Adaptive neural backstepping control of nonlinear fractional-order systems with input quantization

This article addresses the tracking control problem of uncertain fractional-order nonlinear systems in the presence of input quantization and external disturbance. An adaptive backstepping scheme is proposed by combining with radial basis function (RBF) neural networks (NNs), fractional-order disturbance observer (FODO), and backstepping method. The RBF NNs are used to approximate the unknown nonlinearities of fractional-order systems. The FODO is designed to compensate for disturbance and uncertain parameters. The hysteresis quantizer is used to avoid chattering that possibly appears in actual application. The stability of the proposed controller is proved by fractional-order Lyapunov method. In addition, all the signals in the closed-loop system are bounded. The effectiveness of the proposed method is confirmed by the simulation results.


Introduction
Fractional-order calculus has been proposed for more than 300 years (Miller and Ross, 1993); nowadays, it still attracts the interest of researchers due to its unique properties and great potentials in a great many fields (Aguila-Camacho et al., 2014;Anjum et al., 2021;Heymans, 2008;Li et al., 2009;Podlubny, 1999;Xue, 2017).In the field of dynamics, the models can be established more concisely and precisely using fractional-order calculus.In the control field, an increasing number of researchers focus on the study of fractional-order systems, especially in stability analysis and controller design (Aghababa, 2014;Ji et al., 2016;Li and Sun, 2015;Trigeassou et al., 2011;Wang et al., 2012;Yin et al., 2014).
It is known that adaptive backstepping has been widely used in the control of integer-order nonlinear systems, which establishes a series of recursively intermediate variables as virtual control signals for every step.The proof of this method is based on ensuring the negativity of Lyapunov functions for the entire system including the parameter estimates (Liu et al., 2020;Wang et al., 2021;Wu, 2018;Zhou and Wen, 2013).In recent years, some researchers have extended backstepping method to fractional-order nonlinear systems control (Liu et al., 2017;Wei et al., 2015).In Liu et al. (2017), an adaptive fuzzy backstepping controller was designed to deal with a kind of uncertain fractional-order nonlinear system.In Wei et al. (2015), an adaptive backstepping method was used to solve the control problem of a triangular fractional-order systems with non-commensurate orders.
Until now, the quantized control is widely used in linear and nonlinear systems since its theoretical and practical significance in modern engineering.In digital control, the quantization error of control input cannot be neglected, which is treated as an adverse factor to the system performance.The utilization of quantization schemes can not only have higher precision but also achieve low communication rate.To reduce the chattering, a kind of hysteretic quantizer was proposed (Hayakawa et al., 2009).An adaptive backstepping controller with hysteretic quantizer was proposed in Zhou et al. (2014).There are few studies that have investigated the input quantization control of fractional-order nonlinear systems.In Hua et al. (2018), authors studied a fractional-order nonlinear systems with input quantization and proposed an output feedback tracking controller.The fuzzy control of strict-feedback fractional-order nonlinear systems with quantized input is studied in Qiu et al. (2022).Although, the results of the above have made some achievement, the design of quantizer for fractional-order systems has not been fully investigated in the existing literature.In our paper, the hysteresis quantizer will be further studied in fractional-order systems.
In fact, systems are often affected by external interference and uncertain parameters, which will degrade the performance of systems, and even destabilize the systems.Disturbance observer can be used to estimate the disturbance and attenuate their effects.There are many integer-order disturbance observer techniques that have been reported.Two different design methods of nonlinear integer-order disturbance observer were given to handle disturbance (Chen, 2004;Chen et al., 2000).However, there is little research on the fractional-order disturbance observer (FODO) (Sheng-Li et al., 2019;Shuyi et al., 2016Shuyi et al., , 2017)).A class of uncertain fractional-order chaotic systems with unknown disturbance and input saturation is studied in Shuyi et al. (2017).In Sheng- Li et al. (2019), a FODO-based sliding mode control was studied, and the control problem with matched and mismatched disturbances was solved.In our paper, the FODO is designed to estimate both of these disturbance and uncertain parameters for nonlinear fractional-order system.
Inspired by the above discussions, we study the tracking control problem for fractional-order nonlinear systems with disturbance and input quantization.The main contributions of this paper can be highlighted as the following.First, a FODO is designed to compensate for disturbance and uncertain parameters, which can simplify the design process.Compared to integer-order disturbance observer, FODO has more design parameters, which will increase flexibility in controller design.Second, a hysteresis quantizer is considered to avoid chattering that possibly appears in the practical application.Third, a novel adaptive neural backstepping controller is designed for nonlinear fractional-order system with input quantization and disturbance, and guarantees the bound of all the signals in a closed-loop system.Compared with traditional adaptive backstepping control, the control method demonstrates stronger robustness under disturbances.
The organization of the remaining paper is as follows.The problem formulation and preliminaries are given in section ''Problem formulation and preliminaries.''The FODO and backstepping controller design for the system are presented in section ''Main results.''The simulation results can be found in section ''Simulation results.''Finally, we conclude this article in section ''Conclusion.''Notations L(x(t)) Laplace transform of x(t) k Á k arbitrary norm jzj modulus of a complex number z C set of complex numbers R set of real numbers R n n-dimensional vector set R m 3 n m 3 n real matrix set Re(z) real part of complex number z

