Finite-time adaptive prescribed performance DSC for pure feedback nonlinear systems with input quantization and unmodeled dynamics

: This paper presents a new prescribed performance-based ﬁnite-time adaptive tracking control scheme for a class of pure-feedback nonlinear systems with input quantization and dynamical uncertainties. To process the input signal, a new quantizer combining the advantages of a hysteresis quantizer and uniform quantizer has been used. Radial basis function neural networks have been utilized to approximate unknown nonlinear smooth functions. An auxiliary system has been employed to estimate unmodeled dynamics by producing a dynamic signal. By introducing a hyperbolic tangent function and performance function, the tracking error was made to fall within the prescribed time-varying constraints. Using modiﬁed dynamic surface control (DSC) technology and a ﬁnite-time control method, a novel ﬁnite-time controller has been designed, and the singularity problem of di ﬀ erentiating each virtual control scheme in the existing ﬁnite-time control scheme has been removed. Theoretical analysis shows that all signals in the closed-loop system are semi-globally practically ﬁnite-time stable , and that the tracking error converges to a prescribed time-varying region. Simulation results for two numerical examples have been provided to illustrate the validity of the proposed control method.


Introduction
Numerous nonlinear challenges arise in the realms of production and societal life, rendering conventional linear control theories and methods inadequate as tools to address these nonlinear control issues.Since Kanellakopoulos et al. [1] and Swaroop et al. [2] put forward the backstepping and dynamic surface control (DSC) design, the two design methods have been widely used in the controller design of nonlinear systems [3][4][5].In contrast to backstepping design, DSC offers a simpler alternative, circumventing the issue of over-parameterization that is often associated with conventional backstepping and streamlining the overall controller design process.
Unmodeled dynamics are prevalent in practical production and daily life.Modeling errors often arise during the system modeling process, leading to adverse effects on system stability [6][7][8].In the work of Jiang and Praly [9], unmodeled dynamics was estimated by using measurable dynamic signals, and the assumption that unmodeled dynamics was exponentially input-state-practically stable (exp-ISpS).The work conducted by Hua and Zhang [10] effectively addressed a category of strict feedback systems that are characterized by unmodeled dynamics and time-varying full-state constraints through the application of the DSC technique.This approach demonstrated exceptional dynamic performance when handling complex system dynamics and evolving state limitations.In the work of Zhang et al. [11], a dynamic signal was introduced to deal with the dynamical uncertainties associated with adaptive output feedback systems with output constraints and unmodeled dynamics.In the work of Zhang et al. [12], the dynamic signals generated by the auxiliary system were used to process unmodeled dynamics for a class of strict-feedback nonlinear systems with full state constraints.In [13], adaptive triggered control for multiple input multiple output (MIMO) systems with unmodeled dynamics and output constraints was proposed in the work by Hua and Zhang.A class of pure-feedback systems with unmodeled dynamics were studied in the work by Zhang and Lin [14].References [9][10][11][12][13][14] extensively discuss the impact of unmodeled dynamics on the system.However, it is noteworthy that these references do not explicitly consider the influence of input quantization on the overall system.
Signal quantization involves the transformation of a continuous signal into a discrete signal with a limited set of values after passing through a quantizer [15,16].In advanced fields such as medical, aerospace, and military applications, the application of quantization control has become increasingly prevalent.The accurate and efficient quantification of signals plays a crucial role in the advancement of quantitative control methodologies.To date, substantial research efforts have been dedicated to the study of input quantization, illuminating its significance in various domains and the need for robust and effective quantization techniques.So far, there are many studies researches on input quantification.In [17], a class of nonlinear systems with quantization time delay was studied.Under the assumption that the sector region of the quantizer was bounded, the stability problem of strict-feedback nonlinear systems with state quantization was studied in the work of Liu et al. [18].Combining the advantages of the hysteresis quantizer and uniform quantizer, Xing et al. proposed a new quantizer, which is an extension of the existing quantizer [19].In the works of Xia and Zhang [20], a novel controller compensation was designed to eliminate the influence of input quantization on the system.On the basis of the results from previous works [20], in [21], to simplify the difficulty of controller design, the quantizer error was further linearized.References [22,23] primarily focus on addressing a specific category of input quantization problems within the context of finite-time control.
