Adaptive Neural Networks Control Using Barrier Lyapunov Functions for DC Motor System with Time-Varying State Constraints

This paper proposes an adaptive neural network (NN) control approach for a direct-current (DC) systemwith 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 onDC systems. As far as we know, this system is the first studied situations with timevarying 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.


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][2][3][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][12][13][14][15][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][19][20] and MIMO systems in [21][22][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][32][33][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 timevarying constraints.The contributions of this paper are summarized as follows.
(1) 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.(2) To guarantee that the state constraints always remain in the time-varying constrained sets, the asymmetric time-varying BLF is utilized.(3) 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.

Problem Formulation and Preliminaries
Consider the dynamic system with the DC motor without vibration mode as the following form: where  1 () is the motor angular position;  2 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   1 :  + →  and   1 :  + →  such that   1 () >   1 (), ∀ ∈  + .
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.

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  1 is the virtual controller which will be given later on.We consider the timevarying asymmetric BLF: where  is a positive integer.
Obviously, under the premise of Choose the virtual controller  1 as The time-varying gain is given as where   and   ,  = 1, 2 are any positive constants.Make sure that the time derivative  1 is bounded, when k  1 and k  1 are both zero.Substituting ( 15) and ( 16) into ( 14) and noting that we obtain where After finishing it, we get ) Based on (12), we obtain Using Young's inequality, the following inequality holds: Substituting ( 22) into ( 21), V 1 can be further rewritten as The Barrier Lyapunov Function  2 is given as Then, differentiating of  2 with respect to time is given by From the definition of the tracking error  2 =  2 −  1 , it is easy to obtain ż 2 = α 2 − σ 1 , and the derivative of the virtual controller is given as where 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 obtain where  1 is a positive constant.

Complexity 5
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 estimate where  = [  1 ,   2 ]  ∈ Ω  ⊂  3 , and similar to [28], we assume that the approximate error () satisfies |()| ≤  * with the constant  * > 0.
Substituting ( 33), ( 31) can be rewritten as According to Young's inequality, we can easily obtain where  2 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  3 : where Γ = Γ −1 > 0 is a constant matrix and W = Ŵ −  * .The time derivative of  3 is Based on (38), we obtain The adaptive law is given as follows: where  is a positive constant.Substituting (42) into (41), we get Complexity Using Young's inequality, After finishing it, we get Then, the above inequality can be rewritten as where 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, lim →∞ |() −   ()| = 0, all signals of the closed-loop system are bounded, and all state constraints are never violated.
Obviously, we can clearly obtain that the virtual controller  1 is bounded in (15).Based on  2 =  2 −  1 and (56),  2 is bounded.In addition, from (37) and (42), we know the actual controller  and the adaptive law Ŵ 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  1 lies on the design parameters  1 ,  2 , ,   ,  * ,  * ,  1 ,  2 , and Γ −1 .If we fix  > 0, 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.
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  1 () initially from the boundaries   1 () and −  1 () 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 closedloop system bounded.From Figure 5, it can be observed that the control input is bounded by a bounded back and forth reciprocate.

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

Figure 1 :
Figure 1: The trajectories of output  1 and the reference signal   ().