https://orcid.org/0000-0003-3244-3554
Figures
Abstract
In this work, the tracking control problem for a class of nonlinear strict-feedback systems with input nonlinearity is addressed. In response to the influence of input nonlinearity, an auxiliary control system is constructed to compensate for it. To process unknown nonlinear dynamics, radial basis function neural networks (RBFNNs) are introduced to approximate them, and some adaptive updating control laws are designed to estimate unknown parameters. Furthermore, during the dynamic surface control (DSC) design process, first-order low-pass filters are introduced to solve the complexity explosion problems caused by repeated differentiation. After that, an NN-based adaptive dynamic surface tracking controller is proposed to achieve the tracking control. By applying the proposed controller, it can be guaranteed that not only the output of the system can track the desired trajectory, but also that the tracking error can converge to a small neighborhood of zero, while all signals of the closed-loop system are bounded. Finally, the effectiveness of the proposed controller is verified through two examples.
Citation: Zhou M, Zhang X, Deng X (2024) Tracking control problem of nonlinear strict-feedback systems with input nonlinearity: An adaptive neural network dynamic surface control method. PLoS ONE 19(10): e0312345. https://doi.org/10.1371/journal.pone.0312345
Editor: Gang Wang, University of Shanghai for Science and Technology, CHINA
Received: June 26, 2024; Accepted: October 5, 2024; Published: October 24, 2024
Copyright: © 2024 Zhou et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: This study was supported by the Natural Science Research Project of Anhui Province Universities under Grant 2023AH052698, the Guangxi Innovation Driven Development Project (Science and Technology Major Project) under Grant GuiKeAA21077018, the Key Laboratory of Artificial Intelligence and Internet of Things of Guangxi Normal University of Science & Technology under Grant GXKSKYPT2021004, the Anhui Province Young and Middle-aged Teachers Discipline (Professional) Leader Cultivation Project under Grant DTR2023101, and the Key R&D and Achievement Transformation Project of Wuhu under Grant 2023yf124.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
In the past decades, a large number of control methods have emerged, such as sliding mode control method [1], PID method [2], adaptive control method [3], fuzzy control method [4,5], and NN control method [6], among which the NN control method has received widespread attention. Due to the effective approximation of unknown nonlinear dynamics, the NN control method has been used to solve the control problems of various uncertain nonlinear systems. For example, the authors of [7] solved the approximation problem of uncertain nonlinear dynamics caused by the deviation of control parameters from the actual values. In [8], an adaptive NN tracking control strategy with backstepping control technique was designed to achieve the tracking control. Furthermore, the authors of [9] proposed an NN-based exponential feedback control method, in which the output feedback control issue of a calss of strict-feedback nonlinear systems with different time-delay states was solved. Moreover, the fuzzy-neural control method addressed in [10] solved the prescribed performance control problem of constrained wave-rider vehicles.
On the other hand, as an important part of a control system, the occurrence of relevant faults in actuators will impact the system control effect and may even lead to the system instability when serious. Among actuator faults, the input saturation fault has always been a hot topic for scholars. Designing effective control schemes for nonlinear systems with input saturation faults is a key research direction. In [11], the authors considered the fault-tolerant control problem of a class of Takagi-Sugeno fuzzy system with actuator faults and saturation, where an adaptive observer-based fault-tolerant control strategy was addressed. In [12], the authors developed an adaptive DSC strategy with fuzzy-neural control method, where the tracking error can converge in a finite time under the designed control strategy. In [13], the authors proposed a smooth non-affine function that can control the input signal to solve the approximation problem of the input saturation function in the given system. In [14, the authors addressed the trajectory tracking problem of a class of autonomous underwater vehicles with model uncertainties and input saturation, in which the proposed adaptive command filtered backstepping control strategy can ensure the boundedness of all signals. Furthermore, an improved event-triggered command filter backstepping tracking control scheme were further presented by the authors of [15], in which the event-triggered tracking problem of uncertain nonlinear systems under input saturation was solved. Additionally, for nonlinear systems with unknown dead-zone input [16,17], unknown input saturation [18], and unknown control gains [19–21] are also of great interest of researchers. However, it should be noted that in practical control systems, the limited amplitude of actuator output means that actuator saturation can affect the stability and dynamic performance. Therefore, while researching the tracking control problem of nonlinear systems, considering the existence of input saturation faults is of great practical significance.
