Finite-Time Tracking Control for Nonstrict-Feedback State-Delayed Nonlinear Systems with Full-State Constraints and Unmodeled Dynamics

Yao, Yangang; Tan, Jieqing; Wu, Jian

doi:https://doi.org/10.1155/2020/8887925

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

Research Article | Open Access

Volume 2020 | Article ID 8887925 | https://doi.org/10.1155/2020/8887925

Finite-Time Tracking Control for Nonstrict-Feedback State-Delayed Nonlinear Systems with Full-State Constraints and Unmodeled Dynamics

Yangang Yao,¹Jieqing Tan,¹and Jian Wu²

Academic Editor: Dimitri Volchenkov

Received06 Sept 2020

Revised27 Sept 2020

Accepted15 Oct 2020

Published09 Nov 2020

Abstract

The problem of finite-time tracking control is discussed for a class of uncertain nonstrict-feedback time-varying state delay nonlinear systems with full-state constraints and unmodeled dynamics. Different from traditional finite-control methods, a smooth finite-time adaptive control framework is introduced by employing a smooth switch between the fractional and cubic form state feedback, so that the desired fast finite-time control performance can be guaranteed. By constructing appropriate Lyapunov-Krasovskii functionals, the uncertain terms produced by time-varying state delays are compensated for and unmodeled dynamics is coped with by introducing a dynamical signal. In order to avoid the inherent problem of “complexity of explosion” in the backstepping-design process, the DSC technology with a novel nonlinear filter is introduced to simplify the structure of the controller. Furthermore, the results show that all the internal error signals are driven to converge into small regions in a finite time, and the full-state constraints are not violated. Simulation results verify the effectiveness of the proposed method.

1. Introduction

During the past few decades, great achievements have been proposed for uncertain nonlinear systems based on adaptive control technique, especially for pure-feedback systems (e.g., see [1–5]) and strict-feedback systems (e.g., see [6–9]) with the lower-triangular structure. Lately, the authors in [10] introduced a more general nonlinear system named nonstrict-feedback nonlinear systems. By employing the variable separation method, the tracking control problem has been well solved. Since then, many control techniques for nonstrict-feedback systems and extensions to other fields were achieved (e.g., see [11–17]).

It is known to all that many practical systems encounter the effect of the constraints, such as the temperature of chemical reactor and physical stoppages. Thus, the research about the systems with state constraints is very meaningful and necessary on account of the existence of state constraints which may undermine the stability of the system. In order to tackle the problem of state constraints, some effective control techniques (e.g., model predictive control (MPC) [18, 19], reference governors (RGs) [20], one-to-one nonlinear mapping (NM) [21–23], and barrier Lyapunov functions (BLFs) [24–28]) have been presented. Due to the fact that MPC and RGs require strong online computing capability to guarantee constraints, this requirement restricts their applications in engineering design. Therefore, one-to-one NM and the BLFs-based methods become the main methods to deal with the constrained nonlinear systems. There exist many significant results which focus on lower-triangular structure nonlinear systems with different constraints (e.g., input constraints [3], output constraints [24], partial-state constraints [25], and full-state constraints [21–23, 26, 27]). In addition, the rate of convergence is also an essential consideration for most practical systems. The works mentioned above only obtain asymptotic or exponential stability with infinite time, which cannot meet the requirement of finite-time control in most practical control systems. As a consequence, a considerable number of meaningful researches (e.g., see [28–33]) have been proposed on finite-time control for nonlinear systems. However, most of the works are to present finite-time controller by using a backstepping technique together with a nonsmooth fractional feedback design method. In order to achieve a faster convergence rate, the authors in [34] originally proposed a smooth finite-time adaptive NN controller by using a smooth switch between the fractional and cubic form state feedback. Moreover, there are other significant results presented in [35–41], such that two globally stable adaptive controllers were proposed in [35, 36]. To obtain the tracking accuracy, a practical adaptive fuzzy tracking controller for a class of perturbed nonlinear systems with backlash nonlinearity has been designed in [37]. An adaptive fuzzy output-feedback tracking control technique for switched stochastic pure-feedback nonlinear systems has been presented in [38]. The authors in [39] proposed an observed-based adaptive finite-time tracking control technique for a class of nonstrict-feedback nonlinear systems with input saturation. An adaptive finite-time output-feedback controller for switched pure-feedback nonlinear systems with average dwell time has been given in [40]. A decentralized event-triggered controller for interconnected systems with unknown disturbances has been proposed in [41].

