Neural Network Command Filtered Control of Fractional-Order Chaotic Systems

Zhang, Hua

doi:https://doi.org/10.1155/2021/8962251

Computational Intelligence and Neuroscience

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Special Issue

Artificial Intelligence and Machine Learning-Driven Decision-Making

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 8962251 | https://doi.org/10.1155/2021/8962251

Neural Network Command Filtered Control of Fractional-Order Chaotic Systems

Hua Zhang¹

Academic Editor: Wei Xiang

Received16 Sept 2021

Accepted06 Oct 2021

Published21 Oct 2021

Abstract

An adaptive neural network (NN) backstepping control method based on command filtering is proposed for a class of fractional-order chaotic systems (FOCSs) in this paper. In order to solve the problem of the item explosion in the classical backstepping method, a command filter method is adopted and the error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method. Moreover, an adaptive neural network method for unknown FOCSs is proposed. Compared with the existing control methods, the advantage of the proposed control method is that the design of the compensation signals eliminates the filtering errors, which makes the control effect of the actual system improve well. Finally, two examples are given to prove the effectiveness and potential of the proposed method.

1. Introduction

The fractional calculus equations are increasingly used to describe problems in optical and thermal systems, material and mechanical systems, signal processing, system identification, control, robotics, and other applications. The theory of fractional calculus has also been paid more and more attention by scholars at home and abroad, and especially, the fractional differential equations abstracted from practical problems have become the research focus of many mathematicians [1–3]. Bao and Cao [4] proved that fractional-order differential equations can well simulate the dynamics of various chemical materials and special materials in practical applications. The traditional integer-order differential equation can be regarded as a special model of fractional-order differential equation. More information on fractional-order differential integration can be found in [5, 6]. In recent years, many achievements have been made in the study of fractional-order calculus equations, which are used to study the problems of stability analysis and control of fractional-order strictly feedback nonlinear systems. For example, Li et al. [7] proposed a Lyapunov direct method for fractional-order nonlinear systems. Boroujeni and Momeni [8] proposed a fractional-order state observer for fractional-order nonlinear systems. Yang and Chen [9] presented a finite-time stabilization method for fractional-order switching systems. Adaptive control methods had been used in the control of fractional-order nonlinear systems [5, 6, 10]. Huang et al. [11] proposed a fuzzy feedback control method for uncertain fractional-order chaotic systems (FOCSs). Therefore, the research on the stability and control of FOCSs is becoming more and more popular.

The backstepping control was proposed for the first time to obtain asymptotic tracking and global stability of strictly feedback nonlinear systems [12]. The backstepping technique has always been a powerful tool for the design of controllers for nonlinear dynamic systems [13, 14]. However, the classical backstepping control has two obvious disadvantages: one is the explosion of terms caused by the repeated derivation of virtual control signals [15–17], and the other is certain functions in the systems with uncertain parameters must be linear [18–20]. Li et al. [20] presented a fuzzy adaptive output feedback control method to improve the tracking effect of the systems. In addition, Shao-Cheng Tong et al. [21] proposed a backstepping method based on command filters to solve the problem of item explosion caused by repeated derivation of the virtual controller. Then, the command filtered backstepping control was extended to the adaptive case in [22]. In each step of the control design, the output of the command filter was used to approximate the differential coefficient of the virtual control signal, which eliminates the problem of item explosion. Note that the error compensation mechanism was designed to eliminate errors generated by the command filters [23]. However, the above works considered only the cases of systems with unknown parameters which are constant and systems without unknown parameters. In addition, the above research results only consider the case of integer-order systems, but the research results of the backstepping control method based on command filtering are relatively few for fractional-order strict feedback systems. In this paper, a backstepping control method based on command filtering is presented for a class of fractional-order strictly feedback nonlinear systems.

Based on the radial-basis-function neural network (NN), an adaptive control method based on approximation theory was proposed to deal with nonlinear systems with uncertain parameters [24–28] or NN [29–31] approximation. The neural network control based on the backstepping control method of command filtering is used to solve nonlinear problems in uncertain systems. It can solve the nonlinear system which does not meet the matching condition and the certain functions are nonlinear [32–35]. For a class of FOCSs, an adaptive controller via the backstepping technology similar to [36, 37] was designed in [38]. However, because it is difficult to solve the fractional-order derivative of quadratic function, the adaptive backstepping control technology of integer-order nonlinear system cannot be directly applied to the fractional nonlinear system. For FOCSs, an adaptive backstepping control method was proposed in [39], and the stability of the controller was analyzed by the integer-order Lyapunov method. Therefore, for fractional-order nonlinear systems, how to design an adaptive backstepping controller via Lyapunov stability theory is an urgent problem to be solved.