As discussed in [22–24], the backstepping control method has been widely used to tackle various control issues of complex nonlinear systems. However, this method requires repeated differentiation of the designed virtual control law, which may lead to the “complexity explosion” problem. In order to solve this drawback, many scholars have explored research from different perspectives. In [25,26], the authors introduced a DSC method to replace the traditional backstepping control method, and proposed an adaptive DSC strategy to solve the control problem of nonlinear systems with sensor faults. In [27], the authors designed an adaptive backstepping DSC scheme that effectively eliminated the “complexity explosion” problem caused by repeated derivation of the virtual control law. It is worth noting that the authors of [27] introduced an unknown function in the design of the controller, which may lead to the occurrence of oscillation phenomena. In [28], although the proposed controller performed well for the given system, the introduction of the sign function in the controller design may result in control signal divergence. To this end, in [29], a class of sliding mode filter was introduced to improve the DSC method, where the authors proposed an adaptive fuzzy DSC method and solved the adaptive tracking control problem of a class of uncertain non-affine nonlinear systems. Meanwhile, the authors of [30] further combined the DSC method with nonlinear filters, and proposed a filter-based adaptive DSC strategy. Moreover, the authors of [31] introduced the nonlinear function approximation technique into the DSC design, and improved the control effect while simplifying the controller design process. Similarly, the authors of [32,33] addressed uncertain nonlinear dynamics of given systems from the perspectives of fuzzy logic system and NN, where the proposed tracking control methods not only achieved excellent tracking performance but also ensured strong robustness.
In light of the above analysis, it is not difficult to find that there are not many discussions on the nonlinear strict-feedback systems with input saturation. Therefore, this work discusses the tracking control problem of a type of strict-feedback nonlinear system with input saturation nonlinearity. The impact of input saturation is handled by constructing a type of auxiliary dynamic systems. By introducing the DSC method and RBFNN approximation technique, the parameter adaptive updating control laws and an NN-based adaptive DSC controller are proposed. Overall, the main contributions of this work include:
- An auxiliary dynamic system is constructed to compensate for the input saturation nonlinearity of the studied strict-feedback nonlinear systems. Compared with [13–15], the introduction of auxiliary control systems not only reduces the difficulty of handling input saturation nonlinearity, but also reduces the difficulty of controller design.
- Different from [25–27], this work simultaneously considering dynamic surface control method and neural network approximation technique, the problems caused by repeated differentiation and unknown nonlinear dynamics are effectively solved. Furthermore, it effectively overcomes the issue of computational complexity.
- An NN-based adaptive dynamic surface tracking control strategy is proposed for the tracking problem. It can be ensured that the system tracking error can converge to a small neighborhood of zero, and all signals of the closed-loop system are bounded.
The following arrangements of this work are as follows. In Section 2, the problem description and some useful assumptions and lemmas are provided. In Section 3, as the important part of this work, involves the design and stability analysis of the adaptive tracking controller. Section 4 presents two simulation examples to illustrate the theoretical analysis results; and finally, brief conclusions of this work are presented in Section 5.
2. Problem description and preliminaries
Problem description
Consider a class of strict-feedback nonlinear systems with input saturation nonlinearity, which is described as
(1)
where
(i = 1,⋯,n) and y∈R represent the state vector and output of the system, Q(u(t)) is the control input,
(i = 1,⋯,n) represent the unknown nonlinear functions,
(i = 1,⋯,n) are the control gains and represent known time-varying continuous functions, and di(t)(i = 1,⋯,n) represent unknown external disturbances. For the convenience of subsequent analysis, without causing confusion, this work abbreviates
,
and di(t) as fi, gi and di, respectively.
