Fixed-Time Adaptive Neural Network Tracking Control for Output-Constrained High-Order Systems Using Command Filtered Strategy

This paper proposes a ﬁxed time adaptive neural command ﬁltered controller for a category of high-order systems based on adding a power integrator technique. Different from existing research, the presented controller has the following distinguishing advantages: i) ﬁxed-time control framework is extended to the tracking control problem of high-order systems. ii) the error compensation mechanism eliminates ﬁlter errors that arise from dynamic controllers. iii) growth assumptions about unknown functions are relaxed with the help of adaptive neural networks. iv) more general systems: the developed controller can apply to high-order systems subject to uncertain dynamics, unknown gain functions and asymmetric constraints. Stability analysis shows that all states are semi-globally bounded, and the convergence rate of tracking error is independent of initial conditions. The main innovation of our work is that the presented controller considers simultaneously ﬁlter errors, ﬁxed-time convergence, growth conditions and asymmetric output constraint for the tracking control issue of high-order systems. Finally, the simulation results validate the advantages and efﬁcacy of the developed control scheme.


Introduction
This paper focuses the following class of nonlinear systems [1]: 1, x n = ρ n (x n )u a n + ϕ n (x n ), y = x 1 (1) where x i , x n represent system states, and they constitute these state vectorsx i = [x 1 , x 2 , ..., x i ],x n = [x 1 , x 2 , ..., x n ]; ρ i (x i ), ρ n (x n ), ϕ i (x i ), ϕ n (x n ) are unknown bounded nonlinear functions, u denotes the final control signal; a i and a n are positive odd integers and at least one of their values is greater than 1, which leads to (1) being a high-order system [2]. It can be observed that when both a i and a n are equal to 1, (1) is a standard strict feedback system [3]. The high-order nonlinear system can be viewed as a general form of strict feedback systems, so it has a wider range of applications. x i+1 and u can be regarded as affine variables of the i-th subsystem and the n-th subsystem, respectively. Due to the existence of affine variables whose powers greater than 1, the Jacobian linearization of higher-order systems may be uncontrollable. As a result, how to develop a viable controller for higherorder systems has become a challenging task. An effective tool called a power integrator [4,5] was adopted to analyze the stability of higher-order systems, which promotes the development of controllers for this type of system. For example, a large number of controllers with different convergence rates (asymptotical convergence rate [6,7], finite-time convergence rate [8,9] and fixed time convergence rate [10,11]) are proposed and used to solve the stability control problem of high-order systems.
Different from asymptotical convergence control schemes [12], finite-time controllers have distinct advantages, including faster response speeds, higher tracking accuracy, and stronger anti-disturbance ability [1,13]. The finite-time stability framework has been widely employed to solve the control problems of high-order systems because of its excellent characteristics in recent years [1,8,14,9,15,16,13,17]. Among them, Cai et al. proposed a finite-time stabilization controller for a class of uncertain high-order systems and considered various practical difficulties (dead-zone, saturation and unknown control coefficients) that the controller may encounter [14]. Sun et al. extended the finite-time theory to a fast finite-time theorem [15,16] that is able to less conservatively estimate the convergent time of state variables, and then the theorem was used to address the stabilization control issue of high-order systems that suffer from asymmetric output constraint [17]. It should be emphasized that the convergence time of closed-loop systems contain the information of initial states under the finite-time stable framework [18].
Hence, when the initial states are unavailable or far from the equilibrium point [17], the transient characteristics of controllers will be seriously affected [19]. Fortunately, fixedtime stable theorems have gained remarkable attention because they ensure that the convergence speed of system states is only associated with the main control parameters [20,21]. That is, the initial state information no longer determines the convergence time of closed-loop systems. For example, the fixed-time stable criterion has begun to deal with the stabilization control problem of linear systems [22] and highorder systems [10]. In addition, Sun et al. expanded the application range of traditional fixed-time convergence theories such that they can cope with the tracking control problem of nonlinear systems with unknown dynamics [23]. It can be found that these researches in [4,6,10,14,15,17] concentrate on the stable control of high-order systems. Therefore, how to extend the fixed-time stable theory to the tracking control problem of high-order systems containing unknown dynamics