Fault decomposition-based convergent FE and FTC for Lipschitz nonlinear systems

The problem of fault estimation and fault-tolerant control for Lipschitz nonlinear systems subject to actuator and sensor faults is investigated in this paper. Different from the lower triangular matrix linear transformation method in the literature, a fault decomposition technique is proposed to design a set of relaxed iterative observers, so as to derive the iterative estimates for the state and multi-fault. It can be proved that in certain condition, the obtained mean sequence of estimates converge to the true values of state and multi-faults as the number of iterations increases. A perturbation coefficient matrix-dependent LMI condition that guarantees the states of the obtained error dynamics to be uniformly ultimately bounded is proposed, which can degenerate into the traditional ones in the literature by tuning the perturbation coefficient matrix. Based on the obtained final estimation of multi-faults, an output feedback FTC is designed to stabilize the Lipschitz nonlinear system. The longitudinal dynamics of an aircraft is applied to test the proposed method.


Introduction
As the increase in the safety and reliability requirements for actual control systems, such as flight control systems [1][2][3], the fault-tolerant control (FTC) has become a hot topic in the industrial and theoretical areas and some excellent works have emerged, see [4][5][6][7][8][9][10][11][12][13][14][15][16] and [17][18][19]. Meanwhile, as one of the important means of FTC, the research of fault diagnosis (FD) has become more and more frequent. In the field of fault diagnosis, fault estimation (FE) technique can obtain more exact fault information such as size and shape so as to perform desired fault detection alternately, which has become an important research topic. In recent years, the field of fault estimation has been developed by the efforts of scholars. A good many of observer-based fault estimation techniques have emerged, such as the robust observers (RO) [4,20], adaptive observers (AO) [21][22][23], sliding mode observers [24,25] and other estimation observers [7,10,[26][27][28][29][30]. However, the observer design for control systems, especially for control systems with multi-faults, and the convergence of fault estimation need to be further explored and improved.
In the existing research results of observer-based fault estimation, almost all focus on how to improve and perfect the design structure of fault estimator. For example, the robust estimators in [4,20] only involve the output errors, which are easy to implement but have in general low estimation accuracy. The adaptive estimators in [22,31] cover the output errors and their derivatives, which in certain conditions have a good estimation accuracy, but the output in disturbances free is required. In [27], an intermediate estimator was proposed based on the intermediate variable. The essence of this method is the robust observer design technique under a special linear transformation. Then based on a combination of adaptive estimator and intermediate estimator, in [7], a lower triangular matrix factor-based estimator was proposed. Compared with the estimation method in [27], a relaxed and adjustable lower triangular matrix linear transformation is introduced to design the fault estimator in [7]. The essence of the method in [7,10] is a reasonable deformation of the adaptive estimator, and the output in disturbances free is not required. However, to further improve the estimation accuracy, it is necessary to break through the observer design structure based on linear transformation in the literature such as [7,10,27], which motivates the current work.
On the other hand, in the existing research results of observer-based fault estimation, the traditional robust estimator, adaptive estimator and intermediate estimator, see [4,7,10,20,22,27,31] and so on, can only ensure that the constant fault estimation residual converges to zero (vector) without external disturbances, while the time-varying fault estimation residual is uniformly ultimately bounded or meets the given performance index without external disturbances. In the field of fault diagnosis, the convergence of time-varying fault estimation is still a challenge. Iterative fault estimation technology in [4] can realize the convergence of time-varying fault estimation from a certain point of view. For the linear system in [32] and T-S fuzzy system in [33], the obtained mean sequences of state and fault estimates, in certain conditions, converge to the true values as the number of iterations increases. However, the study for the convergence of time-varying multi-fault (actuator and sensor faults) estimation was not fully investigated. The occurrence of the sensor fault may result in poor regulation or tracking performance and even affect the stability of the control systems, which has caused great concern by the related scholars, see works [15,27,31,[34][35][36][37][38][39][40] and references therein. Nevertheless, linear systems [34], or systems without uncertainties [31,[35][36][37][38][39][40], are still the mainstream of the models mentioned above, which may not be able to better characterize actual control systems. Furthermore, almost all of the sensor fault estimation methods in the past (see [15,27,31,[36][37][38]) required an initial assumption about the fault derivatives [i.e.,ḟ s (t) is norm-bounded or belongs to L 2 [0, ∞)]. Therefore, it is necessary to study the time-varying fault estimation convergence and FTC for control systems with multifaults, which further motivates the current work.
Inspired by the above aspects, this paper focuses on the problem of convergent multi-fault estimation and output feedback FTC for Lipschitz nonlinear systems involving faults and disturbances. A fault decomposition technique is firstly introduced to obtain a desired system model. Based on the technique, a set of iterative estimation observers are proposed, by which the obtained mean sequences of estimates of state and multi-faults, in certain conditions, converge to the true values as the number of iterations increases. Then, based on the final multi-fault estimates, an output feedback FTC is designed to stabilize the Lipschitz nonlinear system in consideration. A numerical example tests the proposed method. The main contributions of this paper can be summarized as follows: (1) Different from the linear transformation methods in [7,10,27], a fault decomposition technique is applied to act on the system with multi-faults and disturbances, so as to obtained a desired system model. Then, compared with the existing observers in [7,10,27], for the desired system model, a set of relaxed iterative estimation observers are construct to obtain the convergent mean sequences of estimates of state and multifaults. (2) It can be proved that in certain conditions, the obtained mean sequences of state and multi-fault estimates converge to the true values, which was not fully investigated in the latest work [7,10]. A relevant iterative estimation algorithm is exhibited subsequently. (3) By reconstructing the perturbation function, a perturbation coefficient matrix-dependent LMI condition with less conservativeness is proposed to guarantee the states of the error dynamics to be uniformly ultimately bounded. The LMI condition can reduce to the existing ones in the work [7,27] by tuning the perturbation coefficient matrix.

