Adaptive tracking control for a class of uncertain switched stochastic nonlinear systems
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Abstract
This paper pursues an adaptive fuzzy control scheme for a class of nonlinear systems with stochastic switching. A general controller and adaptive mechanism are designed by utilizing Lyapunov function approach and backstepping technique. It is demonstrated that the presented control method can guarantee that all the signals in the closedloop system are semiglobally uniformly ultimately bounded (SUUB) and the tracking error is convergent to a neighborhood of the origin. Finally, the simulation results verify the feasibility of the control strategy presented in this paper.
Keywords
Adaptive fuzzy control Stochastic switched nonlinear systems Backstepping technique1 Introduction
As is well known, a common Lyapunov function (CLF) guarantees that the switched system is stable under arbitrary switching [1]. As an impactful approach for the stability analysis, CLF has been widely employed for control synthesis of switched linear systems [2, 3, 4, 5, 6]. For instance, [7] used the classic quadratic Lyapunov function and solved the stabilization problem for a class of stochastic nonlinear strictfeedback systems. Based on CLF method for a class of switched nonlinear systems, [8, 9] have investigated three state feedback control methods; however, the nonlinear functions of the above control systems are known. Additionally, the backstepping technique is used for the global stabilization problem for switched nonlinear systems in strictfeedback form under arbitrary switchings [8]. The adaptive backstepping approach is a recursive design methodology for controller design. It constructs associated Lyapunov functions and feedback control laws, and its main purpose is to design the adaptive laws and virtual control functions to counteract the unknown nonlinearity of system [10]. In recent years, in view of several classes of switched nonlinear systems, some backstepping control design methods have been proposed. Nevertheless, few of them take into account the uncertainties that exist extensively in practical switched nonlinear systems [6, 11].
In the last few decades, as a typical hybrid system, switched systems have been a great concern with their increasing significance in engineering practice, such as multiagent systems, aircraftcontrol systems and circuit and power systems [12, 13]. Switched systems present switching between a set of subsystems resting with changing environmental factors. The system detects and breaks down various parameters in the changing environment, and then switches to the subsystem matching with the environment. So far, the controller design and stability analysis of switched systems have proposed remarkable results [2, 14, 15, 16, 17, 18, 19, 20]. In the actual control system, the dynamic characteristics of controlled objects such as production process, production equipment and transmission system are difficult to describe by accurate mathematical model. With the change of working environment, the components of the control system may be aged or damaged, and the characteristics of the controlled object also change. All these factors lead to some inevitable errors between the mathematical model of the controlled object and the actual object. For example, in large power systems, due to the large dimension, many systems contain unmodeled dynamic, uncertain parameters and random noise. In the actual operation, the system will also be affected by various harmonic and load disturbances. These random uncertainties of the power system bring about security risks to the normal operation of the power system. All of the above systems can be described by switching system. Since stochastic switched systems integrate the characteristics and difficulties of stochastic systems and switched systems, it is very difficult to analyze the stability and application of stochastic switched systems.
Obviously, stochastic disturbance is considered as one of the unstable sources of control systems which usually exists in many practical systems [21, 22, 23]. So, for a deterministic nonlinear system, the control for stochastic nonlinear system is much more difficult. Therefore, the research on control design and stability analysis for nonlinear stochastic systems is a significant and challenging subject, and it has been a topic of great concern in the last few years [24, 25, 26, 27, 28]. Specifically, some control methods based on adaptive backstepping technique for deterministic nonlinear systems [29, 30, 31, 32] have been successfully generalized to nonlinear stochastic systems [33, 34, 35, 36, 37, 38, 39]. For instance, an outputfeedback backstepping controller was developed for a class of stochastic nonlinear systems in [40], the state feedback controller is designed for nonlinear stochastic systems with Markovian switching [41] and [42] presented the backstepping control design approaches. Nevertheless, these methods are only suitable for those nonlinear stochastic systems with known nonlinear dynamic models. Adaptive outputfeedback control methods for a class of uncertain nonlinear stochastic systems were proposed by utilizing the fuzzy logic system (FLS) and the stability of the control systems was discussed in [35]. The results of [35] were extended to a class of uncertain largescale nonlinear stochastic systems. The approaches [17, 20] decreased the adjustable parameters. The presented controller in [43] has a simple structure because the unknown virtual control signals were directly approximated via FLS. From the above, adaptive fuzzy control approach plays an important role in dealing with uncertain nonlinear systems.
