<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M1" height="15.7437pt" version="1.1" viewBox="-0.0657574 -11.3994 26.6531 15.7437" width="26.6531pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.012,0,0,-0.012,13.299,4.134)"><path id="g50-30" d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"/></g></svg> Fault Estimation and Fault-Tolerant Control for T-S Fuzzy Systems with Actuator and Sensor Faults Using Sliding Mode Observer

Dhahri, Slim; Ben Alaïa, Essia; Ben Hmida, Faycal; Sellami, Anis

doi:https://doi.org/10.1155/2022/3647171

Mathematical Problems in Engineering

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Advances in Observer Design and Observation of Nonlinear Systems

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Research Article | Open Access

Volume 2022 | Article ID 3647171 | https://doi.org/10.1155/2022/3647171

Fault Estimation and Fault-Tolerant Control for T-S Fuzzy Systems with Actuator and Sensor Faults Using Sliding Mode Observer

Slim Dhahri,¹Essia Ben Alaïa,²Faycal Ben Hmida,³and Anis Sellami³

Academic Editor: Omar Naifar

Received03 Sept 2021

Revised05 Dec 2021

Accepted07 Jan 2022

Published01 Apr 2022

Abstract

This paper investigates the problems of the robust fault estimation (FE) and fault-tolerant control (FTC) for the Takagi-Sugeno (T-S) fuzzy systems with unmeasurable premise variables (PVs) subject to external disturbances, actuator, and sensor faults. An adaptive fuzzy sliding mode observer (SMO) with estimated PVs is designed to reconstruct the state, actuator, and sensor faults simultaneously. Compared with the existing results, the proposed observer is with a wider application range since it does not require the knowledge of the upper bound of faults that some FE methods demand. Based on the FE information, a dynamic output-feedback fault-tolerant controller (DOFFTC) is designed to compensate the effect of faults by stabilizing the closed-loop systems. By using the filtering method, sufficient conditions for the existence of the proposed SMO and DOFFTC are derived in terms of linear matrix inequalities (LMIs) optimization. Finally, a nonlinear inverted pendulum system is given to validate the proposed methods.

1. Introduction

In industrial applications, the increasing demand of higher performance, safety, reliability, maintainability, and survivability represent a major concern. So, it is important to support research on fault estimation (FE) and fault-tolerant control (FTC) for a class of nonlinear systems specifically Takagi-Sugeno (T-S) fuzzy models [1–5]. Many approaches have been developed in recent decades, such as sliding mode observer (SMO) [6–12], unknown input observer [13–15], adaptive observer [16, 17], and descriptor observer [18, 19].

The sliding mode (SM) scheme is a powerful tool to overcome uncertainties and external disturbances in dynamic systems, due to its good robustness, simple structure, and strong applicability. Therefore, it has a good application prospect in the field of FE and FTC and has attracted more and more attention from both academia and industry. Fruitful works can be found regarding this issue. For instance, in [8], a sliding mode observer (SMO) is designed for the estimation of actuator faults in T-S fuzzy models with digital communication channel, but the sensor faults were not considered. While in [9], the estimation of simultaneously actuator and sensor faults is realized for T + S fuzzy systems using nonquadratic Lyapunov function. However, the FTC problem was not treated. While, in [20], a FTC design for T-S fuzzy systems was established using unknown input observer approach. Moreover, a FTC scheme based on SMO for T-S fuzzy systems with local nonlinear models is proposed in [6]. Sufficient conditions are derived to calculate observer and controller gains which are solved using linear matrix inequalities (LMIs). In [21], an adaptive sliding mode FTC design is developed for a class of uncertain T-S fuzzy systems affected by multiplicative faults. Unfortunately, these previous results needed that the premise variables of the T-S fuzzy systems are measurable and the upper bounds of faults are known. Therefore, how to design a suitable SMO to overcome the above drawbacks?

Motivated by the above discussion, an adaptive fuzzy SMO is developed for the T-S fuzzy systems with unmeasurable premise variables in order to estimate the state, actuator, and sensor faults. Then, based on FE a dynamic output-feedback fault-tolerant controller (DOFFTC) is designed to compensate the fault effects by stabilizing the closed-loop system. Finally, the simulation result of a nonlinear inverted pendulum system is given to illustrate the effectiveness of the proposed method.

Compared with the existing results, the advantages of our work can be summarized as follows. First of all, we investigate the problem of fault estimation and fault-tolerant control based on the dynamic output-feedback controller for T-S fuzzy systems with external disturbances and actuator and sensor faults. Whereas, many researchers have considered only actuator faults [22, 23] or senor faults [24 + 26]. Then, the sliding mode observer is designed using adaptive law to avoid the hypothesis concerning knowledge of the upper bound of faults, which give less conservative results and offer more freedom in comparison with [27 + 29]. Moreover, authors in [6, 24] assume that the premise variables of the T-S fuzzy systems are measurable. Whereas, in this work, we consider the unmeasurable premise variables. The gains of the observer and controller are computed separately to avoid the coupling and reduce the computation complexity. Whereas, in [25], a single step solving algorithm is needed.

