Adaptive Neural Network Finite-Time Control of Uncertain Fractional-Order Systems with Unknown Dead-Zone Fault via Command Filter

: In this paper, the adaptive ﬁnite-time control problem for fractional-order systems with uncertainties and unknown dead-zone fault was studied by combining a fractional-order command ﬁlter, radial basis function neural network, and Nussbaum gain function technique. First, the fractional-order command ﬁlter-based backstepping control method is applied to avoid the computational complexity problem existing in the conventional recursive procedure, where the fractional-order command ﬁlter is introduced to obtain the ﬁlter signals and their fractional-order derivatives. Second, the radial basis function neural network is used to handle the uncertain nonlinear functions in the recursive design step. Third, the Nussbaum gain function technique is considered to handle the unknown control gain caused by the unknown dead-zone fault. Moreover, by introducing the compensating signal into the control law design, the virtual control law, adaptive laws, and the adaptive neural network ﬁnite-time control law are constructed to ensure that all signals associated with the closed-loop system are bounded in ﬁnite time and that the tracking error can converge to a small neighborhood of origin in ﬁnite time. Finally, the validity of the proposed control law is conﬁrmed by providing simulation cases.


Introduction
Over the past several decades, control problems of uncertain nonlinear systems [1], nonsmooth nonlinear systems [2], strict-/nonstrict-feedback systems [3,4], and pure-feedback systems [5] have been widely studied, and to achieve the specified control objectives, various control laws have been constructed by scholars. It should be pointed out that the order of the above-mentioned systems is integer order, namely, the so-called integer-order systems. In fact, some systems, such as hyper-chaotic economic systems and heat conduction and viscoelastic structures [6,7], cannot be modeled by integer-order systems. Therefore, as the extension of integer-order systems, the control problems of fractional-order systems have been developed by many scholars. Currently, whether it is the solution problem of fractional calculus or the control problem, the research results of fractional calculus can be found in many literatures [8][9][10][11][12].
Because fractional-order systems break through the limitation of integer-order systems, they can better describe the historical information of control objects [13,14], which have attracted more and more attention in recent years [15][16][17]. An adaptive control law based on neural network was presented in [15], which guarantees that the tracking error of the switched fractional-order nonlinear systems can converge to a small neighborhood of the origin under arbitrary switching. In [16], an L 1 adaptive control law for the control lems of fractional-order systems [39][40][41], they did not consider the existence of unknown dead-zone fault. Inspired by the above discussion, the objective of this paper is to address the finite-time control for fractional-order systems with unknown dead-zone fault and uncertain dynamics. Based on the application of the fractional-order command filter, the radial basis function neural network, and the Nussbaum gain function technique, an adaptive neural network finite-time control law was developed. The main contributions of this paper are as follows: (1) A class of uncertain fractional-order systems with unknown dead-zone fault is investigated. Compared with [30,39,40], the model considered in this paper is more general.
(2) A fractional-order command filter is introduced to obtain the filter signals and their fractional-order derivatives, which avoids the computational complexity problem existing in the conventional backstepping recursive procedure.
(3) To deal with uncertain nonlinear functions in the step of recursive design and unknown control gain caused by the unknown dead-zone fault, the radial basis function (RBF) neural network and Nussbaum gain function technique are applied in this paper. Then, the virtual control laws, adaptive laws and finial adaptive neural network finite-time control law are designed.
(4) By using the designed adaptive neural network finite-time control law, it can be guaranteed that all signals associated with the closed-loop system are bounded in finite time, and the tracking error converges to a small neighborhood of origin in finite time.
The rest of this paper consists of the following sections. The problem formulation and preliminaries are given in Section 2. In Section 3, the main design processes of the control law are provided, and the stability analysis is also shown in this section. In what follows, we give the simulation results and brief conclusions in Sections 4 and 5, respectively.
Notations: Throughout this paper, R, C, and N represent, respectively, the sets of real numbers, complex numbers, and integers; R n represents the set of n− dimensional real vectors; | · | stands for the absolute value of a constant; · is the induction norm of a matrix or the Euclidean norm of a vector; C T stands for the transpose of matrix C or vector C; and min(X) or max(X) represent the minimum value or maximum value of X.