According to the above discussion, an NN adaptive backstepping control method based on command filtering is proposed for FOCSs in this paper. The NN is used to deal with unknown functions in the system. The command filters are proposed to solve the problem of item explosion caused by repeated derivation of virtual controllers, and compensation signals are designed to eliminate the errors caused by the command filter. Based on NN approximation theory, an adaptive backstepping method based on command filtering is presented. The results show that this control method can adjust the tracking error to the small neighborhood of origin and ensure that all the closed-loop system signals have the limit boundedness of half blade uniformity.

The main advantages of this approach over current results can be summarized as follows:(i)The command filtered adaptive NN backstepping control can solve two problems of classical backstepping of a class of fractional-order nonlinear systems and reduce the calculation burden.(ii)An error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method, and the tracking error is controlled in the small neighborhood of zero.

The rest of this article is organized as follows: Section 2 is about the fuzzy logic system, fractional calculus description, and the fractional-order nonlinear systems. In Section 3, detailed controller design and stability analysis are given. In Section 4, simulation results are given to verify the effectiveness of the backstepping adaptive control method based on command filtering. Finally, Section 5 is the conclusion of this paper.

2. Preliminaries and Problem Description

2.1. Preliminaries of Fractional-Order Calculus

In recent several decades, the relatively common two kinds of fractional-order calculus are Riemann–Liouville (RL) fractional-order calculus and Caputo fractional-order calculus. The nice property of the Caputo fractional-order differential equation is that its integral value at zero makes sense. Therefore, this paper will adopt the definition of Caputo fractional-order calculus.

The fractional-order calculus operators can be regarded as a broader concept of integral calculus operators.

Thus, the definition of the fractional-order integral can be expressed as [1]where is the gamma function and . Consequently, the fractional-order derivative operator is represented by [1]

The Laplace transform of (2) can be expressed aswhere is the Laplace transform of .

In the following sections, we examine only the case of .

Definition 1 (see [1]). The Mittag–Leffler function can be expressed aswith , and . The Laplace transform of (5) can be described as

Lemma 1 (see [1]). If there is a complex number and two real numbers () and , where satisfies the following condition:then, the following formula is true for all integers :when and .

Lemma 2 (see [1]). If () and are two arbitrary real numbers, and there exists a positive constant satisfying the following condition:then the following inequality holds:where , , and .

Definition 2 (see [7]). Suppose is a continuous function. If the continuous function is strictly increasing and , then it is said to be class K.

Lemma 3 (see [7]). If an equilibrium point of fractional-order nonlinear system is origin, which is Lipschitz continuous. There exist a Lyapunov function and class K functions to satisfyThen, is asymptotically stable; i.e., .

Lemma 4 (see [2]). Assume that , then

2.2. Preliminaries of Radial-Basis-Function NN

Let be a continuous unknown nonlinear function defined over a compact set . Then, there exists a radial-basis-function NN to appropriate the unknown nonlinear function as follows:where () is a continuous mapping, is the NN input vector, , (), () is the weight vector, is a regressor, and is the number of NN nodes. The regression variable selected in this paper is the Gaussian radial-basis-functionwhere , , = () represent the centers of the Gaussian function, and represent the widths of the Gaussian function.

Therefore, according to the above notation, the optimal estimate of can be expressed asin which and is the optimal constant weight vector and satisfies the following condition:

Letin which is the parameter estimation error. According to the properties of the radial-basis-function NN, it is assumed that the optimal approximation error remains bounded. For any , there exists a sufficient number of NN nodes such thatwhere is a known positive constant and .

Then, one can obtainin which .

2.3. FOCSs

Consider the following FOCSs:where is the system commensurate order, is the state vector of the system, , are unknown smooth nonlinear functions, is an unknown external disturbance, and and are control input and control output of the fractional-order strictly feedback nonlinear systems.