 1.
This paper studies the tracking control problem of switched nonlinear uncertain systems, which is different from the available methods on switched nonlinear systems. The stochastic disturbance is considered and all system functions studied in this paper are unknown completely. Therefore, compared with existing work, the controlled system is more general and the control design is more challenging.
 2.
There are two kinds of adaptive fuzzy backstepping control approaches proposed in this paper for a class of switched nonlinear uncertain systems. We propose a design approach with multiple adaptive laws in the first place. After that, another approach with only one adaptive law is presented in order to avoid too many parameters. In addition, we use the norm of the unknown weight vector of FLS basis function rather than the weight vector elements themselves as the estimated parameter at each step, which significantly reduces the number of adaptive parameters. Therefore, the presented control design approach becomes more practical to use.
The remainder of manuscript is organized as follows. The preliminaries and problem formulation are addressed in Sect. 2. A novel adaptive fuzzy control scheme is introduced in Sect. 3. A simulation example is developed in Sect. 4, Finally, conclusions are given in Sect. 5.
2 Preliminaries and problem formulation
The following notations are used in this paper. \(R_{+}\) means the set of all nonnegative real numbers, \(R^{n}\) represents the real ndimensional space, and \(R^{n\times r}\) stands for the set of all \(n\times r\) real matrices. \(\Vert X \Vert \) indicates the Euclidean norm of a vector x. \(C^{2,1}\) represents the set of all the functions \(V(x,t)\) which belong to \(C^{2}\) with respect to x and belong to \(C^{1}\) with respect to t. \(\operatorname{Tr}(A)\) means a trace of the matrix A.
2.1 Stochastic stability
Definition 1
([44])
Remark 1
The term \(\frac{1}{2}\operatorname{Tr} \{ h^{T}\frac{\partial ^{2} \mathcal{V}}{ \partial x^{2}}h \}\) is called Itô correction term, \(\frac{\partial ^{2} \mathcal{V}}{\partial x^{2}}\) will be more difficult to construct the common virtual control function and the unified adaptive mechanism for uncertain switched stochastic systems than that of deterministic system.
Lemma 1
([45])
Lemma 2
([34])
Lemma 3
(Young’s inequality [46])
2.2 Problem formulation
Assumption 1
([47])
The tracking target \(y_{v}(t)\) and its time derivatives up to the nth order are continuous and bounded.
Remark 2
When we do not consider the unknown functions and the tracking control problem, system (3) will be reduced to system (1) in [17], So, the system studied in this note is more general.
Assumption 2
([48])
Remark 3
In the existing researches on purefeedback nonlinear systems, it is usually considered that the sign of \(h_{n,r}u_{\tau (t)}\) is known. Therefore, Assumption 2 is reasonable, it is a meaningful work for the stochastic nonlinear systems.
2.3 Fuzzy logic systems
In the process of controller design and stability analysis, the FLS is adopted in order to approximate the unknown functions.
 \(R_{j}\):

IF \(\bar{x}_{1}\) is \(\varGamma _{1}^{j}\) and … and \(\bar{x}_{n}\) is \(\varGamma _{n}^{j}\), then y is \(P^{j}\), \(j=1,2,\ldots,\aleph \),
Lemma 4
([49])
Remark 4
Lemma 4 shows that real continuous function \(f(\bar{x})\) can be expressed as a linear combination of bounded error ϵbased function vectors \(\zeta (\bar{x})\). That is, \(f(\bar{x})=\Im ^{T} \zeta (\bar{x})+\xi (\epsilon )\), \(\vert \xi (\epsilon ) \vert < \epsilon \), it plays an important role in the whole process of adaptive laws design. It is noted that \(0<\zeta ^{T}\zeta \leq 1\).
