Finite-Time Adaptive NN Backstepping Dynamic Surface Control for Input-Delay Fractional-Order Nonlinear Systems

This article focuses on a finite-time adaptive dynamic surface control (DSC) approach for a kind of nonstrict fractional-order nonlinear systems (FONSs) with input delay. An auxiliary compensation function is presented by using the integral function of the input signal to handle input delay problem. To overcome the problem of inherent computational complexity, a fractional-order filter is applied to approach the virtual controller and its fractional-order derivative in each step of the backstepping procedure. By using backstepping technology and neural network (NN), the finite-time adaptive DSC approach is developed, and finite-time stability criteria based on fractional-order Lyapunov method are introduced to prove the finite-time convergence of the tracking error into a small region around the origin. The effectiveness of the presented control scheme is demonstrated by two examples.

In fact, the above control methods for FONSs above can only guarantee the desired performance of the control systems as the time approaches infinity. Many contributions for finitetime control methods have been developed and reported, especially with respect to the controller design for the The associate editor coordinating the review of this manuscript and approving it for publication was Haibin Sun . nonlinear integer-order systems, to illustrate the advantages of finite-time control in terms of converge precision and robustness [18], [19], [20]. The authors in [21] present the finite-time fuzzy controller for fractional-order tumor systems with chemotherapy, and the stability analysis is given under Lyapunov theory. For the nonlinear FONSs, a finitetime adaptive NN controller is developed by using the eventtriggered mechanisms in [22]. In [23], the output feedback controllers with finite-time performance is presented for nonlinear fractional-order interconnected systems. To ensure the signal variables arrive at a domain within the fixedtime level, a dynamic sliding mode control (SMC) scheme for nonlinear FONSs represented by interval type-2 fuzzy models is developed in [24]. In [25], a kind of unknown nonlinear FONSs under actuator faults is researched, and a finite-time adaptive fuzzy backstepping SMC method is developed. In [26], a finite-time backstepping controller with compensation approach based on command filter is developed for uncertain nonlinear FONSs subject to external disturbances, and the convergence of tracking error in finite time can be guaranteed. For multi-agent FONSs with disturbances and unknown dynamics in [27], a distributed adaptive backstepping NN finite-time control is proposed, and the finite-time convergence can be achieved. However, the above-mentioned design methods for FONSs does not consider the influence of input delay.
In practical systems, the phenomenon of input delay is an unavoidable problem when send control signals to actuators due to the time spent in signal transmission. If the influence of input delay is ignored, the system performance will be deteriorated or even be disastrous since the feedback to change the system states can not be given timely. To deal with the problem of input delay in FONSs, many effective schemes have been developed recently. The authors in [28] propose an observer-based feedback control scheme for a class of fractional-order systems with input delay, in which the Smith predictor is employed to transform the system into a fractional-order system without input delay. For the uncertain FONSs with input delay and external disturbances considered in [29], a fractional-order adaptive learning controller is developed by using function approximation technique, and a augmented controller is designed to compensate the input delay effects. In [30], the authors propose an adaptive control scheme by using fractional-order command-filter for uncertain FONSs with subjected to input delay, in which a fractional integral is introduced to deal with the input delay. The authors in [31] develop an adaptive NN dynamic surface controller for FONSs subject to time delayed input, and the input delay problem is solved by a fractional-order auxiliary system. Furthermore, none of results about the adaptive control strategy of FONSs with input delay in finite time.
Motivated by the above-mentioned literates, the finite-time adaptive control designing procedure for FONSs under input delay will be investigated. The innovations of the proposed scheme are as follows: 1) Compared with the finite-time controller by using adaptive backstepping in [22] and [26] and adaptive SMC in [24] and [25] for FONSs, the input delay is further investigated by introducing the integral function of the input signal, and a fractional-order filter is applied to approach the problem of inherent computational complexity, which makes the proposed control scheme more general for application in practical engineering.
2) Compared with FONSs subject to input delay in [28], [29], [30], and [31], the finite-time convergence can be guaranteed by proposed adaptive neural backstepping DSC and all the singals of the closed-loop system are bounded, in which the faster convergence rate and higher tracking precision can be achieved.
T and χ i = χ i1 χ i2 · · · χ in T are the center and the width of the Gaussian function, respectively. Lemma 6 (Young's Inequality [40]): For ∀ (d, e) ∈ R 2 the following inequality holds

