Approximation-Based Fixed-Time Adaptive Tracking Control for a Class of Uncertain Nonlinear Pure-Feedback Systems

This paper examines approximation-based ﬁxed-time adaptive tracking control for a class of uncertain nonlinear pure-feedback systems. Novel virtual and actual controllers are designed that resolve the meaninglessness of virtual and actual controllers at the origin and in the negative domain, and the suﬃcient condition for the system to have semiglobal ﬁxed-time stability is also provided. Radial basis function neural networks are introduced to approximate unknown functions for solving the ﬁxed-time control problem of unknown nonlinear pure-feedback systems, and the mean value theorem is used to solve the problem of nonaﬃne structure in nonlinear pure-feedback systems. The controllers designed in this paper ensure that all signals in the closed-loop system are semiglobally uniform and ultimately bounded in a ﬁxed time. Two simulation results show that appropriate design parameters can limit the tracking error within a region of the origin in a ﬁxed time.


Introduction
Nonlinear pure-feedback systems [1,2] are more common than general strict feedback nonlinear systems or nonlinear systems with input affine structure. Such systems have been studied widely in recent years because they can reflect more accurately the working conditions of actual engineering systems owing to the nonaffine structure of their control inputs and state variables.
In 1988, David S. Broomhead proposed the radial basis function (RBF) neural network [3], which is used widely in pattern recognition, signal processing, and control system theory and application because of its simple structure and generalizability. Many scholars have used it to address the uncertainties of nonlinear systems based on its approximation ability, which has been demonstrated by many researchers [4][5][6][7]. Kanellakopoulos et al. [8,9] proposed the backstepping method and the backstepping adaptive control scheme for a class of strict feedback nonlinear systems in 1991.
e adaptive control of RBF neural networks has attracted extensive attention [10][11][12][13][14][15][16][17][18], resulting in the stability analysis method of RBF neural networks based on the Lyapunov method [19][20][21]. Finite-time stability technology can be traced back to the 1960s [22], and it developed rapidly in the 1990s, owing primarily to the improvement of finitetime Lyapunov theory [23] and homogeneous systems [24]. Much has been achieved in recent years on finite-time stability [25][26][27][28][29][30]. A finite-time control system ensures that a nonlinear system converges in a finite time, but the convergence time is related to the initial state of the system. For this reason, fixed-time control systems have been proposed [31][32][33][34] that ensure that the upper limit of the fixed convergence time is no longer related to the initial state of the system but only depends on the design parameters. Since Polyakov et al. first proposed fixed-time stability control [33], nonlinear fixed-time control has developed rapidly and has been studied by many scholars. For example, in [35], a fixed-time control with generalized directional topology was proposed for nonlinear multiagent systems; in [36], a prescribed performance fixed-time recurrent neural network control was proposed for a class of uncertain nonlinear systems; in [37], a fast fixed-time nonsingular terminal sliding mode control was proposed to solve the problem of chaos suppression of power systems; and in [38], a fixedtime observer was proposed to detect distributed faults of nonlinear multiagent systems.
In [35][36][37][38][39][40], strict feedback nonlinear systems are considered primarily, and the fixed-time control problem of more common nonlinear systems is not solved. In this paper, the fixed-time control problem is solved based on nonlinear pure-feedback systems, and the sufficient condition and design steps for semiglobal fixed-time stability are provided. e principal contributions of this paper are as follows: (1) e fixed-time control algorithm proposed in [35][36][37][38][39][40] does not solve the problem of the nonaffine structure of the control input u(t). is paper applies fixed-time control theory in nonlinear pure-feedback systems to solve this problem. (2) e controller designed in [40][41][42][43] has a power function similar to z 2q− 1 , 0 < q < 1. Not selecting q properly results in singularity. A novel fixed-time controller is designed in this paper to solve this problem. (3) An RBF neural network control algorithm is introduced to approximate the unknown functions f i (·) to overcome the difficulty of modeling accurately and solving the problem of interference in nonlinear pure-feedback systems. e remainder of this paper is organized as follows. Section 2 presents the problem description and preliminaries. Section 3 proposes fixed-time adaptive neural tracking control using backstepping, adaptive neural networks, and Lyapunov functions for a class of unknown nonlinear pure-feedback systems to solve the problem of fixed-time tracking control for nonlinear systems with nonaffine structure. In Section 4, all signals in the closed-loop system are proved to be semiglobally uniform and ultimately bounded. In Section 5, the proposed control scheme is proved to be effective through simulation experiments. Section 6 draws conclusions.

