Fuzzy observer-based command filtered tracking control for uncertain strict-feedback nonlinear systems with sensor faults and event-triggered technology

For the trajectory tracking problem of nth-order uncertain nonlinear systems with sensor faults, a fuzzy controller based on command filtered and event-triggered technology is designed to improve the tracking error of the system. Concurrently, a fault-tolerant control scheme is introduced to effectively solve the problem of sudden output sensor failure. Additionally, the proposed controller can also greatly avoid complexity explosion problem of derivations of virtual control laws, which makes the design of the controller simpler. Furthermore, an effective observer is designed to solve the problem of system state immeasurability. Therefore, the proposed control scheme makes the design of the controller more convenient and flexible. According to Lyapunov stability theory, it is proved that all closed-loop signals are uniformly and ultimately bounded. Finally, two simulation examples of second-order nonlinear system and single-link robot show the effectiveness of the proposed scheme.


Introduction
In previous studies, many researchers have paid more attention to adaptive control for uncertain nonlinear systems. Because the optimization of controller design and the improvement of performance are of great significance to adaptive control, thus backstepping method is widely used in the design of nonlinear controller because of its many advantages. The control goal is to make the parameter estimation error converge when the parameter is identified under some operations or continuous external signals. The nonlinear control laws in traditional adaptive backstepping control were derived from the established model [8]. Then, a series of backstepping applications were developed for various classes of nonlinear systems, such as uncertain chaotic systems [34], nonlinear systems with dynamic uncertainties [26], nonlinearly parameterized systems with periodic disturbances [3] and pure-feedback systems [23]. Additionally, some real applications were also considered by using backstepping method, such as a sliding-mode torque control for induction motor drive [25] and a servo control for permanent magnet synchronous motor [12].
Although backstepping approach is effective for controller design for nonlinear systems, it is obliged to say that the calculations of derivations of virtual control laws will be very complex if the system order is too high. Therefore, command filtered technique was proposed to solve the complexity explosion problem and be more convenient in the design of controller [6]. On the basis of command filters, the fuzzy approximations were introduced in [32] and [9]. By introducing command filters, the calculations of partial derivatives were not required to make the control law and adaptive law simpler. In [15], a finite-time controller was designed by using command filtered technique, so as to ensure both tracking performance and closed-loop stability in finite time. In [31], a control problem of nonlinear systems with unmodeled dynamics and input saturation was solved by combining with command filters. The above results illustrate the effectiveness of command filtered technique.
In the process of information transmission, network resources are usually limited, so saving communication resources have gradually become an important indicator in the controller design process. In order to make full use of communication resources, researchers proposed a variety of event-triggered schemes, such as distributed strategy based on small gain method [4] and switching threshold strategy [30]. Even the system actuator fails, it can still reflect the superior performance of the event-triggered technology [29]. In addition, event-triggered technology also has certain contributions in uncertain nonlinear switching systems [16], singularly perturbed systems [10] and nonlinear systems with input delays [19]. The event-triggered technology can not only be applied to different systems to improve the controller performance of the system, but also it can be combined with other methods to achieve better control effect. The event-triggered technology was combined with the adjustable time performance function, so that the system controller can be designed to achieve smooth switching between fixed threshold and relative threshold [17]. From the existing results, it can be seen that the event-triggered technology can further improve the design performance of different types of controllers for nonlinear systems, and its diversity and performance advantages are also verified through extensive applications.
When the system states are unknown, fuzzy observer is effective for controller design. The fuzzy observer can estimate the system states to make the design of the controller more flexible and convenient. So far, fuzzy observers are widely used. In [13], a fuzzy observer was designed based on finite-time adaptive backstepping method and dynamic surface control. In [14], a fuzzy output feedback fault-tolerant control was proposed to solve the problem of unknown faults and immeasurable states in the system. In [7], the tracking error was controlled within the preset range by using Tan-barrier Lyapunov. In [33], a finite-time output feedback control scheme was presented for uncertain nonlinear systems with actuator faults and disturbances. By both considering input dead zone and saturation for actuator, the control problem was solved by using fuzzy observer and Nussbaum gain method [28]. To deal with control problem for nonlinear systems with time-varying disturbance, an observer-based adaptive dynamic surface controller was designed [2]. Furthermore, some different types of fuzzy observer were also developed and the existing results verify the diversity and practicability of fuzzy observers: (1) High-gain fuzzy observers were designed to achieve a strong robustness for the uncertainties of nonlinear functions [38] and [39]. (2) Fuzzy disturbance observers were proposed to further realize a better robust performance to disturbances of nonlinear systems [1] and [21]. (3) Reduced-order fuzzy disturbance observers were developed to reduce the complexity of observer design [22] and [18].
