Adaptive tracking control for switched strict-feedback nonlinear systems with time-varying delays and asymmetric saturation actuators
Introduction
In the past decade, the control design and stability analysis of switched systems has received increasing attention due to various practical applications such as robot system [1], aerial traffic control system [2], switching power system [3] and sensor networks [4], [5]. Most of the results on switched systems have been available by using the multiple Lyapunov functionals approach for constrained switching and the CLF approach for arbitrary switching. Note that arbitrary switching can cover the case that the switching mechanism is either unknown or too complicated to be useful in the stability analysis or control design, the study on switched systems under arbitrary switching has been of great interest. As pointed out in [6], [7], a switched system under arbitrary switching is asymptotically stable if all subsystems share a CLF. In contrast to plenty results on the control design of switched linear systems (see in [8], [9], [10], [11] and the reference therein), fewer results have focused on the control problem for switched nonlinear systems. Under arbitrary switching, the stabilization problems for switched nonlinear systems were presented by using the CLF approach and the backstepping technique in [12] and [13]. Nevertheless, since it is difficult to find common intermediate controllers for all different uncertain subsystems, the preceding works [12], [13] did not take into account uncertainties in systems. To relax the restriction from finding common intermediate controllers, some adaptive tracking control schemes were developed for several classes of uncertain switched nonlinear systems under arbitrary switching in [14], [15], [16], [17]. However, the aforementioned results [12], [13], [14], [15], [16], [17] are derived in the case of no time delay.
As well known, time delay widely exists in a variety of practical systems, and often gives rise to the instability of system. Many researchers have paid much attention to stability analysis and control design for time-delay systems [18], [19], [20], [21], [22], [23]. Especially, there has been a considerable research interest in control synthesis of uncertain switched time-delay systems which possesses the main ingredients of multi-modes of operation, nominally inherent time-delay, and model uncertainties. By using the Lyapunov–Krasovskii method, some results were concerned with various switched time-delay systems in [24], [25], [26], [27], [28], [29], while the main drawback in [24], [25], [26], [27], [28], [29] is that the delay in each subsystem must be independent of the subsystem number (or the switching signal). Though the considered delays depend on the subsystem number in [30], the stabilized system is investigated for the situation in which all subsystems have the same constant delay. Unfortunately, no enough attention has been paid to switched nonlinear time-delay systems in which the time delays are absolutely dependent on the subsystem number. Due to the existence of such delays, it is very hard to construct the common Lyapunov–Krasovskii functional for all subsystems, and then the aforementioned results [24], [25], [26], [27], [28], [29] will be invalid. On the other hand, the classical Krasovskii method often requires the time-varying delay τ(t) to satisfy some conservative conditions, such as 0 ≤ τ(t) < ∞ and . However, the aforementioned restrictions from the Lyapunov-krasovskii method can be avoided by using the Lyapunov–Razumikhin method. Firstly, the common Lypunov–Razumikhin function would be constructed as a simple quadratic function for all subsystems. Besides, the Lyapunov–Razumikhin technique applies to the bounded time-varying delay (i.e., 0 ≤ τ(t) < ∞) only. In [31], the Lyapunov–Razumikhin method was employed to study the robust stability of switched nonlinear systems with time delays. So far, there have still been no works available about the use of the Lyapunov–Razumikhin method for the adaptive control design of switched nonlinear time-delay system.
Additionally, saturation is unavoidable in every practical actuator, thus it is one of the most common nonlinearities in many practical control systems. Actually, the presence of saturation nonlinearity often restricts the performance of closed-loop system, especially the stability. In recent years, there has been an increasing interest on the analysis and synthesis for control systems with actuator saturation nonlinearities [32]. Since the symmetry of saturation function facilitates the analysis of the closed-loop system, symmetric saturation has been studied well in many works (e.g., see in [33], [34], [35]). But fewer works [36], [37], [38], [39], [40] are available to handle the asymmetric saturation nonlinearity since it is difficult to represent the asymmetric saturation nonlinearity with a smooth model. In [39], a Gaussian error function-based continuous differentiable model was proposed to approximate the asymmetric saturation nonlinearity. Based on the smooth model in [39], a more suitable model was developed to describe the unknown asymmetric saturation actuator in [40] by using the mean value theorem. To the best of our knowledge, there have been no reported works dealing with the switched nonlinear time-delay system with asymmetric saturation actuators.
