Funnel-Based Adaptive Neural Fault-Tolerant Control for Nonlinear Systems with Dead-Zone and Actuator Faults: Application to Rigid Robot Manipulator and Inverted Pendulum Systems

Tis study addresses an adaptive neural funnel fault-tolerant control problem for a class of strict-feedback nonlinear systems with actuator faults and input dead zone. To guarantee the boundedness of the tracking error, a modifed transformation for funnel error is devised and incorporated into the control design process. To manage unknown nonlinear functions, radial basis function neural networks (RBFNN) are employed in designing an adaptive neural funnel fault-tolerant controller through the backstepping technique. Te proposed controller guarantees the output tracking error stays within a predefned funnel, and all signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, simulations of a rigid robot manipulator system and an inverted pendulum system are conducted to validate the practicality and efectiveness of the proposed control method.


Introduction
In recent years, there has been a growing interest in addressing the control challenges of nonlinear systems, as many modern control systems demonstrate nonlinear behavior [1][2][3].In addition, studies on nonlinear systems have attracted a lot of attention recently, and numerous control approaches, including sliding mode control, adaptive backstepping control, and intelligent control, have been proposed to control design for nonlinear systems [4][5][6].An adaptive backstepping control method can overcome many of the technical limitations of classic adaptive control, such as the matching condition and the relative-degree constraint.Fuzzy logic systems (FLSs) and neural networks (NNs) have been established to solve this problem [7].Te interest in neural networks (NN) and fuzzy control of nonlinear systems has grown signifcantly as a result of their ability to fnd unknown nonlinear functions.As a result, various publications in this feld have been published [8,9].For instance, the issue of adaptive control for nonlinear systems using fuzzy logic systems has been studied in [7].Te problem of adaptive control for stochastic nonlinear systems has been reported by employing fuzzy approximation capabilities and integrating an output feedback mechanism.[10], and an innovative adaptive NN-based decentralized control strategy has been developed in [11] for interconnected nonlinear systems.For nonlinear systems that are subject to input saturation, the authors in [12] presented a composite adaptive control strategy.For nonlinear switched systems with unmodeled dynamics, an adaptive fuzzy control strategy has been presented in [13].However, none of the aforementioned articles addressed the controlled system's fault tolerance problem.
Actuators in real-world control systems may fail while they are in service.Tese defects have the potential to temporarily worsen control performance, infuence system instability, and potentially precipitate disastrous outcomes [14].Te fundamental prerequisite for system dependability is fault-tolerant control (FTC), and system performance improvement is crucial in light of this.Te passive and active approaches to the FTC design can be broadly classifed into two groups [15].Although the passive technique is typically used to manage whole and partial actuator faults because its passive control rules are set, it also has a limited ability to address unknown actuator problems.Active techniques, as opposed to passive ones, involve recreating the controller live and are better equipped to handle unknown actuator problems.Research interest in fault-tolerant control has grown recently due to concerns over dependability and safety, and a growing number of relevant advances have been made [16][17][18].In the presence of actuator faults, adaptive fault-tolerant control techniques for nonlinear systems have been reported in [19].For nonlinear systems with unmeasured states, the authors in [20] reported the adaptive fault-tolerant control issue based on observers.An adaptive controller has been reported for a class of nonlinear systems that are subjected to command-flter and actuator failure [21].An adaptive fault-tolerant control methodology, utilizing the approximation method, has been detailed for nonlinear systems in nonafne forms incorporating nonlinear faults through an event-triggered mechanism [22].
On the other hand, dead zones are one of the most signifcant nonsmooth nonlinear phenomena that arise in real-world applications.It has the potential to seriously impair system control capabilities and potentially cause instability [23].As a result, it presents a challenge to controller designs that must produce accurate tracking results for nonlinear systems [24].Some control strategies have recently been put out to address the impact of dead-zone nonlinearity [25][26][27].Te challenge of adaptive fuzzy control for nonlinear systems has been explored, taking into account dead-zone nonlinearity in the system input [28].An adaptive control strategy employing fuzzy approximation is introduced for nonlinear systems, addressing unknown functions and dead-zone nonlinearities through the incorporation of a fuzzy observer [29].In [30], the researchers devised a tracking control strategy using an observer-based adaptive fuzzy approach for a specifc class of nonlinear systems characterized by strict feedback form, unmeasurable state variables, and dead zones.Moreover, a recently published work in [31] presents an adaptive output control scheme for stochastic nonlinear systems in pure-feedback form, incorporating neural networks and accounting for input dead zones.
Te funnel control has emerged as one of the most successful control mechanisms in recent years [32][33][34].Te primary objective of funnel control design, as outlined in [35], is to regulate both the transient and steady-state responses of nonlinear systems.Ensuring the boundedness of tracking errors involves transforming the tracking error into a modifed form using an improved funnel error function, an integral part of the control design process [36].Notably, in the realm of nonlinear systems, funnel control has been successfully implemented without the need for intricate techniques [37].In the context of multiple-input multiple-output (MIMO) linear systems with input saturation, a funnel control strategy has been introduced, boasting stringent relative-degree one dynamics and stable zero dynamics [38].Addressing nondiferentiability, [39] introduces a funnel error transformation and an adaptive controller utilizing the properties of funnel and fuzzy approximation to ensure both steady-state and transient performance in tracking errors for nonlinear systems.
Building upon the aforementioned research, this study introduces an adaptive fault-tolerant control approach employing a funnel for a nonlinear system encountering actuator faults and input dead zones.Te handling of unknown functions is facilitated through radial basis function neural network (RBFNN) approximation.Subsequently, an adaptive funnel fault-tolerant controller is developed by incorporating funnel control within the framework of the backstepping method.Te primary contribution of this work is as follows: (i) In comparison to previous results [6,7,9], this work investigated the adaptive fault-tolerant control design problem for a nonlinear system with actuator faults and input dead-zone.Designing the suggested control method with actuator faults and dead-zone input considerations makes it more versatile for use in real-world engineering.Te unknown functions included in the nonlinear systems are modeled using radial basis function neural networks.Te suggested control strategy not only ensures nonlinear system stability but also minimizes the impact of dead-zone and actuator faults on the performance of the control.(ii) Funnel control is employed to regulate nonlinear systems experiencing actuator faults and input dead zones.To address the nondiferentiable issue in [40] and guarantee that the output tracking error always remains inside a predetermined funnel border, a funnel variable is constructed.Te proposed control methodology guarantees that the output tracking error remains within a predetermined funnel.Furthermore, utilizing Lyapunov stability analysis and the backstepping method ensures the semiglobal uniform ultimate boundedness (SGUUB) of all signals in the closed-loop system.
Te paper is organized as follows: In Section 2, an overview of the system is provided, along with an outline of preliminary concepts.Controller design and the stability assessment of the closed-loop system are discussed in Section 3. Section 4 demonstrates the efectiveness of the controller through an illustrative example.Lastly, Section 5 serves as the conclusion of the paper.

