Adaptive neural control for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis

Abstract

This paper aims at addressing the adaptive neural control problem for switched nonlinear systems with output hysteresis and unmodeled dynamics. The switching law in this study is arbitrary. In our model, the unmodeled dynamics are assumed to be of input-to-state practical stability (ISpS). With the help of this assumption, a dynamic normalizing signal is constructed to dominate the unmodeled dynamics. And then, a direct adaptive neural state-feedback control scheme is developed with the help of approximation-based backstepping. The stability analysis shows that the system output is convergent to an adjustable small region of zero asymptotically, and furthermore, all the closed-loop signals are bounded. Finally, we further present two simulation examples to verify the effectiveness of our control scheme.

Introduction

Adaptive control has attracted numerous attentions over the past years for its ability to provide online estimate of unknown parameters. Backstepping is one of the most promising adaptive control design approaches [1]. In contrast with other approaches, backstepping has the advantage to avoid cancellations of useful nonlinearities flexibly. With the help of adaptive backstepping, some restrictions (e.g., matching conditions and extended matching conditions, see [2], [3], [4], [5]) arising in the early stage of adaptive control are removed. By further combing with the universal approximators, such as neural networks (NNs) and fuzzy logic systems (FLSs), the approximation-based backstepping is able to handle the adaptive control design problems when there is no prior knowledge about the structural information of nonlinear functions. More recently, the progress of adaptive backstepping has witnessed many remarkable results reported [6], [7], [8], [9], [10], [11], [12], [13], [14]. Among these results, the adaptive compensation for nonsmooth nonlinearities (e.g. saturation [6], dead-zone [9] and hysteresis [10]) is one of the most popular topics.

Hysteresis is common in many physical systems and devices, e.g., electro-magnetic fields, piezoelectric actuators and other areas. The hysteresis nonlinearity may degrade the control performance. In contrast with dead-zone and saturation, hysteresis is more complicated since its output not only depends on the input, but also the change of input. For this reason, the compensation for hysteresis is more difficult as well. With the joint efforts of numerous researchers, many results on the compensation for hysteresis have been reported over the past year. For example, in [15], [16], [17], [18], several mathematical models were developed to describe the hysteresis nonlinearity. Among these models, Bouc-Wen hysteresis model [17], [18] may be the most widely accepted one. With Bouc-Wen hysteresis model, further investigations on adaptive compensation for hysteresis nonlinearity have been carried out [10], [19], [20], [21]. To list a few, in [20], an effective hysteresis compensation method was developed with the idea of decomposing the Bouc-Wen hysteresis into a combination of linear term and disturbance-like term. Moreover, any other adaptive inverse-based compensation methods were also reported in [10]. Note that [10], [19], [20], [21] only investigated the compensation for input hysteresis not the output hysteresis, while the hysteresis phenomenon may also exist in the output channel of control plants. The output hysteresis will lead to the control problem of time-varying unknown control gain (also referred as unknown high-frequency gain in the field of adaptive control). For this reason, the compensation for output hysteresis is more complicated that those for input hysteresis. Moreover, the control scheme designed by [10], [19], [20], [21] may suffer from poor control performance, since they need to assume the unknown control gain are time-invariant or slow time-varying. So far, there are few results reported on the compensation for output hysteresis. In addition, the aforementioned results are limited in non-switched nonlinear systems and they cannot be applied to the switched nonlinear systems directly.

Switched nonlinear system is a special type of hybrid system [22]. A single nonlinear switched system may contain multiple subsystems. During the run time of a switched nonlinear system, there exists a switching law orchestrating which subsystem is active. It is well known that the control or stabilizing problem of switched nonlinear systems is more complex than those of non-switched nonlinear systems, since it needs to face the dual challenges of control design and switching design. In order to deal with these challenges, some useful tools, e.g. common Lyapunov function (CLF) [23], multiple Lyapunov function (MLF) [24] and average dwell time (ADT) [25] have been developed so far. With the help of these tools, many excellent results on control design or stability analysis of switched nonlinear systems were reported [26], [27], [28], [29], [30], [31]. In [26], some ADT-based switching conditions to guarantee input-to-state stability (ISS) properties of switched nonlinear systems are presented. As a major approach to establishing the ISS property of switched interconnected nonlinear systems, some MLF-based small-gain conditions were further given in [27], [28]. By using backstepping technique, [29], [30], [31] successfully addressed the adaptive control problem for switched nonlinear systems with actuator nonlinearity (e.g., dead-zone, saturation and hysteresis). However, these works are restricted since they rely on the assumption that the control plant is free from unmodeled dynamics.