Assumption 1. For the FOCSs, if is bounded and unknown, then there exists an unknown constant () such that for all .

3. Adaptive NN Backstepping Control Design

Letin which is a known reference signal and () are the command filters.

The command filtering mechanism is defined as follows:where is a positive constant and and () are the input signal and the output signal in the command filtering mechanism, respectively.

It is worth noting that command filtering produces errors that add to the burden of getting better tracking results. To solve this problem, an error compensation mechanism is presented to overcome the errors generated during the filtering process.

Then, the error compensation mechanism is defined aswhere are design parameters and () are the error compensating signals.

The compensated tracking error signals are designed asin which .

Our goal is to design the control input of the nonlinear systems such that the control output of the system to track and the compensated tracking error of the nonlinear systems eventually converge to a sufficiently small region of origin.

Then, the virtual control functions () are constructed aswhere () are design parameters and will be determined later.

The controller is designed as follows:in which is the estimation of , is a design parameter, and will be determined later.

Then, the backstepping control algorithms of the fractional-order nonlinear system are shown as follows.

Step 1. Consider the first Lyapunov function as follows:According to Lemma 4, differentiating can gainin which and .
Substituting (25) into (30), we can obtainwhere and are design parameters.
Choose the Lyapunov function as follows:The design of adaptive law is as follows:where and are positive design parameters.
Because is a constant, then we haveThen, according to Lemma 4, we can haveSubstituting formulas (31) and (33) into the above inequality, the following can be obtained:According to Young’s inequality, then we can obtainAccording to formulas (36) and (37), it can be obtainedin which and are two positive constants.

Step 2. Consider the second Lyapunov function as follows:According to Lemma 4, differentiating can gainin which and.
Substituting (27) into (41), we can obtainwhere and are design parameters.
Choose the Lyapunov function as follows:The design of adaptive law is as follows:where and are positive design parameters.
Because is a constant, then we haveThen, according to Lemma 4 and formulas (42) and (44), we can haveAccording to formulas (45) and (46), the above equation can be reduced toin which and are two positive constants.

Step 3. .
Consider the th Lyapunov function as follows:According to Lemma 4, differentiating can gainin which and .
Substituting (27) into (49), we can obtainwhere and are design parameters.
Choose the Lyapunov function as follows:The design of adaptive law is as follows:where and are positive design parameters.
Because is a constant, then we haveThen, according to Lemma 4, we can haveFrom formulas (50), (52), and (54), we haveAccording to Young’s inequality, then we can obtainSubstitute the above formula into (55), the following can be obtainedin which and are two positive constants.

Step 4. n.
Consider the th Lyapunov function as follows:According to Lemma 4, differentiating can gainin which and .
Substituting (28) into (59), one obtainswhere and are design parameters and is the estimation error.
Choose the Lyapunov function as follows:The design of adaptive law is as follows:where , , , and are positive design parameters.
Because is a constant, then we haveThen, according to Lemma 4, we can haveFrom formulas (60) and (65), we haveBy substituting (62) and (63) into the above inequality, it can be obtainedAccording to Young’s inequality, we can obtainSubstitute the above formulas into (67), one obtainsin which and are two positive constants.
The stability analysis of fractional-order nonlinear systems is shown below:

Theorem 1. Consider system (20) under Assumption 1. The control input is constructed as (29) with (26)–(28), and the adaptation laws are designed as (34), (44), (52), (62), and (63); then, there exist appropriate design parameters, such that the tracking error tends to an arbitrarily small region of the origin.

Proof. According to (70), there exists a function such that the following equation holds on:The Laplace transform of equation (71) is applied to obtainin which and are Laplace transforms of and , respectively, and is the initial condition. According to (6) and (71), it can be represented aswhere is the convolution operator. If sine and are both nonnegative functions, then can be obtained. Hence, the following inequality holds on:Because , , for all and and according to Lemma 2, then there exists a positive constant such thatAccording to (75), one can obtainConsequently, for every , there exists a constant , such that holds on, then one can obtainIn addition, according to Lemma 1, one can obtainin which the integer is chosen as 1. According to (78), for every , there exists a constant , such that holds on, one obtainsThen, we can select appropriate design parameters such that . Therefore, from (74), (77), and (79), the following inequality can be obtained:According to (80) and the definition of , all signals in the closed-loop system are bounded and the tracking error tends to an arbitrary small region of the origin, i.e., for all , such that .

