Abstract

This paper focuses on a single neural network tracking control for a class of nonlinear strict-feedback systems with input dead-zone and time-varying output constraint via prescribed performance method. To release the limit condition on previous performance function that the initial tracking error needs to be known, a new modified performance function is first constructed. Further, to reduce the computational burden of traditional neural back-stepping control approaches which require all the virtual controllers to be necessarily carried out in each step, the nonlinear items are transmitted to the last step such that only one neural network is required in this design. By regarding the input-coefficients of the dead-zone slopes as a system uncertainty and introducing a new concise system transformation technique, a composite adaptive neural state-feedback control approach is developed. The most prominent feature of this scheme is that it not only owes low-computational property but also releases the previous limitations on performance function and is capable of guaranteeing the output confined within the new form of prescribed bound. Moreover, the closed-loop stability is proved using Lyapunov function. Comparative simulation is induced to verify the effectiveness.

1. Introduction

In recent years, actuated by practical requirements and theoretical developments, numerous adaptive back-stepping control schemes have been proposed for uncertain nonlinear systems in lower-triangular form including strict-feedback and pure-feedback systems [1]. Specially, fuzzy logic systems (FLS) or neural networks (NNs) based control schemes have attracted great concern due to their inherent approximation capabilities as well as relaxing linearly in parameters assumption [2, 3]. The benefit of applying FLS or NNs is that the problem of spending much effort on system modeling can be elegantly overcome. Although there has been significant progress in aforementioned literatures, the problem of complexity (POC) is the main drawback of traditional adaptive back-stepping schemes caused by two reasons [4, 5]: the first one is the reduplicative derivations of virtual controllers, and the other one is that there exist numerous NNs/FLS. In the existing literature, several so-called dynamic surface control [6], sliding mode differentiator (or tracking differentiator) [7], and command filter techniques [8] have been incorporated into adaptive back-stepping control design to avoid the repeated derivations problem. Meanwhile, those techniques have been extensively employed to multifarious practical systems such as flexible-joint robot system [9], flight control [10], and static Var compensator [11]. By using those techniques, the repeated differentiations of virtual control laws can be avoided. However, numerous NNs employed to construct the virtual and actual control laws are still necessary for high-order systems. To thoroughly refrain from this problem, single neural (SN) approximation-based adaptive back-stepping control has been proposed for strict-feedback nonlinear systems where the control gains in each subsystem are equal to one [4]. After that, Pan and Sun have extended SN control structure to a general class of uncertain strict-feedback [5] and pure-feedback nonlinear systems [12], respectively. Meanwhile, some researchers have also utilized the single neural control technique (SNC) to discrete-time nonlinear systems [13, 14]. However, the works described in [4, 5, 12–14] only focus on the stabilization of interested systems, and the output-constraint problems are rarely considered. Meanwhile, it must be pointed out that violation of the output constraint may lower the system performance or even lead to systems instability [15, 16].

To manage the problem of output constraint, a number of attempts regarding model predictive control [17], barrier Lyapunov function (BLF) [18–24], and prescribed performance control (PPC) [25–28] techniques have been investigated from both academia and industries. Using the property of BLF, a constrained back-stepping control approach is proposed for nonlinear strict-feedback systems to ensure that the static constraint is not transgressed [20]. More specifically, some constrained adaptive neural/fuzzy back-stepping schemes are investigated for nonlinear systems subject to external disturbance and unknown functions [19, 22, 29]. Apart from the technique about BLF, Bechiloulis et al. have also proposed an alternative approach named PPC to conquer the problem of output constraint [25].

The dead-zone characteristic on behalf of the most important input nonlinearities widely exists in actuators and sensors among numerous industrial processes, which also severely deteriorate the system performance [30–34]. Thus, many researchers have devoted themselves to improving the performance of the control systems in the presence of dead-zones. Some researches apply an inverse dead-zone model to minimize the influence of the dead-zone; see reference [30]. On the other hand, by exploring the bounds of the dead-zone slopes and regarding the coefficients of the dead-zone as a system uncertainty, some robust adaptive neural/fuzzy control schemes are proposed of nonlinear systems with nonsymmetric dead-zone inputs [31, 35–37]. Na [38] and Chen [39] both proposed an adaptive neural prescribed performance control for nonlinear system with dead-zone input. Unfortunately, aforementioned PPC approaches still suffer from the POC and use numerous NNs, which result in a complex control structure. Moreover, there also exists restriction that the initial tracking errors must be known in advance. However, actually, the initial tracking error is hard to obtain in the presence of uncertainties and additional disturbances. Thus, how to concurrently tackle such restriction and aforementioned problems needs further investigations.

To simultaneously solve the aforementioned problems, a systematic design procedure is developed to derive a composite single neural network control scheme of nonlinear strict-feedback systems subject to dead-zone input, system uncertainties, and time-varying output constraint. In the controller design, to release the limit condition on previous performance function that the initial tracking error needs to be known, a modified performance function is first constructed. Meanwhile, depending on a new system transformation technique, the prescribed performance of the constrained system can be equivalent to ensure the stability of the transformed system. Regarding the coefficients of the dead-zone as a system uncertainty, an adaptive NN approach is developed via single NN control structure to stabilize the transformed system. Finally, stability analysis is conducted by using Lyapunov theory. The main contributions of this paper are summarized as follows:(1)A new modified prescribed performance function is constructed to stipulate the predefined region of the tracking error. Thus, the previous restriction on initial tracking error is removed.(2)In construct to previous output-constraint neural/fuzzy back-stepping control approach [15, 25, 38, 40, 41], the proposed control scheme does not need numerous NNs/FLS to construct virtual and practical control law in each step, only one neural network including one adaptive laws is required to approximate the lumped unknown function, thus deriving a low-computational control scheme.(3)By regarding time-varying input-coefficients of input dead-zone as a system uncertainty and using a new system transformation technique in the control design, an integrated single NN adaptive controller, which is capable of arbitrarily prescribing the system performance and dealing with dead-zone input nonlinearity, simultaneously, is first presented for a class of nonlinear strict-feedback systems.