In this work, it is assumed that the system control input Q(u(t) is subjected to actuator saturation fault [15,34], which is described as
(2)
where Umax and Umin represent the upper and lower bounds of input saturation faults, respectively.
The main control objective of this work is to design an adaptive NN dynamic surface tracking controller for the system (1), such that the system’s output can track the given reference trajectory, and the tracking error converges to a small neighborhood of zero, while ensuring that all signals of the closed-loop system are bounded.
Assumption 1
The external disturbances di (i = 1,⋯,n) are bounded, that is, there exist positive constants diM and satisfy |di|≤diM.
Assumption 2
The given reference trajectory yd and its the ith derivative exists and is smooth, bounded, and there exists a positive constant Ξ1, such that .
Assumption 3
The control gains gi (i = 1,⋯,n) are bounded, and there exists a positive constant g0, such that 0<|gi|≤g0. Without loss of generality, we assume gi>0.
RBFNN
To deal with the unknown nonlinear dynamics generated in control design, an RBFNN is introduced for approximation processing. Considering a continuous unknown nonlinear function h(Z), and applying an RBFNN, then we have
(3)
where Z∈ΩZ⊂Rn is the input vector of the NN, W = [w1,⋯,wl]T∈Rl represents the weight vector, and l>1 is the number of nodes, S(Z) = [s1(Z),⋯,sl(Z)]T∈Rl represents the radial basis function vector, and si(Z)(i = 1,⋯,l) is usually chosen as a Gaussian function, that is
(4)
where χi = [χi1,⋯,χin]T∈Rn and bi represent the center and the width of radial basis function, respectively.
If the ideal weight vector W* is selected to have the minimum approximation error [8,33], then we have
(5)
and
(6)
where ε(Z) is approximation error and satisfies
with
being a positive constant.
Lemma 1 [35]. For any X∈Rn, Y∈Rn, p>0 and q>0, if satisfies (p−1)(q−1) = 0, then the following inequality holds
(7)
Lemma 2 [36. For any b∈R and η>0, the hyperbolic tangent function tanh(∙) satisfies
(8)
3. Controller design and stability analysis
Tracking controller design
In order to achieve the tracking control problem of the strict-feedback nonlinear systems with input saturation nonlinearity, this section combines DSC method and NN control method to design relevant parameter adaptive updating control laws and tracking controller. In the first n−1 step, RBFNNs are used to approximate the unknown nonlinear dynamics, and corresponding parameter updating control laws are designed for the unknown parameters. In the final step, the adaptive NN dynamic surface tracking controller is proposed. For the convenience of analysis in the following sections, the time variable t will be ignored.
To handle input saturation nonlinearity in the system (1), the following auxiliary dynamic system is designed.
(9)
where λi(i = 1,⋯,n) are the states of the auxiliary system, ci>0(i = 1,⋯,n) are design parameters.
Hence, we can define the following error coordinate transformation, that is
(10)
where zi(i = 1,⋯,n) represent the error surfaces,
(i = 2,⋯,n) are filter’s output with input being virtual control laws αi−1 which will be provided in subsequent analysis, λi(i = 1,⋯,n) are the auxiliary states defined in (9), and yd is the given reference trajectory.
Step 1.
According to (1), and (10), the time derivative of z1 can be expressed as
(11)
Noting f1 is an unknown nonlinear function, and an RBFNN is introduced to approximate it, then we get
(12)
where ε1(Z1) is approximation error and satisfies
with
.
Substituting (12) into (11) gets
(13)
Due to and |d1|≤d1M, then
and let
, where D1 is an unknown positive constant.