Furthermore, due to the fact that unmodeled dynamics can severely degrade the closed-loop system performance, dealing with the effects of unmodeled dynamics is essential for practical nonlinear control systems. Therefore, several results were proposed by employing backstepping or DSC in [4, 21–23, 42–47]. Generally, unmodeled dynamics was disposed by introducing a dynamic signal in [4, 21–23, 42–46] or a Lyapunov function description in [47].

In addition, time delays frequently occur in some practical engineering systems. As stated in [48], their existence can deteriorate the transient performance and even can destroy the stability of the control systems. Thus, the research on nonlinear time-delay systems has become one of the hot topics and some meaningful results have been achieved during the past decades [49–53]. For uncertain nonlinear time-delay systems, the effective controller was developed originally in [50] by combining the backstepping technique with Lyapunov-Krasovskii functionals. Soon afterward, this method was extended to nonlinear strict-feedback time-delay system with unknown control gain functions [51] and uncertain multi-input/multi-output nonlinear systems with time delays [52]. Later, some improved control schemes based on [50] were proposed (e.g., see [35, 53, 54]).

Although many significant research results on adaptive neural network control for uncertain nonstrict-feedback systems have been obtained in [11–17], their considered systems did not include unmodeled dynamics or full-state constraints. In [21–28], the effective controllers have been designed for the lower-triangular structure nonlinear systems with state constraints and unmodeled dynamics, but their considered systems did not include state delay and their control methods may be invalid to nonstrict-feedback systems on account of subsystem function which contains the whole state variables. Furthermore, the above-mentioned control methods only obtain asymptotic or exponential stability with infinite time. To the best knowledge of the authors, finite-time tracking control for a class of uncertain nonstrict-feedback time-varying state-delayed nonlinear systems with full-state constraints and unmodeled dynamics has not been fully discussed in the literature, which is still open and remains unsolved. In this paper, we are committed to solving the problem mentioned above. The main contributions of the paper are summarized as follows:(i)In contrast to the existing results reported in [21–28, 47] where the control methods have been proposed for nonlinear strict-feedback or pure-feedback systems with state or output constraints and unmodeled dynamics, a generalization of the results is proposed for a class of nonstrict-feedback state delay systems with state constraints and unmodeled dynamics of which the subsystem function contains the whole state variables. To the best of authors’ knowledge, it is the first time to develop an adaptive DSC method for uncertain nonstrict-feedback state delay systems with state constraints and unmodeled dynamics.(ii)Different from the finite-control methods in [31–33], a smooth finite-time adaptive control framework is introduced by employing a smooth switch between the fractional and cubic form state feedback reported in [34], so that the desired fast finite-time control performance can be guaranteed. Moreover, unmodeled dynamics is coped with by introducing a dynamical signal and the uncertain terms produced by time-varying state delays are compensated for by constructing appropriate Lyapunov-Krasovskii functionals. The results show that all the error signals are driven to converge into small regions in a finite time, and the full-state constraints are never violated.

The remainder of this paper is organized as follows. In Section 2, the problem formulation and preliminaries are presented. Adaptive DSC design and stability analysis are given in Section 3. Simulation results verify the effectiveness of the proposed control approach in Section 4, followed by Section 5, which concludes this paper.

Notation. In this paper, denotes a set of real numbers, denotes a set of nonnegative real numbers, denotes a set of real matrices, denotes a set of n-dimensional real vectors, denotes the least upper bound, denotes 2-norm of a vector or matrix, denotes an absolute value of a real number , denotes an exponential function of , and denotes the natural logarithm of .