As we all know, the controller design process of actual systems becomes more difficult due to the presence of sensor or actuator failures. Therefore, more attention should be paid to the safety and stability of the system when sensor or actuator failure occurs. Once a fault occurs, the system performance will be reduced. In view of such problems, a cubic absolute Lyapunov stability theory was proposed to estimate the fault information online [36]. In addition, the effect of sensor fault was eliminated through the signal compensation mechanism [35], so as to achieve a desired effect. For uncertain nonlinear systems with mismatched nonlinear fault functions and external disturbances, a faulttolerant controller was designed by recursive observers with small output estimation errors [37]. For Lipschitz nonlinear systems, a fault-tolerant control strategy was applied to an adaptive integral sliding mode, so as to eliminate the influence of actuator fault [11]. Furthermore, the problems of system instability and tracking performance degradation caused by sensor failure were solved in [24] and [5]. The above results show that the occurrence of sensor fault in the system is inevitable, and eliminating the impact of this problem is also a major mission in nonlinear system control.
From the mentioned results, we can conclude that there are some challenging problems that have not been solved. When there exist sensor faults and complexity explosion problem in high-order strict-feedback systems, the following difficulties need to be faced: (1) How to ensure the stability of nonlinear system when the output signal is affected by sensor failure; (2) How to use error feedback information to design state observer; (3) How to design an effective controller to avoid complexity explosion problem and save the communication burdens of actuator simultaneously.
Based on the above design difficulties, in the case of sudden sensor failure, it is more challenging to ensure that the controller design can make strict-feedback nonlinear systems have good tracking performance. Driven by the discussions, the tracking control problem of nthorder uncertain nonlinear systems with sensor faults is studied in this paper. The main contributions of this paper are summarized as follows: (1) This research is the first solution to the problem of sensor fault and differential explosion under the backstepping framework. The related works [7,14,28,36] and [35] exist complexity explosion problems in the design procedure of virtual control laws; thus we introduce command filtered technique to solve this problem. Here, we choose firstorder low-pass filters to make the control scheme simpler; (2) A new type of fuzzy observer is designed to estimate the unknown system states. Different from the existing fuzzy observers [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], we introduce an estimation term into the observer to compensate the effects of sensor faults. Furthermore, a more effective cubic Lyapunov function is introduced in the controller design process. The estimation accuracy of sensor fault coefficient can be improved by reasonably selecting parameters; (3) A novel event-triggered strategy is designed to deal with the control problem of nonlinear systems with sensor failures, that is, the problem of network resource bandwidth can still be considered when sensor failure occurs in the system. Comparing with the existing strategies [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], the proposed strategy has a smooth switching property rather than the relevant existing strategies, which is an improved switching threshold strategy to save the communication burdens of actuator.
To sum up, we proposed a novel control scheme for nonlinear strict-feedback systems with the sensor faults. Different from the traditional control scheme for dealing with sensor faults, we introduce command filtered control approach to optimize the complexity of the controller for high order systems, so that complexity explosion problem is eliminated successfully. Additionally, in the case of increasingly tight network resources, the proposed control scheme can solve this problem by using a novel switching event-triggered strategy. Considering that when the proposed scheme is applied to the actual system, most of the system states are normally unknown; then a fuzzy observer is adopted to estimate the unknown system states to make the proposed controller be of great significance in practical systems. The proposed controller is applicable to some practical nonlinear systems with strict-feedback forms, such as linear stepping motor [39] and single-link robot systems [20].