is still unclear and open [24]. Fuzzy logic systems [25], neural networks [26][27][28] or state observers [11,29] are viewed as promising approximation tools and widely employed to identify and deal with the unknown dynamic that is contained in practical engineering [30]. For instance, fuzzy logic systems were utilized to overcome growth assumptions imposed on unknown functions contained by high-order systems [31]; an adaptive neural backstepping controller was presented to achieve the position tracking of a linear induction motor where the recurrent neural network is mainly used to model uncertain friction force [32]. Notice that the backstepping technologies [13] exist differential explosion problem [30]. Therefore, the backstepping-based controller may be extremely complex when the order of practical dynamic systems are more than three [33,34]. To simplify the structure of backstepping controllers, dynamic surface control (DSC) approaches appeared. Recently the dynamic surface controller has been utilized to solve the tracking control issue of output-constrained high-order systems [35].
It should be pointed out that limiting the output of control systems is of great significance in many practical control environments, for example, industrial robots [36], unmanned vehicles [12,37] and permanent magnet synchronous motors [38]. Because of physical limitations, each joint of the manipulators should rotate within a prescribed angle range [39,40]. Unmanned vehicles must drive in the specified lane to avoid collisions [30,41]. Ling et al. proposed an adaptive DSC algorithm for a 4-order under-actuated high-order systems with asymmetric output constraint [42], which can be regarded as a pioneering work. It is noteworthy that the DSC algorithm mainly uses the output of filters to approximate derivatives of virtual control signals such that the repeated derivative operations of backstepping controllers are avoided [43], however the DSC does not consider filtering errors. A command filtering technique [33] is used to conquer the shortcoming of the DSC methods because the technology provides a filtering compensation mechanism to eliminate filtering errors. By applying the command filtered scheme [26] an adaptive backstepping controller was presented for a class of n-order nonlinear systems [11,44]. Thereafter, a fixed-time command-filtered controller was presented to deal with the tracking control of surface vehicles, which proves the practical application prospects of the command-filtered technology [45]. Because the power contained in high-order systems is difficult to handle, the command filtering technology has not been applied to the control design of high-order systems, which is a strong motivation of this paper.
As far as our knowledge goes, filter errors, the fixedtime convergence, growth condition assumptions and asymmetric output constraints are not considered in the control field of uncertain high-order systems simultaneously [35]. Therefore, this paper aims to handle the trajectory tracking problem of high-order systems. The error compensation system is employed to overcome the problems of differentiation explosion and filter error. With the help of the fixed-time convergence theory, the fixed-time stability analysis of the closed-loop system is completed. Adaptive neural networks are utilized to remove growth conditions imposed on uncertain functions [31]. A coordinate transformation composed of newly defined output and reference trajectory is used to conquer the difficulty of asymmetric output constraints. Unlike the available research achievements, the proposed controller has the following novelties: 1 This paper proposes a fixed time adaptive command filtered control approach for high-order systems with unknown dynamic knowledge and asymmetric output constraints. The main challenge of our work is how to take filter errors, the fixed-time convergence, growth condition hypotheses and asymmetric output constraints into consideration simultaneously during the process of track-ing controller design for high-order systems in comparison with available researches [9,39]. 2 Compared with the existing literature [42,35], a new error compensation system and fixed-time stable theory are introduced into the controller design, so the presented controller not only overcomes the weaknesses of dynamic control schemes but also improves the transient performance of itself. 3 Different from the refs [8,10,46], the developed controller is also applicable even though high-order systems contain unknown functions that may not satisfy nonlinear growth conditions.
The rest of this paper is outlined as follows. Section 2 provides preliminaries, including Lemmas, Assumptions, and a brief description of a class of neural networks. The design of the proposed controller and its fixed-time stable analysis are given in Section 3. A comparison simulation example is provided to elaborate on the superiority of the developed control method in Section 4. Section 5 summarizes the current research results and gives the future development direction of the presented control scheme.