Problem Formulation and Preliminaries
This section will introduce the problem formulation for uncertain fractional-order systems, and some preliminaries, such as the fractional calculation, Nussbaum gain function technique, and some lemmas are given for the subsequent analysis.

Problem Formulation
Consider the uncertain fractional-order systems with unknown dead-zone fault, which is described as x n + f n−1 (x) + γ T n−1 ϕ n−1 (x) C D α t x n = g n (x)u F (t) + f n (x) + γ T n ϕ n (x) y = x 1 (1) where α is the fractional order; x = [x 1 , · · · , x n ] T ∈ R n , u F (t) ∈ R, and y ∈ R are the state vector, the control input, and the output of system, respectively; g i (x) and f i (x), i = 1, · · · , n, represent the known nonzero smooth functions and uncertain nonlinear functions, respectively; γ i and ϕ i (x), for i = 1, · · · , n, stand for the unknown constant vectors and known nonlinear function vectors, respectively. For convenience, the functions g i (x), f i (x) and ϕ i (x) are denoted by g i , f i , and ϕ i , respectively. In this paper, the control input u F (t) is subjected to the dead-zone fault, where "F" is the first letter of "Fault". Based on [42], u F (t) is given as where k d > 0 represents an unknown bounded constant and is defined as the slope of the dead zone; b l > 0 is the left breakpoint of dead-zone, and b r > 0 is the right breakpoint of the dead zone.
By applying the mean value theorem, the control input (2) can be rewritten as and there exists u F (t) ≤ |u(t)| ≤ U, where U represents the maximum value allowed by the system; φ(t) is a bounded function that satisfies |φ(t)| ≤ φ, and φ(t) is shown as For the system (1), the control goal of this paper is to construct an adaptive neural network finite-time control law u(t) such that all signals of the closed-loop system are bounded in finite time, and the system output y = x 1 can track the reference signal y d in finite time.
To achieve the desired control objective, some assumptions are provided as follows.
Assumption 1. The reference signal y d and its fractional-order derivative C D α t y d are smooth and bounded. Assumption 2. The smooth functions g i , i = 1, · · · , n are bounded and the signs are identical; namely, there exist positive constants g i,min and g i,max such that g i,min ≤ |g i | ≤ g i,max . Remark 1. Assumptions 1 and 2 are common in the control law design of fractional-order systems and can be found in most existing results [18,20,28]. Assumption 2 implies that the time-varying control gains g i are either strictly positive or strictly negative with the same sign. Moreover, the purpose of introducing positive constants g i,min and g i,max is to analyze the boundlessness of all signals and the stability of the system.

Fractional Calculation
Definition 1 ([43]). The αth Caputo derivative of a smooth function f (t) is described as where C D α denotes the Caputo fractional operator with q − 1 < α < q for q ∈ N; Γ(·) is the Gamma function, which is given as Γ(q) = +∞ 0 s q−1 e −s ds.
For the Caputo fractional operator, the following properties hold.
where λ 1 and λ 2 are constants; x 1 (t) and x 2 (t) are smooth nonlinear functions. Remark 2. In the following analysis, only the case of 0 < α < 1 is considered. In addition, the notation C D α for the Caputo operator is replaced by D α .

κ(t)can be maintained.
To facilitate the analysis of finite time problems, the following lemmas are provided.

Lemma 6 ([30]
). For any continuous function F(x) over a compact set Ω ∈ R n , there exists an RBF neural network (W * ) T Φ(x) such that where W * ∈ R l is the optimal weight vector, l > 1 is the neural network node number, ε(x) is the approximation error and there exists |ε(x)| ≤ ε * , and Φ(x) = [ϕ 1 (x), · · · , ϕ l (x)] T ∈ R l represents a Gaussian-like basis function vector with where ι i = [ι i1 , · · · , ι in ] T and are the center of the basis function and the width of the Gaussian function, respectively.

Control Law Design Process and Stability Analysis
For this section, the adaptive neural network finite-time control law for uncertain fractional-order systems with unknown dead-zone fault (1) is proposed. This can not only ensure that all signals of the closed-loop system are bounded in finite time, but it also makes the output of the system track the reference signal in finite time.

