Simultaneous Multiplicative and Additive Actuator Faults Estimation-Based Sliding Mode FTC for a Class of Uncertain Nonlinear System

. In this article, an adaptive sliding mode fault tolerant control (FTC) is improved in the case of uncertain nonlinear system which is afected by both multiplicative and additive faults in actuator. Especially, when the nonlinear system is modeled by Takagi–Sugeno (T-S) fuzzy system with local nonlinear model. Te main contribution of this paper is developing a model of multiplicative faults, which ofers a more realistic dynamic evolution of the actuator degradation. Te degradation process is modeled by Wiener process and estimated by the maximum likelihood estimation (MLE). Sliding mode observer (SMO) is conceived to realize the additive actuator faults using convex multiobjective optimization. On these bases, the estimated multiplicative and additive actuator faults are used to design the adaptive sliding mode controller (SMC). Finally, the proposed fault-tolerant control scheme is demonstrated by the results of inverted pendulum system simulation.


Introduction
During the past few decades, the felds of fault estimation (FE) technique and fault tolerant control (FTC) have been the result drawing an intensive research interests due to the increasing demands for system's performance, safety, and reliability. Active FTC and FE to a category of nonlinear systems particularly T-S fuzzy models [1][2][3] have a signifcant position in recent control implementation, as well as in supervision and reliability of actuators. In recent decades, a variety of methods have been developed, using adaptive observer [4][5][6] or SMO [7][8][9]. Te sliding mode scheme has an excellent application prospect in fault estimation and fault tolerant control due to its simple structure, strong applicability, and good robustness. Several publications have appeared in recent years regarding this issue. In [10], the authors studied Takagi-Sugeno fuzzy systems with uncertainties and multiplicative and additive actuator faults and then developed an adaptive sliding mode FTC design. However, in [11], a FTC design predicated using SMO for T-S fuzzy systems is developed. In [12], the authors used a nonquadratic Lyapunov function to estimate simultaneously actuator and sensor faults for T-S fuzzy systems. In a recent paper [13], the authors developed the FE and FTC for the T-S fuzzy systems subject to actuator and sensor faults. For nonlinear systems, popular fault detection methods have been elaborated in an efective and precise manner. In [14,15], the authors modelled fault as an additive occurring in sensors or actuators. Te main disadvantage of the previous techniques is that they regard sensor and actuator faults as additive. Nevertheless, some actuator and sensor errors, in addition to component faults, are frequently found in the multiplicative form. As a result, multiplicative faults and the system's inputs and outputs are mixed. Estimating the magnitude and characteristics of multiplicative faults has become an increasing attract in control theory due to the practical importance of decoupling their structure efects or parameter in the model or system and improving fault tolerant control concept for nonlinear systems. We can use stochastic process to model the actuator degradation [16][17][18][19]. Te actuator environment can infuence the deterioration process and strongly depends on several factors [20] (shock, temperature, and load variation) of the monitored system. Degradation process for industrial systems is infuenced by both external and internal factors including operating conditions and dynamic environment [21]. Stochastic dynamics are the common characteristics involved in actuator increasing degradation process in actuator. Tis results in system uncertainties and measurements errors. For the past few decades, extensive research studies have been conducted in the area of stochastic degradation modelling [22][23][24]. Degradation models can be divided into shock-based degradation model [25], progressive degradation model [26,27], and combined degradation model [28]. Te degradation such as the wear out of engineering devices, the fatigue, and the corrosion of metals can be caused by multiple degradation processes and induces total failures of actuator.