2. Problem Formulation and Preliminaries

2.1. Problem Statement

Consider a class of uncertain nonstrict-feedback state-delayed nonlinear systems with unmodeled dynamics for in the following form:where is the state vector, is the unmodeled dynamics, and denote the system input, the system output, and the unknown time-varying delays, respectively. , and are the unknown smooth functions. Let and be the unknown uncertain disturbances. All the states are required to remain in the sets , where are positive constants.

Remark 1. System (1) is called a nonstrict-feedback form in which the system function and its bounding function contain all the state variables [10]. Apparently, strict-feedback and pure-feedback structures are the special cases of system (1). The methods proposed in [21–28, 31–33, 47] cannot be directly applied to system (1) on account of its nonstrict-feedback structure.
The control objective of this paper is to construct an adaptive NN controller to make sure that the output follows the desired trajectory in a finite time, while every state is never violated.

2.2. RBFNN Approximation

In this paper, for , the unknown smooth nonlinear functions will be approximated on a compact set by the following RBFNN:where denote input vectors, weight vectors, and NN node number, respectively. are the NN inherent approximation errors which are bounded over the compact sets; that is, , where are unknown constants and are known smooth vector functions with being chosen as the commonly used Gaussian functions, which have the formwhere is the center vector and is the spreads of the Gaussian function. The optimal weight vector is defined aswhere is the estimate of .

2.3. Key Definition and Lemmas

Definition 1 (see [21]). The unmodeled dynamics is said to be exponentially input-state-practically stable (exp-ISpS), that is, for system , if there exist functions of class and a Lyapunov function , such thatand there exist two constants and a class function , such thatwhere and are known positive constants and is a known function of class .

Lemma 1 (see [21]). If is an exp-ISpS Lyapunov function for a system , that is, (5) and (6) hold, then, for any constant , any initial instant , any initial condition , and any continuous function , such that , there exist a finite , a nonnegative function defined for all , and a signal described bysuch that for and with .

Lemma 2 (see [11]). Let be the basis function vector of an RBFNN and be the input vector, where and . For any positive integer , let , and the following inequality holds:

Lemma 3 (see [55]). For any real numbers and , an extended Lyapunov condition of finite-time stability can be given in the form of fast terminal sliding mode as ; then, is in fast finite-time convergent with a finite settling time .

Lemma 4 (see [56]). For , if , where are odd integers, then , where and .

Lemma 5 (see [34]). Consider the dynamic systemwhere ( and are positive odd integers), and are positive constants, and is a positive function. Then, for any given bounded initial condition , one has that , .

Lemma 6. (see [57]). For , and , then .

To obtain the control objective, the following assumptions are needed.

Assumption 1. The unmodeled dynamics is exp-ISpS.

Assumption 2. There exist unknown nonnegative continuous functions and nondecreasing continuous functions such thatwhere .

Remark 2. From Definition 1 and Assumption 1, we have . According to Lemma 1, there exists a positive constant such that . This inequality will be used to cope with the uncertain terms in the following controller design.

Assumption 3. The sign of is known, and there exist some unknown positive constants and such that . Without loss of generality, this paper assumes that .

Assumption 4. The reference trajectory and its derivatives about time and are in a bounded region , and there exists a known constant , such that .

Assumption 5. The unknown continuous functions satisfy the following inequality:and the time-varying state delays satisfy the inequalities and , where are unknown positive smooth functions and and are unknown constants.

3. Adaptive DSC Design and Stability Analysis

3.1. Adaptive DSC Design

Similar to traditional backstepping, the backstepping-design procedure with n steps is developed to construct the adaptive neural controller in this part.

By using the backstepping technique, the proposed adaptive DSC scheme contains n steps as follows.

Step 1. Define the first surface error ; the time derivative of is defined as

The virtual control law and the update law for are designed aswhere are positive design parameters, is an estimate of , , is defined in Assumption 3. is defined aswhere and are the positive odd integers, , and is a small positive constant.

Consider the BLF candidate as

Obviously, is positive definite and continuously differentiable. Based on Assumptions 2 and 5 and Young’s inequality, we obtain the time derivative of as follows:where

Note that is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:where is an NN, , and is any given.