B. PROBLEM DESCRIPTION
Consider the following FONS with input delay: . . x n T ∈ R n are the state vectors, y ∈ R is the output of systems, f i (·) is the unknown smooth function, and g i (·) is the known function, i = 1, 2, . . . , n. u is the controller input to be constructed, and τ ∈ R + represent the input delay.
The control target is to design a finite-time adaptive neural control strategy for system (5) such that: 1) the output y tracks the desired trajectory y r (t) in finite time; 2) all the signals in the closedloop system remain boundedness. VOLUME 11, 2023 Assumption 1: For ∀k c 1 > 0, there exist positive constants A 0 , A 1 and A 2 , such that |y r | ≤ A 0 < k c 1 , D α y r (t) ≤ A 1 , and D 2α y r (t) ≤ A 2 . In addition, there exists a compact y r = { y r D α y r D 2α y r T y 2 r + (D α y r ) 2 + D 2α y r 2 ≤ δ y r }, δ y r > 0, such that y r D α y r D 2α y r T ∈ y r .

III. ADAPTIVE NEURAL CONTROL DESIGN AND STABILITY ANALYSIS A. CONTROL DESIGN
In this section, the backstepping technology and DSC method will be using to design finite-time adaptive neural control for the systems (6). First, coordinate transformations can be given as follows: where χ 1 is the tracking error, χ i is the dynamic surface error, ζ i is the fractional-order dynamic surface filter output error, a i−1 is the virtual controller, and a i.l is the output of the fractional-order dynamic surface filter, which is defined as: where κ i is a constant. The virtual controller a i , actual controller u and parameter adaptive law D α W i are designed as follows: B. STABILITY ANALYSIS Theorem 1: Consider the nontriangular FONS (6) with the Assumptions 1. The actual controller (11) with virtual controllers (9) and (10) designed with the parameter adaptive laws (12) can confirm that all the signals in the closed-loop system are bounded, and the tracking error into a small region around the origin in a finite time, i.e. for ∀t ≥ T , error χ i will stay around the compact set χ i , i = 1, 2, . . . , n, where Proof: Step 1: From system (6) and the change of coordinates (7), the Caputo fractional derivative of where f 1 (x 1 ) is approximated via the NN W * T 1 1 (x 1 ) according to Lemma 5. W * 1 denotes the optimal parameter vector and satisfies f 1 (x 1 ) = W * T 1 1 (x 1 ) + ε 1 . Assume that there exists constantε 1 > 0 such that ε 1 ≤ε 1 .
The Lyapunov candidate function is constructed as: where γ 1 > 0 is a design parameter.W 1 = W * 1 − W 1 is the parameter estimation error with parameter estimation W 1 .
Combine with (6), Lemma 1 and Lemma 2, one can obtain the Caputo fractional derivative of V 1 The Young's inequality is applied and yields: where b 1 > 0. Substituting (9), (12) and (17) into (16), D α V 1 is presented as: 5208 VOLUME 11, 2023 Step i (i = 2, 3, . . . , n − 1): The derivative of χ i is presented as follows: Assume that ∃ε i > 0 such that ε i ≤ε i . The Lyapunov candidate function is selected as where Using the Young's inequality, yields: where b i > 0. Based on formulas (7), (8), Lemma 5 and DSC technology, one can obtain where (24) Similar to step 1, the D α V i in (21) can be rewritten as Step n: From (6) and (7), one can obtain D α χ n = D α x n − D α a n.l + u − u (t − τ ) = W * T n n (x) + ε n + u + g n (x) − D α a n.l (26) where f n (x) is approximated via the NN W * T n n (x), and W * n is the optimal parameter vector satisfying f n (x) = W * T n n (x) + ε n . Assume that ∃ε n > 0 such that ε n ≤ε n . Choose the following Lyapunov candidate function where γ n > 0.W n = W * n − W n and W n is used to estimate W * n . The derivative of V n is It is true that and D α ζ n = D α a n.l − D α a n−1 = − ζ n κ n +H n (·) where Substitute (11), (12) and (29) into (28), one gets Define sets y r and i as For ∀δ r > 0, the set y r from Assumption 1 and i are compact, respectively. Thus, y r × i is also compact. So there exists a constant j > 0, such that H j ≤ j . Then, it holds: whereq j > 0. From (33) and (35), (32) can be rewritten as where q j+1 i = 1, 2, . . . , n; j = 1, 2, . . . , n − 1.
Based on Lemma 4, choose (38) Use the same process, the following inequality hold: Using the previous results (38) and (39), (36) can be rewritten as . According to Lemma 3, it holds that Then Using the Caputo fractional integral and V = V 1−ρ n , one gets Furthermore, with the definition of V n , one obtain This illustrates the tracking error converges to χ i after the finite time T . The boundedness of χ 2 i ,W i and ζ i+1 can be obtained according to (27) and V n ≤ M (σ (1 − ϑ)) 1 / ρ . As χ 1 and y r (t) are bounded, and x 1 is bounded due to x 1 = χ 1 + y r (t). From (9), a 1 is a function of χ 1 and W 1 . Then, a 1 is also bounded and there is the supremumā 1 of a 1 . Due to χ 2 = x 2 − a 2.l and ζ 2 = a 2.l − a 1 , one can know that a 2.l and x 2 are bounded. Meanwhile, notice the boundedness of In order to easily understand the presented concept in this paper, the block diagram representation is presented in Fig. 1.
Step 1: Choose appropriate vector L and confirm matrix A is a strict Hurwitz matrix.
Step 3:Choose appropriate parameters satisfying γ i > 0 and ς i > 0 the parameter adaptation law W i in (13).
Step 4: Choose appropriate parameters satisfying b n > 0 and c n > 0 the adaptive controller (12).
Remark 1: In fact, the larger parameters c i , ς i , b i , γ i can guarantee the smaller radius of the region of the tracking error converges. However, the enlarged parameters c i , ς i , b i , γ i may consume the excessive control effect in the control process. To guarantee the robustness based on the aforesaid argument, the control parameters c i , ς i , b i , γ i should be selected as appropriately large within the feasible ranges. In the future, we will focus on how to make parameter selection less conservative.
Remark 2: The integral function of the input signal is adopted to deal with input delay, the finite-time convergence can be guaranteed by proposed adaptive neural backstepping DSC with the ∀t ≥ T , in which the faster convergence rate and higher tracking precision can be achieved.

