#### 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 .

Define , and (37) can be rewritten as

As done previously, we add and subtract in equality (38) to remove this interconnected term , then (38) can be changed to

The virtual control is designed as where .

Using and (40), (39) can be written as Consider (41) and (36), we can obtain that where .

*Step n*. The derivative of is shown as where can be written asDefine , then, we can get

As done previously, we add and subtract in equality (45) to remove this interconnected term and consider the dead-zone input (3), then (45) can be changed toThe desired control law is chosen as where .

Applying (42) and (40), can be rewritten as It must be noted that is unreliable due to the unknown functions , , and with . The expression of in (48) is substituted into (47), then we havewhere and .

Hence, we use NN to approximate the lumped unknown uncertainty .where satisfies ; is a certain positive constant.

Therefore, the actual control law and adaptive law are shown as where and are positive design constants.

Consider (47), (49), and (51); (46) can be converted towhere .

##### 3.4. Stability Analysis

Lemma 13. *For updated law (52), there exists a compact set where , such that , in case that .*

*Proof. *Consider the following Lyapunov candidate . The time derivative of function along with (52) is derived asIt follows that as long as . Therefore, if .

Theorem 14. *Consider system (1) satisfying Assumptions 1 to 4, controller (51), and the updated law (52). Then, we know that the output time-varying output constraint can be achieved and the signals , in the closed-loop are bounded.*

*Proof. *Consider the following Lyapunov candidate:By using (28), (35), (41), and (53), the derivative of (56) is shownSince is bounded as presented in Lemma 13, it follows that , where denote the upper bound of the approximation error. We note that . Consider the following facts, we have we haveBy choosing and such that , , and and defining , , . The derivative of can be reformed as If , , , and , will become negative. Thus, we can conclude that the transformed error , tracking errors , and are uniformly ultimately boundedness. Then, the boundedness of can ensure the time-varying output constraints.

##### 3.5. Further Design

It must be noted that the control design described in (51) needs the high-order derivatives of the reference signal . In practice, high-order derivatives are hard to obtain in the implantation of the controller. Motivated by [46, 47], we can employ a high-gain observer (HGO) or high-order sliding mode observer (HOSM) to estimate , . As indicated in [46], the prominent features of those observers lie in the fact that they can achieve finite-time observer error convergence and thus can be utilized in almost any feedback with separation principle being trivially fulfilled. If we use HGO to estimate the high-order derivatives of , the actual control law (51) and updated law (52) can be modified as where and are the estimation of and , respectively.

By using HGO to estimate the high-order derivative of , the actual controller and updated law have been changed to (61). The stability of the closed-loop system can be demonstrated the same as Section 3.4. From Lemma 7, it has with . Employing the same procedure of Lemma 13, we can easily prove that there also exists a compact set where , such that , in case that .

Using the property of Gaussian radial basis function (12), (53) can be rewritten aswhere .

Substituting (63) into the derivative of Lyapunov function, we havewhere .

Since is bounded, it follows that where denote the upper bound of the approximation error. We note that . Using Lemma 5, we can also get that there exist a constant such that . Until now, it can be easily concluded that there is a constant with such that .

Consider the following fact: we have The same analysis with Theorem 14, the signals , and in the closed-loop are bounded.

*Remark 15. *Compared with the previously proposed output-constraint [40] or prescribed performance control [41, 44, 45] adaptive back-stepping approach, the proposed structure is extremely simple and the computational burden is really low since there is only actual controller required to be implemented and there exists single neural network to approximate the lump uncertainty. It also be noted that the issue of explosion of the complexity inherent in the traditional back-stepping approach is completely removed without employing DSC, command filter, or differentiator technique.

*Remark 16. *In contrast to the results [4, 5, 12] using cascade low-pass filter to approximate the high-order derivatives of the reference without considering the influence of estimated error in the closed-loop stability analysis, the HGO is applied to approach the derivatives of the reference and the estimated error effect is also taken into account in the control design and stability analysis.

#### 4. Simulation Studies

In this section, the effectiveness of the proposed method is illustrated by one-link manipulator with a brush DC (BDC) motor. One-link manipulator actuated by a BDC motor can be expressed as [48]where , , and represent the link angular position, velocity, and acceleration, respectively. denotes the motor current. represents the input control voltage. The parameter values with appropriate units are given by , , , , , and .

Setting , , and , (67) can be written as where , , , and .

The dead-zone model is assumed as

The control goal is to derive the output to follow the desired trajectory with dead-zone input nonlinearity shown in (69) while ensuring predefined time-varying output constraints (5) with and . Here, we define , , and .

System (68) satisfies Assumptions 1, 2, and 3. Neural network contains 128 nodes (i.e., ), , with centers evenly spaced in their corresponding scopes and widths . The design parameters of the controller are chosen as , , , , and . The parameters of HGO are selected as , , , and . In the simulation, the initial conditions are set as , , .

To verify the proposed controller, a comparative simulation study [49] has been conducted in this section. For fair comparison, the control gains and initial conditions of control scheme in [49] are the same with this approach. The comparative simulation results are depicted in Figures 1, 2, 3, and 4, where , , , and denote the simulation results of comparative study [49]. In Figures 1 and 2, we plot the controller tracking performance and tracking errors trajectory along with their bounds. Figures 1 and 2 imply that the proposed control method can ensure the output and its tracking error both confined within their bounds. However, the tracking performance and tracking error of comparative study are given an unsatisfactory effect. The control input and norm of NN weights are depicted in Figures 3 and 4, respectively. It is easy for us to find that the control input and norm of NN weights are bounded. Apparently, as opposed to the output tracking performance achieved by control scheme [49], the proposed control approach has a satisfactory result which can ensure the output and tracking error without overstepping their bounds.

#### 5. Conclusion

In this paper, a new concise adaptive neural tracking control scheme has been developed for a class of strict-feedback system subject to input dead-zone and time-varying output constraint. Firstly, to release the limit condition on previous performance function that the initial tracking error needs to be known, a new modified performance function is constructed. By utilizing a system transformation technique, after we transform the original constrained nonlinear system into an unconstrained one, a composite adaptive neural state-feedback control approach is investigated. All unknown functions from to steps are integrated to the step such that only one neural network is required to approximate the lump uncertainty, thus deriving a low-computational scheme. Meanwhile, the output constraint and input dead-zone are both considered in this scheme. Finally, the closed-loop stability is proved using Lyapunov technique. Comparative simulation is used to demonstrate the effectiveness of this scheme.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work is partially supported by the Natural Science Foundation of China (Grants nos. 61374032 and 61573286).