4. Simulation Results

In this section, two examples will be given to demonstrate the effectiveness of the adaptive neural network backstepping control based on the command filtering.

Example 1. The fractional-order Duffing’s Oscillator chaos system is as follows:in which and are unknown functions. Let the fractional-order and the initial conditions . In addition, it can be seen from Figures 1 and 2 that the nonlinear system is unstable when both the controller and the disturbance signal of the system are zero.
The known smooth reference signal and the unknown external disturbance signal are chosen as and , respectively. The parameters to be designed are selected as follows: , , , , , and . In the process of designing the controller, is used to represent to avoid the chattering phenomenon.
Next, let us design the radial-basis-functions. According to the property of the radial-basis-function NN, the input variable is in the first radial-basis-function and six Gaussian functions evenly distributed within the interval are designed. The second radial-basis-function uses and as input variables. Same as the first radial-basis-function, each input variable corresponds to six Gaussian functions evenly distributed on . The initial conditions are presented to be and .
When the above initial conditions are met, the simulation of the nonlinear system (81) is shown in Figures 3–7.
As can be seen from Figures 3 and 4, the state variables and of the system are tracked by signals and and the tracking effect is relatively well. As shown in Figure 5, the tracking error rapidly converges to a relatively small neighborhood of the origin. From Figure 6, the control input is large at first, then decreases rapidly, and when the system reaches stability, it is controlled in a relatively small neighborhood. As shown in Figure 7, the adaptive laws of the system are bounded and converge rapidly to the neighborhood of the origin. Consequently, the effectiveness of the adaptive neural network backstepping control method based on command filtering for a class of classical fractional-order nonlinear strictly feedback systems.

Example 2. The fractional-order Arneodo’s chaos system is as follows:where . If the input and disturbance of the system satisfy and , the state variables of the system are shown in Figures 8 and 9.
The known smooth reference signal and the unknown external disturbance signal are and , respectively. The parameters to be designed are selected as follows:, , , , , , , , and .
In Example 2, there are three radial-basis-function neural networks. In the first radial-basis-function NN, the input variable is and four Gaussian functions evenly distributed within the interval are designed. The second radial-basis-function uses and as input variables. Same as the first radial-basis-function, each input variable corresponds to four Gaussian functions evenly distributed on . The third radial-basis-function uses , , and as input variables. Same as the before radial-basis-functions, four Gaussian functions in every input variable are evenly distributed on . The initial conditions are presented to be , , and . When the above initial conditions are met, the simulation of the nonlinear system (81) is shown in Figures 10–15.
The simulation results of Example 2 are shown above, which is similar to the result analysis of Example 1, and the final results are the same as that of Example 1.

5. Conclusion

In this paper, an adaptive neural network backstepping control method based on command filtering is proposed for a class of fractional-order strictly feedback nonlinear systems. The command filtering technology is used to deal with the explosive of terms in the traditional backstepping technology, and the error compensation mechanism is introduced to overcome the shortcomings of the dynamic surface method. The simulation results of the fractional-order nonlinear strictly feedback system show the effectiveness of the proposed method. The future research will be adaptive neural network control of fractional-order nonlinear strictly feedback system with disturbance observer based on command filtering.

Data Availability

All the datasets generated for this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

