Adaptive Asymptotic Tracking Control for a Class of Uncertain Input-Delayed Systems with Periodic Time-Varying Disturbances

In this paper, the problem of adaptive asymptotic tracking control for a class of uncertain systems with periodic time-varying disturbances and input delay is studied. By combining Fourier series expansion (FSE) with radial basis function neural network (RBFNN), a hybrid function approximator is used to learn the functions with periodic time-varying disturbances. At the same time, the dynamic surface control technique with a nonlinear filter is used to avoid the “complexity explosion” problem in the process of traditional backstepping technology. Ultimately, all closed-loop signals are guaranteed to be semiglobally uniformly bounded, and the given reference signal can be asymptotically tracked by the output signals of system. A simulation example is given to verify the effectiveness of the proposed control scheme.


Introduction
In recent years, with the deepening of the research on uncertain nonlinear control systems, more and more scholars have carried out indepth research on this problem and achieved remarkable results. e uncertainties appear in nonlinear systems in strict-feedback form [1][2][3][4][5][6], purefeedback form [7,8], multiple-input-multiple-output (MIMO) form [9][10][11], time-delay form [12,13], switched form [14][15][16][17][18][19], and discrete-time form [20,21]. Aiming at different forms of system, different methods are proposed to solve the problem of system control. For example, the adaptive gain scheduling backstepping sliding mode control method [1] is proposed for a class of strict-feedback systems. e problem of predetermined performance control for a class of pure-feedback uncertain nonlinear systems with input saturation [7] is studied. For a class of uncertain MIMO nonlinear systems with given tracking performance, the error-driven nonlinear feedback design method [10] is proposed to improve the dynamic performance of fuzzy adaptive moving surface control. e dynamic output feedback fault-tolerant controller [12] is designed to solve the problems of fault estimation and fault-tolerant controller design for a class of discrete-time fuzzy systems. e adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems [14] is studied. For a class of uncertain strict-feedback nonlinear systems, the output feedback adaptive neural tracking control [15] is studied.
In particular, the study of uncertain nonlinear systems with periodic perturbations is also a challenging problem [22][23][24][25][26]. Since RBFNN does not use periodic perturbation functions as inputs, it is difficult to deal with such systems in traditional ways. erefore, some new methods are proposed to solve this problem. In particular, the study of uncertain nonlinear systems with periodic perturbations is also a challenging problem [27] is studied.
is article [28] investigates the sampled data stabilization problem of a class of switched nonlinear systems. To guarantee that all states of the closed-loop system (CLS) are bounded, a new allowable sampling period is deduced.
It is well known that the traditional backstepping technology has the problem of "complexity explosion." To avoid the problem, many scholars have introduced dynamic surface control technology and achieved some results [29][30][31][32][33][34]. For a class of fractional-order nonlinear systems with uncertain parameters, the design of adaptive dynamic surface control [30] is studied. For a class of single input single output strict-feedback fractional-order uncertain nonlinear systems, the dynamic surface control method based on fractional-order filter [32] is proposed to avoid the inherent "complexity explosion" problem in the backstepping design process. e problem of adaptive fuzzy dynamic surface control [33] is studied for a class of nonstrict feedback nonlinear systems with unknown virtual control coefficients and full state constraints. e dynamic surface control technology [34] is used to solve the "complexity explosion" problem and combines the Nussbaum function to solve the problem of unknown control direction.
It is worth noting that, for uncertain systems with periodic perturbations, the asymptotic tracking control problem is not well solved. It is because the asymptotic tracking control problem of uncertain systems with periodic perturbations has not been well solved. Aiming at this difficulty, the control method proposed in this paper solves the asymptotic tracking control problem of this kind of system well. e main work is as follows: (1) in this paper, the problem of adaptive asymptotic tracking control for a class of nonlinear uncertain systems with periodic timevarying disturbances and input delay is studied. To solve this problem, we propose a new adaptive control scheme. (2) To ensure the feasibility of the control scheme, the FSE-RBFNN approximator is used to approximate the unknown disturbance function. (3) An adaptive dynamic surface control technique with a nonlinear filter is designed to avoid the "complexity explosion" problem in the traditional backstepping design process. (4) Based on the designed adaptive controller, all closed-loop signals are ultimately guaranteed to be semiglobally uniformly bounded. It is proved that the output signal can asymptotically track the given reference signal. Finally, a simulation example is given to verify the effectiveness of the proposed control scheme.