Finite-time control has witnessed substantial advancements in recent decades [24,25].A defining characteristic of finite-time control is its ability to bring the system's state to equilibrium within a finite duration, after which the system remains in equilibrium.In the work of Bhat and Bernstein [26,27], the Lyapunov theory and homogeneous system theory of finite-time stability were first proposed, and the flutter phenomenon caused by the controller was solved.On this basis, many finite-time stability problems of nonlinear systems were discussed.In the works of Chen and Yang [28] and Yang and Sun [29], the finite-time stability of time-varying delay systems was successfully addressed through the application of Lyapunov-Krasovskii functions.However, it is important to note that the discussions in these works did not cover specified performance and input quantization aspects.In the work of Liu et al. [30], the finite-time control methods of nonlinear systems were systematically and perfectly presented.The finite-time problem was investigated for a class of strict-feedback nonlinear systems in the work of Li et al. [31].References [32,33] predominantly delve into finite-time control methodologies applied to diverse systems by using the backstepping design.It is worth noting that, while effective, the backstepping technique can give rise to computational complexity issues [34][35][36].Additionally, the virtual control law is derived repetitively in the virtual control design, leading to an increase in the structural complexity of the controller as the system's order rises.As a result, addressing these challenges becomes imperative for the widespread applicability and scalability of finite-time control techniques.References [37,38] adopted DSC to avoid the "complexity" problem in conventional backstepping, and they brought about new ideas for the design of finite-time controllers.MIMO non-strict feedback systems were studied, but they still had the problem of singularity in the work.Reference [38] adopted a DSC method and presented a new form of first-order filter.However, the symbolic function was used in the design of the virtual controller, which made the virtual control law non-differentiable and led to theoretical errors in the following process.In a study conducted by Liu et al. [39], the investigation focused on finite-time control for nonlinear systems.Notably, Young's inequality was employed as a substitute for a conventional inequality theorem commonly found in the finite-time literature.This innovative approach aimed to enhance the efficiency and analytical tractability of finite-time control methodologies in the context of nonlinear systems.
The prescribed performance function (PPF) requires that the system satisfies the conditions for instantaneity and stability performance.In the research conducted by Liu et al. [40], an adaptive PPF tracking controller was formulated, leveraging the backstepping technique as its foundational design methodology.By using the properties of the bounds of the hyperbolic tangent function, the error transformation was carried out, which provided a new idea for the study of PPFs in the work of Shi [41].Xia et al. [42] used linear mapping to transform the error and realized the prescribed performance tracking control.
Inspired by the above literature, this paper focuses on the finite-time adaptive prescribed performance DSC for pure-feedback nonlinear systems featuring unmodeled dynamics and input quantization.A novel control strategy is proposed, and the primary contributions can be summarized as follows: (1) This paper introduces a novel finite-time adaptive tracking control scheme for a specific class of pure-feedback nonlinear systems characterized by input quantization and dynamical uncertainties.A hybrid quantizer, amalgamating the advantages of a hysteresis quantizer and a uniform quantizer, is employed to address the input signal.(2) The proposed control scheme incorporates the hyperbolic tangent function and a performance function to ensure that the tracking error adheres to prescribed time-varying constraints.The singularity problem of differentiating each virtual control in the existing finite-time control scheme [22,32,37] is removed by constructing an appropriate unknown continuous function at each step of recursion.(3) Theoretical analysis demonstrates the semi-global practical finite-time stability (SGPFTS) of all signals in the closed-loop system, with the tracking error converging to a prescribed timevarying region.To validate the efficacy of the proposed control method, simulation results from two numerical examples are provided, illustrating its applicability and robustness in real-world scenarios.
The remaining sections of this paper are organized as follows.Section 2 provides the problem statement and some preparatory work, such as radial basis function (RBF) neural networks and quantization design.Sections 3 and 4 present the error transformation and adaptive controller design.Section 5 outlines the main results of this paper.The portion based on MATLAB/SIMULINK is covered in Section 6.Finally, the conclusion and prospects for future work are discussed in Section 7.