System statement and control objective
Consider the following uncertain nonlinear systems: ..n represent the systems states, y ∈ R denotes the output of the system, and g i (x i ) are unknown smooth nonlinear functions with g i (0) = 0. Definition 1 [5,24]: (Scaling measurement sensor fault): If the system has a sensor fault at time T f , then the measuring variable ζ f (t) ∈ R is expressed as The form of sensor fault in (2) is called a partial loss fault. According to this definition, the possible sensor fault in nonlinear system ( 1) can be written as follows: where β = 1 indicates that the sensor has not failed.
The control goal is to design a fault-tolerant controller with event-triggered strategy to ensure that the output of the systems can track the trajectory of the desired reference signal. Concurrently, all the signals in the closed-loop system are bounded. Before the design of the controller, the following assumption, lemma and proposition need to be established: Assumption 1 [32]: The reference signal y r (t) and its first-order time derivativeẏ r are continuous and bounded.
for any x ∈ R and σ > 0 is a constant.
Note that sgn (·) denotes a symbolic function in this paper.

Fault-tolerant fuzzy observer design
In this section, the design process of fuzzy observer is introduced. The fuzzy observer has the following advantages: (1) The known nonlinear functions in the system are not required in the controller. (2) When the system states are unknown, the observer is used to estimate the system states. To realize fuzzy observer design, we need to introduce fuzzy logic system (FLS) firstly. From [13] and [14], the knowledge base of FLS contains the following IF-THEN rules: where x = [x 1 , ..., x n ] T ∈ R n is the input of FLS, and g ∈ R is the output, N represents the number of IF-THEN rules, U z i and V z are called fuzzy sets and related to their fuzzy membership functions ϕ U z i (x i ) and ϕ V z (x i ) . Then, FLS can be described as follows: Next, define the following function: Finally, f (x) can be written as where Lemma 2 [27]: For a continuous function g (x) within compact set , then there is a FLS (8) such that where ε > 0 represents an approximation error.

Command filtered controller design with event-triggered technology
In this section, the controller design is based on command filtered and event-triggered technology. On this basis, the backstepping recursive method is used to complete the derivation. It is worth noting that sensor failures often occur suddenly, and the impact on the original system is also random. Due to the randomness of the failure, the failure time T f and parameters β are unknown. Therefore, the change of tracking error z 1 caused by sudden sensor failure has become one of the difficulties in the process of controller design, and thus we need to introduce the following transformation: If the sensor fault becomes more and more serious with the extension of time, the stability and tracking performance of the original system will be greatly reduced, that is, the tracking error z 1 will deviate from the true value after the fault occurs. Therefore, from the perspective of control theory, in the case of sudden sensor failure (26), a control signal should be sought to reduce the error between the actual measured value and the actual fault output value. To solve this problem, a cubic absolute Lyapunov function [36] is used to compensate the error between them, and fault tolerant control is added to the fuzzy observer to ensure the stability of the closed-loop system. Due to the change of tracking error caused by sensor failure, the following coordinate transformations are designed: In order to design the controller by backstepping approach, we define where X i is an error variable and ς i is called a compensating signal.ᾱ i is the output of the first-order command filter with respect to the virtual control law α i , which is defined as where ν i is a positive constant.
Step 2: For the 2nd subsystem, the following Lyapunov function is selected: where τ 2 > 0 is a design parameter. Then,V 2 is given bẏ According to X 2 = z 2 − ς 2 and z 2 =x 2 −ᾱ 2 ,Ẋ 2 is obtained in the following form: Take subsystem (49), the specific form of X 2Ẋ2 is obtained as follows: Then, a compensation signal ς 2 is generated bẏ where b 2 is a design parameter. According to (50) and (51), it gives that Next, the virtual control law α 3 is designed as By introducing (53) into (52), the final form of X 2Ẋ2 can be written in the following form: Substituting X 2ι y f ≤ X 2 2 /4 + ιy f 2 , (46) and (54) in (48) produceṡ where ϑ is defined by Design the adaptive laws · κ 2 and · ι as follows: where ρ 2 , τ 2 and σ r are positive parameters, and m is defined by Combining with (58), and the processing ofκ T 2κ 2 and X 2κ T 2 φ 2 (x 2 ) is similar to Step 1, thus the results ofV 2 can be stated as follows: where Step i : (i = 3, ..., n − 1) For the ith subsystem, the following Lyapunov function is selected: where τ i > 0 is a design parameter. Then, the compensation dynamicς i , virtual control law α i+1 and adaptive law · κ i are designed as follows: where b i is a positive design parameter. Similar to Step 1 and 2, it yields thaṫ where D j = b j − 1/2 > 0 j = 3, 4, ..., n − 1.