Preliminaries
This section first introduces the basic concept of a radial basis function neural network (RBFNN) and then provides some necessary assumptions and lemmas. These tools play an important role in the design process of the controller.
For the convenience of description, we give the definition of these symbols: ℜ stands for the real number,ℜ + stands for the positive real number, and ℜ i is i-dimensional real space throughout this paper. Some variables about functions will be omitted if no confusion arises.

Brief description of neural networks
This paper utilizes a RBFNN [47] to identify some unknown bounded function ϕ(x) ∈ ℜ 1 , and it is mainly composed of an input layer, a hidden layer and an output layer. Assume thatx = [x 1 , x 2 , ..., x N a ] T ∈ ℜ N a , so the total number of nodes of the input layer is N a . A brief definition of the hidden layer is described as follows [48]: where F o (x) is the output of the output layer. It is a approximation value of ϕ(x) ∈ ℜ 1 ;η = [η 1 , η 2 , ..., η L a ] T ∈ ℜ L a denotes positive weight vector, and L a is also equal to the total number of neurons contained in the hidden layer; .., M L a (x)] T ∈ ℜ L stands for a activation function vector. M j , j = 1, 2, ..., L a can be viewed as a neuron model, and its detailed definition is given as follows where b j ∈ ℜ + presents the Gaussian width;c j ∈ ℜ N a denotes a center position vector. All center position vectors form the following matrix: where c i, j , i = 1, 2, ..., L a ; j = 1, 2, ..., N a is a designed positive constant.

Lemmas and Assumptions
Lemma 1 [49]. For arbitrary small positive constant δ , then their is a neural network such that whereη * denotes the optimal weight vector. Lemma 1 implies that the RBFNN in this paper is able to approximate a unknown bounded function ϕ(x) ∈ ℜ 1 with arbitrary accuracy if we choose suitable weight parameters. Lemma 2 [13]. Let p ≥ 1 be a odd integer, and then for given real functions g 1 , g 2 , their is the following inequality: Lemma 3 [14]. Let p ∈ [0, +∞). For any real variables g 1 , g 2 , ..., g n , one obtains the following results: Lemma 4 [50]. Let p 1 , p 2 be positive constants. There are real functions g 1 , g 2 such that where δ > 0 denotes a specific real number. Lemma 5 [51]. Let g 1 , g 2 be real variables, For given real numbers p 1 > 1, p 2 > 1, there exists a Young's inequality: where δ is a positive constant, and p 1 and p 2 additionally satisfy the equality: Lemmas 2-5 are employed to extend application range of related inequalities.
Lemma 6 help us to analysis the stability of the proposed controller. Lemma 7 [41]. Let B L,1 and B R,1 be designed positive smooth functions. For any real variable g, a new barrier function g 1 can be defined as where g belongs to a compact set Lemma 7, as an unified tool, can be used to tackle with the output constraint problem of dynamics systems.
Lemma 8 [52]. Let g be a real function. For given positive constant p, there is a inequality where tanh (•) stands for the hyperbolic tangent function. Lemma 8 is utilized to deal with the derivative of filter outputv i,3 , i = 1, 2, ..., n − 1. Lemma 9 [23]. Let g 1 ∈ ℜ + and g 2 ∈ ℜ + be real functions. For any positive constant p, the following inequality holds: where g 1 is greater than or equal to g 2 . Lemma 9 will be employed to simplify inequalities related to adaptive parameter θ .
Assumption 1 [46]. The control gain function ρ i (x i ), i = 1, 2, ..., n included in the dynamic system (1) not only ensures strictly positive or strictly negative, but also satisfies the following inequality: where ρ i,min and ρ i,max stand for positive parameters. Assumption 1 guarantees that the control signal always acts on the dynamic system (1). For the convenience of stability analysis, suppose that ρ i (x i ) always positive. Assumption 2 [53]. The designed signal y r,1 and its time derivativeẏ r,1 are measured and known.
Assumption 3. The parameters a 1 , a 2 , ..., a n in the dynamic system (1) satisfy the inequality (a − a i + 1) ≤ α 1 (1−α 1 ) , i = 1, 2, ..., n, where α 1 ∈ (0, 1) is a real number; a denotes an odd integer and satisfies inequality a > max {a 1 , a 2 , ..., a n }. Remark 1. Assumption 3 can help us avoid the singularity problem of controllers caused by zero tracking error. It can be inferred from Assumption 3 that the minimum value of α 1 is 0.5. Different from Assumption 3 of reference [42], the parameter α 1 can be determined flexibly by ourselves such that Assumption 3 of this paper is less strict. As a result, the proposed controller is more suitable than Ling's controller [42] in a real industry environment.

