Abstract

Based on fuzzy logic systems and prescribed performance technology, this paper investigated the stable tracking problem of a class of nonlinear systems with unknown dead-zone inputs. Firstly, the original system is transformed into an equivalent system. Then, a Nussbaum-type function is added in the predefined performance controller to estimate the unknown control direction, and the boundedness of all signals in the closed-loop system is guaranteed. In addition, tracking errors will always remain within the prescribed performance boundaries. Finally, a comparative simulation example shows the effectiveness of the method proposed in this paper.

1. Introduction

As is known to all, the dead zone may badly limit the system performance, such as electronic circuits and electric servomotors [1–5]. So, many researchers used different control methods (i.e., backstepping technique, fuzzy systems, neural networks, sliding mode control, and so on) to research the stability of nonlinear systems with unknown control directions [6–9]. For example, based on the terminal sliding mode control, Chen and Ren in [6] proposed an adaptive neural control scheme for permanent magnet synchronous motor (PMSM) servo system with dead-zone input. For uncertain discrete-time systems with nonaffine dead-zone input, Liu and Tong [7] used adaptive neural networks tracking control such that the tracking error converges to a small neighborhood of zero. By using high-dimensional integral Lyapunov–Krasovskii functional theory and direct adaptive control scheme, control of uncertain MIMO systems with time-varying delay and dead-zone input was considered in [8]. In [9], the authors investigated an adaptive backstepping control method for an SISO system with unknown dead-zone input. For eliminating the influence of unknown control directions, one of the commonly applied methods is the Nussbaum-type function technique [10–12]. Liu et al. [10] constructed an adaptive neural controller for discrete-time system, which made all signals of the closed-loop system bounded in the condition of unknown dead-zone input. For a class of MIMO stochastic systems in [11], Li and Tong put forward an adaptive fuzzy output-feedback control method by using Nussbaum gain function and fuzzy logic systems. Xu [12] applied the adaptive neural control scheme to guarantee that the flexible hypersonic flight vehicle system is bounded stable with dead-zone input.

However, control methods in aforementioned literature can only guarantee that the tracking error converges to a small residual set, rather than a small residual set with the prescribed performance bounds. For an uncertain nonlinear system [4, 13–15], in order to solve the above problem, the prescribed performance control (PPC) strategy has been proposed. The idea of this method is converting the original system to an equivalent system and ensuring the boundedness of the states of the equivalent system. For example, based on the backstepping technique and the PPC, Wang and Hu [13] proposed an improved PPC for the longitudinal model of an air-breathing hypersonic vehicle (AHV) with unknown dead-zone input nonlinearity. Yang and Chen [14] developed an adaptive neural PPC for near space vehicles (NSVs) with input nonlinearities including saturation and dead zone. It can be seen from the literature [13, 14] that it has certain practical significance to adopt the PPC method in the practical system. For MIMO nonlinear systems with unknown dead-zone inputs, Shi et al. [4] proposed an adaptive fuzzy PPC scheme such that only three parametric adaptive rules were needed. Cao et al. [16] studied a fractional neural network model with impulse amplitude, time-delay time-varying, and reaction diffusion terms. Many other interesting results about PPC of nonlinear systems can be seen in [17–19].

Inspired by the work of [4, 20], this paper explores the stability problem that the tracking error meets prescribed performance boundary (PPB) with unknown dead-zone input. Compared with related works, the main contributions of this paper are listed as follows: (1) compared with the work of [14, 17, 19], the introduced error transformation variable can be directly differentiated without the participation of the transformation function. (2) The Nussbaum-type function is incorporated into the control method so that the influence of unknown control directions is eliminated. (3) The singular problem in the PPC for nonlinear systems can be avoided by using the proposed method.

The structure of this paper is as follows. In Section 2, the uncertain system model with unknown dead-zone inputs is raised and provides assumption and lemmas. Section 3 includes the designation of fuzzy adaptive prescribed performance control method. The results of simulation are presented in Section 4. In Section 5, a brief conclusion is given.

