Observer-based adaptive fault estimation and fault-tolerant tracking control for a class of uncertain nonlinear systems

This paper considers the problem of adaptive fault estimation and fault-tolerant tracking control for a class of uncertain nonlinear systems with model uncertainty and process fault. By utilising the approximation abilities of fuzzy logic systems, a novel fault estimation fuzzy observer is designed to estimate the unmeasured states, nonlinear functions and process fault. Based on backstepping technologies, a adaptive fuzzy fault-tolerant controller with the adaptive updated laws is established, which can not only ensure the prescribed tracking performances, but also effectively compensate the process fault. It is proved that all the closed-loop signals of systems are uniformly ultimately bounded and the tracking errors can constrain to the prescribed bounds. Finally, simulation results can verify the efﬁciency of the proposed approach.


INTRODUCTION
In the the past years, fuzzy logic systems (FLSs) and neural networks (NNs), as an effective method have been well applied to various uncertain nonlinear systems [1][2][3][4][5][6]. By using the approximation ability of FLSs, in [7], a fuzzy adaptive failure compensation control method was studied for nonlinear systems with unmodeled dynamics and unknown control directions, it can guarantee that all signals were semi-global boundedness. Ref. [8] investigated the problem of composite fuzzy control for uncertain nonlinear systems with unknown dead zone based on disturbance observer. Furthermore, a novel robust adaptive control scheme for uncertain nonlinear systems was considered in [9], based on interval type-2 fuzzy logic system and small gain approach, a composite feedback form was established, which can ensure all the signals were uniformly ultimately bounded (UUB). For a class of strict-feedback nonlinear systems with unknown time delay, the authors in [10] developed a fuzzy adaptive backstepping dynamic surface control scheme based on combination of the backstepping design technique and the dynamic surface control approach, the result showed that system output can track the desired reference signal in an optimal manner.
On the other hand, various constrained models have been applied to different systems. For example, the problem of robust distributed model predictive control (MPC) was investigated in [27] for nonlinear agents systems with control input constraints, which presented a robustness constraint to solve external disturbances and designed a novel robust distributed MPC technology. To handle the full state constrained problem of SISO nonlinear systems, asymmetric barrier Lyapunov functions (BLFs) were employed in [28] to prove that all states can be constrained into the prescribed bounds. Furthermore, work [25] enabled the tracking errors to converge to the predetermined bound by using BLFs. Moreover, in [29], a low-complexity states feedback FTC technique was proposed for uncertain nonlinear systems with unknown control directions. In addition, the problem of constraint control was studied in [30] for strict feedback nonlinear systems, by introducing constraint variable to ensure the transient and steady state performance for the tracking errors.
Specially, in [25], the problem of adaptive FE and FTC for nonlinear systems contains known nonlinear functions and satisfy the Lipschitz conditions. Because most practical systems do not have accurate system models, up to now, the problem of adaptive FE and FTC problems for uncertain nonlinear systems with errors constraint still need to be studied. Based on the above discussions, in this paper, we investigate the problem of adaptive FE and fault-tolerant tracking control of uncertain nonlinear systems with model uncertainty and process fault. Different from the existing results, the main contributions are as follows: (i) The adaptive FE and fault-tolerant tracking control scheme is spread to uncertain nonlinear systems with model uncertainties and process fault, and the improved scheme changes the prescribed bounds into time-varying functions.
(ii) Based on FLSs, an FE fuzzy observer is constructed, which can accurately estimate the unknown states of the systems. By introducing an intermediate estimator and designing FE adaptive law, the time-varying process fault can be accurately estimated. Compared with [25], the systems we considered contain unknown parameter models and also do not have to satisfy the Lipschitz conditions; besides, the proposed method changes the constraint condition from a constant to a time-varying function and makes the tracking errors converge to smaller neighborhoods.
(iii) Based on backstepping methods, an adaptive faulttolerant controller is designed to compensate the effects of process fault and model uncertainties, so that the tracking errors can be constrained to the prescribed bounds and all the signals of closed-loop systems are UUB. Besides, the controller does not need to introduce the derivative of virtual control, so the design of controller is simplified.
The rest section of this paper is organised as follows. The problem description and preliminaries are proposed in Section 2. The adaptive fault estimation fuzzy observer is designed in Section 3. In Section 4, adaptive fuzzy fault-tolerant controller with adaptation updated laws is designed. Section 5 gives the main theorem and the stability analysis. Simulation results are provided in Section 6. Finally, Section 7 draws the conclusion.