. System Description and Preliminaries
Consider the following nonlinear system as follows: 2 Complexity where η i ∈ R represents the state of the system with y is the output of the system, Φ i (•) and ϕ i (•) represent the smooth unknown nonlinear function, u is the system input subject to actuator fault and dead-zone.
Actuator faults involving input dead zones are critical problems with real-world applications due to external environment uncertainties, long system operation, and physical gear mechanism limitations.Te model for actuator fault is described as follows [17]: where ζ(t, t ζ ) ∈ [0, 1] represents the actuation efectiveness, and φ(v) characterizes the actuator input, accounting for dead-zone nonlinearity, t ζ indicates the time when actuation efectiveness is compromised, and t r marks the moment when an uncontrollable additive actuation fault occurs.Te control signal to be designed is denoted as v, while u r (t, t r ) accommodates the uncontrollable additive actuation faults.
Remark 1.When ζ(t, t ζ ) ≠ 0 and u r (t, t r ) � 0, it signifes a partial loss of performance during operation.Tis condition, termed partial loss of efectiveness, implies that the actuator's performance is partially compromised.Conversely, when ζ(t, t ζ ) � 0 and u r (t, t r ) ≠ 0, it suggests that the actuator output u is no longer infuenced by φ(v), i.e., u � u r (t, t r ).Tis condition, known as total loss of efectiveness, indicates that u is fxed at an unknown value u r (t, t r ).Lastly, when ζ(t, t ζ ) � 0 and u r (t, t r ) � 0, meaning u � 0, as seen in [41].When ζ(t, t ζ ) � 1 and u r (t, t r ) � 0, then the actuators work in the failure-free case, as seen in [41].
Te dead-zone model is represented as follows [42]: where v is the dead-zone input signal and R l and R r are uncertain breakpoints on the left and right axes that signify the dead-zone input v. Furthermore, β l and β r are the unknown slopes that characterise the left and right sides of the dead zone.From a practical standpoint, it is necessary to defne β � β r � β l .Ten, φ(v) can be represented as shown in [43] in the following form: where β(t) is the slope of the dead-zone and d(t) is defned as Since β l and β r represent unknown slopes, setting β � β r � β l makes β(t) an uncertain term.In addition, d(t) involves uncertain breakpoints R l and R r and uncertain term β, describing d(t) as an uncertain term.From (5), it is natural to suppose that |d(t)| ≤ d with d � max βR l , βR r  .One of the control objectives is to ensure that the tracking error e 1 � y(t) − y d (t) remains within a specifed funnel.Tis funnel is mathematically defned as , where the funnel boundary is represented as zΘ(t) � Θ ψ (t).In other words, for all t > 0, we want (t, e 1 ) to belong to the set Θ. Te specifc form of Θ ψ (t) is chosen as follows: where κ 0 > 0, κ ∞ > 0, and β > 0 are design parameters and Remark 2. In the work [40], they defne a funnel variable It is evident that χ 1 becomes nondiferentiable when e 1 (t) � 0, thus failing to meet the controller design requirement through backstepping.
To address the nondiferentiability issue associated with the mentioned variable in [40], a new funnel error transformation is defned as where e 1 � η 1 − y d .Te time derivative of χ 1 is given as where 3 .Control objectives.Te control objective of this work is to provide an adaptive funnel fault-tolerant controller for the nonlinear system (1) such that (i) All of the signals in the closed-loop system are SGUUB; (ii) Te tracking error remains inside a defned funnel.
To achieve this objective, we introduce the following assumptions regarding the system and the reference signal.
Assumption 3 (see [17]).Te reference signal y d and its n th order derivative are continuous and bounded.Furthermore, there exists a constant d * such that Complexity Assumption 4 (see [44]).For i � 1, 2, . . ., n, the signs of ϕ i (η i ) are known, and there exist unknown constants c i such that 0 Assumption 5 (see [17]).Te unknown time-varying functions ζ(t, t ρ ) and u r (t, t r ) are constrained within bounded limits.In other words, there exist positive constants ζ min and u max such that Assumption 6 (see [42]).Te measurement of the dead-zone output φ(v) is not available, and the slopes are identical in both positive and negative regions, specifcally, β r � β l � β.Assumption 7 (see [42]).Te parameters of the dead-zone, R r , R l , and β, are unknown but bounded, and their signs are known as R r > 0, R l < 0, and β > 0.
Remark 8. Assumption 3 is frequently employed in tracking control studies, aiming to facilitate subsequent stability analysis [17,44].As outlined in [45], Assumption 4 is justifed by the deviation of ϕ i (•) from 0, satisfying the controllable condition.It is important to note that the values of c i are only necessary for analysis purposes.Assumption 3 is a prevalent condition in fault-tolerant control for nonlinear systems, as discussed in [17].Assumption 4 indicates that the measurement of the dead-zone output φ(v) is unavailable, and the slopes are identical in both positive and negative regions.Meanwhile, Assumption 7 asserts that parameters of the dead-zone, namely, R r , R l , and β, are unknown but bounded, with their signs known.Assumptions 6 and 7 are commonly adopted in [42].
In this paper, radial basis function neural networks (RBFNN) W T P(X) are employed [20] to estimate uncertain continuous function Φ(.) defned within a compact set Ω ⊂ R q to achieve any desired accuracy ϵ > 0 such that represents the ideal weight vector, and P(X) � [P 1 (X), P 2 (X), . . ., P l (X)] T represents the basis function vector which is commonly chosen as Gaussian function as follows: where α i � [α i1 , α i2 , . . ., α iq ] T is the center and η is the width of the basis function.
Remark 10.In the neural network (NN) with q neurons in the input layer, l neurons in the hidden layer, and n neurons in the output layer, the computation for the input layer W T P(X) requires O(l * q) steps for the hidden layer (11).Te output layer is given by  n i�1 P i (Z)W i,j , necessitating O(l * n) computational steps.Here, W i,j denotes the weight vector originating from the k-th hidden neuron and targeting the j-th output neuron.Consequently, the overall computational complexity of the neural network is expressed as O(l * (q + n)).In a special scenario where n � 1 and l � q, the complexity simplifes to O(q 2 ).