2. Preliminaries

Consider the following uncertain nonlinear system, which described aswhere is the system state vector, is an unknown smooth nonlinear function, is an external disturbance, denotes the control input subject to dead-zone type nonlinear, and is an unknown control coefficient, . Here, is described as follows:where , and are unknown positive constants, . Then, we rewrite (2) aswhere

From (3), system (1) is described as

Designing two adaptive prescribed performance control methods is the control objective of this paper. The goal of this paper is that the tracking error satisfies PPB, and all closed-loop system signals are bounded.

In order to meet the objective, we proposed the following assumptions.

Assumption 1. The system state , the reference signal , and its derivative are available and bounded.

Assumption 2. The nonlinear function , the control coefficient , and the external disturbance are unknown but bounded, .

Remark 1. Assumptions 1 and 2 are commonly used to address the problem of unknown control direction [21–23]. And owing to the boundedness of , and , we know that there exists an unknown positive constant , such that .
Let and , then . The error state is defined as , and the following error system is acquired:The following definition and lemmas used in this paper are to overcome the problem of unknown control directions.

Definition 1. A continuous function is called a Nussbaum function, if satisfies

Remark 2. Generally, the selection of the Nussbaum functions for the controller design with unknown control direction is or or and so on. In this paper, we choose the even Nussbaum function to design the PPC controller.

Lemma 1. (see [15]). Let and be smooth functions, for , and be an even smooth Nussbaum function. If the following inequality holds:where and are constants with , is a time-varying parameter which takes values in the unknown closed intervals with , then , , and must be bounded on .

Lemma 2. (see [4]). Let be a symmetric matrix and be a nonzero vector. Let . Then, there exist eigenvalues of such that and .

Remark 3. In this paper, we will construct the time-varying parameter by using Lemma 2. For example, let and is a nonzero vector; we can design as , and then there exist two constants and such thatIn this paper, fuzzy logic systems will be used to approximate unknown nonlinear functions. Generally, the output of fuzzy logic system can be expressed as [24–32]where is the input vector. is a Lipschitz continuous mapping, is a compact subset, and is a real line. , where consists of fuzzy sets . The membership function of the -th rule is , where . (), which represents the centroid of the -th consequent set. Let and , where () is the -th fuzzy basis function and is a continuous mapping, then can be defined asThus, (10) can be rearranged as

3. Synchronization Controller Design

3.1. Prescribed Performance

The steady-state performance of can be preserved by setting the performance constraint condition aswhere are positive constants, which can regulate the error boundaries. is denoted bywhere and . Obviously, is a decreasing function and , .

We abbreviate to and give an error transformation aswhere satisfies

Obviously, if we choose the initial condition as and is bounded, then the error state will satisfy the inequality that . Because when approaches or , will approach infinity, which contradicts the boundedness of . So, here, the work for us is to prove that the error transformation variable is bounded, .

Remark 4. The proposed transformation variable in this paper can be directly expressed by and parameters and , see (15), while the traditional transformation variable in [4, 14] needs to rely on the transformation function to transform. Therefore, the proposed error transformation variable is simpler and is more convenient for the design of the controller .

3.2. Control Design and Stability Analysis

The time derivative of iswhere . So, we obtain the following error transformation dynamic system.

Due to the fact that is unknown, we need to employ fuzzy logic system to approximate . So, the approximate of function can be expressed as , where is the parameter vector and the fuzzy function vector is . For , there is an optimal parameter vector such thatwhere and is the approximation error, which is bounded as , where is a positive constant, .

Let , then . The error transformation dynamic system is rewritten as follows:

Next, we design the controller to guarantee the boundedness of . Let , and another controller as follows:where, , , , , and are the estimates of and , respectively, and the parameters adaptive laws are given aswhere are positive constants.

Now, consider the Lyapunov function:where . By differentiating , one achieves

According to Cauchy–Schwarz inequality,

Considering its adaptive law (21), it is concluded that

Similarly,

So, one getswhere and . For , we define , where . By using Lemma 2, there are two constants and such that . So,