The rest of this paper is organized as follows. In Section 2, we describe the system and the problem studied in this paper. In Section 3, we present the design of the adaptive fuzzy sliding mode observer and the analysis of the stability of the error system. The fault estimation is studied in Section 4. In Section 5, the dynamic output-feedback fault-tolerant controller is developed. Finally, simulation example in Section 6 validates the efficiency of the proposed methods.

1.1. Notations

Throughout the paper, the following notations are used. denotes an identity matrix with dimension of . and denote the n-dimensional Euclidean space and real matrices. The pseudoinverse of a matrix is denoted by . For a real matrix , indicates that is symmetric positive definite, and indicates that is symmetric negative definite. denotes the Euclidean norm or its induced spectral norm. The symmetric terms in a symmetric matrix are denoted by . Finally, the space of square integrable functions is denoted by , that is, for any , .

2. System Description and Problem Statement

Considering the following T-S fuzzy model: Rule : IF is , , and is , THENwhere represents the state vector; is the input; is the measured output; , , , , , and are known matrices with compatible dimensions; and represent the additive faults generated by actuator and sensor, respectively; is the external disturbance which belong to . It is supposed that matrices are of full column rank, i.e., , matrix is of full column rank, is of full row rank, the pairs are observable, and the pairs are controllable. Besides, denotes the unmeasurable premise variable, are the fuzzy sets, is the number of IF-THEN rules, and is the number of the premise variable.

Using the technique [26], the overall model of system (1) is given bywhere and , here is the grade of the membership function of . We assume that , . Then, it easy to see that , for any . Hence, satisfies and .

Lemma 1 (see [27]). For matrices and with appropriate dimensions, we havefor any .

Lemma 2 (see [28]). If the following inequalities hold,we have

Introduce the state such thatwhere is an arbitrary stable matrix. Let ; then, the following augmented T-S fuzzy system is constructed:where

Combining the actuator fault and the sensor fault in the same unknown vector and assuming it is bounded and satisfies , where is an unknown real constant, then system (8) can be re-expressed aswhere .

For designing observers, it is often assumed, in the literature, that the premise variable is available for measurement. In this paper, it is interesting to develop a sliding mode observer for the unmeasurable premise variable T-S fuzzy system. A sliding mode observer for system (9) is in the formwhere is the estimation of unmeasured premise variable, is the state estimation of , is the observer output, is the output estimation error, and are the observer gain matrices, and is the discontinuous vector to be designed.

Define the state estimation error, . Based on (9) and (10), we get the following error dynamics system:where and

Note that when , that is, is treated as an unstructured vanishing perturbation which is supposed to be growth-bounded for such thatwhere is a small Lipschitz scalar. For the simplicity, denotes .

Authors in [29] have proven that the necessary and sufficient conditions for the existence of sliding mode observer when the system includes faults are(A1)(A2)The invariant zeros of are stable

Under A1, there exists a coordinate transformation such that system (8) is transformed intowherewhere , , , , and is nonsingular.

Applying the linear change of coordinates to the error system (11), then we obtainwhere . We suppose the observer gain has been the following form to facilitate the analysis:where . It is noted thatwhich motivates us to consider a coordinate transformation , where

In the new set of coordinates, (16) becomeswhereandwith

Define the matrix . If the observer gain matrices and are chosen as and , where is an arbitrary negative definite matrix, it can be concluded that

According to our choice, is stable. Therefore, is stable from the stability of .

Partitioning the error system conformably to with and yields

Considering the following sliding mode surface aswe now design the discontinuous error injection as follows:where is the unique solution to the Lyapunov equation for with the design matrix :andwith the adaptive law,

is a small positive constant and is designed constant.

Remark 1. The presence of both and in (18) poses the main challenge to the SMO design problem in this paper. To tackle this difficulty, in addition to the design of for the estimation of in the SMO, we use the approach for the stability analysis of the error system.
Define the controlled output of the error system aswhere is a full rank design matrix.
We will utilize the approach to analyse the stability of the error system (18) so that(i) if , (ii), otherwise is the attenuation level to be minimized.

3. Sliding Mode Observer Design

In this section, we will discuss the design method of SMO. The main results are presented as follows.

3.1. Stability Analysis

Theorem 1. Consider system (9) with conditions A1 and A2. The error system (18) is asymptotically stable with a minimal if there exist matrices , , , and small positive scalar such that the following convex optimization problem is solved:
subject towhere and are obtained as and .

Proof. Consider the Lyapunov functional candidate:where , , and . By deviating , we obtainAccording to Lemma 1, we obtainHence, from (33), we haveSubstituting the adaptive law and the discontinuous vector into the , we haveWhen substituting obtained expressions (35) into (34), we obtainFor the case , thenwherewithDenoteUsing the Schur complement, if (32) holds, we havewhich implies that , i.e., .
On the contrary, for the case that , letSubstituting (31) and (39) into (45) yieldswhere and .
Using the Schur complement, we have that if (30) holdswhich means that ; i.e., the error system (23) is asymptotically stable with the performance .
From (43), and are computed as and .

