Adaptive neural network control for nonstrict-feedback uncertain nonlinear systems with input delay and asymmetric time-varying state constraints
Introduction
Over the past several decades, adaptive control has been successfully applied in many fields, such as bioengineering, machinery, navigation, communication. In recent years, adaptive control for nonlinear system has been paid much attention because of the development of adaptive control theory and designing method [1], [2], [3], [4], [5]. The authors in [2], [3], [4] studied adaptive backstepping control problem for a class of strict-feedback nonlinear system based on the ability of universal approximation of neural networks or fuzzy logic system. However, there are many practical systems that are described by the nonlinear systems with nonstrict-feedback form, and it is difficult to design the adaptive control scheme for these systems by applying the traditional backstepping method. To overcome this difficulty, some researchers improved the traditional backstepping method by using the boundedness of basis function of neural networks or fuzzy logic system, and the adaptive control schemes were designed for single-input single-output systems [6], [7], [8], [9], [10], [11], multi-input multi-output systems [12], [13] and interconnected large scale systems[14], [15] with nonstrict-feedback form.
For many real world applications, time delays are usually inevitable due to the finite transferring speed of signal, data or materials. Recently, the system with time delay has been an active research area because it may result in the instability or poor performances [16], [17], [18], [19], [20], [21]. Input delays, as a kind of time delays, usually arise from both physical nature of the plant and communication latencies [22]. In order to study the effect of input delay, a great number of control methods have been reported for linear systems [23], [24]. Actually, the most of practice systems are nonlinear dynamical systems and have inherent model uncertainties including external disturbance and model error. Therefore, the control schemes for uncertainty nonlinear system with input delay are more useful and realistic. As a result, adaptive control for nonlinear system with input delay has been paid much attention, such as [25], [26], [27], [28], [29], [30] and the references therein. In order to eliminate the effect of input delay, an auxiliary system is constructed in [25] and the Pade approximation method is adopted in [26], [27]. However, the input delay in these works was assumed to be constant, and small input delay was further required in the Pade approximation method. Therefore, the authors in [28], [29] introduced an auxiliary signal into the error systems to break through these limitations, while there are some rigid conditions for time-varying delay function. Besides, motivated by the auxiliary system in [25], Ma et al. [30] proposed a similar auxiliary system for the system with time-varying input delay to design the adaptive control scheme, but the Lyapunov function to deal with input delay is complicated, and the time-varying input delay and its derivative also need to satisfy some restrictive assumptions. Thus, how to construct a suitable Lyapunov function and design a suitable controller to relax these limitations is one of great challenges for the nonstrict-feedback nonlinear systems with time-varying input delay.
On the other hand, the state constraints are involved in many actual systems, such as the nonholonomic wheeled systems [31] and uncertain -link robot [32]. In recent years, some excellent results concerning constrained systems were derived by a growing number of researchers [3], [28], [33], [34], [35], [36], [37]. The systems with symmetric and time-invariant state constraints (i.e., the state variables satisfy with positive constant ) were considered in [3], [20] to design adaptive control schemes. As an extension, nonlinear systems with asymmetric and time-varying state constraints were presented in [34]. Meanwhile, the adaptive control problem for nonlinear systems with time-varying output constrained (as particular case of state constraints) was also investigated in [14], [28], [36]. However, all of systems in [3], [20], [28], [34], [36] are the ones with strict-feedback form. For time-varying state constrained system with nonstrict-feedback form, there is only a few results about adaptive control problem [38]. It is worth noting that input delay is overlooked in most of the above works. Although Deng et al. [28] studied adaptive control of uncertain nonlinear systems with input delay, the authors only considered the output constraints rather than full state constraints. In fact, it is difficult to design the adaptive control scheme for nonlinear system with both time-varying full state constraints and input delay, which is another challenge of this paper.
Motivated by the discussions above, in this paper, we study the adaptive control for a class of nonstrict-feedback uncertain systems with input delay and time-varying state constraints. Our main contributions lie into three aspects:
- (i)
The adaptive neural networks control scheme is designed for the nonlinear system with both input delay and time-varying state constraints in this paper, which is difficult because these two factors act on the last equation of our system at the same time. Compared with [28], [29], [30], some rigid conditions for time-varying delay function are released. Due to the using of radial basis function neural networks, some restrictive conditions for feedback gains in [28] are removed in this paper.
- (ii)
Different from [6], [12], [39], the time-varying disturbances with whole state-related upper bounds are involved in our system. A major limitation of the method in [11], [12] is the requirement of monotonic increase of upper bound functions for unknown nonlinear functions, but this limitation is deleted in this paper.
- (iii)
Compared with many existing works with time-varying state constraints, where the state constraints are regarded as known conditions during the stability analysis and it is unreasonable, we provide a more reasonable inference process, and obtain the conclusion that the state variables never violate the time-varying constraints under the designed control scheme.
The rest of this paper is organized as follows: Section 2 is devoted to formulating our problem and giving some useful assumptions and lemmas. The design procedure of adaptive controller and main theorem for adaptive scheme are presented in Section 3. The simulation results are obtained in Section 4. In Section 5, a brief discussion and summary concerning the main results are provided.
Notation: represents the set of real numbers, denotes the set of positive integers and is Euclidean space with dimension . For , let with , and denotes Eucludian norm of real vector. The notation of function is abbreviated sometimes, for example is denoted by or . For function , its -th derivative is denoted by for .
Section snippets
Problem formulation and preliminaries
Consider the following nonstrict-feedback uncertain systems with input delay and asymmetric time-varying state constraintswhere and are state variable and system output, respectively. and are the unknown smooth nonlinear functions, which satisfy and . are external disturbances. denotes the
Controller design and stability analysis
This section is devoted to designing the controller for system (1), which is mainly based on the backstepping method with asymmetric barrier Lyapunov functions.
In order to measure the performance of output , i.e., the distance between reference signal and output, we first define the error systems as followswhere are virtual control functions that will be designed later, denotes the reference
Numerical simulation
In this section, we illustrate the effectiveness of our designed control scheme by numerical simulations. A block diagram of the proposed control scheme is presented in Fig. 1. Example 1 Consider the following nonlinear systemThe state variables and are constrained by , where , and . The
Conclusion
In this paper, the issue of neural networks adaptive control has been studied for a class of nonstrict-feedback uncertain systems with input delay and time-varying state constraints. In order to solve this issue, the asymmetric time-varying barrier Lyapunov function has been introduced in the process of control design. By applying a critical result of neural networks (see Lemma 3), an adaptive control scheme has been obtained and its stability has also been proven. Our method relaxed the
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 jointly supported by the National Natural Science Foundation of China under Grant No. 61673111, and the National Key Research and Development Program of China under Grant No. 2018AAA0100202.
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