Problem Description and Preliminaries
2.1. Problem Description. Consider the following nonlinear pure-feedback system: . . , n, u(t) ∈ R, and y(t) ∈ R are state variables, system input, and system output, respectively. f i (·) are unknown but smooth nonaffine functions, and d i (·) are unknown but bounded disturbances. According to the mean value theorem [44], , and x i0 is a known quantity at the given time t 0 . System (1) can be written as Remark 1. It can be seen from system (2) that the mean value theorem separates the nonaffine structure from f i (·) in system (1). is paper aims to design a fixed-time controller that can meet the fixed-time control requirements in nonlinear purefeedback systems, enabling the system output y to track the reference signal y d in a fixed time. All of the signals of the closed-loop system are semiglobally uniform and ultimately bounded.
Assumption 1 (see [45]). Unknown smooth nonlinear functions h i (·) are bounded, and there are known positive constants b and Without loss of generality, we assume For ease of calculation, vector functions are defined as Assumption 2. Reference signal vector functions y di are known smooth continuous bounded functions.
where Ω di are known compact sets and reference signal y d is an n-order differentiable and bounded function.

Fixed Time
Definition 1. Consider the following nonlinear system: where x ∈ R n and f: R + × R n ⟶ R n , and assume that the origin is an equilibrium point.
Network. An RBF neural network [11,49] is applied in this paper to approximate arbitrary continuous functions. e mathematical expression of an RBF neural network is as follows: where W � [w 1 , w 2 , . . . , w l ] T ∈ R l is the weight vector, l > 1 is the number of nodes of the neural network, Z ∈ Ω Z ⊂ R q is the input of the RBF neural network, q is the input dimension of the RBF neural network, S(Z) � [s 1 (Z), s 2 (Z), . . . , s l (Z)] T ∈ R l is the basis vector function, and s i (Z) is the output of the i th node. A Gaussian function is always chosen as s i (Z), i.e., where r i is the width of the base function and ξ i � [ξ i1 , ξ i2 , . . . , ξ iq ] T is the center of the basis function. With a sufficient number l of nodes selected, an RBF neural network can approximate an arbitrary continuous function φ(Z) in a compact set Ω Z ∈ R q with arbitrary accuracy ε.
where δ(Z) is the approximation error with |δ(Z)| ≤ ε and W * is the given ideal constant weight vector, which is defined as In this paper, let where θ i are the estimates of the unknown constants θ i , W * i are the ideal weight vectors of the RBF neural network, b is a positive design parameter, and ‖ · ‖ is the norm.
Remark 2. b is related to Assumption 1.
Lemma 6 has been proved in [50,51]. Since ∞ k�0 3q(k + 2) q− 1 e (− 2p 2 k 2 )/r 2 is convergent, s is a limited value. In addition, s is independent of the neural network node numbers l and the neural network inputs Z.
where ε 10 is a positive design parameter and coefficients a j , j � 1, 2, ..., n, are calculated using the following equation: Remark 3. One of the main contributions of this paper is to design suitable virtual controllers α i so that nonlinear purefeedback systems meet the requirement of fixed-time control. In [40][41][42][43], virtual controllers designed exhibit similar power functions z 2α− 1 , where 0 < α < 1. If α is not appropriately selected, it will make z unsolvable at the origin and in the negative domain. As 0 < α < 1, power exponents 2α − 1 ∈ (− 1, 1), which leads to the possibility of negative power exponents. For example, if 2α where q 2 is an even number; then, z q 1 /q 2 is unsolvable at the negative domain. For example, if q 1 /q 2 � 1/2, z 1/2 is unsolvable at the negative domain, that is, (− 1) 1/2 does not exist. e controller designed in this paper overcomes the