IV. SIMULATION EXAMPLE
In this section, two examples and comparing results will be provided to show the effectiveness of the presented algorithm.

Consider the following nonlinear FONSs
where f 1 (x 1 ) = sin (x 1 ) and f 2 (x) = −x 2 e −x 2 1 x 2 1 are the unknown functions, g 1 (x) = −x 1 1 + x 2 2 and g 2 (x) = 0.1 cos (x 2 ) are the known functions. Input delay is τ = 0.05, and y r (t) = 0.65 sin (t) is the reference signal.   In order to illustrate the advantages of the developed control scheme, the robust adaptive backstepping control (RABC) presented in [41] for FONS is compared here. Fig. 2 shows the trajectories of system output y, tracking signal y r (t) and tracking errors by RABC and the proposed VOLUME 11, 2023   finite-time controller. Fig. 3 depicts the state x 2 , and the trajectories of the norm of parameters estimation W 1 and W 2 are illustrated in Fig. 4. Fig. 5 shows input u.
From Fig. 2 -10, it displays that the proposed control strategy can ensure that all signals of the controlled system are bounded. Furthermore, the tracking objective can be obtained within finite time.

V. CONCLUSION
A finite-time adaptive neural DSC scheme for FONSs suffering from input delay has been proposed. The fractionalorder filter is applied to approach the virtual controller and its fractional-order derivative to overcome the difficulty of inherent computational complexity by using the backstepping approach. The finite-time stability criteria based on the fractional-order Lyapunov method is presented, which can ensure that the tracking error converges to a small region around the zero in finite time and all the singals of the closedloop system are bounded.