Problem formulation
Consider the following nonlinear system described by where the system state vector x(t) ∈ R n is assumed to be unmeasurable; u(t) ∈ R m is the control input; f a (t) ∈ R m represents the unknown additive actuator fault; w(t) ∈ R q is the exogenous disturbance input; y(t) ∈ R p is the control output vector; f s (t) ∈ R l represents the unknown additive sensor fault; the nonlinear vector function described by N (t, x(t)) could depict modeling uncertainties; and the system matrices A, B, C, D, E and F are constant matrices of appropriate dimensions. Here, the pairs (A, B) and (A, C) are controllable and observable, respectively.
Before starting the work, some assumptions in the following are required: Assumption 1ḟ a (t), f a (t) and w(t) are normbounded, i.e., there exists three positive scalars α, β and

Assumption 2
The nonlinear term N (t, x(t)) is known and satisfies a Lipschitz constraint, i.e., where h is the Lipschitz constant independent of x(t) and t.
Remark 1 The scalars α, β and δ in Assumption 1 can be unknown. Similar assumptions can be seen in [22,32]. In the actual systems, some nonlinear terms can be assumed as Lipschitz, so Assumption 2 is a common assumption in [10,27].

Remark 2
In some literature that considers both actuator and sensor fault estimation, the conditions: rank(F) = l, rank(B) = m, l + m < p = rank(C) must be satisfied to be able to provide fault estimation, as can be seen in [31,37]. In contrast, the conditions in Assumption 3 are less conservative.

Iterative observer-based fault estimation
In this section, a fault decomposition technique is to be introduced to design a set of iterative estimation observers for the Lipschitz nonlinear system, so as to obtain the estimates of state and multi-faults with a good accuracy. Relevant stability analysis will be discussed subsequently.

Fault decomposition-based initial estimation observer design
If the state matrix of the system is stable (else = 0), we introduce a virtual fault decomposition where is a learning matrix to be designed.

Remark 3
In fact, if the state matrix of the system is stable, the state curve of the system can roughly describe the shape of the system fault to a certain extent. Therefore, the fault decomposition in (3) is reasonable. Furthermore, under certain conditions, one has the following inequalities: On the other hand, when the system is controllable, one can easily design a feasible output feedback control to make the state matrix of the system stable.
From (1) and (3), one haṡ Then, a virtual system can be obtained as Denote , according to Assumption 3 and (5), the virtual system described in (5) can be re-expressed as According to the virtual system (6), an integrated estimation observer based on fault decomposition is designed as: whereχ(t),ŷ(t) andf s (t) are the estimates of χ(t), y(t) and f s (t), respectively. In this case,x(t) andf a (t) can be obtained aŝ Remark 4 Compared with the estimation observers based on the deformation d(t) = f (t) − y(t) in [7] and d(t) = f (t) − x(t) with = RC in [10], the estimation observer in this paper is formally more general and applies richer state estimation information. On the other hand, in [27], the learning matrix contained in the intermediate variable was designed as wE T , while the learning matrix of the virtual fault decomposition in this paper is obtained by solving LMIs. Obviously, the designed estimation observer in this paper outperforms the estimators mentioned in the above literature.