aforementioned defect and promotes the application of fixed-time control in more common nonlinear systems. Substituting α 1 into (26) yields where (18) is utilized to approximate f 1 (Z 1 ) and introduce inequalities (19) and (20), following which (30) can be rewritten as where σ 1 � η 2 1 /2 + ε 2 1 /4bk 13 . e adaptive law θ 1 is then defined as where λ is a positive design parameter. Combining with Assumption 3 and substituting (32) into (31), one can obtain where According to (21), (22), and (28), if |z 1 | ≥ ε 10 , then (33) is rewritten as According to (21), (22), and (28), if |z 1 | < ε 10 , then (33) is rewritten as Remark 4 (see [52]). Based on (28), when |z 1 | < ε 10 , there is an additional term in (36): bk 11 /2) . Note that if |z 1 | < ε 10 , then this additional term is obviously limited by some smaller constant ε 11 , so the structure of (35) is retained, while the constant term C 1 only slightly increases. Owing to page limitations and to avoid repetitive discussions, we will omit this part in the rest of the analysis.
Step 2. According to Construct a Lyapunov function as e derivative of V 2 is written as where is a smooth function that is used to overcome the design difficulty of _ θ 1 zα 1 /z θ 1 . e virtual controller α 2 is defined as where k 21 , k 22 , k 23 , and η 2 are positive design parameters.
In (42), S z 2 ,1 and S z 2 ,2 are defined as where ε 20 is a positive design parameter. Choose the adaptive law θ 2 as Combining (18)- (20) and Assumption 3, substituting (42) and (44) into (40), and adopting the same design method as in Step 1 yields (45) as a rewriting of (40). where According to (21) and (22) and Remark 4, we consider only |z 2 | ≥ ε 20 . en, (45) can be written as It can be seen from (47) that defining the design smooth function M 1 (Z 2 ) to overcome the design difficulty of (zα 1 /z θ 1 ) _ θ 1 is one of the difficulties of designing the controllers in this paper.
From Lemmas 4-6 and (32), it follows that with the result that Substituting (50) into (47) yields Step 3. where Constructing a Lyapunov function as where Define virtual controller α k as where k k1 , k k2 , k k3 , and η k are positive design parameters. In (56), S z k ,1 and S z k ,2 are defined as where ε k0 is a positive design parameter. Define the adaptive law θ k as Combining (18)- (20) and Assumption 3 and substituting (56) and (58) into (54) enable (54) to be rewritten as where In accordance with (21) and (22) and Remark 4, we consider only |z k | ≥ ε k0 . en, (59) can be written as Lemmas 4-6 and (58) yield erefore, the smooth function M k− 1 (Z k ) can be defined as with the result that Substituting (64) into (61) yields Step 4. According to z n � x n − α n− 1 , we have Constructing a Lyapunov function as V n � V n− 1 + z 2 n /2 + b θ 2 n /2c yields 8 Complexity where Define an actual controller u as where k n1 , k n2 , k n3 , and η n are positive design parameters. In (70), S z n ,1 and S z n ,2 are defined as where ε n0 is a positive design parameter. e adaptive law θ n is defined as _ θ n � c 2η 2 n S T n Z n S n Z n z 2 n − λ θ n .
Combining (18)- (20) and Assumption 3 and substituting (70) and (72) into (68) enable (68) to be rewritten as where According to (21) and (22) and Remark 4, we consider only |z n | ≥ ε n0 . en, (73) can be written as e treatment of M n− 1 (Z n ) is similar to that of (62) and (63), with M n− 1 (Z n ) being defined as Complexity 9 with the result that

Stability Analysis
Theorem 1. If system (1) satisfies Assumptions 1-3 and uses the virtual controllers (27), (42), and (56), the actual controller (70), and the adaptive laws (32), (44), (58), and (72), all signals in the closed-loop system are semiglobally uniform and ultimately bounded, and the upper limit of the fixed convergence time is irrelevant to the initial state.