In the following, the authors point out the main focus of the present study on developing efective and robust active FTC for a class of uncertain nonlinear system. Tis study ofers a FE-based sliding mode FTC technique for nonlinear uncertain system afected by both multiplicative and additive faults. Te major contributions of this article are as follows: (i) In this study, for more simplicity and to fnd a model for the description of the interactions between the control system behavior and the actuator stochastic degradation process, the deterioration of the entire control system would be supposed to lie in the actuator loss of efciency. In practice, when an actuator operates dynamically in a random environment, its capacity decreases overtime which is related to degradation process. Terefore, the multiplicative faults model was conceived based on not only the degradation process but also the capacity of actuator. Actually, it is straight forward to think multiplicative faults should be estimated in order to conceive a fault tolerant control design for dynamic systems. To describe the actuator degradation behavior, we use stochastic Wiener process model which ofers a more realistic evolution of the deterioration. In order to estimate the degradation process, the maximum likelihood method is used. (ii) We propose to conceive an adaptive SMO for T-S systems with the existence of uncertainties to estimate additive actuator faults. To study the stability of the proposed SMO, we use a linear matrix inequality (LMI) and the theories of Lyapunov. To improve the actuator faults estimation accuracy, we use an adaptive update term. (iii) Using FE, we study an adaptive SMC for the T-S fuzzy system having models local nonlinear that complies with the condition of Lipschitz with additive and multiplicative faults. Particularly, it is demonstrated that the suggested sliding mode FTC is used for the parameters setting of the controller in order to obtain the desired performances of actuator even in the presence of both additive and multiplicative faults.
Compared to previous works, many studies have used only actuator or senor faults in system [29]. Ten, most of the previous studies do not take into account multiplicative faults. Furthermore, we use adaptive law to design the SMO which gives more freedom in comparison with [30]; the model used incorporates output disturbance and Itô stochastic noise; and they introduced time delay in the state. However, in our case study, we suppose that the degradation of the total control system lies in the actuator loss of efciency. Terefore, the multiplicative fault model was conceived based on not only the degradation process but the capacity of the actuator. To describe the actuator degradation behaviour, we use the stochastic Wiener process model (continuous models) which ofers a more realistic deterioration. Compared to [31], the authors use only additive faults in both actuator and sensor in the system. Tey transform sensor faults into "pseudoactuator" faults by using an augmented T-S fuzzy system that causes many constraints in the application of the hypotheses; in fact, the total number of actuator faults must not exceed the number of outputs. Tis model may not be practical and conventional in all situations of stochastic degradation process in actuator. Moreover, it cannot adequately capture the dynamics of the actuator's degradation process. Indeed, the model of faults must describe the interaction between the actuator stochastic degradation process and the control system behaviour. Tis includes the dynamics of the actuator's performance, the control system's response to changes in the actuator's performance, and the uncertainty associated with the actuator's stochastic degradation process. Te model must also take into account the physical characteristics of the actuator and the system environment, as well as the impact of external factors such as maintenance and other system parameters. In our study, we develop a model of multiplicative faults, which ofers a more realistic dynamic evolution of the actuator degradation. Te SMO is designed according to adaptive law which ofers less conservative results and gives more liberty in comparison with [32]. Yang et al. in [33] have developed simultaneous multiplicative and additive faults in jump systems, which may be considered as a special class of stochastic systems. Tey use the adaptive backstepping technique to construct the fuzzy logic system based an online adaptive fault-tolerant compensation controller. However, in our study, we use continuous degradation models and we propose to conceive an adaptive SMO for Takagi-Sugeno fuzzy systems having local nonlinear models.
Te outline of the article is organized as follows. Section 2 describes a nonlinear system with local nonlinear models, uncertainties, additive, and multiplicative (loss of efciency) actuator faults. In Section 3, we use Wiener process to model the degradation process in the actuator and the maximum likelihood method to estimate the stochastic model for multiplicative fault. In Section 4, the proposed adaptive SMO is designed to estimate the additive faults of actuator. Section 5 presents the sliding mode FTC design in order to redress the impact of additive and multiplicative faults for the stabilization of the system. A simulation of the inverted pendulum and cart system is used in Section 6 in order to validate and illustrate the efciency of the approach. Finally, Section 7 draws some conclusions.