Controller Design and Stability Analysis
In the following section, an adaptive funnel fault-tolerant control scheme for the nonlinear system (1), which employs the backstepping technique and neural networks approximation, is introduced, beginning with the following coordinate change: where ϑ i− 1 is the virtual control signal to be designed.
Step 1. Te Lyapunov function is considered as follows: where b 1 > 0 is a design parameter, c 1 is defned in Assumption 4, and c 1 represents the estimation error with  c 1 as the estimate of c 1 .Now, diferentiate (14), one has where Given that Φ 1 encompasses unknown nonlinear functions Φ 1 and ϕ 1 , the solution to this challenge involves employing RBFNN to approximate the unknown function Φ 1 .For any ϵ 1 > 0, one has Using completion squares, one has where β 1 > 0 is a design parameter, and We design the virtual controller ϑ 1 as follows: and the adaptation law as where q 1 > 0, μ 1 > 0 represent design parameters.By substituting ( 17)-( 19) into (15), we have Step i (2 ≤ i ≤ n − 1).By using ( 13), one has Te Lyapunov function is selected as follows: By taking time derivative of ( 22), one has where Te RBFNN can approximate the unknown function Φ i (X i ) with accuracy, ensuring that for any given ϵ n > 0, one has By using Lemma 9, one has where c i � ‖W i ‖ 2 /c i , and β i > 0 is a design parameter.Te virtual controller ϑ i is designed as and the adaptation law is designed as where q i > 0, μ i > 0 being the design parameters.Substituting ( 26)-( 28) into (23), we have Step n.Utilizing (13) and computing the time derivative of z n , one obtains Select the following Lyapunov function