Problem Description and Preliminaries
A class of periodic perturbed nonlinear systems with strictfeedback is considered: . , x i ] T is the system state variable, y ∈ R is the output of the system, u is the control input of the system, τ(t) represents the time-varying delay, Remark 1. For the actual control systems, such as industrial control systems, communication systems, and aerospace control systems, the disturbance and time-delay are common problems. ese problems are well solved, and the performance of the controlled systems will have a great improvement. Control objective: an adaptive dynamic surface control algorithm based on a neural network is designed to ensure that the output signal of system (1) can track the reference signal y d (t) asymptotically, and all the closed-loop signals are semiglobally uniformly bounded.
To achieve the above control objectives, we make the following assumptions on system (1): Assumption 1 (see [32]): reference trajectory y d (t) is bounded, and its derivatives _ y d (t) and € y d (t) are continuous and bounded Assumption 2 (see [35]): for i � 1, 2, . . . , n, there exists an unknown constant N i , which makes the following inequality holds |d i (t)| ≤ N i In this paper, the mixed function approximator based on FSE-RBFNN in reference [28] is used to model g i (E i , ω i (t)) as follows: A hybrid neural network approximator is designed is an unknown continuous disturbance vector with known period T, is the inherent NN approximation error with the minimum upper bound ε b > 0, which can be decreased by increasing the NN node number p, cos(2πjt/T), j � 1, . . . , (s − 1)/2, its n-th derivative is smooth and bounded, the upper bound ε ω is the minimum truncation error of δ ω (t).
Lemma 1 (see [36]). For (E i , ω(t)) ∈ Ω, the approximation error δ b (E i , t) in (2) satisfies the following condition: where ε is the minimum upper bound of δ(E i , t) and it can be reduced arbitrarily by increasing the values of s and h. Generally speaking, the parameters W and c are unknown. It needs to be estimated in the controller design. Suppose W and c are, respectively, the estimates of W and c, and the estimated error is Lemma 2 (see [36]). For approximator (2), the estimated error can be expressed as follows: Lemma 3 (see [37]). For ς > 0 and λ ∈ R, the following inequality holds: Lemma 4 (see [38]). Assume that a is an unknown nonzero variable. V(t 0 , t) and R(t) are smooth functions defined in the time interval [t 0 , t), and V(t 0 , t f ) is nonnegative, R(t 0 ) is bounded. In addition, N(R(t)) � e R 2 (t) R(t) is a dynamic gain function, ξ is a normal number, and ξ is a bounded variable. If the following inequality holds it can be concluded that Furthermore, when t f � ∞, the obtained closed-loop system is also bounded.
Lemma 5 (see [38]). For the following equation where a > 0 and d > 0 are the design constants, k(t) is a positive function. If the initial value m(0) is nonnegative, then for any t > 0, m(t) ≥ 0 must be established.

Control Design and Stability Analysis
e main conclusions of this paper can be summarized as the following theorem. (1), including the nonlinear filter (10), the actual controller (20), and the adaptive law (29). Based on Assumption 1 and Assumption 2, for any initial condition satisfying V(0) ≤ c, where c is a positive design parameter, there exists design parameters k i , η i , β i , ξ j (i � 1, . . . , n, j � 1, . . . , n − 1) and invertible matrices Γ W i and Γ c i , such that (i) All the closed-loop signals remain semiglobally uniformly ultimate bounded (SGUUB) (ii) e given reference signal can be asymptotically tracked by the output signal of the system e proof of eorem 1 is divided into two parts: adaptive controller design and stability analysis of the closed-loop system.

Theorem 1. Consider a closed-loop system
In the following, we propose an adaptive NN dynamic surface control scheme based on the backstepping design process. To order to avoid the "complexity explosion" problem, the following nonlinear filters are constructed: where e i � s i − α i is the i-th boundary layer error and ι i and μ are the positive constants. To compensate for the influence of input delay, the following system [35] is introduced: where p i > 1(i � 2, . . . , n) is optional known parameter, and the initial condition of the system is λ(0) � 0. In the next part, an adaptive NN tracking controller is designed based on the backstepping method. Firstly, the following coordinate transformations are introduced: Step 1: according to (12), we can get the following result: e Lyapunov function V 1 is constructed as follows: where η 1 > 0 and β 1 > 0 are the design parameters, the estimation errors θ 1 � θ 1 − θ 1 , N 1 � N 1 − N 1 , θ 1 and N 1 represent the estimation of θ 1 and N 1 , respectively. Γ w 1 > 0 and Γ c 1 > 0 are the invertible matrices of the design. en, we can get According to Lemma 2, we have By using Lemmas 2 and 3, we can get where Combining (23)-(25), the following inequality holds: where en, one has