By using Young’s inequality and Lemma 2, one haswhere .

Substituting (13), (14), and (20) into (17), we can obtain

By utilizing Young’s inequality, the following inequalities can be obtained:

Therefore, we have

According to the inequality and Lemma 4, one aswhere and are defined in Lemma 4.

To deal with the time delay in equation (24), define the Lyapunov-Krasovskii functional as follows:where is a positive constant. Using Assumption 5, we obtain that the derivative of is

From equations (24) and (26), we havewhere .

To move on, introduce the coordinate transformationwhere , and denote the tracking error, the virtual control input, and the boundary layer error for , respectively. is the output of the following first-order filter:where and are the positive design constants and is defined in (15).

Remark 3. From (29), it can be seen that the proposed filter involves both the linear and fractional terms. In particular, when or , filter (29) degrades into the fractional filter used in [58] and the linear filter as widely used in the literature [21–23], respectively. It is the key to ensure the fast finite-time stability of the closed-loop system, which will be detailed in the following analysis.

Remark 4. As mentioned in [34], by designing and properly, both the virtual control input and its derivative are ensured to be inherently continuous in the set . It means that the virtual control input defined in (13) is continuous in the set . From (13), it is not hard to see that and its derivative are the functions of the variables and , respectively. Combining the continuity of and (28) and (29), it can be seen that there exists a continuous function which satisfies

Step 2. Define the surface error ; the time derivative of is defined asThe virtual control law and the update law are designed aswhere are positive design parameters, is an estimate of , is defined aswhere is defined in (15), , and is a small positive constant.
Consider the BLF candidate aswhere is also positive definite and continuously differentiable in the set . Similar to (17), the time derivative of iswhereNote that is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:where is an NN, , and is any given.
By using Young’s inequality and Lemma 2, one hasSubstituting (32), (33), and (39) into (36), we can obtainBy utilizing Young’s inequality, the following inequalities can be obtained:Therefore, we haveAccording to the inequality and Lemma 4, one haswhere and are defined in Lemma 4.
To handle the time delay, define the Lyapunov-Krasovskii functional as follows:where is a positive constant. By using Assumption 5, we obtain that the derivative of isFrom in equations (43) and (45), we havewhere .
Similar to the analysis in Remark 4, there exists a continuous function which satisfies

Step 3. Define the surface error ; the time derivative of is defined as

The actual control law and the update law are designed aswhere are positive design parameters, is an estimate of , , is defined aswhere is defined in (15), , and is a small positive constant.

Consider the BLF candidate as

Similar to (17) and (36), we can obtain the time derivative of as follows:where

Note that is an unknown continuous function and RBFNN can be used to approximate it. Hence, from (2), the following equation holds:where is an NN, , and is any given.

By using Young’s inequality and Lemma 2, one haswhere .

Substituting (49), (50), and (56) into (53), we can obtain

By utilizing Young’s inequality, the following inequality can be obtained:

Therefore, we have

According to the inequality and Lemma 4, one haswhere and are defined in Lemma 4.

To handle the time delay, define the Lyapunov-Krasovskii functional as follows:where is a positive constant. By using Assumption 5, we obtain that the derivative of is

From equations (60) and (62), we havewhere .

3.2. Stability Analysis

In this subsection, we present the stability analysis of the resulting closed-loop system. The main results are presented by the following theorem.

Theorem 1. Consider the nonlinear system (1) with Assumptions 1–5. Let the actual control input and the NN adaptive law be designed as (49) and (50), respectively. If the initial conditions satisfy , in which is any positive constant for and are properly chosen, such that and with for , one has that all internal signals and in the closed-loop system are semiglobally uniformly ultimately bounded and the tracking error will converge into the arbitrarily small regions in a finite time. Meanwhile, each state will remain in the set ; that is, the full-state constraints are never violated.