References

J. Shen and J. Lam, “Non-existence of finite-time stable equilibria in fractional-order nonlinear systems,” Automatica, vol. 50, no. 2, pp. 547–551, 2014.
View at: Publisher Site | Google Scholar
N. Aguila-Camacho, M. A. Duarte-Mermoud, and J. A. Gallegos, “Lyapunov functions for fractional order systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 9, pp. 2951–2957, 2014.
View at: Publisher Site | Google Scholar
H. Liu, Y. Pan, and J. Cao, “Composite learning adaptive dynamic surface control of fractional-order nonlinear systems,” IEEE Transactions on Cybernetics, vol. 50, no. 6, pp. 2557–2567, 2020.
View at: Publisher Site | Google Scholar
H.-B. Bao and J.-D. Cao, “Projective synchronization of fractional-order memristor-based neural networks,” Neural Networks, vol. 63, pp. 1–9, 2015.
View at: Publisher Site | Google Scholar
H. Liu, Y. Pan, S. Li, and Y. Chen, “Adaptive fuzzy backstepping control of fractional-order nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 8, pp. 2209–2217, 2017.
View at: Publisher Site | Google Scholar
L. Heng, L. Sheng-Gang, S. Ye-Guo, and W. Hong-Xing, “Prescribed performance synchronization for fractional-order chaotic systems,” Chinese Physics B, vol. 24, no. 9, Article ID 090505, 2015.
View at: Google Scholar
Y. Li, Y. Chen, and I. Podlubny, “Mittag–leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009.
View at: Google Scholar
E. A. Boroujeni and H. R. Momeni, “Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems,” Signal Processing, vol. 92, no. 10, pp. 2365–2370, 2012.
View at: Publisher Site | Google Scholar
Y. Yang and G. Chen, “Finite-time stability of fractional order impulsive switched systems,” International Journal of Robust and Nonlinear Control, vol. 25, no. 13, pp. 2207–2222, 2015.
View at: Publisher Site | Google Scholar
H. Liu, S. Li, H. Wang, Y. Huo, and J. Luo, “Adaptive synchronization for a class of uncertain fractional-order neural networks,” Entropy, vol. 17, no. 10, pp. 7185–7200, 2015.
View at: Publisher Site | Google Scholar
X. Huang, Z. Wang, Y. Li, and J. Lu, “Design of fuzzy state feedback controller for robust stabilization of uncertain fractional-order chaotic systems,” Journal of the Franklin Institute, vol. 351, no. 12, pp. 5480–5493, 2014.
View at: Publisher Site | Google Scholar
X. Liu, G. Gu, and K. Zhou, “Robust stabilization of mimo nonlinear systems by backstepping,” Automatica, vol. 35, no. 5, pp. 987–992, 1999.
View at: Publisher Site | Google Scholar
M. Krstic, P. V. Kokotovic, and I. Kanellakopoulos, Nonlinear and Adaptive Control Design, John Wiley and Sons, Hoboken, NJ, USA, 1995.
I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Transactions on Automatic Control, vol. 36, pp. 649–654, 1991.
View at: Google Scholar
W. Chen, L. Jiao, J. Li, and R. Li, “Adaptive nn backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays,” IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 40, no. 3, pp. 939–950, 2009.
View at: Google Scholar
A. Boulkroune, M. Farza, and M. M'Saad, “Fuzzy approximation-based indirect adaptive controller for multi-input multi-output non-affine systems with unknown control direction,” IET Control Theory and Applications, vol. 6, no. 17, pp. 2619–2629, 2012.
View at: Publisher Site | Google Scholar
L. Heng, Y. Hai-Jun, and X. Wei, “Adaptive fuzzy nonlinear inversion-based control for uncertain chaotic systems,” Chinese Physics B, vol. 21, no. 12, Article ID 120505, 2012.
View at: Google Scholar
T.-S. Li, S.-C. Tong, and G. Feng, “A novel robust adaptive-fuzzy-tracking control for a class of nonlinear multi-input/multi-output systems,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 1, pp. 150–160, 2009.
View at: Google Scholar
T.-S. Tie-Shan Li, D. Dan Wang, G. Gang Feng, and S.-C. Shao-Cheng Tong, “A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 40, no. 3, pp. 915–927, 2010.
View at: Publisher Site | Google Scholar
Y. Li, S. Tong, and T. Li, “Hybrid fuzzy adaptive output feedback control design for uncertain mimo nonlinear systems with time-varying delays and input saturation,” IEEE Transactions on Fuzzy Systems, vol. 24, no. 4, pp. 841–853, 2015.
View at: Google Scholar
S.-C. Shao-Cheng Tong, X.-L. Xiang-Lei He, and H.-G. Hua-Guang Zhang, “A combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 5, pp. 1059–1069, 2009.