Fractional calculus
There are several different forms for the definition of fractional derivative; among them, the most popular used in engineering applications is the Caputo definition.The Caputo definition form we use is as follows (Podlubny, 1999) where nÀ1\a\n, n is a positive integer, G(z) = Ð ' 0 t zÀ1 e Àt dt denotes the Euler's Gamma function, and G(z + 1) = zG(z).For the differential order a, we only consider it on the set (0, 1).For simplicity of description, c 0 D a t is abbreviated as D a .The Laplace transform of equation ( 1) is given as follows Definition 1. (Gorenflo et al., 2014) The Mittag-Leffler function is defined by the following equation where a .0 and b 2 R, z 2 R.
The Laplace transform can be given as follows where Re(s) .0, l 2 C and jls Àa j\1.
Lemma 1. (Gorenflo et al., 2014) For the above Mittag-Leffer function, if b .0, then Lemma 2. (Li et al., 2009) Assuming that the origin is an equilibrium point of a nonautonomous fractional-order nonlinear system, we get where then equation ( 6) is asymptotically stable.

System formulation
In this paper, a kind of fractional-order nonlinear system with disturbance and input quantization is considered as the following form where a is the system commensurate order, x(t) = ½x 1 (t), x 2 (t), . . ., x n (t) T 2 R n is the system state vector, and we only consider the case of 0\a\1.f i (x i (t)) (i = 1, 2, . . ., n) are unknown smooth nonlinear functions.u(t) is a designed input, and q(u(t)) is the actual quantitative input.y 2 R is the output, and d(t) 2 R is the external disturbance.
Without loss of generality, we can rewrite the system formulation as follows Then, FODO will be designed to estimate both of external interference d(t) and unknown parameter f n (x ^n (t)).
Assumption 1.For system (10), d n (t) and its fractional-order derivatives are bounded with jd n (t)j\d ^n and jD a d n (t)j ł h, where d ^n .0 and h .0 are unknown positive constants.
Remark 1.In comparison to logarithmic quantizer, hysteretic type of quantizer gets more quantization grades, thereby avoiding chattering.h is reviewed as the measure of the density of quantitative, which determines coarseness of quantizer.Some detailed description can be found in Liu et al. (2016).
To make the controller design more suitable, the hysteresis type of quantizer will be rewritten as Liu et al. ( 2016) where h(u(t)) and g(t) are exist functions and satisfy

Radial basis function neural networks
Radial basis function neural networks (RBF NNs) have attracted an increasing amount of attention in the last few years.The ability of the RBF NNs to fit unknown nonlinear function offers powerful tool for the control of the nonlinear systems.In this paper, the following RBF NNs are used to appropriate the unknown nonlinear terms where F = ½f 1 , f 2 , . . ., f l T 2 R l is the weight vector, in which l . 1 is the node number of neural networks.1 2 X & R q is NNs input vector, and q is the input dimension.j(1) = ½j 1 (1), j 2 (1), . . ., j l (1) T means the RBF, which is generally selected as the Gaussian function as follows where i i = ½i i1 , i i2 , . . ., i iq T represents the center of the receptive field and k i denotes the width of the basis function j i (1).By choosing a large enough number of l, RBF NNs can estimate f (1) to arbitrary accuracy in the compact set O 1 2 R l with arbitrary accuracy E (1) .0 (Wang et al., 2016).
Assumption 2. The approximation error E (1) is bounded and satisfies jE (1)j ł e, where e is an unknown positive constant. Then where F Ã is ideal constant parameter vector, which is defined by

Main results
In this section, first, a FODO is designed to compensate disturbance and uncertain parameters.Then, we proposed an adaptive backstepping controller by combining with RBF NNs, FODO, and backstepping method for system (10).

Fodo
To evaluate the unknown term d n (t) in system (10), we design a FODO.First, we introduce an auxiliary variable (Chen, 2004;Sheng-Li et al., 2019) where s .0 is a constant to be designed.Then, the Caputo derivative of u(t) is given as follows The fractional-order disturbance observer is suggested as Compared to Mun˜oz-Va´zquez et al. ( 2019), the FODO (21) uses the fractional-order calculus to estimate the unknown disturbance.Define the disturbance estimation error as follows To verify the feasibility of the above FODO, we choose the Lyapunov function candidate to perform stability analysis of the interference estimation error Based on Lemma 3, the Caputo derivative of V d is given as follows According to equations ( 21), (10), and Assumption 1, we have where ũ = u À û = d n À dn = dn , B 0 = 2s À 1, and B 1 = 1 2 h 2 .To ensure the estimated error dn is bounded, the gain s should be chosen to make 2s À 1 .0. Considering the following design, we choose s .1. So, the estimation yielded by the disturbance observer approaches to the disturbance d n (t) globally exponentially.