3 Main results
In this section, the adaptive fuzzy control scheme of system (3) is proposed by combining the FLS with adaptive backstepping technique and CLF approach. In Sect. 3.1, a specific design process will be given. In each step, we will design a virtual control function \(\sigma _{i}\) via using a proper CLF \(V_{i}\), and the control law \(u_{k}\) will finally be designed. In Sect. 3.2, in order to avoid repetition, a final CLF will be only adopted to prove the design procedure.
3.1 Adaptive control design under multiple adaptive laws
Define: \(\bar{y}_{v}^{(t)}= [ y_{v},y_{v}^{(1)},\ldots,y_{v} ^{(j)} ] ^{T}\), \(j=1,2,\ldots, n\), with \(y_{v}^{(j)}\) denoting the jth derivative of \(y_{v}\).
At step j of the design process, the unknown function \(\hat{f}_{j,r} \) is approximated by a FLS \(\Im _{j,r}(x_{j})\). For this purpose, define a constant \(\varsigma _{j}=\frac{ \Vert \Im _{j,r} \Vert ^{2}}{b _{k}}\), \(j=1,2,\ldots,n\), denote \(\hat{\varsigma }_{j}\) as the estimation of \(\varsigma _{j}\), and the estimation error is \(\tilde{\varsigma } _{j}=\varsigma _{j}\hat{\varsigma }_{j}\).
Now, we give detailed backstepping design process in the following steps.
Remark 5
Note that the FLS is directly used to approximate unknown nonlinear function \(\hat{f}_{1,r}\) rather than only the unknown function \(f_{1,r}\). This method will be used in the remaining design steps.
Theorem 1
Consider a class switched stochastic nonlinear system (3), under Assumptions1and 2, for bounded initial conditions, parameter adaptive laws (35), the control law (46), and the intermediate control signals (47), guarantee that all the signals in the closedloop system are SUUB and the tracking error is convergent to a neighborhood of the origin.
Remark 6
In [43], the adaptive tracking problem for a class of switched nonlinear systems was investigated. By combining the backstepping technique with the approximation scheme of FLS, a design approach with multiple adaptive laws was developed. In this paper, Theorem 1 generalizes the result of Theorem 1 in [43]. Considering the stochastic disturbances, the systems in this paper are more common.
3.2 Adaptive control design under one adaptive law
Theorem 2
Consider a class switched stochastic nonlinear system (3), under Assumptions1and 2, for bounded initial conditions, the control law (60), and the intermediate control signals (61), guarantee that all the signals in the closedloop system are SUUB and the tracking error converging to a neighborhood of the origin.
Remark 7
In [43], the adaptive tracking problem for a class of switched nonlinear systems was investigated. By combining the backstepping technique with the approximation approach of FLS, a design scheme with only one adaptive laws was developed. In this paper, it is noted that Theorem 2 generalizes the result of Theorem 2 in [43].
4 Simulation example
In this section, a simulation example is proposed in order to certify the control performance and the feasibility of the presented method in the previous sections.
Example 1
Remark 8
Example 2
5 Conclusion
This paper studied the adaptive tracking control problem for a class of stochastic nonlinear systems under arbitrary switchings. It was noted that the nonlinear functions and stochastic disturbances of the system were completely unknown. For the sake of releasing the computational burden, the unknown nonlinear function of the system was estimated by employing the approximation property of FLS, then the adaptive backstepping technique was used to construct a class of adaptive fuzzy control. Under arbitrary switching conditions, the presented controller could ensure that all the signals in the closedloop system remained bounded in probability and the system output converged to a small neighborhood of the reference signal. Finally, simulation results further showed the effectiveness of the proposed approaches.
Notes
Acknowledgements
We are thankful to the reviewers for their useful corrections and suggestions, which improved the quality of this paper.
Authors’ contributions
The main idea of this paper was proposed by the first and last authors, while the second and last authors reviewed and modified the paper. Furthermore, all authors read and approved the final manuscript.
Funding
This research work has been supported financially by the National Natural Science Foundation of China (Grant no. 61402265, 61573227), Shandong Provincial Natural Science Foundation (No. ZR2018MF013, ZR2016FM48), the Research Fund for the Taishan Scholar Project of Shandong Province of China, SDUST Research Fund (No. 2015TDJH105) and the Fund for Postdoctoral Application Research Project of Qingdao (01020120607).
Competing interests
The authors declare that they have no competing interests.
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