Problem Formulation
In this study, we refer to a class of nonlinear uncertain systems afected both the additive and multiplicative actuator faults. Consider a nonlinear uncertain system represented by the following equations: u t ∈ R m is the control input, x t ∈ R n represents the state vector, y Lt ∈ R p1 stands for the controlled output, and y t ∈ R p represents the measurement vector of output. Te , and Γ(x, t) are always nonlinear for i � 1, 2, 3, 4. ξ(x, t) ∈ R l denotes the unknown uncertainties vector. In feedback control system, the actuator is a signifcant part in the evaluation of the performance level. Tat is because, considering a degradation process in an electrical or mechanical element of the controlled system, the controlled action is afected as well, which eventually causes a poor performance of the control system.
Let us consider an actuator undergoing a progressive degradation process. It is subjected to both electrical and mechanical degradation that occurs stochastically over time, particularly in the case of the electrical actuator degradation where a rub impact between the rotor and the stator or a bend shaft can occur. It should be noted that rotor faults as well as stator faults are recognized electrical faults. Besides, there are other types of faults that depend on the failure mode. During its functioning, the actuator can be afected by several types of faults. Our study will focus on two types of faults: additive faults and multiplicative faults (loss of efciency). In practice, the actuator operates dynamically in a random environment. Furthermore, its capacity decreases overtime which is related to degradation and depends on environmental factors as well as operational conditions of the feedback control system. Moreover, the wear or natural ageing of the electrical and/or mechanical components of the actuator due to the nondesired impacts of the working condition decreases the efectiveness of actuator in time. Te actuator degradation process is a cause of the physical system performance deterioration. During the initial period of operation, the actuators function fawlessly.
Te actual capacity of actuator K a (t) � K a int where K a int is the initial nominal capacity. If d(t) outlined the actuator degradation, the capacity of actuator (see Figure 1) can be represented by the following equation: Te efciency factor (see Figure 2) can be written as follows: Te minimum efciency is reached when K a (t) � K min , and we defne the minimum efciency factor ε � (K min /K a int ) > 0 such as 0 < ε ≤ β ≤ 1, ∀t ≥ t f and t f is the time occurrence of multiplicative defect.
Te loss of actuator efectiveness is examined to consequence from the dynamic progression of the degradation process β � (K a int − d(t))/K a int .
In this way, the following three cases are defned: (i) K a (t) � K a int : the actuator operates without degradation (d(t) � 0), and the efciency factor is β � 1 (ii) K a (t) � K a int − d(t): the actuator deteriorates, its capacity K a min < K a (t) < K a int , and the efciency factor is β � (K a int − d(t))/K a int (iii) K a (t) � K a min : the actuator operates with its minimum capacity, and then the efciency factor is β � ε � (K a min /K a int ) Te nonlinear system (1)-(3) afected by additive and multiplicative faults at the same time can be described by the following uncertain structure and local nonlinearities as follows: where f a (t) ∈ R q is the additive actuator faults. Multiplicative fault may be rearranged as follows: where F(u t , t) are the multiplicative faults. Ten, the nonlinear system (1)-(3) afected by additive and multiplicative faults at the same time can be expressed in terms of uncertain structures and local nonlinearities as follows:

Mathematical Problems in Engineering
Te multiplicative faults are and the following three case are defned: (i) d(t) � 0: the actuator operates without degradation, the multiplicative faults are neglected F(u, t) � 0, so we have only the additive faults (ii) 0 < d(t) < L (L is a maximum degradation process): the actuator operates with degradation process, and the multiplicative faults are Given the nonlinear system (10)-(12) afected by both the additive and multiplicative actuator faults, respectively, f a (t), F(u, t), and the uncertainties ξ(x, t), our objective to achieve an adaptive sliding mode FTC resides principally on solving the following three problems: (1) First problem: develop and estimate the degradation process to conceive a multiplicative faults model which ofers a more realistic evolution of the deterioration in the actuator. It is very important to be able to estimate the faults before the performance systems degradation. (2) Second problem: estimate T-S fuzzy system states and additive actuator faults, with the adaptive SMO. (3) Tird problem: we need to stabilize the closed loop of nonlinear systems, with the simultaneous occurrence of additive and multiplicative faults, using the robust adaptive sliding mode controller (10)-(12).

Actuator Degradation Models Estimation
In this study, we suppose that the system (10)- (12) is subject to Wiener process. Te actuator degradation is denoted by a random variable d t at time t. In this paper, we suppose that degradation is an increasing Lévy process [34] supported by the following assumptions:

independent and stationary
In this way, Wiener process has been frequently used to conceive a degradation model, particularly, when it was successfully applied to describe the increasing degradation in an actuator. Te Wiener process is one of the most classic processes used in many progressive degradation modeling area; the basic idea is to model the cumulative increasing degradation d W,t by the stochastic Wiener process such that where μ is a linear drift parameter and σ is a difusion coefcient parameter and W t (μ, σ) � μt + σB t . We consider that the Wiener process [24] is used such . If we consider the initial condition d W,t 0 , we can approximate the degradation measure d W,t with the following equation: Here in, we can deduce that In particular, if d W,t 0 � 0, the degradation process Te objective is to estimate the linear drift μ and diffusion parameter σ. We apply the maximum likelihood estimation (MLE) method.

