Abstract
In order to solve the problem of unknown parameter drift in the nonlinear pure-feedback system, a novel nonlinear pure-feedback system is proposed in which an unconventional coordinate transformation is introduced and a novel unconventional dynamic surface algorithm is designed to eliminate the problem of “calculation expansion” caused by the use of backstepping in the pure-feedback system. Meanwhile, a sufficiently smooth projection algorithm is introduced to suppress the parameter drift in the nonlinear pure-feedback system. Simulation experiments demonstrate that the designed controller ensures the global and ultimate boundedness of all signals in the closed-loop system and the appropriate designed parameters can make the tracking error arbitrarily small.
1. Introduction
In recent years, nonlinear systems have been widely studied by researchers at home and abroad. Isidori [1] proposed a precise linearization technique, Krstic et al. [2] proposed backstepping, and Astolfi et al. [3] put forward adaptive control methods. After that, backstepping adaptive control has been widely developed and applied [4, 5]. In the majority of studies on nonlinear system tracking control problems [6, 7], the global or semiglobal asymptotic stability and the gradual stability of tracking error are considered as the ultimate control target of nonlinear systems. However, at present, there is no uniform method to analyze nonlinear systems in the control theory of nonlinear systems.
Since nonlinear systems are different from linear systems, it is difficult to be analyzed in the way that linear systems are discussed. However, nonlinear pure-feedback systems [8, 9] are more general systems compared with lower triangle nonlinear systems, which can reflect the actual physical situation while the nonaffine pure-feedback system is one of the complicated systems. According to the literature [10, 11], there are two nonaffine functions in the pure-feedback system: system control input and system state variables. Liu [12] proposed a novel coordinate transformation in the research on the tracking problem of pure-feedback systems and achieved global asymptotic stability of tracking error using the backstepping method. Zeng et al. [13] proposed an active sliding mode disturbance rejection control of a single-input single-output nonaffine nonlinear system. However, the research of Zeng et al. is only applicable to a second-order system. For systems larger than second-order, this method cannot work. In this regard, Liu [14] proposed nonconventional coordinate transformation based on [12] and introduced a first-order auxiliary controller to solve the nonaffine control input problem of -order system. In engineering applications, system parameters change due to various conditions, so system parameters are variables rather than constants. However, in most of the tracking control theory studies of nonlinear systems [15, 16], we often set system parameters as constants, which reduces the practicality of the tracking control theory of nonlinear systems, so it is necessary to study the influence of unknown parameter drift on system stability. For the uncertainty of parameterization, when there is external disturbance or unmodeled dynamics, parameter drift will occur, and the adaptive controller will deteriorate the system and even lead to system instability [17]. In this case, the boundedness of the estimated value of parameters cannot be guaranteed, causing divergence of other signals in the closed-loop system. A common method for dealing with this situation is to add a robust term (leakage term) to the parameter adaptive law or use a projection operator [18].
In this paper, a sufficiently smooth projection algorithm [19] is introduced to deal with the parameter drift in the designed nonlinear pure-feedback system. Since nonaffine functions exist in the system state variables and the system control input in the nonlinear pure-feedback system, the conventional coordinate transformation [2] will increase the difficulty of designing the controllers, so the nonconventional coordinate transformation proposed by [14] is introduced. Due to the nonaffine structure of the pure-feedback system, it is difficult to design the virtual controllers , and the conventional dynamic surface algorithm [20] is difficult to be directly used in the pure-feedback system. Thus, in this paper, a novel nonconventional dynamic surface algorithm is designed to reduce the complexity of designing the controllers for the nonlinear pure-feedback system. A nonlinear pure-feedback system with unknown parameters is designed, in which unknown parameters are added compared with [21] in order to solve the control problem of parameter uncertainty in the pure-feedback system, which is a further supplement to nonlinear pure-feedback systems. In [21], the unknown parameters are set to , which limits the selection of system parameters, thus constraining the whole system. However, the unknown parameters are in this paper, so the nonlinear pure-feedback system designed is more common. In this paper, backstepping method, nonconventional dynamic surface algorithm, Nussbaum function [22], and sufficiently smooth projection algorithm are combined so that the designed controller can achieve the uniform and ultimate boundedness of all signals in the closed-loop system.
Main contributions of this paper are summarized as follows:(1)The parametric drift concept is first proposed in pure-feedback systems. In order to solve the parameter drift problem of the nonlinear pure-feedback system, a sufficiently smooth projection algorithm is introduced.(2)A novel nonconventional dynamic surface algorithm is designed according to the nonconventional coordinate transformation to eliminate the problem of “calculation expansion” in the pure-feedback system.(3)The virtual controllers , the actual controller , and the adaptive laws are designed so that the tracking error of the nonlinear pure-feedback system with parameter drift converges to a region of the origin, and all signals in the closed-loop system are uniform and ultimately bounded.