Step n: For the nth subsystem, the Lyapunov function is selected as where τ n > 0 is positive parameters andκ n = κ * n −κ n . The derivative of V n over time is given bẏ From (37), we have ς n = 0, then it gives that X n = z n . According to z n =x n −ᾱ n , it gives thaṫ Take subsystem · x n = u+w n ιy f −x 1 +ĝ n x n |κ n , g n x n |κ n =κ T n φ n (x n ) into (68), the specific form of X nẊn is Next, an event-triggered strategy is introduced to save system communication resources. Inspired by the fixed and relative threshold control schemes designed by Xing et al. [30], we proposed a novel threshold strategy to realize a smooth switching mode, that is, a term (2m 1 /π ) arctan |u| is introduced to achieve the purpose of saving communication resources. Then, define the actual control law as follows: Then, define an error between the intermediate control law μ n (t) and the actual control law u (t) as c (t) holds. Note that c = (2m 1 /π ) arctan |u| + m 2 and c <c, m 1 , m 2 andc are all positive design parameters.
According to u (t) = μ n (t)−c (t) , X nẊn is given by Design the intermediate control law μ n as where σ is a positive design parameter. By substituting (72 ) in (71), the final form of X nẊn can be obtained.
From (65) and (73),V n can be given bẏ Then, the adaptive law · κ n is designed as the following form: where ρ n is a positive parameter. Combining with (65), the specific form ofV n is given as follows: From (76), we can get that From c <c, | (t)| ≤ 1 and Lemma 1, it yields that Finally, the following results hold.

Fig. 1 Block diagram of the proposed control scheme
In order to show of the proposed control scheme clearly, we provide a block diagram shown in Fig. 1.
Then, it yields from (81) such thaṫ For ∀t > 0, by integrating both sides of (84), it follows that From (85), we can see that the signals X i ,κ i ,κ i and the observation errors e i (i = 1, ..., n) are bounded. Additionally, it can be realized by selecting appropriate parameters to make the constant b as small as possible. In order to prove that all closed-loop signals are bounded, it is necessary to prove the boundedness of compensation signals.
From X i = z i − ς i , if the compensation signal ς i is bounded, then the tracking error z i must be bounded. To ensure the boundedness of compensation signal ς i , we construct the following Lyapunov function: The time derivative of V ς giveṡ According to the description in [6], we have ||ᾱ i+1 − α i+1 || ≤ i with i being a known constant, hence, the following results hold.
where β ς = min {2b i − 1} and γ ς = n i=1 2 i /2. For ∀t > 0, by integrating both sides of (88), it can be expressed as The results of (86-89) are sufficient to indicate that the compensation signal ς 1 is bounded. Therefore, the system tracking error z 1 is bounded. Combined with Assumption 1, it is known that the system state x 1 is bounded. Since the conclusion that the observation error e 1 is bounded, it implies thatx 1 is bounded. Meanwhile, the compensation signal ς i is bounded, so the error z i is bounded. According to the inherent property of the command filter, since the input of the command filter is a bounded signal, its output is continuous and bounded, thus the boundedness of the virtual control law is ensured. According to coordinate transformation,x i and x i (i = 2, ..., n) are also bounded. To sum up, it shows that all closed-loop signals are ultimately uniformly bounded.