SYSTEM FORMULATION AND PROBLEM STATEMENT
Consider the following uncertain nonlinear systems with process fault described byẋ , n is the system state vector with the states x 2 , … , x n being unavailable, u ∈ R and y ∈ R are control input and measured output of the system, respectively. b and m are known gain, f i (⋅) stands for the unknown smooth system functions with f i (0) = 0, and f 0 denotes the unknown time-varying system fault.
The control objective of this paper is to design a novel adaptive FE and FTC strategy such that the systems output y can track the reference signal y d and all the signals of the closedloop system are UUB. In order to ensure achieve the control goal, in what follows, the following assumptions are introduced for later developments. Assumption 1. The reference signals y d is assumed a known and bounded function with its ith time derivative y (i ) , i = 1, 2, … , n.

Assumption 2.
Fault signal f 0 is assumed to be bounded and its 1th derivativeḟ 0 is also bounded. That is, there exist positive scalarsf 0 andf 0 such that | f 0 | ≤f 0 and |ḟ 0 | ≤f 0 . Remark 1. Assumptions 1 and 2 are quite general assumptions in the literatures [5,12,14]. Particularly, if m = b, the process fault f 0 can degrade into the actuator fault. Duo to the existences of the nonlinear uncertainty, the proposed adaptive FE and FTC method will become a more complex problem.
Lemma 1 [31]. For any nonlinear continuous function f (x) defined on a compact set Ω, there exist a fuzzy logic system W T S (x) such that for any given constant > 0 denotes the basis function vector, and N is the number of the fuzzy rules, S i (x) are chosen as Gaussian functions, that is, where b j , c j = [c j 1 , c j 2 , … , c jN ] T are the vector of the width and center of the Gaussian functions.

ADAPTIVE FAULT ESTIMATION FUZZY OBSERVER DESIGN
In this section, a novel FE fuzzy observer is designed to estimate unmeasurable states, process fault and nonlinear function, moreover, the adaptive FTC scheme is developed based on backstepping approach.
Since the function f i (x i ) is unknown, so the FLSs are used to approximate it. According to Lemma 1, we assumê The optimal parameter vector W i is designed in the given compact sets Ω i and Ξ i where Ω i and Ξ i are the compact sets forŴ i and (x i ,x i ), respectively. Notice that the approximation error is defined as In what follows, design the fault estimation fuzzy observer aṡx with FE adaptive laẇF whereŴ i ,f 0 andF are the estimations of W i , f 0 and F , F is an intermediate estimator which will be given in (11). l i , > 0, i = 1, 2, … , n are corresponding design parameters. (1) and (6), one haṡ Furthermore, the estimation error system (8) can be rewritten aṡ= where Consider the structure of A, and l i is the design constant such that A is a Hurwitz matrix. For any positive definite matrix Q = Q T > 0, there exists positive definite matrix P = P T > 0, which satisfy the Lyapunov equation A T P + PA = −Q. Lemma 2. Consider error system (9), under the conditions of Assumption 1 and 2, the fault estimation fuzzy observer (6) and FE adaptive law (7) can ensure that f is UUB.
Proof: First, we introduce an intermediate estimator soF =f 0 −x n . Define F = F −F , according to the formula (6) and intermediate estimator (11), we havė Then, selecting the following Lyapunov function as Substituting (9) and (12) into (14), we havė Utilising the triangle inequalities where S T (x n )S (x n ) ≤ 1. Moreover, by invoking the triangle inequalities (16) and (15), it can be obtained thaṫ Since the unknown function f i can be approximated well by the FLSs,Ŵ i S i ,̄i is bounded. From (17), it is clear that f is UUB by selecting the appropriate parameters.
Remark 2. In the process of fuzzy observer design, Lemma 1 is applied to approach the uncertain nonlinear functions, which can deal with the systems uncertainties. Lemma 2 provides a rigorous proof of convergence analysis for FE fuzzy observer (6). Furthermore, an intermediate estimator is introduced to estimate the fault, which can improve the accuracy of estimation of fault.