Mathematical Problems in Engineering
Based on (28), the adaptive laws are constructed as follows: where ρ 1 , σ 1 , κ 1 , o 1 , and D 1 are positive design parameters. en, substituting (29) into (28) produces Using Young's inequality, the following inequalities hold: erefore, substituting (31) into (30), we have where Step 2: similar to (13), then the derivative of z i is 6 Mathematical Problems in Engineering e Lyapunov function V i is constructed where η i > 0 and β i > 0 are the design parameters, the estimation errors θ i � θ i − θ i , N i � N i − N i , θ i and N i represent the estimation of θ i and N i , respectively. Γ w i > 0 and Γ c i > 0 are the invertible matrices of the design. Similar to (15), one has Similar to (16), the following inequality can be obtained: Let then according to (2) and Lemma 1, we can get the following conclusions: where en, substituting (37) and (39) into (36), one has Using Young's inequality, we have e design of virtual controller α i is designed as follows: where L st (·) is a dynamic gain function in Lemma 4, the adaptive law of ϖ i is given in (51), and α i is the equivalent unit of virtual controller α i

Mathematical Problems in Engineering
where the definition of v i will give in formula (46), k i is the positive parameter of the design. Substituting (41)-(43) into (40), the following inequality holds: According to Lemma 2, we have By using Lemmas 2 and 3, we can get where Substituting (46) and (47) into (45), the following inequality holds:
(60) en, according to (2) and Lemma 1, we have where E n � [x 1 , . . . , x n , _ s 1 , . . . , _ s n− 1 , λ 2 , . . . , λ n ], ε n denotes the upper bound of δ n (E n , ω n (t)). Combining (58)-(61), the following inequality holds _ V n ≤ _ V n− 1 + z n u + W T n S n E n , c T n φ n (t) z n + ε n z n + d n (t) e controller u is designed as follows: where L st (·) is a dynamic gain function in Lemma 4, the adaptive law of ϖ n will give in (72), and u is the equivalent unit of virtual controller u: where the definition of v n will give in (67), k n is the positive parameter of the design.
Similar to (44), the following inequality holds According to Lemma 2, we have Using Young's inequality, the following inequalities hold: ρ n η n θ n θ n ≤ − ρ n 2η n θ 2 n + ρ n 2η n θ 2 n , Substituting (74) into (73), one has where For boundary layer error e i � s i − α i , i � 1, . . . , n − 1, using the method of differentiation, we have G 1 (·) and G i (·) are continuous functions and defined as e compact sets Ω 1 and Ω 2 are defined as follows: where G 0 is a positive constant. It is noting that the set Ω 1 × Ω 2 is compact in R 4n+1 . ere is an unknown positive number M i on Ω 1 × Ω 2 , such that |G i | ≤ M i . erefore, according to (71) and (72), we use its estimated value M i , i � 1, . . . , n − 1. e Lyapunov function V is constructed as follows: where ξ i is the design parameter. en, the derivative of V is According to Lemma 3, we have Substituting (82) into (81), the following inequality holds: e adaptive law is constructed as follows: where ξ i and ξ i are the positive parameters. Substituting (84) into (83), the following inequality holds: Using Young's inequality, the following inequality holds: Substituting (86) into (85) produces Mathematical Problems in Engineering where By integrating (87), we can get the following inequality: time-varying disturbances are considered in this paper. Secondly, the problem of adaptive asymptotic tracking control for a class of nonlinear uncertain systems with periodic time-varying disturbances and input delay is solved by developing a new adaptive control scheme based on the FSE-RBFNN approximator.

Remark 3.
To show the superior performance of our control scheme, for system (96), the simulation results by using the method proposed in [32] have been shown in Figures 9 and   Figures 2 and 9, it can be seen that the control scheme presented in this paper has a better control performance.

Conclusions
In this paper, the adaptive asymptotic tracking control for a class of periodic time-varying disturbances uncertain systems with input delay is studied. To solve this problem, a dynamic surface technique based on a hybrid approximator is proposed. It is proved that the tracking error asymptotically converges to zero, and all the closed-loop signals are semiglobally uniformly bounded. Finally, the simulation example validates that the proposed design scheme is feasible. e control scheme proposed in this paper is a theoretical result, and from the simulation example, it can be seen that from the implementation and computational point of view, the feasibility of the proposed approach is guaranteed. However, how to test their technique against a realistic system is a more challenging problem, which is the main disadvantage of the proposed method and will be discussed in further research.

Data Availability
No data were used to support this study. Disclosure e manuscript has not been published in whole or in part elsewhere. e paper is not currently being considered for publication elsewhere. All authors have been personally and actively involved in substantive work leading to the report and will hold themselves jointly and individually responsible for its content.  Figure 10: Trajectory of control signal u using method [32].

Conflicts of Interest
e authors declare that they have no conflicts of interest.