State observer-based fuzzy echo state network sliding mode control for uncertain strict-feedback chaotic systems without backstepping
Introduction
Chaos has aroused tremendous interest because of the widespread existence of chaotic behavior in nonlinear dynamical systems of physics, chemistry, biology, engineering, and many other disciplines. However, chaotic systems are highly susceptible to the initial conditions and their dynamics may diverge rapidly due to slightly different conditions, which will lead to high unpredictability. It is believed that the research on chaos control is a complicated and challenging task, which also has a wide range of potential applications. Consequently, massive advanced control approaches have been utilized to control or synchronize chaotic systems, which include linear and nonlinear feedback control [1], [2], time-delay feedback control [3], [4], adaptive control [5], [6], sliding mode control [7], [8], and so on.
It is worth noting that numerous chaotic systems can be rewritten as the strict-feedback form, for these systems, the most popular method is the backstepping control, because this method has a systematic framework, namely a certain of intermediate variables are circularly regarded as virtual control variables. Ref. [9] presented a systematic design procedure to synchronize strict-feedback chaotic systems (SFCSs) and general SFCSs using the backstepping technology. In [10], for fractional-order SFCSs, a backstepping fuzzy synchronization approach was implemented, and the command filter and error compensation mechanism were also introduced. In [11], Rössler system and Duffing oscillator were taken as examples to illustrate the effectiveness and feasibility of the proposed adaptive backstepping synchronization method. Ref. [12] proposed a robust control scheme for cross-strict feedback hyperchaotic systems by combining sliding mode control with the backstepping technology. However, these methods require repeatedly differentiating virtual control inputs, and with the increase of the system order, the complexity of the control law grows drastically [13], i.e., the “explosion of complexity” problem occurs. There are two ways to remedy this problem. The first one is the dynamic surface control (DSC) method, which can be referred to [14], [15], [16]. At each backstepping step, a first-order filter with simple available derivative is designed to estimate the virtual control input. However, filter delay errors are introduced into DSC systems, which results in a large tracking error and degraded tracking accuracy. The second one is the command filter-based contro method [17], [18], which can estimate the derivations of virtual control inputs directly. In [10], [19], error compensation signal was introduced to reduce the filter error, boosting the achievement of robust performance.
The above two methods, DSC and command filter, can settle the “explosion of complexity” problem commendably and have a good control performance for systems with prior information. Nevertheless, in practical applications, chaotic systems are inevitably affected by system uncertainties and external disturbances, which will lead to the fact that system model is completely unknown. Due to the characteristics of accurate approximation ability, fast adaptability, and large-scale parallelism, neural networks have been extensively used in system modeling, such as machine learning [20], mechanics of materials [21], [22], and control engineering.
As we can see, the controller design and the stability analysis will become remarkably complex, if a function approximator is employed to deal with the unknown nonlinearity in each backstepping step, which will produce an approximation error. As the system order increases, the accumulated approximate error will reduce the control performance and even lead to instability. In addition, the traditional fuzzy logic system (FLS) or neural network is limited by the following facts: firstly, a large number of neurons or fuzzy rules are required to improve the approximation ability, which greatly increases the computational burden; secondly, the weight update does not use the internal neurons information, that is, they are not suitable for dealing with time-series related problems ([16], [18], [23], and the reference therein); thirdly, for recurrent neural networks (RNNs) [24], [25], [26], there are several limitations, such as memory fading, slow convergence speed, and complex training algorithm. To solve these problems, an improved function approximator, namely the echo state network (ESN) was introduced [27]. As an advanced RNN, the basic idea of ESN is to use the large-scale randomly connected recursive network structure called reservoir, to replace the middle layers in classical RNNs. The reservoir is fixed once it is selected, and then only the output connection should be trained, which greatly simplifies the training process of RNNs. Furthermore, the stability of ESNs can be guaranteed by presetting the spectral radius of the reservoir weight matrix. Another prominent advantage of ESNs is that it can map the input signal from the low-dimensional input space into the high-dimensional state space through the reservoir. Then, the methods and theories for dealing with linear problems can be used to deal with these complications, which is consistent with the kernel method represented by support vector machine.