Main results
This section first gives the main framework of the new fixedtime adaptive neural command filtered controller, and then provides the stability analysis of the developed controller with the aid of the fixed-time convergence theory [23] and the adding one power integrator tool [4].

Design of a fixed-time controller
Before designing the controller, the following preparations are required.
Preparation 1: According to Lemma 7, the output x 1 and reference signal y r,1 can be redefined as , x 1 ∈ Ξ x 1 e 1,2 = B L,1 B R,1 y r,1 (B L,1 + y r,1 )(B R,1 − y r,1 ) , y r,1 ∈ Ξ y r,1 where B L,1 ∈ ℜ + and B R,1 ∈ ℜ + denote smooth bounded functions which can be designed by ourselves; Ξ x 1 and Ξ y r,1 are compact sets, and their detailed definition are given as follows [42]: Remark 2. Compared with existing log-type barrier function (BLF) [35] and tan-type barrier function [1,54], the designed barrier functions (17) obtain the following strengths simultaneously: i) B L,1 is used to constrain the lower bound of x 1 and y r,1 , and B R,1 is used to constrain the upper bound of x 1 and y r,1 . Therefore the asymmetric output constraint can be achieved by designing B L,1 and B R,1 respectively; ii) the barrier functions (17) unify the fixed-time controller design for high-order system with or without output constraints.
The time derivative of (17) can be expressed aṡ where ∂ e 1,1 ∂ e 1,1 where e 1 and e k denote tracking errors; w i stands for a filter error; v i,2 is a virtual control law designed by ourselves, and v i,3 is a output of the filter (22). Preparation 3: this paper utilizes the following command filter [43] to approximate the virtual control signal of every subsystem: where µ i,1 , µ i,2 ∈ (0, 1) present designed positive parameters, and v i,2 can be viewed as the input signal of the command filter. v i,3 and ↔ v i,3 can replace v i,2 andv i,2 respectively, if µ i,1 and µ i,2 are selected appropriately.
Under Assumptions 1-3, a fixed-time adaptive neural command filtered controller is constructed. The proposed controller mainly consists of the following three parts: Part 1: the control signal corresponding to the j-th subsystem, j = 1, 2, ..., n, can be designed as follows: where v 1,1 =   (23) is further classified, v i,2 , i = 1, 2, ..., n − 1 is a virtual control signal, and v n,2 is the actual control signal u required by the dynamics system (1). Part 2: an adaptive lawθ can update the weight parameters of the RBFNN automatically, anḋ where ξ 1 ∈ ℜ + , ξ 2 ∈ ℜ + , ξ 3 ∈ ℜ + are design constants.
Part 3: a new filter compensation system that is employed to reduce filter errors is designed aṡ where k = 2, 3, ..., n, κ k−1,1 ∈ ℜ + , κ k−1,2 ∈ ℜ + are constants; . Remark 3. The proposed controller not only avoids the "explosion of complexity" of backstepping controllers but also further considers filter errors arise from dynamic surface controllers. Since the filter compensation system (25) is introduced into existing dynamic surface controllers [42,35] for high-order systems, it is theoretically guaranteed that the developed controller can obtain a smaller |x 1 − y r,1 | compared with these research results [42,35].
First, Theorem 1 is provided to illustrate the property of the developed controller.
Theorem 1: Assume that the high-order dynamic system (1) satisfies Assumptions 1-3 and the output constraints (17). If a set of control parameters are chosen properly, the new controller composed of (23), (24) and (25) obtain the following properties: i) all closed-loop signals (χ 1 , χ 2 , ..., χ n−1 , x 1 , x 2 , ..., x n ,θ ) are practically fixed-time stable. ii) the tracking error approaches a neighborhood of origin after the fixed time T a independent of initial conditions. iii) the output is constrained in the given interval (−B L,1 , B R,1 ).