Remark 2. According to Theorem 1, the error system is asymptotically stable with performance. Thus, for some small , we have . In addition, one has .

3.1.1. Sliding Motion Analysis

In this section, the gain parameter in (29) will be determined to demonstrate a sliding motion occurs on in finite time.

Theorem 2. If in (27) is chosen to satisfywhere is positive scalar, then a sliding motion occurs on the surface , for all , where is the finite time at which sliding is established.

Proof. Definewhere the matrix is proposed in Theorem 1; then, the derivative of satisfiesAccording to (35), we haveSince is a stable design matrix and is solution to Lyapunov (28), we haveUsing relation (46), we havewherewhere is the maximum eigenvalue of .
If condition (42) holds, then the sliding mode reaching conditionis guaranteed. Applying the chain rule , the reaching condition (47) can be integrated and rearranged to obtain an estimate for :This proves that the sliding surface is thus reached in finite time .

4. Robust Reconstruction of Actuator and Sensor Faults

In Theorems 1 and 2, one has proven that the error system is asymptotically stable and can thus be driven onto sliding surface at some time instant ; the error system (23) is reduced towherewith is small scalar. Define

Computing the norm of (59) yieldswhere for a matrix denotes the maximum singular value of the matrix. The result (52) follows by keeping in mind that . Therefore, it follows thatwhere and . Thus, for a small , both actuator and sensor faults are estimated by

5. Dynamic Output-Feedback Fault-Tolerant Control Design

In this section, a fuzzy dynamic output-feedback FTC (DOFFTC) will be constructed to guarantee the stability of closed-loop system (2). The following corrected output will be given which is obtained by subtracting the reconstructed sensor faults from the (faulty) outputs:where is estimation of obtained from the FE scheme. System (2) becomeswhere is the sensor fault estimation error. The fuzzy DOFFTC for system (56) is constructed as follows:where is the controller state, is the estimation of , and , , , and denote controller matrices to be obtained later, respectively.

Substituting (63) into (65), we obtain

Then, substituting (64) into (66), we further obtainwhere is the actuator fault estimation error.

The dynamic equation of the closed-loop system is obtained as follows:where , , and

So far, the control purpose in this paper for the closed-loop system (60) is to design the controller gain matrices of (57) such that the corrected output satisfies the performance as follows:for and attenuation level .

Theorem 3. The closed-loop system (60) is asymptotically stable with a minimal in (61) if there exist matrices , , and , , , and , , such that the following convex optimization problem is solved:
subject towherewithThe gain matrices of the DOFFTC are as follows: is obtained as .

Proof. Consider the Lyapunov function , where ; then, the derivative of can be obtained asLetSubstituting (76) into (75), we obtainwhereIt is easy to find that if . Using the Schur complement, is equivalent toLet us define the matrix and its inverse :Due to , we haveWe will also define the matrices:Pre- and post-multiplying (79) by and its transpose and by using the variable change:Inequality (72) can thus be easily obtained.

6. Inverted Pendulum Example

Consider the nonlinear inverted pendulum system from [30]:where , , , , , , , , and . In [30], the above nonlinear system is expressed by two-rule T + S fuzzy model (2) with

With membership functions are and . The membership functions are chosen based on the method of sector nonlinearity [30].

Choosing , , , and and solving LMI optimization problem given in Theorem 1, we can calculate the performance level and the following observer gains:

According to Theorem 3, we obtain the performance level, , and the following controller gain matrices:, , , and . In the corresponding simulation, the parameters associated with the equivalent output error injection have been chosen to be , , , , and and initial conditions , , , and . The considered actuator and sensor faults have, respectively,

And the external disturbances are supposed to be random noises from and 0.1. The simulation results are provided with online simultaneous actuator and sensor faults’ injection. Figures 1 and 2 indicate that the adaptive SMO can estimate existing faults simultaneously with satisfactory precision by rejecting the effects of disturbances.

Simulation result for the output response (considered faulty) is provided in Figure 3. It is observed that the output without FTC does not converge to the output of the fault-free model (i.e., without any fault). However, the output trajectory of with FTC reaches the output of the nominal model. Therefore, the proposed fuzzy DOFFTC design achieves the performance under faults and disturbances, and the stability of the closed-loop system is guaranteed while satisfying the prescribed performance.

7. Conclusion

This paper focuses on the problems of FE and FTC for T-S fuzzy systems with unmeasurable PVs and having external disturbances, actuator, and sensor faults. An adaptive fuzzy SMO is designed for estimating the state, actuator, and sensor faults, simultaneously. Using the FE scheme, a DOFFTC is designed to compensate the faults and to stabilize the closed-loop system. Finally, simulation results show the effectiveness of the proposed methods.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was funded by the Deanship of Scientific Research at Jouf University under grant no. (DSR-2021-02-0366).

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Copyright © 2022 Slim Dhahri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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