Fuzzy logic systems‐based integral sliding mode fault‐tolerant control for a class of uncertain non‐linear systems
2016; Institution of Engineering and Technology; Volume: 10; Issue: 3 Linguagem: Inglês
10.1049/iet-cta.2015.0716
ISSN1751-8652
AutoresLi‐Ying Hao, Ju H. Park, Dan Ye,
Tópico(s)Stability and Control of Uncertain Systems
ResumoIET Control Theory & ApplicationsVolume 10, Issue 3 p. 300-311 Research ArticlesFree Access Fuzzy logic systems-based integral sliding mode fault-tolerant control for a class of uncertain non-linear systems Li-Ying Hao, Li-Ying Hao College of Information Engineering, Dalian Ocean University, Dalian, People's Republic of China Information Science and Technology College, Dalian Maritime University, Dalian, People's Republic of ChinaSearch for more papers by this authorJu H. Park, Corresponding Author Ju H. Park jessie@ynu.ac.kr Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyonsan, 38541 Republic of KoreaSearch for more papers by this authorDan Ye, Dan Ye College of Information Science and Engineering, Northeastern University, Shenyang, People's Republic of ChinaSearch for more papers by this author Li-Ying Hao, Li-Ying Hao College of Information Engineering, Dalian Ocean University, Dalian, People's Republic of China Information Science and Technology College, Dalian Maritime University, Dalian, People's Republic of ChinaSearch for more papers by this authorJu H. Park, Corresponding Author Ju H. Park jessie@ynu.ac.kr Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyonsan, 38541 Republic of KoreaSearch for more papers by this authorDan Ye, Dan Ye College of Information Science and Engineering, Northeastern University, Shenyang, People's Republic of ChinaSearch for more papers by this author First published: 01 February 2016 https://doi.org/10.1049/iet-cta.2015.0716Citations: 43AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study proposes an integral sliding mode control (ISMC) scheme for a class of uncertain non-linear systems subject to actuator faults including outage. It is noted that traditional ISMC method cannot handle actuator outage. To tackle the problem, matrix full-rank factorisation technique and adaptive mechanism are incorporated. Based on the above technique, two novel integral sliding surfaces using construction methods I and II are then introduced and existence conditions of sliding modes are given in terms of linear matrix inequalities, in which less conservativeness and better robustness against actuator faults are obtained using construction method II than I. The fuzzy logic systems are applied to approximate the bounds of unknown non-linear functions. Furthermore, an integral sliding mode controller, without an fault detection and isolation mechanism, is synthesised to guarantee the asymptotic stability and the robustness of the closed-loop system against actuator faults and non-linearities from the every beginning. Finally, simulation results for a model of B747-100/200 aircraft confirm the effectiveness of the proposed control method. 1 Introduction In recent years, due to the increasing demands for safety and reliability in dynamics systems, fault-tolerant control (FTC) has received considerable attention [1–12]. With the increasing complexity and non-linearities of system dynamics [13–15], some works about fault tolerant strategies for non-linear systems have been found in [16–21]. In the literature [16], a reliable filtering problem against sensor failures is investigated for a class of continuous-time systems with simultaneous sector-bounded non-linearities and varying time delays. Two fault-tolerant design schemes for single-input-single-output non-linear system are proposed in [17], which has been extended to multi-input-multi-output non-linear system based on the backstepping technique in [18]. Moreover, the literature [19] has designed an adaptive FTC scheme for a class of unknown non-linear systems in strict-feedback form. Moreover, two adaptive output feedback FTC schemes have been designed for a class of unknown non-linear large-scale systems [20] and uncertain stochastic non-linear systems with unmodelled dynamics [21]. On another research front, sliding mode control (SMC) [22–27] has attracted much recent attention in the field of FTC [28–35]. A contributing factor to its popularity is its inherent robustness to matched uncertainty. Traditional sliding modes, as employed in the above-mentioned work, consist of the initial reaching phase and the reduced order sliding motion. Once the system trajectory maintains the sliding mode, the closed-loop system is inherently robust to faults in actuators which can be well modelled as matched uncertainty. However, this robustness is only achieved during the sliding motion. To achieve a sliding mode throughout the entire system response, the concept of integral sliding modes (ISMs) has been proposed in [36, 37]. Recently, few papers have been devoted to the problem of designing ISM controller for systems with actuator faults. Notable exceptions are the works [38–41]. The actuator degradation and time-varying additive actuator fault are dealt with in [38]. Furthermore the main contribution of [39–41] is to cope with the total failure of certain actuators by incorporating the ISM ideas with control allocation approach. It should be pointed out that the efficiency level of actuator in [39–41] is obtained by an fault detection and isolation (FDI) mechanism, which might give incorrect fault information. Inspired by the above considerations, this paper proposes an ISM fault-tolerant scheme for a class of uncertain non-linear systems with actuator faults including outage. First, the matrix full-rank factorisation technique and adaptive mechanism are combined to cope with actuator outage, while traditional ISM method cannot do it. Second, two novel construction methods of integral sliding surface are given. From the comparisons of two stability conditions of sliding dynamics, better robustness against actuator faults are obtained using construction method II than that of construction method I. Moreover, the fuzzy logic systems (FLSs) are applied to approximate the bound of unknown non-linear function. Based on some adaptive laws, the non-linear part of ISM controller is designed to make compensation for actuator faults, non-linearities and unmatched uncertainties. Finally, the above-mentioned conclusions have been made. The main contributions of this paper are stated as follows: (a) Two construction methods of ISM surface are proposed. The first method does not consider the fault effect, while the second one takes fault