Considering the degradation increment
. Te density function is given by the following equation: is given by the following equation: Te log-likelihood function for the ith item can be expressed by the following equation: Since the measurements d i,j are independents, we can express l(ρ) � ln(Δd 1 , We fnd the MLE ρ � [σ, μ] by maximizing with respect to σ and μ, the partial derivatives of the log-likelihood function.
As a result, we write the partial derivative of loglikelihood function compared to μ as and compared to σ as Ten, we obtain the expression as follows: Te measurements data Δ i,j are generated by MATLAB with the parameters μ � 0.4, σ � 0.2. We compute equations (23) and (24), and the estimated parameters are obtained as follows: μ � 0.398 and σ � 0.213.
A design procedure for multiplicative fault development and estimation is described as follows: (i) Step 1: we defne the expression of the efciency factor using the degradation process and the actuator capacity to conceive the multiplicative faults model. (ii) Step 2: we use the Wiener process to model the stochastic degradation in actuator. (iii) Step 3: the maximum likelihood method is used to estimate the linear drift and difusion parameter.

Additive Actuator Faults Estimation
It was shown that using T-S fuzzy system with local nonlinear models concept was well suited to the study of a several class of systems. Te nonlinear system (10)- (12) afected by additive and multiplicative faults at the same time is written by the T-S fuzzy system with uncertainty and models local nonlinear where M i , D i , C i , B i , and A i are matrices with known real values. F(u, t) is the multiplicative fault. We suppose that (A i , B i ) and (A i , C i ) are, respectively, controllable and observable. f a (t) and F(u, t) represent the additive and multiplicative fault in control channel. Te functions μ i (ζ t ) (fuzzy normalized membership) must satisfy the properties of sum convex We will use the following assumptions in this paper.
Assumption 1. We assume that the uncertainties and faults are unknown and bounded. For the faults f a (t), F(u, t) and the uncertainties ξ(x, t), there exist known positive Mathematical Problems in Engineering 5 is minimum phase and relative degree one, and we verify that for all complex numbers s when Re(s) ≥ 0 that Ensures that the nonasymptotically stable modes are observable which means it is detectable.
Te constant of Lipschitz c > 0 is unknown and Γ(x, t) is globally Lipschitz if M � R n . Edwards and Spurgeon [35] have studied an SMO to estimate faults and stats taking into account the following required assumptions.
Te following lemma and defnition are utilized to achieving the principal results.

Defnition and Notation
Lemma 5. For the two matrix Y and Z, the next condition carries We develop a novel adaptive SMO defned as where y(t) and e y (t) denote, respectively, the output of the observer and estimation error, x(t) represents the observer state. G n,i and G l,i are suitable gain matrices. In this way, the signal υ(t) of the robust adaptive sliding mode is written as follows: where P e � P 2 e y (t), η(t) � ρ + ϱ with ϱ is a scalar positive, P 2 ∈ R p×p is positive defnite and symmetric and ρ is the adaptive term can be updated given by the following equation: α > 0 is a gain. Under condition in equation (30), we can use a change of coordinates as follows: Te matrices A i , B i , D i , M i , C i , G l,i , and G n,i become where A 11,i ∈ R n p ×n p , n p � (n − p), D 1,i ∈ R n p ×l , M 2,i ∈ R p×(q) and C 2,i ∈ R p×p is nonsingular. where According to equation (37), the nonlinear gain is described by the following equation:

Sliding Motion Stability.
In addition, another change of coordinates is expressed by the following equation: where L i is discussed later. In the new coordinates system, we obtain If Te observer gain matrices are where A s 22 is the stable design matrix. By referring to equation (42), the error system from equations (39)-(40) can be handled as follows: where Our aim is to estimate the actuator faults and the states variables in the presence of multiplicative faults. Defne where H is the weight matrix and it is supposed that H � di ag (H 1 , H 2 ). We defne the measure of performance in worst case as follows: (48)