Remark 5
In some sensor fault estimation references, such as the generalized coordinate transformation [27,31], or other techniques [36][37][38] are applied to construct estimation observers, which are based on the premise that ḟ s (t) ≤ (orḟ s (t) ∈ L 2 [0, ∞)) satisfies. In the paper, the constraint on sensor fault derivatives can be loosened. In this case, the magnitude of E must be relatively small to ensure the accuracy of the sensor fault estimates, which leads to the conservativeness of the proposed approach.
, e T f a (t)] T , then the corresponding error dynamic can be gained by: where Then, the estimation error dynamic (8) can be further calculated aṡ

Remark 6
The lower triangular matrix transformation technique is a contribution in our published literature [7], which essentially does not change the original eigenvalues of the system. In contrast, the eigenvalues of GĀ in this paper can be optimized by selecting the learning matrix reasonably, which can bring more flexibility to the design of observer gain matrices compared with the method in [7]. However, it should be noted that because of the structural characteristics of the matrix G, it is hard to find the optimal matrix through LMI technique.

Perturbation coefficient-dependent observer gain design
The following subsection is the stability analysis for the estimation error dynamic (9) and the solution of observer gains via perturbation coefficient-dependent LMIs.
Lemma 1 Under Assumptions 1-3, the state of the estimation error dynamic described in (9) is uniformly ultimately bounded, if for given scalars γ 1 > 0, γ 2 > 0, μ > 0 and an adjustable parameter matrix H , there exist matrices , L and P > 0 such that the following inequality holds: Proof Consider the Lyapunov function as then one haṡ Furthermore, the following inequalities always hold: Then, setw(t) = Hg(t), H is an adjustable parameter matrix to be designed to ensure that g(t) ≤ w(t) is satisfied. In this case, (13) can be inferred as Then, one can havė On the other hand, if (10) holds, by Schur complement, it follows that It further follows thaṫ Then, based on the design g(t) = H −1w (t), and combined with Assumption 1 and Remark 3, one has Therefore, with ϑ 2 = α 2 + δ 2 , which implies that the state of the estimation error dynamic described in (9) is uniformly ultimately bounded according to stability theory. The proof is completed.

Remark 7
It should be noted that by reconstructing the perturbation function asw(t) = Hg(t), the inequality (13) can be converted to (14). In this case, for the given positive scalars μ and γ 2 , one can tuning the matrix H to improve the stability conditions in (10). For example, one can appropriately increase the norm of the matrix H to reduce the norm of g(t), so that the estimation error can converge to a smaller range according to the inequality in (19).

Remark 8
In fact, when H = I , the stability condition in Lemma 1 reduces to the form obtained in [7,10]. It implies that for the given positive scalar γ 2 , one can appropriately increase the norm of the matrix H > 1, the obtained stability condition in (10) is to be better than the ones in [7,10].
Theorem 1 Under Assumptions 1-3, the state of the estimation error dynamic described in (9) is uniformly ultimately bounded, if for given scalars γ 1 > 0, γ 2 > 0, μ > 0, adjustable parameter matrices H 1 , H 2 with large norms, and a parameter matrix ‫ג‬ ∈ R n×m , there exist matrices , X, P n > 0 and P m > 0, such that the following LMIs hold: Then, the learning matrix and estimation observer gain can be computed as Proof Set P as defined in (20) and the variables: The proof can be completed via the steps in the proof of Lemma 1. The proof is completed.

Iterative estimation observers design
Based on the initial estimate from the fault decomposition based estimation observer (7), we design the following iterative estimation observers.
the sensor fault f s (t) and where L and are the gain matrices of appropriate dimensions to be designed. Denote , e T f ak (t)] T , then the corresponding iterative error dynamics can be gained by: and (23) and (24) can be inferred as and with 1 = GĀ + L¯ C and 2 = GD + L¯ E, for k ∈ N + . In fact, the kth error dynamics are affected by the (k − 1)th bounded functionḋ(t) estimation errors.