Other parts of this paper are arranged as follows: In Section 2, problem statement and preliminaries are presented. In the third section, aiming at the tracking control problem of the nonlinear pure-feedback system with parameter drift, the virtual controllers , the actual controller , and the parameter adaptive laws are designed and semiglobal uniform ultimate boundedness of all signals in the closed-loop system are proved. In Section 4, the correctness of the proposed control scheme is proved by simulation study. The conclusion is presented in Section 5.
2. Problem Statement and Preliminaries
2.1. Systems Description
Consider the following nonlinear pure-feedback system:where , with , , and are system state variables, system control input, and system output, respectively. The nonlinear functions are smooth nonaffine known functions; the nonlinear function is the smooth nonaffine known function with control input coefficient , among which is a unknown nonzero constant; are known smooth functions, where ; are unknown parameters; and are unmodeled dynamics and external disturbances related to state variables.
In system (1), the control coefficient is added compared with [11, 12]. The unknown parameters are added in system (1) compared with [21], where represents the th order subsystem and represents the number of unknown parameters in the th subsystem. In [21], the number of parameters in each subsystem is equal, while in actual engineering systems, the number of parameters in each subsystem cannot be equal. The unknown parameters designed in this paper just solve this defect, that is, different is set according to different .
For easy analysis and calculation, are simplified as and is simplified as .
In order to avoid the control problem of system (1) in the design of the controller, the following assumptions are made.
Remark 1. Since the control coefficient is an unknown nonzero constant, the positive and negative signs are unknown, i.e., the control direction is unknown.
Assumption 1. The following equation is made workable at any time:
Remark 2. Assumption 1 is weaker than the assumed condition , in [11].
Because the input is characterized by being nonaffine. Therefore, the first-order auxiliary system of the controller is designed as follows:where is the undetermined function and .
Assumption 2. Unknown parameters are bounded and there is a bounded set where is a known positive constant.
Assumption 3. are bounded and their known upper bounds satisfywhere are positive constants correlated with .
Assumption 4. Smooth nonaffine functions meetwhere are positive constants.
Lemma 1. If smooth bounded functions meet assumption 4, are bounded.
Proof. According to mean value theorem, the following equation is obtained:where is the arbitrary point of interval. It can be known from assumption 4 that are bounded and are also bounded; thus, are bounded with .
Assumption 5. The smooth reference trajectory is bounded and the th order is derivable.
2.2. Preliminaries
In this paper, a sufficiently smooth projection algorithm [19] is introduced to the nonlinear pure-feedback system, as shown in (8)–(11).where and are arbitrary positive constants.
If , the algorithm has the following properties:(1)(2)(3) is -order differentiable
Definition 1. If the continuous function satisfies (9), it is called the Nussbaum function [22].where is adopted as the Nussbaum function in this paper.
Lemma 2 (see [23]). If and are smooth functions defined in and with , is even smooth Nussbaum function, if the following inequality holds:where and are certain appropriate constants, then , and are bounded in .
3. Main Results
The backstepping design is based on the following nonconventional coordinate transformation [14]:where , , , and are the tracking error, smooth reference trajectory signal, and error state variables, respectively.
Remark 3. In the conventional coordinate transformation [2], are made as the virtual controller of the th order subsystem, as shown in (15).Unlike the standard dynamic surface algorithm [20], the nonconventional dynamic surface algorithm designed in this paper is as follows.
Define the first-order low-pass filter formula.where , are the filter time constants.
The filter error equation is defined as follows:where , are the th virtual controllers. The nonconventional coordinate transformation cooperates with the nonconventional dynamic surface algorithm to solve the problem of “calculation expansion” caused by the application of the nonconventional coordinate transformation in the nonlinear pure-feedback systems.
Remark 4. The first-order low-pass filter of standard dynamic surface algorithm [20]The filter error of standard dynamic surface algorithm
Step 1. According to (1) and (11), we obtainDesign the virtual controller aswhere is a positive design constant.
According to (14), the following equation is obtained:By deriving with respect to time, we can obtainwhere is a continuous function, the maximum of is set to be .
The Lyapunov function is defined asAccording to (17), (20) and Young’s inequality, we havewhere , is a positive constant.
Step 2. According to (1) and (11), we haveDesign the virtual controller aswhere is a positive design constant, is the estimation of , and is a nonlinear damping term which is used to process the disturbance term in (23).
Design the adaptive law asSubstituting (24) into (23) yieldswhere .
According to (14), we can obtainBy deriving with respect to time, we can obtainwhere is a continuous function and the maximum of is set to be .
The Lyapunov function is defined asAccording to (22), (25), (27), (28), and Young’s inequality, we haveAccording to the properties (2) of the sufficiently smooth projection algorithm [20], the following equation is obtained:where and is a positive constant.