To avoid the problem of complexity explosion caused by repeated differentiation, a first-order low-pass filter 1/(τ1s+1) with time constant τ1 being introduced, then we obtain
(14)
where the virtual control law α1 as the input and the newly introduced state
as the output.
Defining the filtering error β1 as
(15)
Combining (14) and (15), we have
(16)
Considering , and substituting (15) into (13) yields
(17)
Choosing the Lyapunov function V1 as
(18)
where and
,
and
represent the estimation of
and D1, a1>0 and r1>0 are design parameters. Then, the time derivative of V1 can be expressed as
(19)
Design the virtual control law α1, parameter adaptive updating control laws and
as
(20)
(21)
(22)
where k1>0, σ1>0 and υ1>0 are design parameters.
Substituting (17), (20)-(22) into (19), and considering Lemma 2, we have
(23)
Further, using Lemma 1 and considering (15) and (16), we have
(24)
(25)
where
is a non-negative continuous function.
Substituting (24) and (25) into (23), we get
(26)
Step i (i = 2,⋯,n−1). According to (1), (i = 2,⋯,n−1) and (10), the time derivative of zi is described as
(27)
Noting fi is an unknown nonlinear function, and an RBFNN is introduced to approximate it, then we have
(28)
where εi(Zi) is approximation error and satisfies
with
.
Substituting (28) into (27) yields
(29)
Due to and |di|≤diM, then
and let
, where Di is an unknown positive constant.
To avoid the problem of complexity explosion caused by repeated differentiation, a first-order low-pass filter 1/(τis+1) with time constant τi being introduced, then we have
(30)
where the virtual control law αi as the input and the newly introduced state
as the output.
Defining the filtering error βi as
(31)
Combining (30) and (31), we have
(32)
Considering and substituting (31) into (29) yields
(33)
Choosing the Lyapunov function Vi as
(34)
where
and
,
and
are the estimation of
and Di, ai>0 and ri>0 are design parameters. So the time derivative of Vi can be expressed as
(35)
Design the virtual control law αi, parameter adaptive updating control laws and
as
(36)
(37)
(38)
where ki>0, σi>0 and υi>0 are design parameters.
Considering Lemma 2, and substituting (33), (36)-(38) into (35), we have
(39)
Further, using Lemma 1 and considering (31) and (32), we have
(40)
(41)
where
is a non-negative continuous function.
Considering the result of in step i−1, we have
(42)
Substituting (40)-(42) into (39), we obtain
(43)
Step n. In this step, the actual control law will be given. According to (1), , Δu = Q(u(t))−u(t) and (10), the time derivative of zn is given as
(44)
Noting fn is an unknown nonlinear function, and an RBFNN is introduced to approximate it, then we have
(45)
where εn(Zn) is the approximation error and satisfies
with
.
Substituting (45) into (44) has
(46)
Due to and |dn|≤dnM, then
and let
, where Dn is an unknown positive constant.
Choosing the Lyapunov Vn as
(47)
where
and
,
and
represent the estimation of
and Dn, an>0 and rn>0 are design parameters. So the time derivative of Vn is
(48)
Design the adaptive NN dynamic surface controller u(t), adaptive updating control laws and
as
(49)
(50)
(51)
where kn>0, σn>0 and υn>0 are design parameters.
Considering the result of in step n−1, we have
(52)
Considering Lemma 2 and , and substituting (46), and (49)-(52) into (48), we have
(53)
Stability analysis
According to the above analysis, the main results of this work can be summarized as follows.
Theorem 1.
Under the Assumptions 1–3, considering the strict-feedback nonlinear system with input saturation nonlinearity (1), using the virtual control laws (20) and (36) with adaptive updating control laws (21), (22), (37) and (38), and the adaptive NN dynamic surface controller (49) with adaptive updating control laws (50) and (51), then it can be obtained that the output of the system output can track the reference trajectory and the tracking error can converge to a small neighborhood of zero, and all signals of the closed-loop system are bounded.