2) For ∀t ∈ [t m , t m+1 ), we recallē(t) = μ n (t) − u(t), then one has From (72),μ n is a function that consists of bounded signals (x 1 ,x 1 , x 2 ,x 2 ,x n−1 ,x n , X n ,κ 1 ,κ n−1 ,κ n ) and the functions g 1 (x 1 ),Ẋ n ,ᾱ n−1 ,ᾱ n , (72) is differentiable because y is differentiable. From the design process, it can be seen that g 1 (x 1 ),Ẋ n ,ᾱ n−1 ,ᾱ n , · α n−1 ,α n and · κ n are all composed by bounded signals of the closed-loop system. Generally, φ 1 (x 1 ) , φ n−1 (x n−1 ) and φ n (x n ) are chosen as Gaussian function form, which means they are bounded. Furthermore,φ n (x n ) is also bounded. Therefore,μ n must be bounded. Additionally, there must exist a constant k > 0 such that |μ n | ≤ k. Becausē e(t m ) = 0 and lim t→t m+1ē (t) = c, it shows that the lower bound of t * must satisfy t * ≥ c/k. The demonstration results show that Zeno behavior is successfully avoided.

Simulation results
In this section, the effectiveness and feasibility of the proposed control scheme are verified by two examples.
(1) Numerical simulation: Choose the following nonlinear system to verify the performance of the designed controller: where x 1 and x 2 in (91) are system state variables, y is the system output, and u is the actual input signal of the system, g 1 (x 1 ) = 0.5 sin (x 1 ) cos (x 1 ) and g 2 (x 2 ) = 0.5 sin (x 1 ) cos (x 2 ) . The reference signal y r of the system is selected as y r = sin (t) . The selection of initial conditions are chosen as follows: For the approximations of g 1 (x 1 ) and g 2 (x 2 ), the fuzzy membership functions are chosen as The event-triggered strategy is given by with c = (2/π ) arctan |u| + 3 and c <c = 10. The compensation dynamicς 1 is presented in (34), and the virtual control law α 2 and the actual control law u are shown in (36) and (72) with b 1 = 9.8, b 2 = 1 and σ = 1. The adaptive laws · κ 1 , · κ n and ·l are described as (45), (75) and (58) with n = 2, τ 1 = 1, τ 2 = 1, ρ 1 = 60, ρ 2 = 60, d = 1, and σ r = 2. The observer parameters are w 1 = 3.5 and w 2 = 3.5. The filter parameter is ν 2 = 0.01. Then, we choose the parameters of ϑ as r 1 = 0.05,ξ = 3 andζ = 1. When the sensor fault occurs, the output of the system is y f = 0.8y.
By using the selected parameters, the tracking control of system (91) can be realized. In order to verify the advantages of the proposed controller, we compare the simulation results with the output feedback fault-tolerant controller in [35]. By selecting appropriate parameters for the controller [35], the tracking performance results are shown in Fig. 2. Figure 2 shows the output variables of two schemes start from an initial value of 1.5, and the output of the scheme [35] follows up with the reference signal at about 3 s. By contrast, the proposed scheme achieves this goal within 0.5 s. When the sensor fault occurs at 17.5 s, it shows that both schemes can track the reference signal well. Meanwhile, the sensor fault has less impact on the proposed scheme. From the perspective of tracking errors, Fig. 3 shows the tracking error of the proposed scheme starts from the initial value of 1.5 and converge into a small neighborhood near zero at about 0.5 s, and the scheme [35] needs more time to do so, which means the transient performance of the proposed scheme is better than that of the scheme [35]. Additionally, the overshoot and the sensor fault effect of the scheme [35] are both larger, and the steady-state performance of the proposed scheme is also better. Therefore, it is concluded that the proposed scheme can track the reference signal in shorter time and has better performance when sensor fault occurs. Furthermore, the proposed scheme has the advantage of saving communication resources rather than the scheme [35]. Figure 4 shows the intermediate control law μ n and the actual control law u, and it displays that the controller only needs a few trigger events after 0.5s to achieve the control goal. The corresponding trigger times are shown in Fig. 5. Figure 5 shows that the triggering times are 95 within 30 s under the proposed switching strategy. On the contrary, the controller [35] needs to transmit the control signal continuously and real timely. Moreover, the adaptive parameters of the proposed scheme are shown in Fig. 6; it shows that the adaptive parametersι,κ 1 andκ 2 are bounded and feasible. Notice that vectorsκ 1 andκ 2 are displayed in 2-norm form. The simulation results demonstrate that the proposed scheme is effective for numerical systems. In next section, an experimental simulation for practical systems is carried out.