Due to the above difficulties, for uncertain strict-feedback nonlinear systems, Park et al. proposed another interesting control method that is not based on backstepping technology in [28], whose main idea is that the state-feedback control problem can be translated into an output-feedback control problem of a straightforward normative system by a series of coordinate transformation. Consequently, some methods suitable for normal systems can be extended to strict-feedback nonlinear systems. As far as the authors know, up to now, the results on output feedback control is very limited for uncertain SFCSs.
Inspired by the aforementioned discussion, and considering the advantages of ESN and classic FLS, in this paper, a fuzzy echo state network (FESN) by composing an FLS and a leaky ESN through certain threshold functions is proposed to approximate the lumped uncertainty. Then a state observer-based FESN sliding mode control for uncertain SFCSs without backstepping is designed. Compared with the aforementioned works, the main contributions are as below. 1) Through corresponding coordinate transformation, the uncertain SFCS is translated into a new straightforward normative system, then a new controller is designed, which is different from the existing controller design based on backstepping method, so the “explosion of complexity” problem and circular problem previously encountered are all circumvented. Consequently, the complexities of the new controller structure, stability analysis, and computational burden are also significantly reduced. 2) An FESN combined with the advantages of FLS and ESN is proposed, which can improve the approximation accuracy and figure out the existing problems of RNNs. Besides, only one FESN is needed to approximate the lumped uncertainty. 3) A simple state observer is constructed to approximate the new unmeasured state variables of the transformed system, and the observer error convergence is guaranteed.
The rest of this paper is organized as follows. In Section 2, the system description and the corresponding coordinate transformation are presented. In addition, the working principle and the structure of the FESN are also introduced. The detailed controller design and the stability analysis are given in Section 3. Section 4 provides comparative numerical simulations to show the viability and efficiency of the designed controllers. Some conclusions are summarized in Section 5.
Section snippets
System description
Consider the following general uncertain SFCS with additional control inputin which is the state vector, are sufficiently smooth nonlinear functions, is the output variable, is the control input. Remark 1 In several researches based on nonlinear dynamics (such as bifurcation and chaos) and the related control or synchronization problems, some chaotic systems, including Duffing oscillator [29], Van der Pol oscillator
Sliding mode and observer design
In this section, an adaptive FESN sliding mode controller based on the state observer will be constructed for the system (5) without using the backstepping technique. For the sake of analysis, define the tracking error as , then we have
Since is unknown, it can be estimated by the FESN (8). According to Assumption 2, we havewith being the optimal parameter, and . The sliding mode surface is designed aswhere
Simulation results
In this part, comparative simulation examples are given to verify the validity and simplicity of the proposed adaptive FESN controller. Consider the following controlled Chua–Hartley's system [35]
The initial state values are and . When , the system (30) shows rich dynamical behavior, which is illustrated in Fig. 2. To highlight the good extrapolation quality of the proposed method, during the actual control process,
Conclusions
In this study, a novel adaptive FESN sliding mode control without backstepping technology for uncertain SFCSs is proposed. Unlike the already existing control methods using the backstepping technique, the proposed technique makes the state-feedback problem convert to the output-feedback problem of the standard system by a series of coordinate transformations, and thus, whether the controller design or stability analysis or simulation experiments become straightforward. Since state variables of
CRediT authorship contribution statement
Jiayan Li: Writing – original draft, Software. Jinde Cao: Writing – review & editing, Project administration. Heng Liu: Conceptualization, Funding acquisition.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 61967001), and the Xiangsihu Young Scholars Innovative Research Team of Guangxi Minzu University (Grant No. 2019RSCXSHQN02).
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