Proof. Construct the overall Lyapunov function candidatewhere and are defined in (35) and (44), respectively.
From (28), (29), and (47), the derivative of isDefine a compact set as with being a positive constant. If , together with Assumption 4 and (65), it can be obtained that there exists a positive constant , such that on the compact set . Then, applying Young’s inequality to (65) yieldswhere are positive constants.
According to the above analysis, we can obtain the derivative of the overall Lyapunov function candidate aswhereHere, we choose , such that .
From the definition of in (15), (34), and (51), the following two cases should be considered. Case 1: When , substituting into (67) gives Noting (69), we can have with , which further implies that all the internal signals are uniformly ultimately bounded. Case 2: When , substituting into (67) giveswhereBy virtue of [[59], Th.5.2], there always exists a finite-time , such that for all . Thus, for all , one has , and it then comes from Lemma 3 that the fast finite-time stability of the closed-loop system can be ensured with a finite settling time . Furthermore, it is readily seen that . Therefore, . Then, the internal error signals , and will converge into the following compact sets:in a finite-time with . It is readily seen that the regions (73) can be made as small as possible by adjusting with proper control parameters.
Then, we will prove that the full-state constraints are never violated. According to [[60], Lemma 1], we can conclude from (70) and (71) that , for all . Noting that from Assumption 4 and , we have that . To get , we need to show the boundedness of . From (73), one has that is bounded and is also bounded. With the proper choices of and , is a continuous function of , and . Then, there exists an upper bound , such that . From and , we get that . Similarly and iteratively, we have that and for are bounded, which together with ensures that . Therefore, each state will remain in the set . The proof is completed.

4. Simulation Results

Example 1. Consider the following nonlinear system:where , and the dynamic signal . The desired tracking trajectory . u is the control input. The design parameters of the controller are taken as . There are 68 nodes with the center placed on and the width of Gaussian functions is in the first RBF vector. There are 85 nodes with the center placed on and the width of Gaussian functions is in the second RBF vector. With the initial conditions, . Simulation results are shown in Figures 1–6. The profiles of the system output y and the desired signal are shown in Figure 1, which indicates that the output follows the specified desire trajectory . From Figure 2, we know that all state constraints are not violated.

(a)

(b)

(a)

(b)

Example 2. A Spring-Mass-Damper system is provided in this part. The system model is as follows:where are the position, the velocity, and the force applied to the object, respectively. Let . Assuming that the controlled system (75) gives unmodeled dynamics and time delay, let , and the dynamic signal . Then, system (75) can be rewritten asThe desired tracking trajectory . The design parameters of the controller are taken as . There are 68 nodes with the center placed on and the width of Gaussian functions is in the first RBF vector. There are 85 nodes with the center placed on and the width of Gaussian functions is in the second RBF vector. With the initial conditions, . Simulation results are shown in Figures 7–12.

(a)

(b)

5. Conclusions

The problem of finite-time tracking control for a class of uncertain nonstrict-feedback state-delayed nonlinear systems with full-state constraints and unmodeled dynamics has been proposed in this paper. Unmodeled dynamics is dealt with by introducing a dynamical signal and the uncertain terms produced by time-varying state delays are compensated for by constructing appropriate Lyapunov-Krasovskii functionals. By utilizing a smooth switch between the fractional and cubic form state feedback, novel smooth finite-time NN control laws have been provided for nonlinear systems with full-state constraints. Based on a modified DSC method and adaptive NN control, together with the BLFs, the fast finite-time control performance of the closed-loop nonlinear systems can be ensured, while the full-state constraints are never violated. Theoretical proofs and experimental simulation show that all the internal signals in the closed-loop system are uniformly bounded, and the tracking error signals can converge into compact sets in a finite time with sufficient accuracy, respectively. To extend this control scheme to solve the finite-time tracking control problem for some more complicated systems, such as MIMO nonlinear systems, switched nonlinear systems are also the direction of our future efforts.

Data Availability

This paper is a theoretical study and no data were used to support this study.

Conflicts of Interest

The authors declare that they do not have any financial or nonfinancial conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61603003 and 61472466).

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Copyright © 2020 Yangang Yao 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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