Problem statement and preliminaries
Consider the following pure-feedback nonlinear systems where xi = [x 1 , ..., x i ] T ∈ R i (i = 1, 2, ..., n) denotes the state vectors of system (2.1); ∈ R n 0 represents the unmodeled dynamics; Q( , x, t) is an unknown Lipschitz function; y ∈ R represents the system output; u ∈ R is the control input; f i ( xi ) denotes an unknown smooth function; q(u) is the quantized signal, and d i ( , x, t) is the unknown disturbance.
The control objective is to design a controller u and quantizer q(u) so that the output can track the desired trajectory and the tracking error can converge to a preset area in finite-time.Meanwhile, all signals of the closed-loop system exhibit SGPFTS.Definition 1. [9] For system ˙ = Q( , x, t), if there exist functions ᾱ1 and ᾱ2 of class K ∞ and a Lyapunov function V 0 (ξ) such that and if there exist two known constants c > 0, d ≥ 0 and a class K ∞ function Λ(•) such that then the unmodeled dynamics is said to be exp-ISpS.

Radial basis function neural networks
Using radial basis function neural networks θ * T i φ i (Z i ) to approximate the continuous function where , and the basis functions ϕ i, j (Z i ) are selected as Gaussian functions as follows: .
(2.10) φ i j and µ i j are respectively, the width and center of the Gaussian functions, and i = 1, . . ., n; j = 1, . . ., l i , l i is the number of nodes of the neural networks.The ideal weights θ * i are defined as follows: (2.11)

Quantizer design
In this paper, a new quantizer that combines a hysteresis quantizer with a uniform quantizer not only avoids quantization signal chatter, it also keeps the upper bound of quantization error constant.The new quantizer was proposed by Wang et al. [21], and it is described as follows: where (2.14) stands for the maximum integer which is less than or equal to d.The design parameter ῡ ≥ |q h (u th ) − u th | determines the quantization density of the uniform quantizer, u th which is a designed normal number and the threshold for switching between two quantizers.The quantization error can be obtained by the new quantizer as follows: It is obvious that the quantization error ∆ q of the new quantizer is bounded for any u, and that there exists a positive constant d such that ∆ q ≤ d, where d = max {v, δu th + (1 − δ)u min }.According to ∆ q (u) = q(u) − u, we can conclude that q(u) = u + ∆ q (u). (2.16)

Error transformation
Define the PPF (t) as follows: where (t) is a positive decreasing smooth function, and 0 , ∞ and k 0 are the positive constants.The tracking error is defined as follows: In this section, the control signal u is designed to make the following inequality hold: According to the control target in this paper, the tracking error is in the prescribed region described by Eq (3.3). Let Inverse transformation of the coordinates is given as ) Therefore, Eq (3.6) can be rewritten as follows: (3.9)