Based on the foregoing discussion, it is interesting and challenging to design an effective control scheme for switched nonlinear systems in the presence of unknown time-varying delays and asymmetric saturation actuators under arbitrary switching. Therefore, this paper will first be concerned with the problem of adaptive tracking control for such systems in which the time-varying delays are absolutely related to the subsystem number. Some special techniques and methods, such as Lyapunov–Razumikhin method, DSC technique, variable separation technique and NN approximation-based approach, are used for the control design of such systems under some appropriate assumptions. The main contributions of this paper are emphasized as follows:
- (1)
It is the first time to explore the tracking control for switched strict-feedback nonlinear time-delay systems with asymmetric saturation actuators under arbitrary switching. Moreover, a common adaptive tracking control scheme is presented for all subsystems. As a result, all signals in closed-loop system are SGUUB and the satisfactory tracking performance can be achieved by properly adjusting the control parameters.
- (2)
In comparison with the existing results on the switched nonlinear time-delay systems (see in [24], [25], [26], [27], [28], [29], [30]), this paper concentrates on the time-varying delays which absolutely depend on the subsystem number. Besides, by employing the Lypunov–Razumikhin method, the difficulty from finding an appropriate CLF for all subsystems can be avoided and the restrictions on time-varying delays are also relaxed.
- (3)
Owning a simple form and containing only one adaptive parameter facilitate the online computation of the proposed control scheme. Consequently, it is more convenient to implement the control scheme in practice.
The rest of this paper is organized as follows: Section 2 gives the preliminary results and describes the considered system. The main results are presented in Section 3. In Section 4, two simulation examples are given to verify the effectiveness of the proposed design. Finally, this paper is concluded with a short discussion in Section 5.
Section snippets
Preliminaries and problem statement
Notations:(1) Throughout this paper, | · | and ‖ · ‖ denote the absolute value and the Euclidean norm, respectively; max { · } and min { · } represent the functions of minimum and maximum, respectively; λmax (A)(or λmin (A)) represents the maximum (or minimum) eigenvalue of matrix A; If a scalar continuous function ϕ(θ), θ ∈ [0, tf) is strictly increasing and satisfies then ϕ(θ) belongs to class-; Furthermore, if it is defined for all θ ≥ 0 and ϕ(θ) → ∞ as θ → ∞, then ϕ(θ) is said to
Adaptive control design and stability analysis
To begin the procedure of DSC design, the following coordinate transformation is introduced. where is the output of the filter with the virtual control law being shown later and χi being a positive design parameter. Moreover, the boundary layer error si is defined as for .
It is evident that the backstepping design procedure for uncertain switched nonlinear system (1) will involve n design steps. Additionally, a
Simulation studies
To demonstrate the effectiveness and applicability of the proposed design methodology, two simulation examples are provided in this section.
Example 1 Consider the following switched nonlinear time-delay system described as
where
Conclusions
A common adaptive tracking control scheme has been developed for a class of switched strict-feedback nonlinear systems in the presence of both unknown time-varying delays and asymmetric actuator saturation under arbitrary switching. The presented design methodology has the following advantages: (1) The tracking control problem for a class of uncertain switched strict-feedback nonlinear time-delay systems with both asymmetric saturation actuators and arbitrary behavior has been tackled. (2) The
Acknowledgments
The authors are grateful to the editors and reviewers for their valuable comments and kind help. This work is partly supported by National Natural Science Foundation of China under Grant numbers 61304071, 61403139 and 71271132, and the Fundamental Research Funds for the Central Universities.
Zhaoxu Yu received the B.S. degree in Mathematics from Jiangxi Normal University in 1998, the M.S. degree in Applied Mathematics from Tongji University in 2001 and the Ph.D. degree in Control Science and Engineering from Shanghai Jiaotong University in 2005. He is currently an Associate Professor with the Department of Automation in East China University of Science and Technology. His research interest includes nonlinear control, adaptive control and stochastic system.
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Zhaoxu Yu received the B.S. degree in Mathematics from Jiangxi Normal University in 1998, the M.S. degree in Applied Mathematics from Tongji University in 2001 and the Ph.D. degree in Control Science and Engineering from Shanghai Jiaotong University in 2005. He is currently an Associate Professor with the Department of Automation in East China University of Science and Technology. His research interest includes nonlinear control, adaptive control and stochastic system.
Yan Dong received the B.S degree in Automation from Changzhou University in 2015, and is currently working toward the M.S. degree in control theory and control engineering from East China University of Science and Technology. His current research interest is nonlinear control.
Shugang Li is a Professor in the School of management at Shanghai University, Shanghai, China. He received his Ph.D. degree in control engineering from Shanghai Jiao Tong University in 2004. His current research areas of interest are information system and information management, data mining, soft computing and artificial intelligence.
Fangfei Li received her B.S. degree from Liaoning Normal University in Dalian, in 2004, M.S. degree from University of Shanghai for Science and Technology in Shanghai, in 2007 and Ph. D degree from Tongji University in Shanghai, in 2012. Since 2012 she is serving as an Associate Professor at Department of Mathematics, East China University of Science and Technology in Shanghai. Her research interests include Boolean networks, impulsive hybrid systems, systems biology, etc.