduce the actuator resources. In the event-triggered strategy, the control signal is updated only on some discrete trigger time to implement the aperiodic signal update. The trigger time is calculated based on some condition of the system state, which is also known as a trigger condition. This strategy makes the system have no complete transmission state throughout the time period and reduces the calculation workload and the use of communication channels [36, 37]. In [38], an adaptive control method is utilized to solve the unknown system parameters, and the new triggering mechanism is proposed to increase the executive efficiency of the controller. An event-triggered observer was designed for the estimation of the system states, and the dynamic event-triggered sliding mode controller is designed for a class nonlinear dynamic systems [39]. The sliding mode control method combines with event-triggered strategy, and a robust trajectory tracking controller is designed for uncertain EL systems [40]. In [41], a fully distributed event-triggered finite-time consensus controller is designed for EL systems, which can enable each agent to complete consistency tracking after a settling time.

Inspired by the above researches, the purpose of this paper is to design a robust track control strategy for EL systems with uncertainties. The ATBLF method is employed to constraint tracking errors, which can enhance the robustness of the tracking control. The adaptive NN is used to estimate the uncertainties of EL systems, and the self-structure mechanism is designed to reduce the calculation burden. The compensator is designed to estimate NN approximate error and the disturbances, which can improve the tracking accuracy. An event-triggered strategy is adopted to save actuator resources. The main contributions are summarized as follows:(1)To ensure the robustness of the tracking control, the control strategy design is divided into two layers. The ATBLF method is introduced to construct virtual control law at the kinematic level, it makes the position error guaranteed in a certain boundary, and the robustness of tracking is enhanced. In terms of kinetics, the adaptive NN is employed to estimate the uncertainty of EL systems. The NN approximate errors and unknown disturbances can compensate by a designed compensator, which ensures tracking stability(2)In order to improve the practicability of the control systems, a self-structure mechanism is developed to adjust NN approximation performance, which can appropriately find optimal NN structures and avoid excessive calculation burden. In addition, an event-triggered strategy is adopted to reduce the communication bandwidth and effectively save communication resources

This paper is organized as follows. The problem formulation of EL systems is introduced in Section 2. The main results of the design of SSNN and the robust tracking control strategy on the EL systems are in Section 3. Section 4 presents numerical simulation results. The conclusions of this paper are presented in Section 5.

Notation: and denote the largest and smallest eigenvalue, respectively. and denote dimensional column vectors and the real matrices, respectively. and represent the Frobenius norm and the Euclidean norm. represents a block-diagonal matrix.

2. Problem Formulation

2.1. System Model

Consider the uncertain EL systems with external disturbances, which iswhere , , and denote the position, velocity, and acceleration vectors, respectively; denotes the gravitational force, denotes the Coriolis and centripetal torques, denotes a symmetric inertia matrix, is the external disturbances of the systems caused by the environment and human beings, and is the control input.

Property 1 (see [42]). , , and in the dynamic system are all bound, and the matrix is skew-symmetric; i.e., for any .

2.2. Control Objective

The reference trajectory is defined as , which is time-varying twice-differentiable, and the tracking errors are defined as . The goal is to design a robust tracking controller for the EL systems to track the reference trajectory and to keep the tracking error constraints within a time-varying asymmetric bounded range as follows:where and denote the constraint bounded functions on the tracking error , and the initial condition satisfies .

Assumption 1. The disturbance and its first derivative are bounded, such that and , where and are unknown positive constants.

Remark 1. Disturbance may occur in the form of variable friction or load, which is often variable and unpredictable, and the energy is limited. If it is infinite energy, it will destroy the control system. Therefore, Assumption 1 is reasonable.

3. Main Results

In this section, the design process of the robust tracking control strategy for EL systems based on BLF and SSNN is introduced. The SSNN is developed to estimate unknown model dynamics. The TABLF is applied to deal with error time-varying constraint problems. The compensator is designed to estimate unknown disturbances and NN estimate errors. An event-triggered strategy is adopted to reduce actuator communication pressure.

3.1. Self-Structuring Neural Networks

In this article, the radial basis function (RBF) NN is applied to approximate unknown nonlinear dynamics. The RBFNN is composed of the output layer, hidden layer, and input layer, and its structure is shown in Figure 1.

Figure 1 

Structure of the NN.

The RBFNN output is expressed aswhere is the activation function vector, is the input vector, there are neurons here, and is the ideal NN weight vector. represents the NN approximation error, where the activation function is selected as the Gaussian function:where and represent the center and width of the Gaussian function, respectively.

About RBFNN, the more the neuron nodes are selected, the more accurate the approximation. However, more neurons mean that the system has a more computational burden, and some neurons are invalid when the nonlinear function is not complex. Therefore, we design a self-structuring mechanism for NN to change the approximation structure, which can determine whether to split neurons or eliminate neurons depending on the complexity of the actual nonlinear function. The aim is to split more effectively activation neurons and delete less activated neurons to obtain good approximation performance of NN.