Step . According to (1) and (11), we can obtainDesign the virtual controller aswhere is a positive design constant, is the estimation of , and is the nonlinear damping term which is used to process the disturbance term in (33).
Design the adaptive law asSubstituting (34) into (33) yieldswhere
Based on the filter error equation (14), we can obtainBy deriving with respect to time, the following equation is obtained:where is a continuous function and the maximum of is set to be .
The Lyapunov function is defined as
According to (35), (37), (39) and Young’s inequality, we haveBy the properties (2) of the sufficiently smooth projection algorithm [20], the following equation is obtained:where
Step . According to (1) and (11), we can obtainDesign the virtual controller aswhere is a positive design constant, is the estimation of , is the nonlinear damping term which is used to process the disturbance term in (42).
Design the adaptive law asSubstituting (43) into (42) yieldswhere
According to (14), we can obtainBy deriving with respect to time, the following equation is obtained:where is a continuous function and the maximum of is set to be .
The Lyapunov function is defined as
According to (44), (46), (48), and Young’s inequality, we haveBy the properties (2) of the sufficiently smooth projection algorithm [20], the following equation is obtained:where
Step . According to (1) and (11), we can obtain
Remark 5. The control coefficient is included in , so it is unnecessary to analyze the unknown parameter independently, which is also the advantage of introducing nonconventional coordinate transformation in this paper.
Design the actual controller aswhere is a positive design constant, is the estimation of , and is the nonlinear damping term which is used to process the disturbance term in (51).
The adaptive law of the parameter is designed asThe adaptive law of parameter is designed to beSubstituting (52) into (51) yieldswhere
Theorem 1. For system (1), under the condition that Assumptions 1–5 are satisfied, the first-order low-pass filter (13), the control law (52), and the parameter adaptive laws (26), (35), (44), (54), and (55) are adopted to make the tracking error of the system converge to a region of the origin, and all signals in the closed-loop system are uniformly and ultimately bounded.
Proof. The Lyapunov function is defined as
According to (50), (55), (57) and Young’s inequality, we can getBased on the property (2) of the sufficiently smooth projection algorithm, we can obtainwhere the parameter is designed to satisfyMakethen (59) can be rewritten towhere
When both ends of (62) are multiplied by the exponent , we can getAfter the integration of both ends of (63) in , we can getSince the adaptive law utilizes a sufficiently smooth projection algorithm, it can be known from the properties of a sufficiently smooth projection algorithm (1) that is bounded and thus is also bounded, which means the is bounded.
It is noted that (64) can be written asAccording to Lemma 2, it can be known that in the interval , , , and are bounded, and it can be known from Lemma 1 that are bounded and satisfies assumption 4; thus, are bounded. When tends to , all signals in the closed-loop system are uniformly and ultimately bounded. The upper bound of is set to be , then (64) can be written asFor arbitrarily given , there is ; make it when , , . Appropriate parameters are selected so that the tracking error can be adjusted to small region of the origin:
4. Illustrative Example
Consider the following second-order nonlinear pure-feedback system:
Corresponding to system (1), we have
Set the design parameters in the above control scheme as
The control coefficient is set to be .
The simulation results are shown in the following figures. Figure 1 shows the system output and the reference signal . It can be seen that the system output can track the reference signal well even when the parameters , , and control coefficient change. Figure 2 shows the control input , indicating that the system input curve is asymptotically bounded. Figure 3 shows the estimated curve of the parameter . Figure 4 shows the estimated curve of the parameter . Figure 5 shows that functions and are bounded. Figure 6 shows the system state variables and . Figure 7 shows the error between the system output and the reference signal .
It can be seen from Figures 1–7 that system state variables and are bounded, actual controller is bounded, and are bounded, and are bounded, and are bounded, and error is bounded, so all signals in the closed-loop system are bounded.
5. Conclusion
In this paper, the adaptive tracking control for a class of nonlinear pure-feedback systems is studied to solve the parameter drift problem. Different from the conventional backstepping method, the nonconventional coordinate transformation is introduced and the nonconventional dynamic surface algorithm is designed to solve the problem “calculation expansion” in the pure-feedback system, and the sufficiently smooth projection algorithm is introduced to solve the unknown parameter drift problem in the pure-feedback system so that the designed controller can make all signals of the closed-loop system globally bounded. Finally, the correctness of the algorithm in this paper is verified by simulation. The nonlinear pure-feedback system considered in this paper has the defect, that is, are known functions, but are often not accurately modeled in engineering applications, so are smooth unknown functions. In future, the radial basis function (RBF) will be used to approximate the unknown functions , so that the adaptive control for a class of nonlinear pure-feedback systems with parameter drift is more extensive and practical.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Science Foundation of China under grant 61663032.