Adaptive Neural Network Finite-Time Control Law Design
We define the following coordinate transformation: where y 1,d = y d , y i,d for i = 2, · · · , n, and y i,d = ψ i,1 is the output of the fractional-order second-order command filter (see Lemma 7) with the virtual control law υ i−1 as the input. The compensated tracking error z i is defined as where s i is the compensating signal to be designed.
Design the adaptive lawsŴ 1 andγ 1 as where η 1 > 0 and δ 1 > 0 are design constants. Substituting (34) and (35) into (33), and considering Lemma 10, gives Step i (i = 2, · · · , n − 1): Considering (1), (24), and (25), the fractional derivative of Similarly, an RBF neural network is introduced to approximate the unknown nonlinear function f i . Then we obtain Design the compensating signal s i as where λ i > 0 and i > 0 are design constants. Substituting (38) and (39) into (37) yields Design the Lyapunov function candidate as where A i and B i are the designed positive constants; Design the virtual control law υ i as where c i > 0 and β ∈ (0, 1). Substituting (43) into (42) has Design the adaptive lawsŴ i andγ i as where η i > 0 and δ i > 0 are design constants. Substituting (45) and (46) into (44), and considering Lemma 10, one has Step n (i = n): In this step, the adaptive neural network finite-time control law is derived. Considering (1), (3), (24), and (25), the fractional derivative of z n is given as The unknown nonlinear function f n in (48) is approximated by using the RBF neural network, that is Design the compensating signal s n as where λ n > 0 and n > 0 are the design constants. Substituting (49) and (50) into (48) yields where G n = k d g n and ε n = g n φ(t) + ε * n . Considering the boundlessness of g n , k d , and φ(t), there exist |G n | ≤ G * and |ε n | ≤ ε * n with unknown constants G * > 0 and ε * n > 0. Design the Lyapunov function candidate as where A n and B n are the designed positive constants; W n = W * n −Ŵ n and γ n = γ n −γ n , whereŴ n , andγ n are the estimations of W * n and γ n , respectively. The fractional derivative of V n is given as Design the adaptive neural network finite-time control law u(t) as .

Stability Analysis
Based on the virtual control laws, adaptive laws, and adaptive neural network finitetime control law designed above, the main results can be summarized as follows.

Theorem 1.
Consider an uncertain fractional-order system (1) that is subject to unknown dead-zone fault (2). Under Assumptions 1 and 2, if the compensating signals are selected as (28), (39), and (50), the virtual control laws are designed as shown in (32) with adaptive laws (34) and (35), and (43) with adaptive laws (45) and (46), and the adaptive neural network finite-time control law is designed as shown in (54) with adaptive laws (58) and (59). Then, all signals of the closed-loop system are bounded in finite time and the tracking error e 1 can converge to a small neighborhood of origin in finite time.
Proof. Design the following Lyapunov function as Invoking (36), (47), and (60), the αth fractional-order derivative of V is By applying Lemma 8, the following results can be obtained: where ν 1 and ν 2 are positive constants.
, respectively. Thus, the following inequalities hold By substituting (67) and (68) into (66), and applying Lemma 9, the following result is satisfied by choosing appropriate parameters satisfying λ i > i , η i > ν 1 , and δ i > ν 2 , that is where a, b and D 1 are respectively given as Next, we verify our results in three steps.
Step 1. Considering (69) and the definition of V, it can be easily obtained that bV β ≥ 0.
By applying Lemma 5, there exist a positive constant G * such that max(G n N (κ) + 1) . κ(t) = G * for t ∈ [0, t 0 ). Therefore, (69) can be written as Step 2. Based on the results of Step 1, from (71), we have Applying Lemma 1 and Lemma 3, then there is a positive constant ς such that which means that V is bounded, and it further implies that the signals z i , W i , and γ i are also bounded. Noting W i = W * i −Ŵ i and γ i = γ i −γ i , then the boundlessness ofŴ i and γ i can be also obtained.
Step 3. From the definition of z 1 = e 1 − s 1 , if z 1 and s 1 are finite-time stable, then the tracking error e 1 is also finite-time stable. Considering (71) and the fact that aV ≥ 0, then we have By applying Lemma 4, it can be held that and the setting time T f 1 is where µ 1 ∈ (0, 1) and V 0 = V(0). According to the definition of V, one gives Now, we show that the compensated signal s 1 is finite-time stable. Choose the following Lyapunov function candidate: Invoking (28), (39), and (50), the αth fractional-order derivative of Y is