Proof. According to Lemma 1 and considering (53), we have
(54)
(55)
Further, the following compact set is defined as
According to Assumption 2, it can be inferred that Ω1 is a compact set. Therefore, we can obtain that all variables in the non-negative continuous functions Bi(∙)(i = 1,⋯,n−1) are bounded within the compact set Ω1×Ωi. This also implies that Bi(∙) are bounded within the compact set Ω1×Ωi, and further meaning that |Bi(∙)| must exist a maximum value. Without loss of generality, let .
Considering and substituting (54) and (55) into (53), we have
(56)
Hence, the Eq (56) can be simplified as
(57)
Multiplying both sides of (57) by exp(ϑt) and integrating on (0,t] yields
(58)
As can be seen from (58), variables zi, ,
and βj(i = 1,⋯.n, j = 1,⋯.n−1) are all bounded. Furthermore, it can be concluded that αj(j = 1,⋯.n−1) and u(t) are also bounded.
Combining (18), (34), and (47), it can be seen that
(59)
Owing to |x1−yd|=|z1+λ1|≤|z1|+|λ1|, to prove that |x1−yd| is bounded, we only need to prove that |λ1| is bounded. The proof of the boundedness of |λ1| is shown below.
Taking the Lyapunov function . Considering (9), then the time derivative of Vλ is
(61)
According to Assumption 3, we can see that 0<gi≤g0(i = 1,⋯.n). Meanwhile, owing to u(t) being bounded and Q(u(t)) being the output of saturation fault, so it can be further obtained that Δu is bounded, and thus |gnΔu| is bounded.
Hence, (61) can be simplified as
(62)
where |gnΔu|2≤A0.
Multiplying both sides of (62) by exp(μt) and integrating on (0,t] gets
(63)
It can be inferred from (64) that |λ1| is bounded. Taking into account (60) and (64), there is
(65)
It is not difficult to observe from (65) that by selecting appropriate parameters, the system tracking error |x1−yd| can be ensured to converge to a small neighborhood of zero. This completes the proof.
Remark 1.
Observing (65), to adjust the size of the system tracking error, we can decrease the values of ω and A0, and on the other hand, we can also increase the values of ϑ and μ. By selecting these parameters appropriately, the tracking error can be ensured to be arbitrarily small.
Remark.
For the parameters ω, ϑ and μ, on the one hand, decreasing ω can be achieved by increasing am and rm or by decreasing σm and υm, and increasing ϑ can be achieved by increasing km, σm and υm or by decreasing τj, and increasing μ can be achieved by increasing cm. On the other hand, however, the changes in km, cm, am, rm, σm and υm(m = 1,⋯,n, j = 1,⋯,n−1) further affect the amplitude of virtual control laws, parameter update control laws, and actual controller signals. Therefore, a reasonable balance needs to be made between tracking performance and control signals.
4. Simulation analysis
In this section, two simulation cases are provided to demonstrate the effectiveness of the controller designed in this work.
Case 1 (numerical case). Consider the following a type of strict-feedback nonlinear systems
(66)
The auxiliary dynamic system is given as
(67)
where
and
. For the input saturation nonlinearity (2), taking Umax = 6 and Umin = −6. The initial states of systems (66) and (67) are considered as x1(0) = 0.1, x2(0) = 0, and λ1(0) = λ2(0) = 0, respectively. The desired reference trajectory is given as yd = 0.5(sin(t)+sin(0.5t), and simulation time is set as t = 40s.
The RBFNN for , the number of nodes is set as 7, and its center χi is evenly spaced in the interval [−6,6]. The RBFNN for
, the number of nodes is set as 7, and its center χi is evenly spaced in the interval [−6,6]×[−6,6]. The width of both NNs is selected as bi = 5.