Controller design
Define the error variables where t i (t) are virtual controllers, y d (t) is the designed output signal.
Then, apply the RBF NNs (15) to approximate nonlinear function.We get the approximate of unknown smooth function f i (x(t)) (i = 1, 2, . . ., n À 1) as follows Define Fi = F Ã i À Fi , where F Ã i is the optimal parameter vector.As far as we know, the derivative of parameter F Ã i is zero, we can obtain The RBF NNs error is defined by According to Assumption 2, we set E i xi (t) ð Þł e i (e i .0).Then, we can get The design process of fractional-order backstepping controller will be introduced as the following steps.
Step 1: The Caputo derivative of e 1 is given as follows The Lyapunov function is chosen as Let the virtual control law t 1 (t) be and the fractional-order adaptation law be where c 1 .0, g 1 .0, and r 1 .0 are designed parameters.By applying Lemma 3, Assumption 2, equations ( 31), ( 33), (34), and Young's inequality, the Caputo derivative of V 1 is given as follows Step i(2 ł i ł n À 1): The Caputo derivative of e i (t) is given as follows The Lyapunov function is chosen as Let the virtual control law t i (t) be and the fractional-order adaptation law be where g i .0, r i .0, c i .0 are designed parameters.By applying Lemma 3, Assumption 2, equations ( 36), ( 38), (39), and Young's inequality, the Caputo derivative of V i can be described as follows Step n: The Caputo derivative of e n (t) is given as follows The Lyapunov function is chosen as The disturbance observer is designed as where c n .0 is designed parameter.Let the input control law u(t) be where s .0 is designed parameter.
According to equation ( 14), we have By applying Lemma 3, Assumption 2, equations ( 41), ( 43), ( 44), and Young's inequality, the Caputo derivative V n can be described as follows We can rewrite equation ( 46) as Theorem 1.Consider fractional-order system (10) under Assumption 1, and the design of disturbance observer (43), the controller (44), the virtual controller (33), ( 38), and adaptive law (34) and ( 39), then all the signals in the closed-loop system remain semiglobally uniformly bounded, and the tracking error eventually converges to an arbitrary small region.
Proof.According to equation ( 46), there exists a non-negative function m such that Taking Laplace transform on equation ( 48), we obtain where V n (0) is the initial condition.Using Definition 1, we can solve equation ( 49) as follows which yields that Using Lemma 5, we can obtain According to Li and Sun (2015), we can get that there is a constant t 0 .0 for all t 2 (t 0 , ') Therefore, from equation ( 53) and V n , we can conclude that all signals in system (10) keep bounded and tracking error e gradually approaches arbitrary small range.This completes the proof of Theorem 1.

Simulation results
In this paper, a fractional-order backstepping controller design method with FODO and RBF NNs is proposed.In this section, we use a fractional-order nonlinear system to verify the effectiveness of the design method Let the fractional-order be a = 0:95, and the initial condition be x(0) = ½0, 0 T .The design parameters are chosen as s = 30, r 1 = 1, c 1 = 14 , c 2 = 20, c = 0:2, and u min = 0:01.
By assuming there exist an external disturbance d(t) = 0:2 sin (2t) in system (54), we can rewrite equation (54) as system (10).Such that we can get d n = x 3 2 À0:5x 2 1 1 + x 4 1 + 0:2 sin (2t).On the corresponding compact sets, select the centers and widths of RBF NNs on a regular lattice.In this simulation, a three-layer NN with five nodes per layer is used, and the center of each of them spaced evenly in the interval ½À2, 2. Therefore, NN F T 1 j 1 contains 125 nodes and the widths of Gaussian functions equal to 2. The simulation results are shown in Figures 2-7.
To display the availability of the presented FODO, the simulation experiment of backstepping control method without FODO will also be performed to make a comparison.The same parameters are chosen for this controller.Figures 8-9 show the simulation results.
Figure 2 displays output signal y and desired reference signal y d .Figure 3 shows the state x2 and the virtual control law t 1 .Figure 4 shows that the quantized input signal is bounded.From Figures 5 and 6, we can see that the disturbance estimation error is bounded.Therefore, the FODO designed above can estimate the unknown disturbance and uncertain parameters well.The tracking error of e is shown in Figure 7.It can be seen from Figures 8 and 9 that the backstepping controller without FODO exists the evident chattering phenomenon.Figure 9 shows that the tracking error of x 2 is larger than the method proposed in this paper.It is concluded that the simulation system (54) is bounded synchronization under the designed FODO-based fractional-order backstepping controller.Therefore, the proposed adaptive neural backstepping controller for fractional-order systems with disturbance and input quantization is effective.

Figure 5 .
Figure 5.The disturbance d n (t) and the approximation output of dn (t).