Research articleFuzzy adaptive finite-time output feedback control of stochastic nonlinear systems
Introduction
In many practical systems, there will be random disturbances, which may be the source of system instability. How to design and analyze the controller in a stochastic system has become particularly important and significant progress has been made in this field [1], [2], [3], [4], [5], [6]. In [7], the problem of robust sliding mode control was studied for stochastic systems with randomly occurring nonlinearities, time delays and uncertain discrete-time. The moment stability and sample-path stability were analyzed for switched stochastic systems in [8]. A unified optimal and exponentially stable filter was developed in [9] for stochastic systems with linear discrete-time. In [10], the result of the stability of the linear stochastic system with state and control related noise in a finite time was reported, considering that the filter simultaneously estimated the state and the unknown in the sense of unbiased minimum variance input without any assumptions of directly feeding the matrix.
The backstepping control algorithm is a popular design tool for system stability analysis. The application of backstepping technology has solved the stability problems of various types of nonlinear systems with random disturbance and there have been many remarkable results [11], [12], [13], [14], [15]. The authors have achieved asymptotic tracking and interference attenuation for stochastic nonlinear strict-feedback systems described in the form of strict parameter feedback and affected by other external interference inputs in [16] and this was the first time that the backstepping scheme was applied to stochastic systems. A nonlinear pure feedback system with random disturbance and unknown dead zone was considered, and its tracking control problem was solved in [17] and the proposed method ensured that the system achieves semi-globally, uniformly bounded in probability. The adaptive approach to the tracking control problems the output feedback canonical systems corrupted by Wiener noise with unknown covariance were addressed [18]. In the traditional backstepping control method, repeated derivation of the virtual function may cause “computational explosion”, which is also its inherent problem. In order to solve the problem of “computational explosion”, the algorithm of dynamic surface control (DSC) was considered firstly. The purpose of this algorithm is to estimate the derivative of the intermediate variable function by using the output of the filter. Subsequently, there have been many promising results [19], [20], [21], [22]. In [23], an adaptive DSC method was reported based on neural networks for nonaffine nonlinear systems with unknown time delay and input delay nonlinearity. The trajectory tracking problem of marine vehicles with unknown time-varying disturbance, actuator rate limit and input saturation was studied and an n-DOF (degree-of-freedom) nonlinear control law was developed by using disturbance observer and nonlinear DSC technology in [24]. In [25], the problem of an adaptive neural tracking control was addressed for uncertain nonlinear systems with time-delay, random disturbance, input and output constraints. Because it does not need to calculate a large number of derivatives of intermediate variable signals, the proposed DSC control method overcomes the inherent computational complexity of classic backstepping control methods.
Compared with asymptotic stability, it provides fast response speed, high tracking accuracy and strong anti-jamming ability. The finite-time convergence problem has become a meaningful and interesting topic and many excellent results have been achieved [26], [27], [28], [29], [30], [31], [32]. The result of the optimal finite-time stabilization was considered for nonlinear systems in [33]. In [34] the tracking problem was solved for the distributed Euler–Lagrange dynamic network and a finite-time algorithm was developed. For non-strictly feedback nonlinear switching systems with incalculable states, arbitrary switching, and unknown functions composed of the entire state, the problem of finite-time switching control was studied in [35]. For the non-linear quantization system in [36], whose state could not be measured, the finite-time quantitative feedback control was studied for the first time, which solved the problem of finite-time tracking. The application of the finite-time method improves the tracking performance of the system. In the process of system stability analysis, the handling of unknown items has always attracted attention. For unknown functions, two intelligent methods, fuzzy logic system (FLS) [19], [37] and neural networks (NNs) [38], [39], [40], [41], were widely used. For the processing of unknown states, the construction of the state observer was particularly important [42], [43]. However, to our knowledge, the stability analysis of stochastic nonlinear systems with unmeasured states has not yielded results in a finite-time frame. This is the motivation for this work.
In this paper, we study the finite-time control problem for a class of stochastic nonlinear systems with unknown states. The fuzzy logic system is adopted to estimate the unknown function and the designed state observer is used to estimate the unmeasured state. The application of DSC in backstepping control alleviates the calculation problem of virtual control function. The finite-time method improves both the tracking speed and convergence performance. The novelties of this study are listed as follows:
(1) A DSC method is introduced to solve “computational explosion” problem caused by the repeated derivation of intermediate variable function. The DSC method is applied in each step of backstepping scheme and the output of the first-order filter is used to approximate the derivative of the virtual control function. Compared with traditional schemes, the uniqueness of this scheme is based on the finite-time frame in that it uses DSC technology in the backstepping control step.
(2) Nonlinear systems with random disturbance and unknown state are considered within a finite-time frame, which is different from [44]. Constructing a fuzzy state observer can estimate the unmeasured state. A new output feedback controller is developed to guarantee that all the signals of the closed-loop system are bounded within a finite time and the error of tracking is able to converge to arbitrarily small residual sets within a finite time.
Section snippets
Preliminaries and system description
The form of a class of stochastic nonlinear systems is given as follows: where the state vector is , represents the control input and is the output. and denote continuous, smooth and uncertain functions, where . is an -dimensional standard Brownian motion. is defined on a complete probability space with the incremental covariance , is an
Fuzzy state observer design
Define , , is an unknown parameter vector, so from Lemma 5, we can know , where is the estimation of , the estimated error is , we have where is compact set and is compact set. The fuzzy minimum estimation error is defined as follows where with is an unknown positive constant. The entire system (1) can be expressed as:
Main results
In this section, an algorithm based on DSC and backstepping technique is presented. We first give the coordinate transformation as follows where is the reference output signal and represents the output signal of the filter.
is the output of the first-order filter when is the input, is an virtual control function and is a time constant. can be introduced as with the filter errors
Step 1: According to
Simulation examples
In this section, we present two simulation examples to illustrate the effectiveness of our proposed control scheme.
(i) Numerical Example: a strict-feedback stochastic nonlinear system is given: where and are the states, the output is and is control input. and .
The form of the state observer is given as follows It is
Conclusion
An adaptive finite-time algorithm is presented for nonlinear systems with random disturbance and unknown state. The FLS scheme is adopted to estimate unknown functions while the unmeasured state is estimated by using a designed state observer. By applying the dynamic surface control method, the significant computation load due to the multiple derivations of the intermediate variable function is solved. The development of a finite-time output feedback controller ensures that all the signals of
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 in part by the Fundamental Research Funds for the Central Universities 2021YJS140, and in part by the National Science Foundation of China under projects 61761166011, 61773072 and 61773051, and in part by the Education Department of Liaoning Province under the general project research under Grant No. LJ2020001.
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