Then, the rigorous proof of Theorem 1 can be performed in n steps.
Step 1: we design a Lyapunov candidate function L 1 = e a−a 1 +2 1 a−a 1 +2 + χ 2 1 2 , and then its first derivative can be computed aṡ With the help ofẋ 1 x 1 ρ 1 (x 1 )v a 1 1,2 and coordinate transformations (21),L 1 can be further described aṡ Remark 4. Since the residual term d 2 contains the information of v 1,2 , directly using v 1,2 to eliminate d 2 will cause algebraic loop problems. So we try to use the control signal of the next subsystem to cancel d 2 . Different from existing research results [35,42], the error compensation system will further handle the filtering error contained in d 2,2 , which will help improve the tracking accuracy of the controller.
In accordance with Lemma 1, a RBFNN η * T 1 M 1 (x 1 ) is able to achieve the following approximation: where the absolute value of the error function δ 1 (x 1 ) is less than or equal to the given accuracy δ 1,max ∈ ℜ + . Remark 5. The related control algorithms [8,10,46] require that unknown function ϕ 1 (x 1 ) must be bounded and satisfy the growth assumption. It can be inferred from Lemma 1 that if ϕ 1 (x 1 ) is a bounded function, then there exists a ideal neural network to approximate it with given accuracy.
The proposed controller is applicable even though ϕ 1 (x 1 ) does not satisfies growth assumption such that it is more suitable in practical control filed.
Remark 7. Many existing researches [14,1,9,40,2,16,15] are concentrate on the finite-time stable control problem for high-order systems. It is hard for finite-time controller to obtain the accurate convergence time, since its convergence time linked to initial states and tracking error of close loop systems. Stability control can be regarded as a special case of tracking control. In contrary, this paper use the fixed-time convergence theory to design a tracking controller for highorder systems. Therefore, the controller designed here is an upgraded version of these controllers [14,1,9,40,2,16,15].

Simulation Verification
This section illustrates the efficacy and superiority of the proposed controller by a comparative simulation.
The simulation focuses on the tracking control, so these finite-time stabilization controllers [14,1,9,40,2,16,15] fail to suitable. To further illustrate the advantages of the fixedtime control framework, two different initial conditions are considered: Condition 1:  Fig. 1 implies that the output signals under different initial conditions all enter a stable state after 0.44 seconds. Therefore, simulation results validate the effectiveness and superiority of the presented controller, compared with the existing finite-time stabilization controllers.

Conclusion
This paper has investigated the adaptive tracking control problem for a category of high-order nonlinear systems. Both the output constrain and the uncertainty of dynamics systems are taken into consideration. Adaptive RBFNNs are used to identify unknown functions that may not satisfy the growth conditions. A new error compensation system is employed to overcome filter errors. As a result, a fixed time adaptive neural command filtered controller is proposed. Simulation results validate the effectiveness and superiority of the presented controller, compared with the existing finite-time stabilization controllers. Future research will further consider the impact of the actual environment on the output signal of the proposed controller, such as input saturation, dead zone, and control channel bandwidth constraints.