Approximation-Based Robust Adaptive Backstepping Prescribed Performance Control for a Huger Class of Nonlinear Systems

Xu, Jihui; Wang, Xiaolin; Zhang, Lei

doi:https://doi.org/10.1155/2019/6234965

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 6234965 | https://doi.org/10.1155/2019/6234965

Approximation-Based Robust Adaptive Backstepping Prescribed Performance Control for a Huger Class of Nonlinear Systems

Jihui Xu,¹Xiaolin Wang,¹and Lei Zhang¹

Academic Editor: Sergey Dashkovskiy

Received01 Mar 2019

Revised24 May 2019

Accepted25 Jun 2019

Published25 Aug 2019

Abstract

This paper proposes an innovative adaptive neural prescribed performance control (PPC) scheme for large classes of nonlinear, nonstrict-feedback systems under input saturation constraint. A restrictive hypothesis under which the upper and lower bounds of control gain functions exist a priori is first relieved by constructing appropriate compact sets within which all state trajectories are held. A novel asymmetry error transformed variable is then introduced to cope with the nondifferentiable obstacle and complex deductions corresponding to traditional PPC schemes. To efficiently manage the input saturation constraint, a new auxiliary dynamic system with a bounded compensation tangent function term is established as the strictly bounded assumption of the dynamic system is canceled. It is rigorously proven that all signals in the closed-loop systems are semiglobally uniformly ultimately bounded under both Lyapunov and invariant set theories. The tracking errors converge to a small tunable residual set with prescribed performance under the effect of the input saturation constraint. The effectiveness of the proposed control scheme is thoroughly verified by two simulation examples.

1. Introduction

The approximation-based adaptive control of uncertain nonlinear systems is a significant theoretical challenge that has garnered a great deal of research interest in recent years [1–3]. Many researchers have utilized fuzzy logic systems or neural networks (NNs) for this purpose [4–10]. When combined with the backstepping methodology, approximation-based adaptive approaches can achieve global stability for many classes of nonlinear systems [11–15]. However, as the traditional backstepping controller repeatedly differentiates the virtual controllers at each step, its complexity drastically increases as the order of systems increases. The dynamic surface control (DSC) technique is designed to mitigate this problem by introducing a first-order low-pass filter at each step. The DSC technique has allowed scholars to construct approximation-based adaptive control schemes for many nonlinear strict-feedback and nonstrict-feedback systems [16–21]; however, these control schemes are all based on the assumption that the control gain functions must be bounded. To relax this restrictive hypothesis, two adaptive NNs control schemes were developed for strict-feedback and nonstrict-feedback systems by assuming that control gain functions are continuous and are bounded on a compact set [22, 23]. However, these schemes do not consider the simultaneous occurrence of prescribed performance and input saturation constraints due to inherent difficulties in the design.

The prescribed performance constraint is unavoidable in many industrial control systems, such as precise microinstruments and robotics, and may result in degradation, hazards, or system failure [24, 25]. Thus, it represents an issue worth of careful attention during the control system design process [26]. The prescribed performance control (PPC) concept has been extensively employed in controller design for systems of various forms [27–35]. For example, in [27], an adaptive control scheme for SISO strict-feedback nonlinear systems capable of guaranteeing prescribed performance bounds is considered. In [30], an improved prescribed performance control scheme is proposed for a strict-feedback nonlinear dynamic system based on the backstepping technique. Recently, an adaptive NN-based decentralized control scheme under a prescribed performance constraint was recently presented for uncertain switched nonstrict-feedback interconnected nonlinear systems [35]. These control schemes share the common assumption that bounded control gain functions are required, which is very restrictive [27–35].

The input saturation constraint also appears in many actual systems, such as industrial robots and numerical control machines [36–38]. If it is ignored during control system design, it can cause inaccuracy, instability, or even deteriorate the performance of the entire closed-loop system. The input saturation problem has received much scholarly interest in recent years [39–43]. For example, an adaptive neural control method is proposed for a class of strict-feedback stochastic nonlinear system in the presence of input saturation constraint in [40]. Moreover, in [42], an adaptive fuzzy prescribed performance control scheme is presented for a class of nonstrict-feedback systems subject to input saturation constraint. Recently, an efficient fuzzy controller for a larger class of stochastic nonlinear systems is constructed to manage the input saturation constraint in [43]. However, previously published techniques [39–43] depend upon the assumption that the control gain functions are always bounded, which is overly restrictive; the upper and lower constants bounds of the control gain functions may be difficult to acquire in some practical systems or may even be nonexistent [41–43].