Complexity
By diferentiating (31), one has where Te RBFNN can approximate the unknown function Φ n (X n ) with accuracy, ensuring that for any given ϵ n > 0, one has Furthermore, we have where c n � ‖W n ‖ 2 /c n , and β n > 0 being a design parameter.We defne the actual control law as By using Lemma 9 and Assumption 5, one has Using Assumptions 5-7, Lemma 9, and ( 36), one has We defne the adaptation law as where q n and μ n are positive design parameters.Using ( 35)-( 40) into (32), we have Theorem 11.Under Assumptions 3-7, the nonlinear system (1) with actuator faults (2), input dead-zone (3), the virtual control signals (18), (27), real controller (36), and adaptive laws ( 19), ( 28), (40), with bounded initial conditions.Te proposed control strategy ensures that the tracking error consistently remains within a specifed funnel boundary, while also ensuring the SGUUB) behavior for all signals in the closed-loop system with the initial condition of 6 Complexity Substituting ( 42) into (41), we have where From ( 43), one has which implies that V(t) is bounded by λ/ς.Consequently, all signals in the closed-loop system exhibit SGUUB behavior.Furthermore, (45) implies that Furthermore, substituting (45) into (7) results in: which implies that Furthermore, one has which shows that by carefully adjusting the design parameters, the tracking error can be minimized and still remain within the specifed limit.Figure 1 illustrates the block diagram representing the presented control method.

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Remark 12. Te derivation of control laws in backsteppingbased methods sometimes involves repeated diferentiation of virtual control inputs, presenting challenges such as the explosion of complexity, particularly in higher-dimensional systems.To address this issue and enhance applicability, command flters are employed [47,48], ofering a practical solution to manage the computational complexities associated with this control methodology.
Remark 13.Te existing literature has dealt with diferent control challenges in various scenarios, including strictfeedback nonlinear systems with constraints on states and input delays [49], and pure-feedback stochastic nonlinear systems with constraints using adaptive fuzzy control schemes [50].In addition, research has focused on stochastic nonlinear time-delay systems with multiple constraints using neural network-based adaptive control schemes [51].In comparison, this paper stands out by proposing a novel funnel-based adaptive fault-tolerant control scheme.Te distinctive feature of this approach lies in its focus on addressing nonlinear systems afected by dead-zone nonlinearity and actuator faults.Recognizing the practical implications of actuator faults on system performance and stability, the proposed scheme provides a valuable contribution to the feld by ofering an efective solution to these specifc challenges.