Unmodeled dynamics are widespread in many practical devices owing to the existence of measurement noise and modeling errors, etc. The closed-loop system may perform poorly and even become instable if the control scheme without consideration of unmodeled dynamics [32]. So far, there are mainly two approaches to dealing with unmodeled dynamics. The first approach is to dominate the unmodeled dynamics with some designed dynamic signals [33]. And the other approach is to exert small-gain constraints on the nonlinear input-to-output gains (definition see [34]) of the control plant and the unmodeled dynamics, such that, the composition of such nonlinear gains is less than unity [35]. Following these principles, many results have been reported on switched systems [36], [37], [38], [39], [40], [41] or non-switched systems [42], [43], [44], [45], [46], [47] with unmodeled dynamics. For example, by combining MLF and ADT, the authors in [38] addressed adaptive control problems with the first approach mentioned above. However, the conrol scheme in [38] may be infeasible when there is no priori information about the switching law. More recently, any other novel adaptive control schemes were further proposed in [39], [40] by using small-gain and CLF approaches. The advantage of [39], [40] is that their approaches allow the switching law is arbitrary. However, the implementation of these small-gain-based approaches may be not an easy task when the switched nonlinear systems simultaneously suffer from unmodeled dynamics and output hysteresis. The main reason is that the nonlinear input-to-output gains are difficult to determine when the system output is coupled with output hysteresis. Therefore, the adaptive control for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis remains an open problem.

Motivated by the discussion above, this study investigates the adaptive neural control problem for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis. The contributions of this study are listed as follows:

• The proposed control scheme can address the adaptive control problem for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis. A dynamics signal is employed in the control scheme to dominate the unmodeled dynamics. In contrast with the existing results in [39], [40], the main difference is that our control scheme can avoid the determination of nonlinear input-to-output gains, which are difficult to determine if the control plant suffers from output hysteresis.

• Different from the previous results on switched nonlinear system with dead-zone or saturation [29], [30], this study consider the more complicated hysteresis nonlinearity. In contrast with [31], where the adaptive control for switched nonlinear systems with input hysteresis has been investigated, this study will consider the output hysteresis. The compensation for output hysteresis is more difficult than the input hysteresis in [31] since the unknown control gain aroused by the output hysteresis is time-varying. To overcome this control difficulty, a Nussbaum function-type gain is employed in the control design.

The rest of this paper is organized as follows. The control problem and preliminaries are introduced in Section 2. And then, an adaptive neural state-feedback control scheme is proposed in Section 3 with the help of approximation-based backstepping. Besides, the corresponding stability analysis is also given in this section to prove that the control objective is achieved. Finally, two simulation examples are given in Section 4 to verify the effectiveness of our scheme.

Terminologies: A function γ: ${\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is said to be of class $\mathcal{K}$ if it is continuous, strictly increasing and is zero at zero. We say that a $\mathcal{K}$-function γ is of class ${\mathcal{K}}_{\infty },$ if it is unbounded. A $\mathcal{KL}$-function β: ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a function with the property that for each fixed t, the function β(·, t) is of class $\mathcal{K}$ and, for each fixed s, the function β(s, ·) is decreasing and tends to zero at infinity.

Notations: Throughout this paper, ∘ denotes the composition operator between two functions; |·| and ‖·‖ denote the absolute value of real numbers and the Euclidean norm of vectors, respectively; $\mathbb{R}$ and ${\mathbb{R}}^{+}$ denote the field of real numbers and the field of nonnegative-real numbers, respectively.