ADAPTIVE FUZZY FAULT-TOLERANT CONTROLLER DESIGN
This section develops a fuzzy fault-tolerant controller for the uncertain nonlinear systems (1) based on the backstepping design method. First, the state transformation is given as follows where y d is the reference signal, i−1 is the virtual control function which is designed in step i − 1, the actual control is designed in step n. step 1: Since z 1 =x 1 − y d , from (4) and (6), we havė Then, define the following Lyapunov function where K 1 is the prescribed bound of z 1 with K 1 = ( 1 − 1∞ )e − 1 t + 1∞ , Γ 1 > 0, r 1 > 0 are positive parameters,̂ * 1 is the estimation of * 1 with̃ * 1 = * 1 −̂ * 1 , which is an unknown constant being defined later. From (19) and (20), the derivative of V 1 iṡ where which is unknown. Next, by Lemma 1, for any givenΔ 1 > 0, Substituting (18) and (22) into (21) results iṅ Design the virtual control function 1 and adaptive laws as where c 1 , 1 and 1 are positive design parameters. Substituting (24) into (23) yieldṡ where 1 = are positive parameters,̂ * i is the estimation of * i with̃ * i = * i −̂ * i . In addition, it is easy to verify that i is a function of Similar to (19)(20)(21)(22), we havė where . By Lemma 1, for any givenΔ i > 0, we Accordingly, design the virtual control function i and adaptive laws as where c i , a i , Γ i , i , r i and i , i = 2, … , n − 1 are the design positive constants. Therefore, we can obtain thaṫ . step n: The actual control input u will be given in this step. Consider the following Lyapunov function where K n is the bound of z n with K n = ( n − n∞ )e − n t + n∞ , Γ n > 0, r n > 0 are positive parameters,̂ * n is the estimation of * n with̃ * n = * n −̂ * n .
Taking the derivative of V n along the trajectory of z n giveṡ where . According to Lemma 1 gives f n = T n n (X n ) + Δ n , |Δ n | ≤Δ n withΔ n > 0 is positive constant. Define * n = ‖ n ‖ 2 , one has Consequently, choose the final control u and adaptive laws as where c n , a n , Γ n , n , r n and n , are design positive constants. substituting (33) into (32) yieldṡ

STABILITY ANALYSIS
In this section, the main result will be given at first, and the stability of the closed-loop systems will be analysed later.

(i) the systems output y can track the reference signal y d and all the closedloop signals of systems (1) are UUB; (ii) the tracking errors can converge to the prescribed bounds, that is,
Proof: To discuss the stability of closed-loop system (1), define the following Lyapunov function as It follows from (17) and (34) thaṫ Note thatW Substituting (37) into (36) yieldṡ i . Furthermore, by choosing i , 1 and 4 so that and define Thus, (38) can be rewritten aṡ Solving inequality (41) from 0 to t gives This means that V (t ) is bounded, thus, all the closed-loop signals of systems (1) are uniformly ultimately bounded. In addition, (42) also implies that 1 2 Furthermore, with 0 < e −Λt ≤ 1. Then, from Λ > 0,̄> 0 and V 0 > 0, one has which means that Therefore, the tracking errors can converge into the prescribed bounds. This completes the proof.
Remark 3. In the existing reference [25], the problem of adaptive FE and FTC for nonlinear systems contains known nonlinear functions and satisfy the Lipschitz conditions. However, it is very difficult to get accurate system model in practical engineering environment. In this brief paper, the problem of adaptive fault estimation and fault-tolerant tracking control is studied for a class of uncertain nonlinear systems with model uncertainty and process fault.

