An Adaptive Control of Fractional-Order Nonlinear Uncertain Systems with Input Saturation

Wang, Changhui; Liang, Mei; Chai, Yongsheng

doi:https://doi.org/10.1155/2019/5643298

Complexity

On this page

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

Research Article | Open Access

Volume 2019 | Article ID 5643298 | https://doi.org/10.1155/2019/5643298

An Adaptive Control of Fractional-Order Nonlinear Uncertain Systems with Input Saturation

Changhui Wang,¹Mei Liang,¹and Yongsheng Chai¹

Academic Editor: Xiaoping Liu

Received15 Sept 2019

Revised23 Nov 2019

Accepted04 Dec 2019

Published20 Dec 2019

Abstract

This paper develops a fractional-order adaptive fuzzy backstepping control scheme for incommensurate fractional-order nonlinear uncertain systems with external disturbances and input saturation. Based on backstepping algorithm, the fuzzy logic system is used to approximate the unknown nonlinear uncertainties in each step of the backstepping, and the fractional-order parameters update laws for fuzzy logic system, unknown parameters, and the external disturbances are proposed. With the aids of the frequency distributed model of fractional integrator for the fractional-order systems in the procedure of controller design, the stability of the closed-loop system is established. To verify the effectiveness and robustness of the proposed controller, two simulation examples are demonstrated at last.

1. Introduction

In the latest two decades, fractional-order calculus as a generalization of derivation and integration of arbitrary order has attracted more and more attentions and interests in both theory and applications due to its unique advantages in describing the hereditary and memory properties of multifarious materials and processes [1–4]. As a powerful tool used to describe most of the real-world behaviors, fractional-order systems can provide more practical value and accurate results in practice [5–11], such as abnormal diffusion, heat conduction electronic, some biological systems, and viscoelastic systems. Sequentially, many researchers have paid close attention to the applications of fractional-order differential equations in both engineering and theory and have drawn some wonderful and meaningful results in the literature [12–17].

As we know that it is difficult to build a precise physical model of the engineering plant because of the uncertainties and noises. Thus, the fractional-order nonlinear system control with uncertainties and disturbances is still a challenging and attractive research field. Due to the inherent approximation capability, neural networks (NNs) or fuzzy logic systems are usually used to approximate the system uncertainties in integer-order system. The scholars in [17] have designed an adaptive fuzzy control scheme for a class of commensurate fractional-order systems with parametric uncertainty and input constraint. In [18], an adaptive backstepping controller is designed for a class of commensurate fractional-order systems with unknown parameters based on the indirect Lyapunov method, in which the control problem of fractional order is converted to the integer-order one [19]. Using the fractional-order extension of the Lyapunov direct method, an adaptive backstepping control method for a class of commensurate fractional-order nonlinear systems with unknown nonlinearity is developed in [20]. In [21, 22], an output feedback control scheme for a class of triangular commensurate fractional-order nonlinear systems is given. For a class of commensurate fractional-order rotational mechanical system with disturbances and uncertainties, a robust adaptive NN control is presented in [23]. Based on dual radial basis function (RBF) NNs, an adaptive commensurate fractional sliding mode controller is proposed to enhance the performance of the system in [24]. In [25], an adaptive NN control scheme is given for a class of commensurate fractional-order systems with nonlinearities and backlash-like hysteresis. In [26], the consensus problem of commensurate fractional-order MIMO systems with linear models is researched via the observer-based protocols. For a class of commensurate uncertain fractional-order nonlinear systems with external disturbance and input saturation, an adaptive NN backstepping control method based on the indirect Lyapunov method is designed in [27]. In [28], the output tracking control problem of a class of incommensurate fractional-order nonlinear systems with unknown nonlinearities and external disturbance is researched, in which the indirect Lyapunov method is used to design the controller. For a class of incommensurate fractional-order nonlinear systems with external disturbances and input dead-zone, a neural network adaptive control scheme is developed in [29]. Although many adaptive control schemes for the uncertain fractional-order system have been considered in above algorithms, the fuzzy backstepping control method for the incommensurate fractional-order systems with input nonlinearity is still short of results.

Input nonlinearity is another important topic when designing the controller, which needs to be considered [30]. In fact, the real actuators cannot be a perfectly good linear characteristic in all operating domains due to physical constraints [31]. Therefore, the nonlinearities of practical actuators can cause severe degradation to the system performance since the theoretical controller is designed from an ideal assumption, which may cause system damage and even put risk on human users [32]. Input saturation often occurs in practical engineering, which is a source of limiting the system performance severely and even leads to the system instability. The study of control input saturation has become increasingly popular in recent years [33–37]. To solve the problem of actuator saturation, two methods are often used. One is to choose a low gain control law that the control input signal can be limited in a region to avoid saturation, and the other is to estimate the saturation region in the presence of actuator saturation. Generally speaking, it is very challenging to design a controller for the nonlinear uncertain systems with the input saturation problem.

Therefore, the problem that nonlinear fractional-order systems with input saturation must be cope with when designing and implementing the control systems. Meanwhile, many real systems are better described by incommensurate fractional-order nonlinear systems, such as lithium-ion battery dynamics [38, 39], ultracapacitor dynamics [40], and quadratic boost converter [41], which means that the order of the fractional-order nonlinear systems should be extended to incommensurate one for the controller design. Motivated by the aforementioned observations, an adaptive fuzzy control method for the incommensurate fractional-order nonlinear uncertain systems with external disturbance and input saturation is presented in this study. The contributions are summarized as follows:(1)Different from [27], the tracking controller is designed for a class of incommensurate fractional-order nonlinear uncertain system with external disturbance and input saturation, in which the indirect Lyapunov method with frequency distributed model is used to analyze the stability.(2)Different from [25, 27], three different fractional-order adaptation laws are designed to update the fuzzy parameters from fuzzy logic system, unknown model constants, and the unknown bound from disturbances, respectively. Thus, the proposed method can handle different situations at the same time.(3)Unlike the previous works [24–27], the orders of the parameters adaptation laws are not fixed the order of the fractional-order nonlinear system in this research, which provides more degree of freedom.

This paper is organized as follows: Section 2 introduces some preliminaries. Section 3 presents the adaptive fractional-order controller design and stability analysis. Section 4 provides the simulation results to illustrate the proposed controller. Section 5 concludes this article.

2. Preliminary

The Caputo fractional derivative is defined as follows [42]:where , and is the classical order derivative operator. When , can be abbreviated as .

Lemma 1 (see [43]). Consider a nonlinear fractional-order systemthe system is exactly equivalent to the continuous frequency distributed model described bywhere . is the infinite dimension distributed state variable.

In the developed control design procedure, fuzzy logic system will be used to approximate any continuous function on a compact set .

Lemma 2 (see [44]). The unknown smooth nonlinear function can be approximated by a fuzzy logic system as follows:where is the weight vector, is the fuzzy basis function vector, N is the number of inference rules, and ε represents the optimal approximation error. The optimal value of the parameter W is given by

3. Adaptive Fuzzy Backstepping Controller

In this paper, we consider a class of incommensurate fractional-order systems presented as follows:where is the system incommensurate fractional order, and are the state vectors, . and denote the unknown parts and known parts of nonlinear functions, respectively. is the unknown constant vector, is the unknown disturbance, is a known constant, and is the system output. Control input expresses an input saturation defined as follows [45]:where ν is the input signal of the input saturation nonlinearity and and are the unknown constants.

For (7), the input saturation function can be approximated by a smooth piecewise function defined as follows:whereand , satisfying

With the theorem of the mean, there exists a constant to makewhere

When selecting , (11) can be represented as

By substituting (8) and (13) into (6), system (6) can be rewritten as

Our target is to design a input ν such that the system output y can follow the desired signal . Some following assumptions for the controller design are given.

Assumption 1. It is supposed that the reference signals and the nth order derivatives are continuous and bounded.

Assumption 2. There exists , such that .
In order to design a fractional-order backstepping algorithm for fractional-order nonlinear system (14), a virtual control law is found for every subsystem. Finally, all the system equations are considered and the control law is established to guarantee the closed-loop system stability. The error equation is defined as follows:where is the virtual control input. The virtual control input and controller ν are designed as follows:where is the estimation of the unknown constant vector (). is the design parameter. is the estimation of the unknown constant defined as follows:where .
is the fuzzy logic system used to approximate the unknown function , wherewhere is the parameter estimation of the ideal parameter , which is described byThe fractional-order adaptation laws are designed as follows:where , and are the design parameters.

Theorem 1. Consider system (6) with input saturation, if the control input and virtual control inputs are chosen as (16), and the adaptation laws are designed as (20), then all the signals in the closed-loop system are globally uniformly bounded with the proper design parameters and , and the tracking error tend to zero asymptotically when .

Proof.

Step 1. Let , then the following equation can be obtained:Based on Lemma 1, (21) will bewhere and .
DefineBased on Lemma 2 and (19), one can obtainDue to (1) and the estimated error from (23), the following equation can be obtained:According to Lemma 1 and (25), the following frequency distributed model can be obtained:where and .
On the other hand, it follows from (14), (24), and error thatSubstituting virtual control input and error equation into (27) givesAccording to Lemma 1, equation (28) will bewhere and .
Let , then the following equation can be obtained:Based on Lemma 1, (30) will bewhere and .
Selecting the Lyapunov function asBased on frequency distributed models (22), (26), (29), and (31), the derivative of is expressed as follows:According to the fractional-order adaptation laws (20), equation (33) can be rewritten as follows:Based on the fractional-order adaptation laws (20) and in equation (34), if and , one can obtain .

Step 2. Let , then the following equation can be obtained:Based on Lemma 1, (35) will bewhere and .
Based on the fuzzy logic system, it follows from the fractional-order system (14), error equation (15), and unknown function (18) thatwhere is the estimated error. Then, the following equation can be obtained:According to Lemma 1 and (38), the following frequency distributed model can be obtained:where and .
Substituting virtual control input and error equation into (37) givesIts frequency distributed model corresponds towhere and .
Let , then the following equation can be obtained:According to Lemma 1, (42) will bewhere and .
Selecting the Lyapunov function asAccording to frequency distributed models (36), (39), (41), and (43), the derivative of V₂ is expressed asBased on fractional-order adaptation laws (20) and in equation (45), if and , one can get .

Step . Based on the fuzzy logic system, it follows from the fractional-order system (14), error equation (15), and unknown function (18) thatwhere is the estimated error for the fuzzy logic system. Then, the following equation can be obtained:According to Lemma 1 and (47), the following frequency distributed model can be obtained:where and .
Substituting virtual control input and error equation into (46) givesIts frequency distributed model corresponds towhere and . is parameter estimation error, and the following equation can be obtained:Based on Lemma 1, (51) will bewhere and .
Let , then the following equation can be obtained:According to Lemma 1, (53) will bewhere and .
Selecting the Lyapunov function asThen, its derivative on the basis of frequency distributed models (48), (50), (52), and (54) is expressed asAccording to fractional-order adaptation laws (20) and in equation (56), if and , one can get .

Step n. Based on error equation , fractional-order system (14), and unknown function (18), one haswhere is the estimated error for the fuzzy logic system, and the following equation can be obtained:Due to Lemma 1, (58) will bewhere and .
Substituting the designed controller ν in (16) into (57) givesthen the following frequency distributed model is obtained:where and . is parameter estimation error, and the following equation can be obtained:Based on Lemma 1, (62) will bewhere and .
Let , then the following equation can be obtained:According to Lemma 1, (64) will bewhere and .
Selecting the Lyapunov function asBased on the procedures in step , the derivative of on the basis of frequency distributed models (59), (61), (63), and (65) is expressed as follows:Based on fractional-order adaptation laws (20) and in equation (67), if , one can get .
According to , if the control input is chosen as (16), and the adaptation laws are designed as (20) then with a proper choice of the design parameters and , one can get . Due to the LaSalle invariance principle [46], , , , and can be close to the set of all points. When , one can obtain , , , and , which is the only equilibrium point. That is the error variables , , , and convergent to zero asymptotically. Therefore, the tracking error tends to zero asymptotically and all the signals are uniformly bounded.

Remark 1. An adaptive fuzzy backstepping algorithm for a class of fractional-order systems with external disturbances and input saturation is proposed in Theorem 1. The fuzzy logic system is used to approximate the unknown nonlinear uncertainties including the fractional-order function in each step of the backstepping, which may bring some computation burden of the control system for the parameter estimation to obtain accurate approximation effect.

Remark 2. There are the sign functions in the controller (16), which can make the chattering phenomenon for the closed-loop control system. However, the sign function used the controller (16) can ensure that the tracking error is bounded, and the chattering phenomenon can be alleviated by replacing the sign function by a continuous function, such as .

Remark 3. The proposed algorithm can be applied to both the incommensurate and commensurate fractional-order systems with input saturation, in which the orders of the parameter estimation laws ( and ) cannot be fixed to the system order (). Therefore, the proposed algorithm brings more degree of freedom and better control performance.

4. Simulation

Two examples are presented in this simulation section to show the effectiveness of the proposed method.

4.1. Example 1

The following three-order incommensurate fractional-order nonlinear system is considered aswhere , , and are the unknown functions. , , and are the known functions. and are the unknown constants. , , and are the unknown uncertain disturbances.

The input saturation is presented as

The reference signal for the system output is chosen as . Three fuzzy systems are used in the presented controller. The first fuzzy system using as its input defines the Gaussian membership functions as selecting the Lyapunov function as

The second fuzzy system uses and as its inputs. The membership functions for the input are chosen as (70), and the membership functions for the input are chosen as

The third fuzzy system employs , , and as its inputs. The membership functions for the input and are chosen as the same as the second one, and the membership functions for the input are chosen as

The design parameters are chosen as , , , , , , , , and . The initial conditions of parameter estimation are , , and .

The simulation results are illustrated in Figures 1–6. The tracking performance of the system output y in comparison with reference signal is shown in Figure 1. It demonstrates that the tracking error has a rapid convergence with the unknown nonlinear terms, unknown parameters, and input saturation, implying a good tracking performance. However, due to the approximation error from unknown nonlinear terms approximated by fuzzy logic system and chattering phenomenon from the in controller, the tracking error converges to a small range of zero. The trajectories of the system states and are indicated in Figure 2, and the norm of parameters estimation of the fuzzy logic system is shown in Figure 3. Figure 4 shows the estimation of the unknown constants , and . Figure 5 displays the estimation of the , and , and system input saturation is presented in Figure 6. Meanwhile, Figures 2–6 demonstrate the boundedness of the signals in the closed-loop control system.

(a)

(b)

4.2. Example 2

To show more results of the proposed method, the following incommensurate fractional-order nonlinear system is considered:where , , and are the unknown functions. , , and are the known functions. , and are the unknown constants. , , and are the unknown uncertain disturbances.

The input saturation is presented as

The reference signal for the system output is chosen as . There are three fuzzy systems used in the presented controller. The first fuzzy system employs as its input and defines the Gaussian membership functions as

The second fuzzy system uses and as its inputs. The membership functions for the input are chosen as the same as (75), and the membership functions for the input are chosen as

The third fuzzy system uses as its inputs, and the membership functions are chosen as (76).

The design parameters are chosen as , , , , , , , , and . The initial conditions of parameter estimation are , , and .

The tracking error shown in Figure 7 converges to a smaller range of zero. Figure 8 shows that the trajectories of the system states and are bounded. The norm of parameters estimation of the fuzzy logic system and the estimation of the unknown constants , and are depicted in Figures 9 and 10. Figure 11 depicts the estimation of the unknown constants , and , and the input saturation is depicted in Figure 12, which is bounded.

(a)

(b)

Above simulation results demonstrate that the system output can track the reference signal in a small range of zero, and all the closed-loop signals are bounded, demonstrating the effectiveness of the proposed control scheme.

5. Conclusion

In this paper, a new adaptive fuzzy backstepping control method for a class of incommensurate fractional-order nonlinear uncertain systems with external disturbance and input saturation is proposed, which can be considered as an extension of the commensurate one in [27]. In each step, the fuzzy logic system is used to approximate unknown nonlinear terms, and the fractional-order parameters update law as well as disturbance estimator are constructed based on the Lyapunov theorem for fractional-order systems with regard to the frequency distributed model. The proposed algorithm guarantees the convergence of the tracking error, and the orders of the adaptation laws are not fixed the order of the fractional-order nonlinear system, providing more degree of freedom compared with [24–27]. Two simulation examples are presented to demonstrate the effectiveness of proposed method, and the results show that the asymptotic tracking problem for a class of incommensurate fractional-order nonlinear uncertain systems with external disturbance and input saturation can be well solved by the proposed scheme. In the future, we will consider to look for suitable real system applications using the proposed method.

Data Availability

The data used to support the findings of this study are included within the article, which are available for researchers.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was supported in part by the Doctoral Program of Shandong Provincial Natural Science Foundation of China (ZR2019BF048), Key R&D Program of Shandong Province in 2019 (Public Welfare Science and Technology Tackling Category) (2019GGX104071), Shandong Scientific Research Projects of Colleges and Universities (J18KA062), Foundation of State Key Laboratory of Automotive Simulation and Control (20171105), and National Key R&D Program of China (2017YFE0300503).

References

T. J. Freeborn, B. Maundy, and A. S. Elwakil, “Fractional-order models of supercapacitors, batteries and fuel cells: a survey,” Materials for Renewable and Sustainable Energy, vol. 4, no. 3, p. 9, 2015.
View at: Publisher Site | Google Scholar
Z. Gao, “Robust stabilization criterion of fractional-order controllers for interval fractional-order plants,” Automatica, vol. 61, pp. 9–17, 2015.
View at: Publisher Site | Google Scholar
Y. Luo, Y. Chen, and Y. Pi, “Experimental study of fractional order proportional derivative controller synthesis for fractional order systems,” Mechatronics, vol. 21, no. 1, pp. 204–214, 2011.
View at: Publisher Site | Google Scholar
I. Petráš and R. L. Magin, “Simulation of drug uptake in a two compartmental fractional model for a biological system,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4588–4595, 2011.
View at: Publisher Site | Google Scholar
I. Podlubny, Fractional Differential Equations, Academic Press, Cambridge, MA, USA, 1999.
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
M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, and R. Castro-Linares, “Using general quadratic lyapunov functions to prove lyapunov uniform stability for fractional order systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1–3, pp. 650–659, 2015.
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
C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu-Batlle, Fractional-order Systems and Controls: Fundamentals and Applications, Springer Science & Business Media, Berlin, Germany, 2010.
I. Pan and S. Das, Intelligent Fractional Order Systems and Control: An Introduction, vol. 438, Springer, Berlin, Germany, 2012.
D. Baleanu, J. A. T. Machado, and A. C. Luo, Fractional Dynamics and Control, Springer Science & Business Media, Berlin, Germany, 2011.
R. Sakthivel, C. K. Ahn, and M. Joby, “Fault-tolerant resilient control for fuzzy fractional order systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 9, pp. 1797–1805, 2019.
View at: Publisher Site | Google Scholar
A. T. Azar, A. G. Radwan, and S. Vaidyanathan, Mathematical Techniques of Fractional Order Systems, Elsevier, Amsterdam, Netherlands, 2018.
X. Zhang and Y. Chen, “Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order α: the 0 < α < 1 case,” ISA Transactions, vol. 82, pp. 42–50, 2018.
View at: Publisher Site | Google Scholar
J. Fei and C. Lu, “Adaptive fractional order sliding mode controller with neural estimator,” Journal of the Franklin Institute, vol. 355, no. 5, pp. 2369–2391, 2018.
View at: Publisher Site | Google Scholar
Y. Wei, Y. Chen, T. Liu, and Y. Wang, “Lyapunov functions for nabla discrete fractional order systems,” ISA Transactions, vol. 88, pp. 82–90, 2019.
View at: Publisher Site | Google Scholar
S. Mirzajani, M. P. Aghababa, and A. Heydari, “Adaptive T–S fuzzy control design for fractional-order systems with parametric uncertainty and input constraint,” Fuzzy Sets and Systems, vol. 365, pp. 22–39, 2019.
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
J. C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, “A lyapunov approach to the stability of fractional differential equations,” Signal Processing, vol. 91, no. 3, pp. 437–445, 2011.
View at: Publisher Site | Google Scholar
Q. Wang, J. Zhang, D. Ding, and D. Qi, “Adaptive mittag-leffler stabilization of a class of fractional order uncertain nonlinear systems,” Asian Journal of Control, vol. 18, no. 6, pp. 2343–2351, 2016.
View at: Publisher Site | Google Scholar
Y. Wei, P. W. Tse, Z. Yao, and Y. Wang, “Adaptive backstepping output feedback control for a class of nonlinear fractional order systems,” Nonlinear Dynamics, vol. 86, no. 2, pp. 1047–1056, 2016.
View at: Publisher Site | Google Scholar
C. Hua, J. Ning, G. Zhao, and Y. Li, “Output feedback NN tracking control for fractional-order nonlinear systems with time-delay and input quantization,” Neurocomputing, vol. 290, pp. 229–237, 2018.
View at: Publisher Site | Google Scholar
S. Shao, M. Chen, S. Chen, and Q. Wu, “Adaptive neural control for an uncertain fractional-order rotational mechanical system using disturbance observer,” IET Control Theory & Applications, vol. 10, no. 16, pp. 1972–1980, 2016.
View at: Publisher Site | Google Scholar
N. Liu and J. Fei, “Adaptive fractional sliding mode control of active power filter based on dual RBF neural networks,” IEEE Access, vol. 5, pp. 27590–27598, 2017.
View at: Publisher Site | Google Scholar
Y. Wu and H. Lv, “Adaptive neural network backstepping control for a class of uncertain fractional-order chaotic systems with unknown backlash-like hysteresis,” AIP Advances, vol. 6, no. 8, p. 085121, 2016.
View at: Publisher Site | Google Scholar
W. Zhu, W. Li, P. Zhou, and C. Yang, “Consensus of fractional-order multi-agent systems with linear models via observer-type protocol,” Neurocomputing, vol. 230, pp. 60–65, 2017.
View at: Publisher Site | Google Scholar
C. Wang and M. Liang, “Adaptive NN tracking control for nonlinear fractional order systems with uncertainty and input saturation,” IEEE Access, vol. 6, pp. 70035–70044, 2018.
View at: Publisher Site | Google Scholar
C. Wang, M. Liang, and J. Gao, “Adaptive fuzzy output tracking control of a class of uncertain fractional order systems subject to unknown disturbance,” IEEE Access, vol. 6, pp. 70655–670665, 2018.
View at: Publisher Site | Google Scholar
C. Wang, M. Liang, and Y. Chai, “Adaptive control of a class of incommensurate fractional order nonlinear systems with input dead-zone,” IEEE Access, vol. 7, pp. 153710–153723, 2019.
View at: Publisher Site | Google Scholar
T. Hu and Z. Lin, Control Systems with Actuator Saturation: Analysis and Design, Springer Science & Business Media, Berlin, Germany, 2001.
F. Yang and Y. Li, “Set-membership filtering for systems with sensor saturation,” Automatica, vol. 45, no. 8, pp. 1896–1902, 2009.
View at: Publisher Site | Google Scholar
F. L. Lewis, J. Campos, and R. Selmic, Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities, vol. 24, SIAM, Philadelphia, PA, USA, 2002.
Y.-Y. Cao and Z. Lin, “Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 1, pp. 57–67, 2003.
View at: Publisher Site | Google Scholar
Y. Zhao and H. Gao, “Fuzzy-model-based control of an overhead crane with input delay and actuator saturation,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 1, pp. 181–186, 2012.
View at: Publisher Site | Google Scholar
N. Kapoor, A. R. Teel, and P. Daoutidis, “An anti-windup design for linear systems with input saturation,” Automatica, vol. 34, no. 5, pp. 559–574, 1998.
View at: Publisher Site | Google Scholar
Q. Zhou, L. Wang, C. Wu, H. Li, and H. Du, “Adaptive fuzzy control for nonstrict-feedback systems with input saturation and output constraint,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 1, pp. 1–12, 2016.
View at: Google Scholar
Y. Li, S. Tong, and T. Li, “Adaptive fuzzy output-feedback control for output constrained nonlinear systems in the presence of input saturation,” Fuzzy Sets and Systems, vol. 248, pp. 138–155, 2014.
View at: Publisher Site | Google Scholar
B. Wang, S. E. Li, H. Peng, and Z. Liu, “Fractional-order modeling and parameter identification for lithium-ion batteries,” Journal of Power Sources, vol. 293, pp. 151–161, 2015.
View at: Publisher Site | Google Scholar
B. Wang, Z. Liu, S. E. Li, S. J. Moura, and H. Peng, “State-of-charge estimation for lithium-ion batteries based on a nonlinear fractional model,” IEEE Transactions on Control Systems Technology, vol. 25, no. 1, pp. 3–11, 2017.
View at: Publisher Site | Google Scholar
L. Zhang, X. Hu, Z. Wang, F. Sun, and D. G. Dorrell, “Fractional-order modeling and state-of-charge estimation for ultracapacitors,” Journal of Power Sources, vol. 314, pp. 28–34, 2016.
View at: Publisher Site | Google Scholar
C. Tan and Z.-S. Liang, “Modeling and performance analysis of the fractional order quadratic boost converter in discontinuous conduction mode-discontinuous conduction mode,” Modern Physics Letters B, vol. 30, no. 6, Article ID 1650014, 2016.
View at: Publisher Site | Google Scholar
S. Das, Functional Fractional Calculus for System Identification and Controls, Springer Berlin Heidelberg, Berlin, Germany, 2008.
J. C. Trigeassou, N. Maamri, and A. Oustaloup, “The infinite state approach: origin and necessity,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 892–907, 2013.
View at: Publisher Site | Google Scholar
R. M. Sanner and J.-J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Transactions on Neural Networks, vol. 3, no. 6, pp. 837–863, 1992.
View at: Publisher Site | Google Scholar
H. Wang, B. Chen, X. Liu, K. Liu, and C. Lin, “Robust adaptive fuzzy tracking control for pure-feedback stochastic nonlinear systems with input constraints,” IEEE Transactions on Cybernetics, vol. 43, no. 6, pp. 2093–2104, 2013.
View at: Publisher Site | Google Scholar
T. Basar, An Invariance Principle in the Theory of Stability, Wiley-IEEE Press, 2001.

Copyright

Copyright © 2019 Changhui Wang et al. 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

646

Downloads

1228

Citations