Convergence analysis
As mentioned in [32], the iterative estimation method is generally sensitive to the disturbance and noise in the iterative process, which is also the limitation of iterative observers. Here, if the system is in disturbances free, then one can have the following result.

Remark 9
The convergence of mean sequence of estimation errors for the state and single fault (sensor fault was ignored) is explored in the work [32,33]. However, Theorem 1 gives the convergence of mean sequence of estimation errors for the state and multi-fault, which is the improvement and breakthrough of the previous work in [32,33].

Remark 10
It should be noted that if the nonlinear function N (t, x(t)) = 0, it would be difficult to prove theoretically the convergence described in Theorem 1, which needs the condition: for ∀k ∈ N. However, the condition is hard to be realized for the nonlinear function N (t, x(t)) = 0. It is a deficiency of Theorem 1, which is worthy of further study and improvement in the future.
Next, we can select the iteration number k suitably such that k i=0x k (t)/(k + 1), k i=0f ak (t)/(k + 1) and k i=0f sk (t)/(k + 1) can be considered as the final estimates of x(t), f a (t) and f s (t), respectively, which can be summarized as the iterative algorithm of state and multi-fault estimation in the following.

Let k = 1 and obtainχ k (t) andf sk (t) by running the iterative estimation observers in (22). Compute
For t final = T > T 0 and a given sufficiently small positive scalar ρ > 0, if then φ k (t) can be considered as the final estimate of χ(t). Accordingly, the final estimates of x(t), f a (t) and f s (t) can be obtained aŝ . else, set k = k + 1, repeat until the above boundary reaches.

Remark 11
In the above iterative algorithm, the value of k is determined by the scalars T = t final , ρ and the iterative errorsχ k −χ i , for i = 0, 1, . . . , k−1. Based on the algorithm, one has Therefore, the value of k is required to satisfy k ≥ λ T /ρ − 1.

Output feedback FTC design
Here, it is assumed that the input matrix can be represented as B = [B T m , B T n−m ] T and B m ∈ R m×m is nonsingular (which is obviously true based on Assumption 3). Now, we introduce a linear transformation Then, the plant in (1) can be transformed to the following form Next, a controller that tolerates both actuator and sensor faults is designed as follows where K ∈ R m× p is the output feedback gain to be designed, andf a (t) andf s (t) are final estimates of f a (t) and f s (t), respectively, which can be obtained online from Sect. 3.

Remark 12
It should be noted thatf s (t) is from the iterative observers in (22), where the sensor fault estimator can not restrain the disturbance. Therefore, the accurate estimation of sensor fault requires a small amount of external disturbance in the output. On the other hand, the output feedback control that can achieve a better control effect requires the disturbance in the output to be small enough. This may be a disadvantage of the observer-based output feedback FTC in (44), which needs to be improved in the follow-up research.
Therefore, the overall closed-loop system for the plant in (43) can be inferred asż

Controller gain design
Lemma 2 Under Assumptions 1-3, the state of the overall closed-loop system described in (45) is uniformly ultimately bounded, if for given scalars γ 1 > 0, γ 2 > 0, μ > 0 and an adjustable parameter matrix N , there exist matrices K and P > 0 such that the following inequality holds: Proof Consider the Lyapunov function as then, one haṡ Further assume N (t, 0) = 0, then from Assumption 2, one has It thus holds that, for two scalars γ 1 and γ 2 2z According to Lemma 1, there exists a scalar α 1 such that e(t) 2 ≤ α 2 1 . In addition, and Then, based on Assumption 1, it can see that there exists a positive scalar β 1 that makes ω(t) ≤ β 1 true.
Set ω(t) = Nq(t), N is an adjustable parameter matrix to be designed to ensure that q(t) ≤ ω(t) is satisfied. Further, (49) can be rewritten as Then, one can havė On the other hand, if (46) holds, by Schur complement, it follows that 11 + 1/γ 2 1 PTT T P + 1/γ 2 2 P RN N T R T P ≤ −μP.
It further follows thaṫ which implies that the state of the closed-loop system described in (45) is uniformly ultimately bounded according to stability theory. The proof is completed.
Theorem 3 Under Assumptions 1-3, the state of the overall closed-loop system described in (45) is uniformly ultimately bounded, if for given scalars γ 1 > 0, γ 2 > 0, μ > 0, adjustable parameter matrices N 1 , N 2 with large norm, and a parameter matrix ∈ R m×(n−m) , there exist matricesK , P m > 0 and P n−m > 0 such that the following LMIs hold: Then, the controller gain can be computed as Proof Set P as defined in (54) and the variables: The proof can be completed via the steps in the proof of Lemma 2. The proof is completed.