SIMULATION STUDIES
In this section, two examples are given to illustrate the effectiveness of the proposed method.
The design parameters are designed as = 4, l 1 = 80, l 2 = 600, c 1 = 60, c 2 = 500, a 1 = 100, a 2 = 100, r 1 = 50, r 2 = 40, Γ 1 = 0.8, Γ 2 = 1, 1 = 2 = 7, 1 = 2, 2 = 5, y d = sin(t ). In addition, the initial values are chosen as x 1 (0) = 0.1, The simulation results are shown in Figures 2-7. From Figure 2, it is seen that, under occurring process fault, the system output y can still track the reference signal y d very well and the estimationx 1 can also track the state x 1 . Figures 3 and 4 display the output errors z 1 and z 2 ; we can see the output errors z 1 and z 2 can converge into the specified performance bound functions. The fault and its estimation are exhibited in Figure 5, from which it shows that proposed method has a good estimate ability of unknown process fault. Figures 6 and 7 show the control input u and adaptive parameters F, Example 2. Consider the following single-link robot systems cited in [20,25], the uncertain nonlinear systems with fault is described asẋ where x 1 = q and x 2 =q denote the angle position and velocity, respectively. The uncertain nonlinear function f 1 (x 1 , x 2 ) = −(1∕(2M ))m 0 gl sin(x 1 ) and the corresponding system parameters of f 1 (x 1 , x 2 ) are chosen as m 0 = 1kg, M = 5kg ⋅ m 2 , g = 9.8m∕s 2 and l = 1m. The bounded functions defined as The design parameters are selected as m = 2, = 5, l 1 = 30, l 2 = 400, c 1 = 60, c 2 = 400, a 1 = 100, a 2 = 150, r 1 = 100, r 2 = 80, Γ 1 = 0.01, 1 = 70, 1 = 20, 2 = 50. What is more, the initial values are chosen as According to Lemma 2 and Theorem 1, the adaptive fuzzy fault-tolerant controller for systems (51) is designed. To illustrate the effectiveness and feasibility of the presented method, the influences of different tracking signals and different faults on the system are considered under the same conditions. Case 1. We first consider the reference signal y d = sin(t ), process fault f 0 = sin(t ) + cos(t ). Simulation results are demonstrated in Figures 8-12. From Figures 8-10, it can be seen that system output y can track ideally reference signal y d and the error signals z 1 and z 2 can be converged into performance bound by using the presented method. Figure 11 shows the response curves of f 0 andf 0 . The control input u, adaptive laws F , 1 , 2 and W 1i are exhibited by Figure 12.
Case 2: This case selects the reference signal and process fault as y d = cos(t ) − 1, f 0 = sin(t − 0.3) + 0.35, respectively. From Figure 13, we can see that system output y can track the reference signal y d very well. The tracking errors z 1 , z 2 and f is presented in Figures 14-16, respectively. From which, it can been seen that the proposed method has a good estimate ability of the systems states and unknown fault. Finally, Figure 17 gives the bounded control input signal and the adaptive laws signals.  Remark 4. From the above simulation results, it can be seen that the method we proposed is feasibility for nonlinear systems with model uncertainties and process fault. For different reference signals and different faults, the system can all achieve good tracking results by the proposed method. Thus, the studied adaptive fault estimation and fault-tolerant tracking control scheme is effective for uncertain nonlinear systems and it can be applied to practical systems.

CONCLUSION
In this paper, the problem of adaptive fault estimation and faulttolerant tracking control is investigated for uncertain nonlinear systems. The considered nonlinear systems contain immeasurable states, fault and unknown nonlinear function. To estimate systems immeasurable states and nonlinear function, a novel fault estimation fuzzy observer is designed based on fuzzy logic systems. By designing an intermediate estimator and FE adaptive law, the time-varying process fault can be accurately estimated Moreover, by utilising adaptive backstepping method, a adaptive fuzzy fault-tolerant controller with prescribed constraint is designed and the corresponding adaptive laws are given at the same time. It can be proved that the tracking error signals can converge into the prescribed bounds and all the signals of closed-loop systems are UUB. Two examples can illustrate the efficiency of the proposed approach.