Control Law Design Process and Stability Analysis
For this section, the adaptive neural network finite-time control law for uncertain fractional-order systems with unknown dead-zone fault (1) is proposed. This can not only ensure that all signals of the closed-loop system are bounded in finite time, but it also makes the output of the system track the reference signal in finite time.

Adaptive Neural Network Finite-Time Control Law Design
We define the following coordinate transformation: According to Lemma 6, an RBF neural network is introduced to approximate the unknown nonlinear function 1 f . Then we have

Control Law Design Process and Stability Analysis
For this section, the adaptive neural network finite-time control law for uncertain fractional-order systems with unknown dead-zone fault (1) is proposed. This can not only ensure that all signals of the closed-loop system are bounded in finite time, but it also makes the output of the system track the reference signal in finite time.

Adaptive Neural Network Finite-Time Control Law Design
We define the following coordinate transformation: where i s is the compensating signal to be designed.

ontrol Law Design Process and Stability Analysis
For this section, the adaptive neural network finite-time control law for uncertain tional-order systems with unknown dead-zone fault (1) is proposed. This can not ensure that all signals of the closed-loop system are bounded in finite time, but it makes the output of the system track the reference signal in finite time.

Adaptive Neural Network Finite-Time Control Law Design
We define the following coordinate transformation: ( ) , Design the compensating signal 1 s as

Control Law Design Process and Stability Analysis
For this section, the adaptive neural network finite-time control law for uncertain fractional-order systems with unknown dead-zone fault (1) is proposed. This can not only ensure that all signals of the closed-loop system are bounded in finite time, but it also makes the output of the system track the reference signal in finite time.

Adaptive Neural Network Finite-Time Control Law Design
We define the following coordinate transformation:  (26) neural network is introduced to approximate the unen we have , n ,  ( ) , the compensating signal 1 s as Substituting (81) into (80) and applying Lemma 9, one has

Control Law Design Process and Stability Analysis
For this section, the adaptive neural network finite-time control law for uncertain fractional-order systems with unknown dead-zone fault (1) is proposed. This can not only ensure that all signals of the closed-loop system are bounded in finite time, but it also makes the output of the system track the reference signal in finite time.

Adaptive Neural Network Finite-Time Control Law Design
We define the following coordinate transformation: where i s is the compensating signal to be designed. (24), and (25), the th  fractional-order derivative of , and ensating signal to be designed.
. Noting Lemma 4, and similar to the proof of z 1 , it can be obtained that the compensating signal s 1 is finite time stable and satisfies (83) and the setting time T f 3 is where µ 2 ∈ (0, 1) and Y 0 = Y(0). Considering (25), (77), and (83), the tracking error e 1 satisfies Observing (85), it can be seen that the tracking error e 1 is the sum of the compensating signal s 1 and the compensated tracking error z 1 . Accordingly, the convergence time T also satisfies this relationship. Moreover, it can be found that the tracking error e 1 depends on parameters λ i , i , c i , η i , δ i , A i and B i , i = 1, · · · , n. It also implies that the tracking error e 1 converge to the specified small neighborhood of origin in finite time within the setting time T = T f 1 + T f 2 + T f 3 by selecting the appropriate parameters. This completes the proof.
Remark 3. Noting (85), the tracking error e 1 can be made arbitrarily small by adjusting parameters λ i , i , c i , η i , δ i , A i and B i , i = 1, · · · , n. We can decrease D * 1 by decreasing the values of parameters η i and δ i or increasing A i and B i , and we can decrease D * 2 by increasing i . We can also increase b by increasing the value of parameter c i , and we can increase c by increasing λ i . Based on the adjustment of D * 1 , D * 2 , b, and c, it can be guaranteed that the tracking error e 1 can converge to the specified small neighborhood of origin in finite time within the setting time. However, it should be emphasized that the change of i simultaneously affects D * 1 and D * 2 , and the change of λ i simultaneously affects D * 2 and c. Moreover, the adjustment of these parameters may be bringing about an increase in the amplitude of the control signal. Therefore, when selecting suitable parameters, a trade-off should be made between the control performance of the tracking and the amplitude of the control signal.