To the best of our knowledge, there are extremely few extant schemes applicable to the control of large classes of nonstrict-feedback systems under both prescribed performance and input saturation constraints where the control gain functions are possibly unbounded. This is yet an open problem with theoretical and practical significance. In this study, we developed a novel adaptive neural PPC scheme for a large class of nonlinear nonstrict-feedback systems with both prescribed performance and input saturation constraints. The main contributions of our work can be summarized as follows.

(1) Unlike other strategies [27–43], this is the first instance in which an adaptive control problem of a large class of nonstrict-feedback systems with continuous, possibly unbounded control gain functions fall under both prescribed performance and input saturation constraints.

(2) As discussed in this paper, we constructed a novel error constraint transformation to overcome the nondifferentiable obstacle and complex deductions existing in traditional PPC schemes. We also obtained a new asymmetry error constraint variable.

(3) By contrast to traditional backstepping techniques, no additional first-order filter or repeated differentiations of intermediate control signals are needed to operate the proposed technique. NNs are utilized to approximate the unknown continuous functions, which contain all states of the whole system.

The remainder of this paper is organized as follows. Section 2 provides the problem formulation and preliminaries. The adaptive neural prescribed performance controller is discussed in Section 3. Section 4 is devoted to stability analysis. In Section 5, two simulation examples are presented to demonstrate the effectiveness of the proposed control scheme. Section 6 concludes the paper.

2. Problem Statement and Preliminaries

2.1. Problem Formulation

Consider the following large class of uncertain nonstrict-feedback nonlinear systems [41]:where is the state of th subsystem and represents the state vector of the whole system , where and is the order of the th subsystem. , , and denote the system input and output of the th subsystem, respectively. are unknown continuous functions with ; are unknown continuous control gain functions and are uncertainties consisting of dynamical coupling terms and external disturbances. The system states are all assumed to be available in the control design process. Input saturation can be written as follows:where means the bound of , is the input to the saturator, and .

Control Objective: the purpose of this study is to establish a novel adaptive neural prescribed performance controller for System (1) to guarantee two performance indicators. All closed-loop signals in the systems are semiglobally uniformly ultimately bounded (SGUUB) and the prescribed output tracking error bound for the tracking error, , is always satisfied. In this paper, the given reference signal satisfies Assumption 1.

Assumption 1. The reference signal is a sufficiently smooth function and there exists a constant such that .

Definition 2 (see [18]). The solution of System (1) is SGUUB if, for any , a compact subset of , and all there exist an and a number such that for all .
Unlike other methods [27–43], we aim here to achieve this objective in the presence of the following assumptions and lemmas.

Assumption 3. The continuous control gain functions satisfy for .

Remark 4. Unlike other scholars [27–43], we assume only the signs of the control gain functions to be known. This effectively relaxes the a priori boundedness assumption. Despite some other techniques available to relax this assumption [22, 23], the case has limited application in practice if the prescribed performance constraint and input saturation constraint are not taken into account. This complicates the control design of nonlinear systems due to the couplings between the output and input constraints.

Remark 5. To the best of our knowledge, the present study marks the first time that the adaptive control problem of a general class of nonstrict-feedback nonlinear systems with both prescribed performance and input saturation constraints has been investigated.

Assumption 6. For , there exist unknown constants satisfying .

Lemma 7 (see [12]). Consider the following dynamic system:where and are positive constants and is a positive function. For , we have , .

Lemma 8 (see [15]). The hyperbolic tangent function is uninterrupted and differentiable. For , we haveHere, RBF NNs [4, 19] are employed to approximate any continuous functions as follows:where is the input vector and is the NN input dimension. is the weight vector, is the NN node number, and is the approximation error satisfying with as an unknown constant. is the basic function vector and is a commonly used Gaussian function, i.e., , with center vector of and width of .

3. Adaptive Neural PPC Controller Methodology

In this section, a novel adaptive neural prescribed performance controller is constructed for a larger class of nonstrict-feedback nonlinear system (1) under the framework of backstepping technology. Before designing the virtual control functions, we need to first introduce the prescribed performance concept. Traditional PPC schemes [25–27, 30, 35] have certain shortcomings such as complex deductions and nondifferentiable obstacles. We attempted to resolve those shortcomings in this study via a novel asymmetry prescribed performance form:where and with and being the initial values of boundary functions and , respectively. The design parameters and are convergence rates of exponential functions. . , , and are properly chosen constants satisfying and . To this effect, and limit the maximum overshoot of tracking error at the transient.