Adaptive control design
) can be changed into the following form: To facilitate the controller design, the following coordinate changes are introduced: where ω i is the output of the nonlinear filter with input i−1 , i = 2, . . ., n. Define some symbols as follows: where λi denotes the estimation of λ i , y j = ω j − j−1 .
Step 1. Define the first dynamic surface as follows: The time derivative of z 1 is given by 1 as the Lyapunov function; the time derivative of V 1 is given by Using Lemmas 1 and 4 and Assumption 2, it follows that ) ) Substituting Eqs (4.6)-(4.8)into Eq (4.5), we have (4.9) where Using Young's inequality yields where a 1 is a positive constant.Therefore, Eq (4.10) can be rewritten as follows: The virtual control law is constructed as follows: ). (4.13) Design the adaptive law of λ1 as follows: where k 1 , a 1 are positive design constants.Substituting Eqs (4.13) and (4.14) into Eq (4.12), we have The non-negative continuous function δ 1 (z 2 , λ1 , y d , ẏd , y 2 , r, k c (t), kc (t)) satisfies Based on Young's inequality, we further get Therefore, we obtain Let x = λ2q 1 , y = 1, a = 1 q > 1, and b = 1 1−q .According to Lemma 4, we obtain Therefore Substituting Eq (4.20) into Eq (4.15), we have (4.21) Inspired by Li et al. in [38], the nonlinear filter ω 2 is designed as follows: i.e., where 1 and ω 2 are the input and output of the filter, respectively, and τ 2 is a positive constant.
Step 2. (2 ≤ i ≤ n − 1).Since z i = s i − ω i , the time derivative of z i is given by żi Substituting Eq (4.22) into Eq (4.24), we obtain From Assumption 2 and Lemma 1, and by using Young's inequality, we get From Eqs (4.26)-(4.28),we know that (4.29) where . Therefore, we have (4.30) Using Young's inequality yields where a i > 0 is a design constant.Therefore, Eq (4.30) can be changed as follows: Construct the virtual control law i as follows: where k i , σ i are positive design constants.Substituting Eqs (4.33) and (4.34) into Eq (4.32), we have The non-negative continuous function δ i zi+1 , ȳi+1 , λi , y d , ẏd , r satisfies Using Young's inequality yields Therefore, we have (4.37) Design a nonlinear filter ω i+1 as follows: Therefore, we obtain (4.41) Step n.Since z n = s n − ω n , the time derivative of z n is given by From Assumption 2 and Lemma 1, and by using Young's inequality, we get From Eqs (4.44)-( 4.46), we know that where Therefore, we have (4.48) Using Young's inequality yields where a n > 0 is a design constant.Therefore, Eq (4.48) can be transformed into the following form: Construct the control signal u as follows: The updating law of λn is designed as follows: where k n , σ n are positive design constants.Substituting Eqs (4.51) and (4.52) into Eq (4.50), we have (4.53) The non-negative continuous function δ n zn , ȳn , λn , y d , ẏd , r satisfies Using Young's inequality yields Therefore, we have (4.55) The nonlinear filter is designed as follows: (4.59)

Main results
Define the whole Lyapunov function V n and two compact sets Ω, Ω k c as follows: ) where P and P k c > 0. The maximum values of the continuous η i (•) and δ i (•) on the compact set Ω × Ω k c are M i and N i , respectively.
Theorem 1.Consider the closed-loop system which consists of the plant (1) under Assumptions 1 and 2, the control law given by Eq (4.51), virtual control laws given by Eqs (4.13) and (4.33), and adaptive laws given by Eqs (4.14), (4.34) and (4.52).If V q n (0) ≤ P is true, then there exist some design constants σ i , τ i and k i such that all signals in the closed-loop system are bounded, and the tracking error can converge to a preset region in finite-time, and the design constants σ i , τ i and k i satisfy the following inequalities: (5.4) Proof.If V q n ≤ P, then z 1 , . . ., z n , y 2 , . . ., y n , λ1 , . . ., λn are bounded.Because y i = ω i − i−1 and y i values are bounded, we can get ω i , and i−1 which are bounded.According to x i = z i + y i + i−1 , we can obtain that x i is bounded.Furthermore, it is known that the control signal u is bounded.
The time derivative of V n is given by Vn where Accordingly, the above equations can be rewritten as follows: Vn ≤ −αV n − βV q n + C, ( where Vn ≤ 0. It can be seen that V q n (t) ≤ P, ∀t > 0. Therefore, Eq (5.6) holds.According to Eq (5.6) and Lemma 2, we have the settling time T r = 1 α(1−q) ln αV 1−q (0)+vβ vβ .It yields that the closed-loop system exhibits SGPFTS for t ≥ T r .