Simulation Analysis
In this section, the simulation cases are given to verify the validity of the control law designed in this paper.
Case 1: Consider a class of uncertain fractional-order systems as follows:   x .  x .     . The parameters selection of the second-order command filter and the initial conditions of the adaptive laws are the same as those described for Case 1.
The simulation results of this case are displayed in Figures 8-13. Figure 8 shows the curves of the system output 1 x and the reference signal d y . It is not difficult to see from Figure 8 that the system (87) can obtain a good tracking performance by applying the proposed control law. The tracking error curve is given in Figure 9. One observes that the tracking error  Figures 11-13. It can be found that the signals of the closed-loop system shown in these figures are bounded, which verifies the validity of the theoretical analysis. However, it is not difficult to observe in Figures 9-11 that there are oscillations in these simulation results. In fact, considering the existence of unknown dead-zone faults and uncertain dynamics in the system, this makes it necessary to make a reasonable trade-off between system tracking performance and control output. The RBF neural network is applied to approximate the unknown nonlinear functions f 1 , f 2 and f 3 . Since there are only two variables in functions f 1 , f 2 , and f 3 , the node number for each RBF neural network is chosen to be 9 with the centers of the basis function ι i (i = 1, · · · , 9) evenly spaced in [−8, 8] × [− 8,8] and the width being = 4.
The other design parameters are ϑ = 0.01, β = 0.95, λ 1 = 10, λ 2 = 5.5, λ 3 = 7.5, The parameters selection of the second-order command filter and the initial conditions of the adaptive laws are the same as those described for Case 1.
The simulation results of this case are displayed in Figures 8-13. Figure 8 shows the curves of the system output x 1 and the reference signal y d . It is not difficult to see from Figure 8 that the system (87) can obtain a good tracking performance by applying the proposed control law. The tracking error curve is given in Figure 9. One observes that the tracking error e 1 can converge to a small neighborhood of zero in finite time. From  Figures 8 and 9, although the system (87) is affected by unknown dead zone fault, the tracking performance of the system can be guaranteed under the designed control law. This also proves the effectiveness of the proposed control law from another perspective. Furthermore, the curves of state variables x 1 , x 2 and x 3 are given in Figure 10, and the curves of the control law u(t) and adaptive laws Ŵ i and γ i (i = 1, 2, 3) are depicted in Figures 11-13. It can be found that the signals of the closed-loop system shown in these figures are bounded, which verifies the validity of the theoretical analysis. However, it is not difficult to observe in Figures 9-11 that there are oscillations in these simulation results. In fact, considering the existence of unknown dead-zone faults and uncertain dynamics in the system, this makes it necessary to make a reasonable trade-off between system tracking performance and control output.   x .   x .  x .

Conclusions
The adaptive finite-time tracking control for uncertain fractional-order systems with unknown dead-zone fault was considered in this paper. The fractional-order command filter was applied to avoid the computational complexity problem existing in conventional recursive procedures, and the neural network approximator was used to approximate the unknown uncertain nonlinear functions. Through the application of the Nussbaum gain function technique, the adaptive neural network finite-time control law was developed to solve the finite-time control problem of the given fractional-order systems. It has been proven that the desinged control law can not only ensure that all signals of the closed-loop system are bounded in finite time but can also ensure that the tracking error converges to a small neighborhood of the origin in finite time. However, it should be pointed out that the control law presented in this paper is only suitable for the systems with known state gains and measurable states. When the nonlinear system under consideration has unknown state gains and unmeasurable states, the proposed control law will not work effectively. Therefore, one of our future research directions is to design feasible control laws to realize the adaptive finite-time control of uncertain fractional-order systems with unknown control gain and partially unmeasurable states.