Remark 13
It should be noted that Theorem 3 gives a feasible approach to convert the matrix inequality in Lemma 2 into LMIs, by which an ideal controller gain can be obtained. However, it is difficult to process the coupling term PBKC using the traditional output feedback control method in [8].

Simulation results
In the following, the longitudinal dynamics of an aircraft [10] is used to test the proposed method.
Here, the states x 1 , x 2 and x 3 represent the angle of attack, the pitch rate and the elevator angle of the longitudinal dynamics, respectively. The measured output is described by

Multi-fault estimation and error convergence
To verify the effectiveness of the proposed iterative observers and the convergence of mean errors (It is assumed that w(t) ≡ 0 in the model), we conduct simulation for this example by establishing For the simulation purpose, here, the actuator fault (cited from [7]) and sensor fault are described as  First, the convergence effect of the mean sequence of estimation errors is shown. Figures 2, 3 and 4 show the trajectories of state, actuator fault and sensor fault estimation error mean sequences under different iteration numbers, respectively. It can be seen that when the fault curve is smooth, the convergence performance of the mean sequence of iterative estimation errors is well reflected in Figs. 2, 3 and 4. However, the mean sequence of iterative estimation errors are  Figures 2, 3 and 4 show that near the nondifferentiable point of the fault function, the condition N gk (t) < h ē k (t) is not true.
Second, when the iteration times k = 1 and k = 7, the faults and their estimated response curves can be obtained, as shown in Fig. 5. It is clear that the estimated response curves when the number of iterations k = 7 can better simulate the fault information when the fault function is derivable, which demonstrates the advantage of observer iteration intuitively.
Third, to further demonstrate the merits of the proposed fault estimation method, we also run the simu-  Figure 6 shows the faults and their estimated response curves obtained by different methods. It is found that the derived iterative observer has better estimation accuracy than the intermediate estimator in [27]. The response trajectories of faults and their estimates presented in the subgraphs are more evident. Here, the disturbance is assumed as w = cos(t). Based on Theorem 1, a set of feasible solutions for the model can be obtained, as shown in Table 1.
Here, H = diag{H 1 , H 2 }. Based on the following observer gains shown in Table 1, the relevant simulation results for the example can be obtained as follows. Figures 7, 8 and 9 show the trajectories of state, actuator and sensor fault estimation errors under different norms of H under γ 2 = 0.11 and μ = 1, respectively. As described in Remark 8, one can appropriately increase the norm of the matrix H to reduce the norm of g(t), so that the estimation error can converge to a smaller range according to the inequality in (19), which can be reflected from the simulation results. Figures 7, 8 and 9 show that when H = I (in [7,10]), the fluctuation range of the corresponding estimation error curve is the largest. Furthermore, it can be seen clearly from the error movement direction in the figures that the mean sequence of estimation errors are well close to zero when the norm of H is sufficiently large.   Based on the obtained controller gain, the simulation result is shown as follows.
The response curves of system states activated by the proposed multi-fault-tolerant controller are given in Fig. 10, which show that the states of the overall closed-loop system can converge to a small range rapidly although the plant is subject to multiple faults and uncertain disturbances.

Conclusion
This paper has studied the problem of iterative observerbased fault estimation and FTC for Lipschitz nonlinear systems subject to multiple faults. A fault decomposition-based observer design technique is proposed to design a set of relaxed iterative observers for the nonlinear system so as to derive the estimates of state and multi-fault. The convergence of estimation for state and multi-fault is given with a theoretical proof. A perturbation coefficient matrix-dependent LMI condition is proposed to ensure the uniformly ultimate boundedness of the state of the obtained error dynamics, by which the estimation error can converge to a smaller range through tuning the coefficient matrix. A numerical example with the longitudinal dynamics of an aircraft is applied to test the proposed method. The future work will focus on the application of the fault decomposition method and the perturbation coefficient matrix-dependent LMI technique to complex control systems. Data availability Data sharing was not applicable to this article as no datasets were generated or analyzed during the current study. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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