The optimization method of NN is proposed. Define a splitting threshold and eliminate threshold , where .

The splitting strategy is to judge whether the neuron with the highest activation function is more than the threshold, which defines the maximum degree ; if , that means the activity does not reach the ideal value; then, the new neurons need to be split. The newly splitting neuron is defined as ; the parameter of the new neuron is

The neuron decay parameter is defined as ; it follows the ruleswhere is a proportion parameter and denotes the inactive bounded function. The elimination strategy is proposed. When the activation function is less than a threshold , the neuron decay parameter will decrease. When , the neuron is pruned.

The logic block diagram of self-structuring strategy is shown in Figure 2.

Figure 2 

Self-structuring algorithm flowchart.

Assumption 2 (see [43]). The ideal NN weight is bounded such that , where are unknown positive constants.

Remark 2. Some existing works [29, 30] show that the more the number of neurons, the better the approximation effect of NN. It is worth noting that not all neurons are effective neurons, which will bring more calculation burden to the control system. Therefore, a self-structuring mechanism with a flexible structure is proposed in this paper. The advantages of SSNN including the structuring of NN can be adjusted online without new membership functions and rules, and the computation can be effectively reduced.

3.2. Controller Design

The design process is divided into two steps.

Step 1. The asymmetrical errors virtual controller is designed.
Define the tracking error vector and aswhere , , and is a filtered control signal to be specified later.
Taking the time derivation of tracking error combined with EL system (1) yieldsand the time-varying error constraint problem can be solved by the BLF method. Consider the asymmetric tan-type BLF (ATBLF) as follows [44]:Computing the time derivative of yieldswhere , the initial state satisfies , and and are the presetting boundaries. Define .

Remark 3. For the formation of asymmetric tan-type BLF, which is shown in (9), we havewhere is differentiable and continuous and the state follows . When there are system states without constraints, such as and , using L’Hospital theory:Then, we proposed the constraint virtual controller aswhere is a positive design constant, , and is a positive gain matrix.
In order to avoid the differential explosion of virtual control law, the DSC method is introduced. The filtered control signal is as follows:where , and is a time constant. Define the filtering error , and take derivatives of :where with being an unknown continuous function which has a maximum value .
Substituting (14) and (15) into (11), the following can be obtained:Consider the following Lyapunov function:And, take the derivative of and combine with (15) and (16) to obtain

Step 2. The robust controller based on SSNN and event triggers is designed.
Consider the following Lyapunov function:According to (7) and (8), EL system (1) can be written asTaking the derivative of , we can obtainCombined with Property 1, one has . Moreover, substituting (21) into (22) yieldsDefine function . However, the parameters , , and are hard to obtain in the practice scene. Hence, the NN is employed to handle the uncertainty model as follows:where the input of NN is selected as , is the NN weight matrix, and denotes the estimated error, which is bounded and satisfied, , where is an unknown positive constant vector. In addition, define the unknown parameters vector , where is the unnecessary systems error. Then, a compensator is designed as follows:The robust control laws are designed as follows:where , , is the NN weight matrix estimate value, and .
The updated law and are given asThe event-triggering mechanism is designed aswhere is the event-triggering errors. The controller update time is defined as , , and designed parameters , , and are positive. When time , the controller holds as . When triggering condition (29) is triggered, the control signal will be updated and it is marked as . Thus, there exist two continuous time-varying parameters and such that , where and . Therefore, one getsThus, substituting (30) into (23), the following inequality holds:In view of and , we haveThe main results of this paper are given as follows.

Theorem 1. EL system (1) with uncertainties and Assumptions 1 and 2 are satisfied. Under the actual controller (29) with control law (14), (15), and (25)-(28), the asymmetric constraint tracking control of EL systems can be achieved. All signals in the closed-loop control system are semiglobally uniformly ultimately bounded (SGUUB), and the position error satisfies design objective conditions (2), which can converge to a neighborhood near of origin, and the interexecution intervals are lower bounded by a nonzero time , provided that the control parameter satisfies

Proof. Consider a new Lyapunov function:Taking the time derivative of (34) and using (31) and (32), we haveThe approximate error and disturbances are bounded, and the following equation can be derived:The following inequalities are based on Young’s inequality theory, which can be derived asAccording to , for a given variable and ; substituting (26) into (35) givesEquation (38) can be expressed aswhereBy integration of (39), we haveTherefore, the equation is held as follows:The opposite solution of equation (43) is obtained:Therefore, remains in the open set , , if the initial state satisfies . Then, the tracking error constraints in the time-varying boundary can be implemented.
Motivated by [38, 45], we can prove that there exists time such that triggering intervals is lower bounded by . Considering , one hasAll the signals mentioned above are bounded, and we can get , where is a positive parameter. We can obtain that and is hold. The lower bound satisfies . That is, Zeno's behavior is avoided.
This completes the proof.