Here, the virtual control law α1, the adaptive NN dynamic surface tracking controller u(t), and parameter adaptive updating control laws and
(i = 1,2) are respectively given as
The simulation parameters are set as τ1 = 0.02, η1 = η2 = 0.05, c1 = c2 = 10, k1 = k2 = 9, a1 = a2 = 20, σ1 = σ2 = 15, r1 = r2 = 5, and υ1 = υ2 = 2. The initial states of parameter updating control laws are given as and
.
The simulation results appear in Figs 1–5. Fig 1 displays the results of tracking performance and tracking error. With the designed adaptive NN dynamic surface controller, the system output effectively tracks the reference trajectory, and the tracking error can converge to a small neighborhood of zero. Obviously, the controller designed in this work achieves favorable control effects, thereby confirming the correctness of the theoretical analysis. Additionally, the curves of system states xi, controller u(t), parameter adaptive updating control laws and
, i = 1,2, are respectively shown in Figs 2–5. From these figures, it is evident that all these signals remain bounded, which further reflects the effectiveness of the designed controller from another perspective.
Further, examining Figs 1–5, despite the given system being affected by input saturation nonlinearity, the designed controller and corresponding parameter adaptive updating control laws ensure that the tracking error of the system converges to a small neighborhood of zero. Furthermore, it is guaranteed that all variables of the closed-loop system are bounded.
Case 2. In order to further demonstrate the effectiveness of the controller proposed in this work, a type of actual system [37], namely a marine surface vehicle, is selected as the research plant of this case. The system model is described as:
(68)
where ϕ represents the yaw angle, u(t) represents the actuator angle, K and T represent hydrodynamic coefficients.
represents an unknown nonlinear function based on
which can be approximated by
(69)
where p1, p2 and p3 are unknown parameters.
Assuming the system is affected by external disturbances and input saturation nonlinearity, and selecting x1 = ϕ and , then the system (68) can be converted into
(70)
where d1(t) and d2(t) represent external disturbances, Q(u(t) is the system input under the influence of input saturation nonlinearity.
Compared to the system (1), in this case, we have and
. Then, only
is approximated by using RBFNN.
According to [37], the initial states are given as x1(0) = x2(0) = 0, K = 8, T = 2, p1 = p2 = p3 = 1. In addition, assuming the external disturbances of the system (68) are d1(t) = d2(t) = 0.5sin(0.1t) and k1 = k2 = 30. The selection of other parameters is the same as in Case 1.
The simulation results display in Figs 6–10. Using the designed controller and parameter adaptive updating control laws, the curves of tracking performance and tracking error are depicted in Fig 6. It can be seen from the figure that the output of the marine surface vehicle system can effectively track the desired reference trajectory, and the tracking error can converge to a small neighborhood of zero. Moreover, Figs 6–10 show the system states, control input and parameter adaptive updating control laws, respectively. From these figures, one can observe that all variables remain bounded.
Furthermore, Figs 6–10 reveal that although the actual system experiences input saturation nonlinearity, the designed controller still achieves a strong control effect, which further proves the effectiveness of the designed controller.
5. Conclusion
This work studies the tracking control problem of a class of strict-feedback nonlinear systems with input saturation nonlinearity. An auxiliary control system is constructed to deal with the input saturation nonlinearity. In addition, DSC method and RBFNN approximate technique are applied for the controller design and the approximation of unknown nonlinear dynamics, respectively. Under the action of first-order low-pass filters, the problem of computational explosion caused by repeated differentiation of virtual control laws is effectively overcome. And then, an adaptive NN dynamic surface controller is designed for the tracking control problem. The simulation analysis verified the effectiveness of the designed controller. It can be ensured that the tracking error can converge to a small neighborhood of zero, and all signals of the closed-loop system remain bounded.
It should be pointed out that the control gains of the given system are assumed to be known, and only the influence of input saturation nonlinearity is considered. This greatly limits the application of the method proposed in this work. Therefore, in our follow work, we will consider the case of unknown gains or high-frequency gains, while also taking into account the effects of different input nonlinearities.
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