#### Abstract

This paper proposes an adaptive neural network (NN) control approach for a direct-current (DC) system with full state constraints. To guarantee that state constraints always remain in the asymmetric time-varying constraint regions, the asymmetric time-varying Barrier Lyapunov Function (BLF) is employed to structure an adaptive NN controller. As we all know that the constant constraint is only a special case of the time-varying constraint, hence, the proposed control method is more general for dealing with constraint problem as compared with the existing works on DC systems. As far as we know, this system is the first studied situations with time-varying constraints. Using Lyapunov analysis, all signals in the closed-loop system are proved to be bounded and the constraints are not violated. In this paper, the effectiveness of the control method is demonstrated by simulation results.

#### 1. Introduction

Due to the requirements of practice and the development of theory, the controller design of uncertain system has become a new research direction and attracted more and more scholars’ attention. The uncertainty of the actual engineering system has been studied in many works [1–4]. The neural networks [5] and fuzzy logic systems [6] have become the two main tools which can effectively deal with the unknown functions in the systems. In [7, 8], these are studies of some actual engineering systems with uncertain parameters. In [9, 10], the NN is used to approximate several random perturbations and unknown functions. In [11–16], several nonlinear system solutions are studied based on neural networks and fuzzy logic systems. In [17], adaptive control schemes based on neural networks were proposed for nonlinear systems with unknown functions. Based on neural networks and fuzzy logic systems, the significant studies proposed the novel adaptive tracking control methods for nonlinear SISO systems in [18–20] and MIMO systems in [21–23]. However, it is worth noting that the constraint problem is worth noting in the above approaches, which lead to the inaccuracy or oscillations of the engineering systems and even cause control systems instability.

In fact, there are constraints in most physical systems with various forms, for example, physical stoppages, saturation, performance, and safety specifications, such as restricted robot manipulation system [24], application to chemical process [25], networked surveillance robots systems [26], and nonuniform gantry crane [27]. In recent years, the barrier Lyapunov functions become the main tools to solve the constrained problem which was proposed for the first time in [28]. Based on BLF, some adaptive control methods were presented for nonlinear systems with output constant constraint in [29, 30] and state constant constraint in [31–34]. As we known, the constant constraint is the special case of the time-varying constraint. Subsequently, the authors in [35, 36] proposed some adaptive control approaches to address the stability problem of nonlinear systems with time-varying constraints.

Motor is the most important electromechanical energy conversion device, which has been widely used in the industrial and agricultural production, transportation, aerospace, and so on. In particular, the motor system with unknown uncertainties has attracted the attention of many scholars, and the control problem of the motor system becomes more and more important. In [37, 38], the authors proposed two adaptive control methods for systems with unknown functions. The authors in [39, 40] presented an adaptive control with time-varying output constraints for DC motor systems. According to the above descriptions, the urgent problem is how to address the stability problem of the DC motor system with time-varying state constraints.

This paper presents an adaptive NN tracking control method for DC motor systems with time-varying state constraints. As far as we know, there is no work dealing with such DC motor systems in the literature at present stage. The contributions of this paper are summarized as follows. The time-varying state constraints are first considered in the DC motor systems; comparing with the existing on DC motor systems, the proposed control method is more general and extensive in the engineering field. To guarantee that the state constraints always remain in the time-varying constrained sets, the asymmetric time-varying BLF is utilized. A novel adaptive tracking controller based on the neural networks and backstepping technique is structured to guarantee that all signals in the closed-loop system are bounded, the tracking errors converge to a small neighborhood of zero and the time-varying state constraints are not transitioned.

#### 2. Problem Formulation and Preliminaries

Consider the dynamic system with the DC motor without vibration mode as the following form:where is the motor angular position; stands for motor angular velocity; is a known inertia, is an unmeasured viscous friction, and is an unmeasured nonlinear friction; represents the unknown disturbance but bounded with ; is the system output; and represents the motor torque. In particular, output is required as follows:where and such that , .

*Remark 1. *From (2), the states of DC systems are constrained by the considered time-varying functions. In [35, 36], the constraint problem is omitted, which is the main factor of the oscillations of the engineering systems. The authors in [39] addressed the stability problem of DC motor systems with constant constraint which is the special case of the time-varying constraint. Comparing with the [40], the authors only consider time-varying output constraint; the proposed adaptive control method tries to stabilize the DC motor systems with time-varying state constraints, which cause the difficulty of controller design.

In this paper, the control objective is to design an adaptive NN tracking controller which adjusts the output of DC motor systems to track desired trajectory of the reference signal in the range of time-varying constraint functions. Meanwhile, all signals in the closed-loop systems are bounded and the time-varying state constrains are not violated.

*Assumption 2 (see [35]). *There exist constants and , , such that , , and , , , .

*Assumption 3 (see [32]). *There exist functions and satisfying and , and positive constants , , such that the desired trajectory and its time derivatives satisfy and , , .

The following lemma is represented for the establishment of binding compensation and performance limits.

Lemma 4 (see [28]). *Let and be open sets. Take into account the systemwhere and in is piecewise continuous and in is locally Lipschitz, united in , on .*

Suppose that there are functions and . In their respective domains, they are continuously differentiable and positive definite, such thatwhere and are class functions. Let , and . If the inequality is established:in , .

Lemma 5 (see [35]). *For all and positive integer , the inequality .*

*Proof. *For , the term can be rewritten asThe proof is completed.