View at: Publisher Site | Google Scholar
W. Dong, J. A. Farrell, M. M. Polycarpou, V. Djapic, and M. Sharma, “Command filtered adaptive backstepping,” IEEE Transactions on Control Systems Technology, vol. 20, no. 3, pp. 566–580, 2011.
View at: Google Scholar
J. Hu and H. Zhang, “Immersion and invariance based command-filtered adaptive backstepping control of vtol vehicles,” Automatica, vol. 49, no. 7, pp. 2160–2167, 2013.
View at: Publisher Site | Google Scholar
W. Assawinchaichote and S. K. Sing Kiong Nguang, “Fuzzy H/sub/spl infin//output feedback control design for singularly perturbed systems with pole placement constraints: an LMI approach,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 3, pp. 361–371, 2006.
View at: Publisher Site | Google Scholar
Q. Shen, B. Jiang, and V. Cocquempot, “Adaptive fuzzy observer-based active fault-tolerant dynamic surface control for a class of nonlinear systems with actuator faults,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 2, pp. 338–349, 2013.
View at: Google Scholar
Q. Shen, B. Jiang, and V. Cocquempot, “Fuzzy logic system-based adaptive fault-tolerant control for near-space vehicle attitude dynamics with actuator faults,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 2, pp. 289–300, 2012.
View at: Google Scholar
M. Chadli and H. R. Karimi, “Robust observer design for unknown inputs takagi–sugeno models,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 1, pp. 158–164, 2012.
View at: Google Scholar
M. Chadli and T. M. Guerra, “LMI solution for robust static output feedback control of discrete takagi-sugeno fuzzy models,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 6, pp. 1160–1165, 2012.
View at: Publisher Site | Google Scholar
J. Yu, P. Shi, W. Dong, B. Chen, and C. Lin, “Neural network-based adaptive dynamic surface control for permanent magnet synchronous motors,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 3, pp. 640–645, 2014.
View at: Google Scholar
J. Yu, P. Shi, H. Yu, B. Chen, and C. Lin, “Approximation-based discrete-time adaptive position tracking control for interior permanent magnet synchronous motors,” IEEE Transactions on Cybernetics, vol. 45, no. 7, pp. 1363–1371, 2014.
View at: Google Scholar
J. Wang, “Adaptive fuzzy control of direct-current motor dead-zone systems,” International Journal of Innovative Computing, Information and Control, vol. 10, no. 4, pp. 1391–1399, 2014.
View at: Google Scholar
M. Chen and S. S. Ge, “Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer,” IEEE Transactions on Cybernetics, vol. 43, no. 4, pp. 1213–1225, 2012.
View at: Google Scholar
J. Yu, Y. Ma, H. Yu, and C. Lin, “Adaptive fuzzy dynamic surface control for induction motors with iron losses in electric vehicle drive systems via backstepping,” Information Sciences, vol. 376, pp. 172–189, 2017.
View at: Publisher Site | Google Scholar
Y. Li, S. Tong, and T. Li, “Composite adaptive fuzzy output feedback control design for uncertain nonlinear strict-feedback systems with input saturation,” IEEE Transactions on Cybernetics, vol. 45, no. 10, pp. 2299–2308, 2014.
View at: Google Scholar
Y.-J. Liu, S. Tong, D.-J. Li, and Y. Gao, “Fuzzy adaptive control with state observer for a class of nonlinear discrete-time systems with input constraint,” IEEE Transactions on Fuzzy Systems, vol. 24, no. 5, pp. 1147–1158, 2015.
View at: Google Scholar
B. Xu, F. Sun, Y. Pan, and B. Chen, “Disturbance observer based composite learning fuzzy control of nonlinear systems with unknown dead zone,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 8, pp. 1854–1862, 2016.
View at: Google Scholar
D. Baleanu, J. A. T. Machado, and A. C. Luo, Fractional Dynamics and Control, Springer Science and Business Media, Berlin, Germany, 2011.
D. Ding, D. Qi, Y. Meng, and L. Xu, “Adaptive mittag-leffler stabilization of commensurate fractional-order nonlinear systems,” in Proceedings of the 53rd IEEE Conference on Decision and Control, pp. 6920–6926, Los Angeles, CA, USA, December 2014.
View at: Publisher Site | Google Scholar
Y. Wei, Y. Chen, S. Liang, and Y. Wang, “A novel algorithm on adaptive backstepping control of fractional order systems,” Neurocomputing, vol. 165, pp. 395–402, 2015.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Hua Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

214

Downloads

793

Citations