Numerical examples
In this section, we present numerical simulations that were conducted by using MATLAB/SIMULINK.We utilize two numerical examples to provide further illustration of the efficacy of the proposed control approach.
As depicted in Figures 1, it is evident that the output signal adeptly tracks the desired trajectory, with the tracking error converging to a predetermined region.Figure 2 illustrates the disparity between the quantization signal q(u) and the control signal u.The small quantization error and the effective discretization of the continuous signal are evident from Figure 3. Furthermore, Figure 4 demonstrates that the updating parameters λ1 , λ2 , λ3 remain bounded.These observations collectively affirm the robust performance and stability of the proposed control methodology.Example 2. Consider a ship's dynamic system; its dynamic equation is described as follows: where v stands for the angular velocity of the ship course, q(u) is input quantization, l 0, M and B represent unknown constants of the ship related hydrodynamic coefficients and ship mass, respectively, and J = 0.2, where we set l = 1.8, M = 0.12, and B = 0.2.Define x 1 = v, x 2 = ḃ.The ship's dynamic system with unmodeled dynamics, input quantization and prescribed performance can be described as follows: Choosing the initial values as x 1 (0) = 0.3, x 2 (0) = 0.2, λ1 (0) = 2, λ2 (0) = 0.5, ω 2 (0) = 0.1, r(0) = 0.1, and (0) = 0.1, and setting the design constants to a 1 = a 2 = 1, σ 1 = 0.1, σ 2 = 0.1, τ 2 = 0.3, k 1 = 2, k 2 = 100, q = 0.6, p = 3 17 , u min = 0.2, and l 1 = l 2 = 10, we chose the desired signal as y d = 1.5 sin(0.6t)and the auxiliary signal as ṙ = −r + 1.5x 4  1 + 0.8.The simulation results are shown in Figures 5-8.It can be ascertained from Figure 5 that this method has good performance in terms of the prescribed tracking.Figure 6 shows the difference between the quantization signal q(u) and the control signal u, and the quantization error is small.Figures 7 and 8 show that the control signal u and adaptive tuning parameters λ1 , λ2 are bounded.Example 3. To validate the effectiveness of the proposed control scheme in this paper, simulations and comparisons of different control methods were conducted on the same pure-feedback system with input quantization and unmodeled dynamics.The proposed finite-time DSC (referred to as FTDSC) in this paper is compared to the traditional DSC.
A pure-feedback system with input quantization and unmodeled dynamics was evaluated: AIMS Mathematics Volume 9, Issue 3, 6803-6831. (6.12) In the proposed FTDSC method in this paper, the virtual control law is chosen as Then, the control law u FT DS C is applied as follows
In Figure 9a, the red solid line represents the desired tracking trajectory y d , the blue dashed line represents the output signal y in the proposed FTDSC, and the green dashed line represents the output signal in the traditional DSC method.Figure 9b is a partial enlarged view of Figure 9a.From the local zoom-in, it can be observed that under both control methods, the output signal y can effectively track the desired trajectory y d .However, from the local zoom-in plot in (b), it is evident that the tracking performance of the FTDSC method is superior to the traditional DSC method, with a faster convergence speed.In Figure 10, the red solid line represents the tracking error y − y d in the FTDSC method, while the black dashed line represents the tracking error y − y d in the DSC method.It can be observed from the local magnification that the tracking error in the FTDSC method is smaller.In Figure 11, in subfigures (a) and (b), the red dashed line represents the quantized signal q(u), and the blue dashed line represents the control signal u.In Figure 12, the blue dashed line represents the quantization error q(u) − u in the FTDSC method, and the red dashed line represents the quantization error q(u) − u in the traditional DSC method.Combining the local magnification chart in Figure 12, it can be observed that the FTDSC method has smaller quantization errors than the DSC method, indicating greater accuracy.

Conclusions
In this paper, a new finite-time adaptive control scheme has been presented for non-affine nonlinear systems subject to input quantization, unmodeled dynamics and prescribed performance.An invertible nonlinear mapping has been formulated by leveraging the characteristics of the hyperbolic tangent function.The development of a novel controller has been achieved through the incorporation of a nonlinear filter and the application of Young's inequality, coupled with an improved DSC strategy.The issue of "complexity explosion" that is inherent in traditional backstepping is successfully mitigated, while the problem of "singularity" in finite time is eliminated.Compared with the previous first-order linear filter, the nonlinear filter reduces the demand on the time constant.Theoretical analysis indicates that all signals within the controlled process are bounded within finite time, and that the tracking error of the system can converge to a predefined time-varying region within a finite duration.Simulation curves substantiate the effectiveness of the proposed design scheme.
Our future work will be aimed at exploring the potential integration of machine learning technologies, such as reinforcement learning or neural networks, to solve complex system dynamics and uncertainties.In addition, it will be an important aspect of our future research work to study the scalability and robustness of the controller under large-scale systems and different operating conditions.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Figure 9 .
Figure 9.The output y and its tracking under two methods.

Figure 10 .
Figure 10.Tracking error of output y under two methods.

Figure 11 .Figure 12 .
Figure 11.The control signals u and quantized control signals q(u) under two control methods.