Proof. (see Appendix A)
Our aim is estimate the additive actuator faults in nonlinear system (25)- (27). Te adaptive SMO donated by equations (33)- (34) has been developed and satisfes the condition of reachability, and then e y (t) � 0 and _ e y (t) � 0. Ten, equation (47) is then expressed as follows: where N � T − 1 L,i x. Te approximate equivalent of the output error injection signal υ eq (t) is where σ > 0 is a scalar to decrease the impact of chattering. We defne the next relation It is clear that where ς max ,i � ‖D 2,i ‖ 2 ξ 0 + (‖A 21,i ‖ 2 + c)ϖ. It remains to conclude from equation (54) that Ten, approximately, for a small ς max , it seems that Terefore, the additive actuator faults estimation is given by Te method for FE with an adaptive SMO is outlined as Step 1: pick out the weight matrix in equation (47). (ii) Step 2: pick out the suitable scalar "0 ≤ λ ≤ 1" and resolve the equation (49) (LMI optimization problem). So, we can obtain P 1 ; W; P 2 and ε, ς and c. . To stabilize the nonlinear system and compensate additive and multiplicative actuator faults efects, the proposed SMC with adaptive law was used to envisage a corrective action. Terefore, we assume that the nonlinear system with uncertainties (6)-(8) can be modeled by T-S fuzzy representation with both local nonlinearities and uncertainties as follows: First, the sliding motion takes place on a sliding surface denoted as S which is defned as follows: We describe the linear switching function S c (t) ∈ R m using the feedback information of the output Te control input may be described as follows: Te linear part denoted as u l (t) and depending on system states as well as both additive and multiplicative actuator faults estimation is expressed as follows: where − q a (f a (t) + F(u, t)) is created to compensate the additive and multiplicative faults infuence. Assume that q a � B + i M i , q F � B + i and K j ∈ R m×n . Te adaptive nonlinear control input part u n (t) is proposed as follows: where ε c and ϱ c are small and positive constants. η c (t) is determined by using ρ c .

FTC Design.
Using the part u n (t) of control input, we need to prove the sliding and the reaching of S in a fnite time. Construct the function of Lyapunov as follows: Referring to equation (58), the derivative of (67) is given by the following equation: Defne Ω c as If the sliding surface S is reached, the condition of reachability is satisfed. Ten, 9 c is select to fulfll An perfect sliding motion is guaranteed to occur in fnite time by the suggested SMC with adaptive law, ∀t ≥ t c S c (t) � _ S c (t) � 0. When the SM is required, we Mathematical Problems in Engineering examine the stability. We suppose that u eq (t) such that _ S c (t) � 0, as (71) Te closed-loop dynamic system with equation (71) is where

Assumption 8. Te condition of Lipschitz is satisfed by
c denotes the constant of Lipschitz.

Theorem 9. Te T-S fuzzy system with local nonlinear models (72)-(74) is stable robustly with a maximization of the
where According to the results, it is possible to obtain the adaptive SMC from K j � Q j P − 1 x .

Proof. (see Appendix B)
A design method for adaptive FTC with an adaptive SMO is outlined as follows: (i) Step 1: select "0 ≤ λ c ≤ 1", resolve the LMI optimization problem (76); so we get the matrices Q j and the scalars δ c , ς c and c c (ii) Step 2: Step 3: concept the Adaptive SMC (65), then the stability of (51), (57) is accomplished.

Case Study
Te sliding mode FTC design according to the sliding mode observer is accomplished by considering the inverted pendulum and cart system. Te objective is to conceive an adaptive stabilization controller that the considered inverted pendulum benchmark system [36,37] consists of a moveable carriage having one degree (see Figure 3). Te carriage is freely rotatable in driving direction on which a pendulum is mounted and actuated by a motor.

Te Nonlinear System Modelling.
We start by examining the model inverted pendulum and cart system where β is the efciency factor. x 4 (t), x 3 (t), x 2 (t), and x 1 (t) represent, respectively, speed of cart, angular velocity of pendulum, cart position, and angle position of pendulum.
Te parameters of the system are shown in Table 1. Te approximation of the nonlinear faulty system may be obtained by T-S fuzzy system. We use in this study the models with local nonlinearity as follows: In this case, Γ(x, t) � x 2 2 (t)lmsin(x 1 (t)) − f c , the fuzzy weights, are defned by the following equations: We assume that the actuator faults as well as the control input are in the same direction B i � M i . Te matrices of the local nonlinear models are

Adaptive Sliding Mode Observer and Controller Design.
We will design an adaptive SMC based on the requirements of the observer information to stabilize the system.