Simulation Results
Two practical examples are presented in this section to validate the performance of the proposed control method.
Example 1.Consider a rigid robotic manipulator system [44] with actuator faults and dead zones described by the following set of equations:

Complexity
where η 1 represents the angular position of the manipulator, η 2 corresponds to the relative angular velocity, m r stands for the load mass, l r represents the length of the manipulator, g v signifes the gravity, and J denotes the inertia coefcient, calculated as Te funnel function is represented as follows: with κ 0 � 3, κ ∞ � 0.15, and β � 0.3.Te model of actuator fault model is defned as where To begin, choose Te initial conditions are set as In addition, trial and error, we assign values to design parameters as q 1 � 50, q 2 � 30, Te centre and width of the RBFNN are chosen as Te simulation results, depicted in Figures 2-6, highlight the efectiveness of the proposed approach.In Figure 2, the successful tracking of the output signal y with the reference signal y d showcases excellent tracking performance.Figure 3 illustrates that the tracking error e 1 consistently stays within the specifed funnel Θ ψ , indicating the reliable tracking performance of the proposed method.Figure 4 show the boundedness of the system state η 2 , while Figure 5 ensures that both the system input u and control input v are bounded.Finally, Figure 6 reveals the bounded nature of the adaptive laws  c 1 and  c 2 .When t < 10, the actuator operates normally, while faults occur after t ≥ 10, as evident in Figures 2, 3, and 5. Te fault efects are also visible in the zoomed-in graphs.Although faults persist after t ≥ 10, the proposed control method efectively minimizes their impact, as illustrated in Figures 2, 3, and 5. Analyzing Figures 2-5, it is evident that the system output y efectively tracks the reference signal y d , and all closed-loop signals maintain bounded behavior.
To assess the efciency of the proposed method in comparison to a previously established approach [7], the following criteria for error assessment, as presented in [52], are employed.Te funnel-based adaptive control scheme is systematically compared with the existing control method [7] where n is the number of observations, y i is the output of the system, and y id is the reference signal.Te comparison results presented in Table 1 clearly indicate that the performance of the proposed control method is slightly superior to the existing method.Tis observation highlights the efcacy of the proposed approach, as evidenced by the evaluation of error assessment criteria.
Example 2. Consider an inverted pendulum system [46] as shown in Figure 7 with actuator faults and a dead zone described by the following set of equations: where θ represents the angle in radians, θ .
is the angular velocity in radians per second, m c � 1kg is the cart mass, m � 0.5kg is the pendulum mass, l � 0.5m denotes half of the pendulum length, g � 9.8m/s 2 is the acceleration due to gravity, and u denotes the system input.
, then the state-space representation is given by where Te funnel function is represented as follows: with κ 0 � 3, κ ∞ � 0.15, and β � 0.3.Te model of actuator fault model is defned as To begin, choose Te initial conditions are set as In addition, by the trial and error, we assign values to design parameters as q 1 � 50, q 2 � 30,   9 shows that the tracking error e 1 consistently stays within the specifed funnel Θ ψ , indicating the reliable tracking performance of the proposed method.Figure 10 confrm the boundedness of the system state η 2 , while Figure 11 ensures that both the system input u and control input v are maintained within prescribed bounds.Finally, Figure 12 reveals the bounded nature of the adaptive laws  c 1 and  c 2 .When t < 10, the actuator operates normally, while faults occur after t ≥ 10, as evident in Figures 9-11.Te fault efects are also visible in the zoomed-in graphs.Although faults persist after t ≥ 10, the proposed control method efectively minimizes their impact, as illustrated in the fgures.Analyzing , it is evident that the system output y efectively tracks the reference signal y d , and all closed-loop signals maintain bounded behavior.
In this example, the error assessment criteria defned in Example 1 are applied.Te results in Table 2      Complexity that the proposed control method exhibits a noticeable superiority over the existing method [7].Tis observation emphasizes the efcacy of the proposed approach, as evidenced by the evaluation of error assessment criteria.

Conclusion
Tis paper addresses the problem of fault-tolerant adaptive neural funnel control for nonlinear systems incorporating actuator faults and input dead zones.To ensure the boundedness of the tracking error, a modifed transformation for funnel error is introduced and integrated into the control design process.Employing radial basis function neural networks (RBFNN) to handle unknown nonlinear functions, an adaptive neural funnel fault-tolerant controller is designed using the backstepping technique.Te proposed controller ensures the tracking error remains within a predefned funnel, and all signals in the closed-loop system are SGUUB.Te viability and efectiveness of the proposed control approach are validated through simulations.Future work will focus on cyber-physical systems with unmodeled dynamics and sensor faults.

Figure 3 :
Figure 3: Te response of tracking error e 1 and Θ ψ for Example 1.

1 Figure 9 :
Figure 9: Te response of tracking error e 1 and Θ ψ for Example 2.

Figure 11 :Figure 12 :
Figure 11: Te of system input u and control input v for Example 2.
that does not incorporate a funnel.

Table 1 :
Comparison of the tracking performance using diferent error calculations for Example 1.

Table 2 :
Comparison of the tracking performance using diferent error calculations for Example 2.