The term in (6) cannot be utilized directly to construct controllers, so we introduce the novel transformed error :with the normalized errorwhere . The time derivative of can be given bywhere and .

Lemma 9. For , if and there exists a constant satisfying , the tracking error satisfies .

Proof. See the Appendix.

Then, the design procedure is given in a step-by-step way as follows.

Step 1. Considering the standard backstepping control design method, we definewhere is a virtual control law to be designed later.
Differentiating and considering (1) and (9) yieldwhere .
Since is an unknown continuous function, we use a RBF NNs to approximate the function:where with being Gaussian functions for , and is the approximation error, satisfying with being an unknown constant.
Consider the following quadratic Lyapunov function candidate:According to (11) and (12), the time derivative of (13) can be given by

Define a compact set , with being a design constant. For the compact set , the following lemma holds.

Lemma 10. The function has a maximum and a minimum in ; namely, there exist constants and such that and .

Proof. the Appendix.

Design the virtual controller law aswhere and are design parameters.

The corresponding parameters adaptation laws and are given bywhere , , and are design parameters; and are estimates of the unknown constants and , respectively, with being the dimension of . According to Lemma 7, we have , for after selecting and .

In view of Young’s inequality, we havewhere is any positive constant. From [22], we have since and . Thus, we further have

Considering (14) and (19), we obtain the time derivative of aswhere .

Let the Lyapunov function candidate bewhere and are the estimation errors of and , respectively.

From (15), (20) and , the time derivative of (21) is

Substituting (16) and (17) into (22) yields

Step (). In the th step, we definewhere and are the virtual control laws of the th step and the th step, respectively.

Considering (1) and differentiating , one haswhere .

Similarly, a RBF NNs is employed to approximate the functions :with , and being the approximation error, satisfying .

Consider the following quadratic Lyapunov function candidate:

It follows from (25) and (26) that the time derivative of is

Design the virtual control laws as

The design process of parameters is similar to Step 1, and the parameters adaptation laws and are given by

The design process of parameters is also similar to Step 1. The functions can be rewritten aswhere , , , and is a continuous function.

Define the compact sets aswhere . Similarly, we can know that the functions have a maximum and a minimum in the compact set ; namely, there exist constants and satisfying

Let the Lyapunov function candidate bewhere and .

According to Young’s inequality, one haswhere is any positive constant. is the dimension of with , for .

From (36), we can rewrite (28) aswhere .

According to (29), (30), (31), and (37), we obtain the time derivative of as

Step (). Define , with being a design parameter, and is defined aswhere is a design parameter.

Considering (1) and differentiating with respect to time yieldswhere . For , there exists a continuous function such that

Define the following compact set

The functions have maximums on . Thus, we have with being the unknown constants.

Choose the actual control law as

The design process of parameters is similar to step and Step 1, and the parameters adaptation laws and are given bywhere , and the design process of other parameters is also similar to step and Step 1.

Similar to the former steps, the function can be expressed aswith being a continuous function.

In line with Lemma 10, the function has maximum and minimum such that

Similarly to the previous steps, define , and a RBF NNs is utilized to approximate the functions . According to (40), (41), and (47), the time derivative of iswhere and .

Consider the Lyapunov function candidate aswhere and .

Substituting (43), (44), (45), and (48) into (49), the time derivative of is

4. Stability Analysis

Choose the Lyapunov function as follows:where is the Lyapunov function for the th subsystem

The main stability result of the proposed scheme is summarized in the following Theorem 11.

Theorem 11. Consider Assumptions 1–6 and the intermediate virtual control laws (15), (29), the actual control law (43), and the adaptive laws (16), (17), (30), (31), (44), and (45). For , , , and , there exist design parameters , , , , , and such that
(i) all the closed-loop system signals are SGUUB;
(ii) the whole system output tracking error remains in a neighborhood of the origin within the preselected transient and steady bounds; and
(iii) the closed-loop system variable is bounded.