Simulation Results.
It is worth mentioning that the results are obtained using online simultaneous additive and multiplicative faults estimation. Figure 4 illustrates multiplicative actuator fault estimation. In our example, we consider an additive linear time varying actuator fault. Te developed adaptive sliding mode observer (33) and (34) can reject the efects of system uncertainties and make an estimation of actuator fault with satisfactory accuracy. Meanwhile, at t � 2, the additive actuator fault ( Figure 5) has been introduced, in order to demonstrate the capacity of the developed estimation method to additive faults.
As shown in Figure 5, it is worth noting that in spite of the existence of uncertainties, the SMO can still track the additive faults f a (t). Hence, the simulation results outline that the suggested fault estimation with the adaptive law for the inverted pendulum and cart system described by T-S fuzzy representation with local nonlinear models accomplishes the frst objective of this article (actuator faults estimation) with an excellent performance in terms of robustness and precision despite the presence of the uncertainties. Figures 6-8 compare nonlinear output referring to three cases: output without faults, output without sliding mode FTC, and output responses with the conceived FTC design.
As can be seen from the comparison with the output responses without faults, the conceived adaptive sliding mode controller (63) is capable of the stabilization of the system.
Zoomed versions of the nonlinear inverted pendulum and cart system output responses illustrated in Figures 6-8 highlight a good satisfactory precision of the proposed adaptive FTC, which ensures the stability of the system subject to both additive and multiplicative faults. More precisely, it can be seen clearly that the integrated adaptive law is an efective way to improve both faults estimation and compensation.

Conclusions
Tis paper addresses two powerful problems: the SMO scheme and the FE-based sliding mode FTC, concerning system described by T-S fuzzy representation and local nonlinear models. We start by the study of the multiplicative faults, and the actuator degradation process is considered as a source of performance deterioration. Te degradation process in the actuator is modeled by Wiener process. Te additive actuator faults are estimated using robust adaptive SMO in the existence of uncertainties so that the proposed observer's stability has been derived using H ∞ performances in order to minimize the uncertainties' efect on the dynamics of estimation error and solved within the linear matrix inequalities (LMI's) optimization design. To compensate the efects of both additive and multiplicative actuator faults and guarantee the system stability, an adaptive FTC with sliding mode control is studied. Te existence conditions are expressed via Lyapunov approach in terms of LMI. Convex multiobjective optimization is employed to simultaneously maximize the nonlinear term Lipschitz constant in the T-S fuzzy modeling in addition to the uncertainties attenuation level, in order to obtain satisfactory gain of the adaptive SMO and controller. Te results of the simulation on the application clearly display the efciency of the proposed additive actuator FE and the adaptive FTC design with uncertainties in system. In our future work, we can use shock models (Poisson process) or combined models to characterize the stochastic deterioration process in an actuator. Indeed, the actuator is a device that operates dynamically in a random environment; its capacity may decrease in an accelerated way. So, we can integrate covariates in the degradation model with accelerated mode deterioration. Te derivative of the output in the designer of observers could be taken into account to increase the performance of the estimation.

ξ(x, t) + r T (t)r(t). (A.9)
We defne the following variable α as (A.10) To maximize c for Γ(x, t), we simultaneously minimize both ε and α. Referring to equation (A.10), we shall write the abovementioned expression (A.9) as follows:  Tus, we conclude that V 0 (t) is negative, if From the Schur complement, we deduce that the relation (A.13) and the relation (49) are equivalent. If equation (49) is satisfed, V 0 (t) is negative.
Tis completes the proof.

B
To guarantee the stability of system, we should investigate the next Lyapunov function V x (t) � x T (t)P x x(t), (B.1) P x ∈ R n×n is defnite positive symmetric. According to (72)-(74), (B.2) Using Assumption 8 and Lemma 5, we obtain

(B.3)
Let us defne J(t) We write equation (B.4) as follows: where (‖θ i ‖c) � c c . A novel variable is defned as Te simultaneous minimization of α c and δ c , i.e., minimization of α c + δ c , can maximize c c according to the nonlinear function of Lipschitz Terefore, it must be proved that J(t) are negative, if (B.7) Using the Shur complement, the following relation is easily obtained as follows: