 Research
 Open Access
Robust adaptive control for fractionalorder chaotic systems with system uncertainties and external disturbances
 Shaoyu Zhang^{1},
 Heng Liu^{2} and
 Shenggang Li^{1}Email author
https://doi.org/10.1186/s1366201818639
© The Author(s) 2018
 Received: 21 July 2018
 Accepted: 25 October 2018
 Published: 7 November 2018
Abstract
This paper studies the robust adaptive control of fractionalorder chaotic systems with system uncertainties and bounded external disturbances. Based on a proposed lemma, quadratic Lyapunov functions are used in the stability analysis and fractionalorder adaptation laws are designed to update the controller parameters. By employing the fractionalorder expansion of classical Lyapunov stability method, a robust controller is designed for fractionalorder chaotic systems. The system states asymptotically converge to the origin and all signals in the closedloop system remain bounded. A counterexample is constructed to show that the fractionalorder derivative of a function is less than zero does not mean that the function monotonically decreases (this property appears in many references). Finally, simulation results are presented to confirm our theoretical results.
Keywords
 Robust control
 Fractionalorder chaotic system
 Fractionalorder adaptation law
1 Introduction
Chaos phenomenon is very common in physical systems (also biology, economics, etc.), and chaotic systems have become one of the hot topics in the nonlinear systems [1–14]. Since Lorenz discovered the first chaotic attractor, the research on the control and synchronization of chaotic systems has been widely used in many fields. In the past 20 years, many chaos control and synchronization methods have been proposed [15–17]. Fractionalorder calculus has almost the same long history as integer calculus. It was found that the complex chaotic behavior appeared if one introduced the fractional differential operator into the integer order chaotic systems. In fact, fractionalorder calculus provides new mathematical tools for many practical systems, especially for chaotic systems in physics, because it is very suitable for describing the dynamic behavior of some physical systems that are very sensitive to the initial state values [18–28]. There are many methods (such as driveresponse control, Lyapunov function method, sliding mode control, generalized synchronization control, active control, nonlinear feedback control) on synchronization and control of fractionalorder chaotic systems. However, some new problems run into the case of fractionalorder chaotic systems. Firstly, as the tiny initial value changes will cause the system’s trajectory shape and affect its stability, it is difficult to control or synchronize the chaotic systems. Secondly, although the Lyapunov second method of fractionalorder systems is proposed in [29], and the control and stability analysis of fractional order nonlinear systems has gradually become a research focus, it is hard to use the squared Lyapunov function in the stability analysis of fractionalorder systems because the fractionalorder derivatives of square functions have a very complex form (as pointed out in [30, 31]). In order to get over it, many efforts have been made. In 2015, Liu et al. [32, 33] proposed a method to realize adaptive fuzzy synchronous control of uncertain fractionalorder chaotic systems with unknown asymptotic control gain. This is really a relatively universal construction method which can be used in many kinds of control and synchronization of chaotic systems [26, 32, 34–50].

To coin a counterexample which shows that the fractionalorder derivative of a function is less than zero does not mean that the function monotonically decreases. Then we correct Lemma 5 in [32] and Lemma 7 in [33] to Lemma 4 in this paper.

To further verify the effectiveness of the new Lemma 4 in the above relatively universal fractionalorder Lyapunov second method, we study the robust adaptive control of fractionalorder chaotic systems with bounded external disturbances and system uncertainties.
2 Preliminaries
In this section we present some notions and lemmas which are needed in this paper.
Definition 1
Lemma 1
([29])
Lemma 2
Lemma 3
([51])
3 Controller design and stability analysis
In the closedloop system, we will design the control input \(u_{i}(t)\) to make sure that the error states converge to the origin and all signals remain bounded. To reach this goal, the following assumptions are needed.
Assumption 1
Assumption 2
Remark 1
In [32], there exists such a result: Suppose that \(f':[0,+\infty)\longrightarrow R\) is continuous. Then \(f:[0,+\infty )\longrightarrow R\) is monotone increasing (resp., monotone decreasing) if \(D^{\alpha}(f)(t)\geq0\) (resp., \(D^{\alpha}(f)(t)\leq0\)) for all \(t\in[0,+\infty)\). However, this result is not proper. In this section we will construct a counterexample to show this and meanwhile give a corrected form (i.e., Lemma 4).
Example 1
Let \(\alpha\in(0,1)\). We are going to construct the needed counterexample in 11 steps.
Step 5. As \(I(t)\) is continuous in \([4,+\infty)\) and \(\lim_{t\rightarrow+\infty}I(t)=0\) (see (28)), there exists \(b>0\) such that \(I(t)\leq b\) (\(\forall t\in[4,+\infty)\)).
Step 11. From Steps 1–10 we know \(h(t)=f(t)\) is differentiable in \([0,+\infty)\), \(h'\) is continuous, and \(D^{\alpha}(h)(t)\leq0\) (\(t\in[0,+\infty)\)), but \(h(t)\) is not monotone in \([0,+\infty)\).
Remark 2
This example shows a difference between fractional and integer order calculus. And the results in [32] are still right because their proofs are valid as long as we replace Lemma 5 in [32] by the following Lemma 4. In many closedloop systems, it is difficult to determine that all signals are bounded. With the following lemma, we can manage to do this. For example, the signals γ̂ and \(\hat{\bar{d}}\) in system (13).
Lemma 4
Assume that \(f'\) is continuous and bounded in \([0,+\infty)\). Then \(f(t)\geq f(0)\) if \(D^{\alpha}(f)(t)\geq0\) (\(t\in [0,+\infty)\)), and \(f(t)\leq f(0)\) if \(D^{\alpha}(f)(t)\leq0\).
Proof
We just prove the first part of this lemma.
Step 4. Since \(D^{\alpha}(f)(t)\geq0\) (\(t\in [0,+\infty)\)), \(f(t)f(0)= I^{\alpha}\circ D^{\alpha}(f)(t)\geq0\) (\(\forall t\in[0,\infty)\)) by Step 3 and the definition of \(I^{\alpha}\), which means \(f(t)\geq f(0)\) (\(\forall t\in[0,\infty)\)). □
And we show the following lemma.
Lemma 5
Proof
Based on the above discussions, now we are ready to give the following results.
Theorem 1
Proof
Thus, we have \(D^{\alpha}V_{i}(t) \leqk_{i} e_{i}(t)^{2}\leq0\). According to Lemma 4, \(V_{i}(t) \leq V_{i}(0)\). In addition, for \(V_{i}(t)=\frac{1}{2}e_{i}^{2}(t)+\frac{1}{2h_{i}} \tilde{\gamma}_{i}^{2}(t)+\frac{1}{2m_{i}} {\tilde{\bar{d}}_{i}}^{2}(t)\), we can know that \(e_{i}(t)\), \(\tilde{\gamma}_{i}(t)\), and \({\tilde{\bar{d}}_{i}}(t)\) are all bounded. From Lemma 5 and (63), we can conclude that \(e_{i}(t)\) will asymptotically converge to the origin. This ends the proof for Theorem 1. □
Remark 3
4 Simulation studies
5 Conclusions
Chaotic systems can be used in many fields. Controlling fractionalorder chaotic systems by using an effective control method is an interesting yet challenging work. In this paper, an example is built to show that being less than zero for a fractionalorder derivative does not mean that the function monotonically decreases. And the robust control of fractionalorder chaotic systems by means of adaptive control is primarily discussed. The proposed method can be divided into three aspects: (1) Based on the fractionalorder Lyapunov second method, analyzing the stability of general fractionalorder chaotic systems; (2) Designing the fractionalorder adaptation laws and uploading the controller; (3) Using the quadratic Lyapunov functions in the stability analysis of fractionalorder systems. The control theorem of fractionalorder systems may be enriched by our results, and the proposed control method can also be extended to other fractionalorder systems.
Declarations
Acknowledgements
The authors are grateful to the referees for their helpful suggestions which have greatly improved the presentation of this paper.
Funding
This work is supported by the National Natural Science Foundation of China (No. 11771263), the Fundamental Research Funds for the Central Universities, and the Natural Science Foundation of Anhui Province of China (No. 1808085MF181).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors conceived of the study, participated in its design and coordination, read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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