Remark 6.
e fixed-time control algorithm in this paper is different from previous control algorithms. e principle differences are as follows: (1) e fixed-time control algorithm proposed in [35][36][37][38][39][40] does not solve the problem of the nonaffine structure of the control input u(t). e fixed-time control algorithm proposed here solves that problem. (2) Some systems' f i (·) structure is complex, which interferes with direct usage of f i (·) for designing controllers. An RBF neural network is used here to approximate unknown functions f i (·), thereby obviating the need to know the structure of f i (x i , x i+1 ) and avoiding the difficult design problem of controllers derived from complex system structures.

Simulation Results
In this section, two samples are studied to verify the effectiveness of the controller designed in the paragraphs above.  Complexity 5.1. Mathematical Example. Consider the following nonlinear pure-feedback system: where x 1 and x 2 are the system state variables, u is the system control input, y is the system output, d 1 (x 1 , t) � 0.7x 2 1 cos(1.5t) and d 2 (x 2 , t) � 0.5(x 2 1 + x 2 2 ) sin 3 (t) 0.5 are the external disturbance terms, and y d � 0.5 sin(1.5t) +cos(0.5t) is the reference signal. It can be seen from system (90) that the state variables and the control input (2u + u 3 )(x 2 1 + x 2 2 ) have the nonaffine structures. e simulation study aims to design a fixed-time controller based on system (90) that ensures that the output signal y can track the reference signal y d .
For a further simulation study, we selected two sets of different data to verify the tracking performance of system (90). It can be seen from Figure 5 that the error between the system output y and the reference signal y d increases with an increase in the α value. It can be seen from Figure 6 that the error between the system output y and the reference signal y d does not change significantly with an increase in the β value.
Remark 7. In [40][41][42][43], because the controller has the power function z 2α− 1 , α can select only specific values; otherwise, the power function z 2α− 1 is unsolvable at the origin and in the negative range. In this paper, α can be any number between zero and one.
It can be seen from Figures 1-6 that the state variables x 1 and x 2 , the actual controller u, and the adaptive parameters θ 1 and θ 2 are bounded, with the consequence that all signals in the closed-loop system (90) are bounded.

Physical
Example. Consider the following electromechanical system [41]:     [53]. y d � sin(0.5t) + 0.5 sin(t) is the reference signal. e system parameters are chosen as follows: M � 0.0642, N � 1.1408, B � 0.0181, L � 0.025, K B � 0.9, and R � 5.0. e simulation study aims to design a fixed-time controller based on system (93) to enable the output signal y to track the reference signal y d .
To verify the superiority of our designed controller, the tracking performance of this controller is compared with those of a fixed-time controller [40] and a finite-time controller [41]. e same design parameters as [40] are used, as follows: k 11 � 10, k 12 � 10, k 21 � 10, k 22 � 10, k 31 � 10, k 32 � 10, k 23 � 10, k 33 � 10, α � 97/101, β � 107/100, e design parameters of the neural network are the same as [41]. Figure 7 shows the system output y and the reference signal y d . As can be seen from Figure 7, the system output y can track reference signals y d effectively. Figures 8-10 show the state variables, the actual controller, and the adaptive parameters, respectively. Figure 11 shows the tracking error comparison between the fixed-time controller in this paper and the fixed-time controller in [40]. Figure 12 shows the tracking error comparison between the fixed-time controller in this paper and the finite-time controller in [41]. It can be seen from the simulation diagram that the performance of the fixed-time controller proposed in this paper is superior.
It can be seen from Figures 7-12 that the state variables x 1 and x 2 , the actual controller u, and the adaptive parameters θ 1 and θ 2 are bounded, with the result that all signals in the closed-loop system (93) are bounded.

Conclusion
A novel virtual controller and a novel actual controller designed in this paper solve the singularity problem of the virtual controller and the actual controller at the origin and in the negative domain. e fixed-time controller designed is applied to nonlinear pure-feedback systems and solves the problem of nonaffine structure. e controller enables the system output to track the reference signal in a fixed time and also to make the tracking error converge to a region of the origin in a fixed time. e simulation results prove the efficiency of the controller designed in this paper. We intend to apply the approximation-based fixed-time adaptive tracking control for a class of uncertain nonlinear purefeedback systems in time-delay systems in the future.

Data Availability
No data were used to support this study.

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