2. Problem Formulation and Preliminaries

2.1. Problem Formulation

Consider uncertain nonlinear strict-feedback systems in the following form:where , , , and are state variables, system input, and output, respectively. are the unknown smooth functions; denote unknown smooth virtual control gain functions. is the output from the controller. According to [30], is a nonsymmetric dead-zone input nonlinearity which is defined as follows:where and denote breakpoints of the input nonlinearity.

To facilitate control design, we need the following assumptions.

Assumption 1. The desired trajectory and its derivatives are bounded and known.

Assumption 2. There exist constants such that with . Without loss of generality, it is assumed that .

Assumption 3 (see [36]). The dead-zone parameters , , , and , are unknown bounded positive constants.

The dead-zone nonlinearity (2) can be rewritten as a slowly time-varying function:where Here, we can easily conclude that , , and there exist positive constants and such that .

Control Objective. The control goal is to design a controller such that the output tracks the desired trajectory well; meanwhile, the output can be always confined within the predefined time-varying constraint. That is to say, the output is needed to satisfywhere and are the bounds of the output.

Assumption 4 (see [15]). The output bounds and are smooth functions and their derivatives from to th are all available.

2.2. Some Preliminaries

Lemma 5 (see [42]). Assume that and its derivatives are bounded. Consider system where is a constant and are selected such that is Hurwitz. Thus, there exist positive constants and such that one has where and are the estimation of .

Neural Network. Radial basis function (RBF) NN is often employed as an effective tool to approximate nonlinear functions. The following NN is applied to approximate where is the input vector; represents weight vector; denotes the node number; and has the formwhere and denote the center and width of the Gaussian function.

RBF network (8) has the ability to approximate any continuous function to arbitrary accuracy over the set where is ideal constant weights and is the approximation error.

Assumption 6. There are ideal constant weights such that for all .

Lemma 7 (see [43]). The Gaussian function described in (9) with being the input vector, where is a bounded vector and is a positive, is given as where is a bounded vector.

Remark 8. Assumption 1 imposes a controllability condition for system (1) and can be found in most existing adaptive back-stepping neural network control approaches.

3. Controller Design and Stability Analysis

3.1. System Transformation

Inspired by [15], the tracking error is defined as By subtracting from both sides of (5), we haveFor clarity, we define and , where and .

Then, (14) becomes Therefore, the output constraints given by (5) are now transformed into tracking error constraints shown as (15). To establish the relationship between and , , we first define . Then, the following novel transformed error is established as where is an error variable.

The inverse transformation of (16) with respect to is shown asTherefore, if is bounded, we haveAdding to every side of (18), condition (5) can be easily got. Therefore, the goal is presently converted to designing a controller to ensure to be bounded.

The derivative of (16) is shown as Therefore, (19) can be rewritten aswhere

Apparently, we have the transformed systems shown as

Remark 9. It should be noted that , , , , , , and are definitely known and can be used in the controller design.

Remark 10. From (20) and (18), we know that , . The expression of can be rewritten as Define a new function with ; the derivative of with respect to can be described asIt is easy for us to obtain that function will get minimum in the set as long as . By considering Assumption 4, therefore, we can conclude that , which also means that there exists a time instant such that with .

3.2. Constrained Functions Design

To study the constrained character of the tracking error , the following positive functions are chosen as where , , , , , , , , and are positive design constants and , ; denotes the hyperbolic cotangent function. Then, the control objective can be achieved if the following conditions hold:where .

Remark 11. From the definition of , apparently, satisfies and and . Thus, the initial and ultimate tracking accuracy of (26) are confined within the bounds and , respectively. It is worth mentioning that traditional prescribed performance function [25, 44, 45] requires the initial tracking error to be precisely known in advance and needs the initial value of performance function satisfying strict condition such as . However, in practical, the precise initial tracking error is uneasy to obtain in real systems with consideration of the uncertainty. Fortunately, the proposed prescribed performance can naturally release this limit condition.

Remark 12. In contrast to previous prescribed performance functions with constant parameters and , where the designed and are time-varying functions. It implies that the proposed constrained function owes more degrees of freedom to adjust the prescribed performance bounds of the tracking error. This is quite beneficial for the controller design.

3.3. Controller Design

Before the controller design, we define , , and for , where with are design control gains, is a positive integer, and is a nonnegative integer. Moreover, let , , , , , and , where , , , , , and with . Here, , , and denote the th derivative of , , and , respectively. Meanwhile, we define the virtual tracking errors with , where are virtual control inputs which will be defined later. Let with . denotes the neural input vector which will be used in the following derivation.

Step 1. Consider the transformed system (22), ; the first virtual control input is chosen as where .
Applying and (27), we haveApplying (27) to , one obtains where .

Step 2. The derivative of is shown as follows:From (27), (28), and (13), one get ; thus we have Defining and using (31), (30) can be rewritten as To remove the term in (28), a related term is added and subtracted in equality (32), then we have Consider the virtual control inputwhere .
Applying and (34) to (33) yields Utilizing (34) and (29), can be changed towhere .

Step i (). The derivative of is as follows:where .