Proof. In view of (23), (38), and (50), the time derivative of isBy completion of squares, it holds thatwhere and are unknown positive constants.
Then, (53) can be rewritten aswhere .
From , we can further have on compact set . Considering , choose the control parameters ,,, , with being a positive constant.
Invoking (55), we can obtainwhere .
Note that can be made arbitrarily small by decreasing , , and , increasing , , and in the meanwhile. Thus, we can have by selecting proper design parameters. It follows from and (56) that on the level set . Subsequently, the compact set is an invariant set and the signals , and of the closed-loop system are SGUUB. The property (i) of Theorem 11 is proved.
Multiplying (56) by and integrating over yieldswhere .
According to (21) and (57), we haveFrom and Lemma 9, we have , for . Now let us consider the Lyapunov function candidate for the whole systems as . From (57), it can be derived thatwhere and . Then, we further havewhere is a positive constant.
Similarly, from (21) and (60), we can obtainFrom and Lemma 9, we have , for . Consequently, by appropriately choosing the design parameters, the tracking control error can be shown to converge to a small neighborhood of the origin and its prescribed performance is satisfied. Thus, property (ii) of Theorem 11 is proved.
Furthermore, there exists a scalar to satisfy with and . Define , we can haveIf , we can obtain . Thus will lie in the compact set , for . This completes the proof.

5. Simulation Analysis

In this section, to illustrate the effectiveness of the proposed control scheme, two simulation examples are considered.

Example 12. Consider the following uncertain nonlinear systemswhere , , , and , , , . , , , and . The input saturations and are described byChoose the reference signal as the van der pol oscillator system described as follows:where and . Based on Theorem 11, the virtual stabilizing control laws and actual control laws are chosen aswith adaptation lawsand performance functionsChoose the initial conditions as , , and . The simulation results of Example 12 are shown in Figures 1–5.
Figure 1 shows that the system outputs and follow the desired trajectories and as closely as possible, and the control inputs and are shown in Figure 2. The prescribed performance bounds , , , and the tracking errors , are illustrated in Figures 3-4, from which we can see the output tracking errors achieve preselected transient and steady bounds. Finally, Figure 5 is given to explain the boundedness of adaptive parameters , , , , , , , and .

Example 13. Consider the following uncertain nonlinear systemswhere , , , and , , , . , , and . and are described byThen, the virtual stabilizing control laws and actual control laws are constructed aswith adaptive lawsand performance functionswhere the initial conditions , , , and . The simulation results of Example 13 are shown in Figures 6–9.
Figure 6 shows that the system outputs track the desired trajectories, perfectly, and Figure 7 reveals that the system inputs are bounded. From Figures 8-9, the tracking errors , satisfy the asymmetry prescribed bounds and the transformed errors , are bounded, from which we can see that a fairly good tracking performance is achieved.

6. Conclusion

This paper proposed a novel adaptive neural PPC scheme for large classes of nonlinear nonstrict-feedback systems with prescribed performance under the effect of input saturation. Compared with previously published methods [27–43], the restrictive assumption that the upper and lower bounds of control gain functions must be positive constants or coefficients is relieved under the proposed method. The innovative error transformation proposed in this paper also overcomes the nondifferentiable obstacle and complex deductions corresponding to traditional PPC schemes. A novel auxiliary system with a bounded compensation term was also proposed to relax the strictly bounded assumption of the dynamic system. The stability of the closed-loop system was proven under Lyapunov and invariant set theories. Tracking errors were made arbitrarily small by appropriately selecting the design parameters. Finally, the simulation results show that the proposed control laws do achieve the desired performance. In the future, we will address control problems for uncertain nonlinear systems with an unbounded or nondifferentiable time derivative of disturbance; this will allow us to utilize the PPC approach combined with set-invariance mechanisms.

Appendix

Proof of Lemma 9. From and (8), we have .
Case 1 (). According to (7) and , we can obtain (i) , for ; (ii) , for . Clearly, there is a contradiction between (ii) and .
Case 2 (). According to (7) and, we can obtain (i) , for ; (ii) , for . Clearly, there is a contradiction between (ii) and .
Thus, if and , we will obtain for . From (8), we can have for . This completes the proof.

Proof of Lemma 10. Observing , we have , so that the function can be expressed by with being a continuous function of and . Note that is a compact set since and are compact sets. Furthermore, it can be seen from (A.1) that all the variables of are included in . Thus, we haveThis completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

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

Acknowledgments

This work is supported by the National Social Science Foundation of China (Grant no. 14GJ003-173) and the National Natural Science Foundation of China (Grant no. 71601183).

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Copyright © 2019 Jihui Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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