(2) Experimental simulation Consider the following single-link robot system [20]: where χ is the angle, u is the input torque, M is the moment of inertia, g is the acceleration of gravity, m is the mass of linkage, l is the length of linkage, and the where x 1 = χ, x 2 =χ . Define g 2 (x 2 ) = −mgl sin γ /2M. The reference signal y r is selected as y r = sin (t) . For the approximation of g 2 (x 2 ), we choose the same fuzzy membership functions as example 1.
The simulation operation is the same as that of numerical simulation, which means the sensor fault occurs at 17.5 s to verify the tracking performance of the proposed scheme and the scheme [35]. Figures 7 and 8. show that the tracking performance and tracking error comparisons of two schemes. It can be seen that the simulation results are similar to numerical simulation. To deal with the control problem of practical system, the proposed control scheme is also better than the scheme [35]. In the aspect of event-triggered strategy, the control laws μ n and u are shown in Fig. 9. Figure  10 shows the triggering times of the proposed eventtriggered strategy are 85 times in 30 s. The adaptive parametersι andκ 2 are shown in Fig. 11 to illustrate the effectiveness of the adaptive laws. The simulation results also show that the control scheme proposed can achieve better results when applied to practical systems. Additionally, we can see that the controller [35] contains a large number of partial derivatives. When the system order is high, the complexity explosion prob- lem is very difficult to handle. However, the proposed scheme is out of this problem by introducing command filters. To sum up, the simulation results and discussions verify the superiority and feasibility of the proposed scheme.
Finally, in the process of parameter selection, the impact factors of the parameters on tracking performance are summarized as follows: (1) The larger sensor fault effect (the smaller value of parameter β ) results in the more instantaneous changes of control signal when the sensor fault occurs; (2) The parameters b 1 , b 2 , w 1 , w 2 , ν 2 , r 1 , σ r ,ξ andζ have a certain impact on tracking performance, and the increases of the parameters ρ 1 and ρ 2 lead to the overall upward movements of system output, and thus these parameters need to be adjusted carefully and appropriately. Notice that b 1 and b 2 must satisfy b 1 > 1 and b 2 > 3/4; (3) Within a certain range, the upper bound of the event-triggered thresholdc can affect the results of μ n and u. The larger the upper bound ofc leads to the more triggering times shown in Figs. 3. and 9. It is worth noting that the selected value of parameter c could not exceed the upper boundc. In addition, the parameters m 1 and m 2 also affect the triggering times, and the tradeoff between tracking performance and triggering times should be considered when selecting the values of m 1 and m 2 . (4) The rest parameters τ 1 , τ 2 and d are not sensitive on tracking performance, and thus we can set them to the value of 1.

Conclusion
An adaptive tracking control scheme has been designed for uncertain nonlinear systems with sensor faults. Based on backstepping method, the stability of the system has been analyzed by using Lyapunov theory. The fuzzy observer, command filtered technique, faulttolerant control and event-triggered strategy have been combined to ensure that all closed-loop signals are uniformly and ultimately bounded. Then, two examples have been simulated to illustrate the effectiveness and flexibility of the proposed controller. Finally, the future research directions are summarized as follows: (1) The proposed design is not limited to nonlinear single-input and single-output systems. Further research on control problem for nonlinear multiinput and multi-output systems can be carried out.
In addition, the proposed control scheme is also suitable for other forms of nonlinear systems, such as switching systems, stochastic systems, and timedelay systems; (2) Other types of fuzzy observers can be tried to improve the performance of the proposed control scheme, such as high-gain fuzzy observer or reduced-order fuzzy observer; (3) In order to improve the transient and steady-state performance, the prescribed performance control can be combined with the proposed control scheme. Additionally, the finite-time control can be also considered to realize a faster convergence speed of tracking error. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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