#### 3. State Feedback Adaptive Controller Designs

This paper presents an adaptive tracking controller based on a backstepping technique with the asymmetric time-varying BLF for the DC motor systems. The detailed designs process is shown in this section.

Denote , , where is the virtual controller which will be given later on. We consider the time-varying asymmetric BLF:where is a positive integer.

The time-varying barriers are chosen asand is defined as

Based on Assumptions 2 and 3, there are positive constants , , , and , such that

Through the change of error coordinates,

According to (12), (7) can be rewritten as

*Remark 6. *According to (10), we know that when , we obtain , , and . When , we obtain , , and . From the above, we can get (13) based on (12).

Obviously, under the premise of , is definite continuously differentiable. The time derivative of is

Choose the virtual controller as

The time-varying gain is given aswhere and , are any positive constants. Make sure that the time derivative is bounded, when and are both zero. Substituting (15) and (16) into (14) and noting thatwe obtainwhere

After finishing it, we get

Based on (12), we obtain

Using Young’s inequality, the following inequality holds:

Substituting (22) into (21), can be further rewritten as

The Barrier Lyapunov Function is given as

Then, differentiating of with respect to time is given by

From the definition of the tracking error , it is easy to obtain , and the derivative of the virtual controller is given aswhere .

According to (26), (25) can be rewritten as

Based on (23), we get

Substituting (26) into the above formula, we obtain

Using Young’s inequality and noting , we obtainwhere is a positive constant.

Based on (30), (29) can be rewritten as

For convenience, we define

In fact, since the parameters of and are not available, is unknown in practice. In order to solve the uncertainty of this parameter, we designed NN, as shown below to estimatewhere , and similar to [28], we assume that the approximate error satisfies with the constant

Substituting (33), (31) can be rewritten as

According to Young’s inequality, we can easily obtainwhere is a positive constant.

Based on (35), (34) can be rewritten as

The actual controller is given as

Substituting (37), we obtain

Design the Lyapunov function candidate :where is a constant matrix and .

The time derivative of is

Based on (38), we obtain

The adaptive law is given as follows:where is a positive constant.

Substituting (42) into (41), we get

Using Young’s inequality,

After finishing it, we get

Then, the above inequality can be rewritten aswhere

Theorem 7. *Consider the unknown DC motor control system (1), based on the assumptions of Assumptions 2 and 3, Lemma 4, actual controller (37), and the adaptive law (42). The following properties guaranteed that the tracking error singles will remain in a compact neighborhood of zero, that is, , all signals of the closed-loop system are bounded, and all state constraints are never violated.*

*Proof. *With both sides of (46) multiplied by , we obtainIntegrating (48) over , we haveBased on (7), (24), and (39), we can obtainThen, we havewhere Therefore, we know thatBased on the above inequality, the following inequality is obtained:whereSimilar to the derivation of , we can obtain the conclusion thatwhereFrom Assumption 2, we can be known that , and from the definition of , we have . In fact, from and , we obtainBased on the above inequality, we know , . Therefore, the output signals are bounded.

Obviously, we can clearly obtain that the virtual controller is bounded in (15). Based on and (56), is bounded. In addition, from (37) and (42), we know the actual controller and the adaptive law are bounded. Therefore, all the closed-loop system signals are bounded.

The proof is completed.

*Remark 8. *In the above analysis, it is apparent that the boundedness of lies on the design parameters , , , , , , , , and . If we fix , it is clear that decreasing might result in small and increasing might result in large ; thus, it will help to reduce . This represents that the tracking errors can be made arbitrarily small by selecting the design parameters appropriately.

#### 4. Simulation Results

To illustrate the validity of the proposed adaptive NN control method, a simulation example is provided. Specifically, the following the DC motor system is described bywhere the inertia is kg·m^{2}, denotes an unmeasured viscous friction with , is an unmeasured nonlinear friction with , and is the external interference with . The desired reference signal is given as . The virtual controller, the actual controller, and the adaptation law are chosen as follows:

The angular position and the angular velocity of motor systems are bounded by and with , , , and . The NN contains 20 nodes and the centers , . The design parameters of the proposed control method are chosen as , , , , , , , and and the initial condition of the system state is chosen as , , and .

For the DC motor system, using a method of controlling the program can be obtained by the simulation results shown in Figures 1–5. Figures 1 and 2 show the output trajectory. Figure 1 shows the output and the reference signal tracking effect; the figure shows that the two curves almost coincide; that is to say, the tracking error converges to zero. Figure 3 shows the tracking error trajectory of initially from the boundaries and repulsion, but eventually converging to zero. Figure 4 shows a bounded and adaptive law of locus. According to Figure 4, we can see that the track adaptation law is bounded. Thus, we can conclude that a good tracking performance can make all the signals in the closed-loop system bounded. From Figure 5, it can be observed that the control input is bounded by a bounded back and forth reciprocate.

#### 5. Conclusion

In this paper, we propose an adaptive tracking control method for a DC system with full state constraints. The asymmetric time-varying BLF is employed to guarantee that the states always remain in the time-varying constrained sets. In the asymmetric system, neural networks and a backstepping technique are used to construct an adaptive control and adaptation laws to ensure that all signals in the closed-loop system are bounded and the state constraints are not transitioned. The performances of the adaptive control method based asymmetric time-varying BLF are verified by a simulation example.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (61603164, 61473139, and 61622303) and the project for Distinguished Professor of Liaoning Province.