estimation information into account. Thus a less conservativeness and better robustness against actuator faults are obtained by using the second construction method than the first one. (b) Comparing with the ISM fault tolerant schemes in [39–41], the ISMC method in this paper can cope with actuator outage without using an FDI mechanism, which the above-mentioned works rely on. (c) The scheme proposed in this paper has certain advantages compared to the authors' works [42, 43], which are based on traditional SMC methods. On one side, the ISM ensures that sliding is achieved for all time, thus guaranteeing robustness to actuator faults from the every beginning, while the works in [34, 35] cannot do it. On the other side, an important advantage is that the analysis of the closed-loop system is less complex and less conservative than the works in [42, 43], which requires a 'synthesis-followed-by analysis' procedure. However, the synthesis and analysis in this paper is totally integrated. Then, the stability analysis proposed in this paper allows a more effective synthesis procedure to be employed to compute the parameters involved in the control law. The remainder of this paper is organised as follows. In Section 2, the control problem and some useful lemmas are presented. In Section 3, two construction methods of integral sliding surface are given and an adaptive ISM controller is designed to account for actuator faults, non-linearities and uncertainties. The simulation results are given in Section 4. Finally, the conclusion is drawn in Section 5. The notations used throughout this paper are fairly standard. denotes the transpose of matrix , means a unit matrix with appropriate dimensions, denotes the set of all real matrices, is the -dimensional Euclidean space. Matrices on the main diagonal of a block diagonal matrix is denoted as , Euclidean norm is represented by , is defined as , where . 2 Problem statement and preliminaries 2.1 Problem statement Consider a class of mismatched uncertain non-linear systems with uncertainties and subjected to actuator faults (1)where is the system state vector, is the regulated output vector, is the control input vector, is an unknown smooth non-linear vector function and is the external disturbance in . is of the form , where is unknown, but bounded as for all . Moreover it is assumed that in this paper. , , , , , and have corresponding appropriate dimensions. For all possible faulty modes , the following fault model is adopted as follows [8, 10] (2)where and . Let and denote the known lower and upper bounds of actuator effectiveness level , is the stuck function. For , define the sets in the following form The faults considered in this paper include actuator outage, loss-of-effectiveness and stuck, and the following Table 1 shows the fault types more clearly. Remark 1.The framework developed in this work is applicable to a larger class of systems than the ones in [9–11, 17, 30], in which the non-linearity term is not considered, which might be very restrictive in reality. Though the non-linearities are considered in [17–21], the stuck faults are constants and the design process needs the triangularity condition. In [44], this condition is removed, however, the system uncertainties and disturbances are both matched ones, i.e. . Moreover, a more general model (2) is given in this paper than the ones in [38–40], in which just one or two of the above fault types are coped. Therefore, the ISMC fault tolerant technique of this paper suits for wider application. Table 1. Fault type Fault type normal 1 1 0 loss of effectiveness 0 1 0 outage 0 0 0 stuck 0 0 1 The following assumptions are essential to make in the FTC design: Assumption 1.For all , , all pairs are completely controllable. Assumption 2.Loss-of-effectiveness failures could occur on all actuators simultaneously. Up to actuators undergo stuck or outage fault such that the remaining actuators can still complete a desired control task. Assumption 3.There exist unknown positive constant and such that the unparameterisable actuator stuck fault and external disturbance are piece-wise continuous bounded by , . Assumption 4.For all , . Remark 2.Assumptions 1 and 2 ensure the internal stabilisability of FTC system [9] and the existence of a feasible solution to the actuator failure accommodation problem [10], respectively. Assumption 3 is natural and quite common [9]. To compensate the stuck faults or outage completely [10], the actuator redundancy condition in Assumption 4 is made in the robust FTC literature [18]. The purpose of this paper is to construct a suitable integral sliding manifold and design an ISM FTC law that can maintain closed-loop stability and satisfy the performance index no larger than in Definition 1 in spite of actuator faults and failures. Definition 1 ([11]).Let the following closed-loop system be described by (3)For any , if the following inequality holds, where is a given positive number, then the closed-loop system (3) is said to be with an adaptive performance index no larger than . 2.2 Fuzzy logic systems An FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine and the defuzzifier [45]. The knowledge base is composed of a collection of fuzzy if-then rules of the following form where and are the input and output of the FLS, respectively. and are fuzzy sets and is the number of the fuzzy rules. Through singleton function, centre average defuzzification and product inference, the FLS can be expressed as follows (4)where , is the membership function of linguistic variable , is the point at which reaches its maximum, and we assume that . Then, the FLS (4) can be rewritten as follows where and with Lemma 1 ([45]).Let be a continuous function, which is defined on a compact set . Then, for any constant , there exists an FLS (4), such as where is the ideal constant parameter. Then, considering the system (1) in the presence of smooth non-linear vector function , according to Lemma 1, one has (5)where the fuzzy-basis function and constant parameter are defined according to Lemma 1, and is the approximation error, which is bounded by the unknown constant . In general, it is assumed that the reconstruction error are bounded such that (6)Another lemma will be introduced to facilitate control system design, which will be used in the later control law design. Lemma 2 ([46]).The closed-loop system (3) is stable and satisfies if there exist a positive scalar and a positive definite matrix such that 3 ISM controller design In this section, two new integral sliding surface construction methods will be introduced based on the matrix factorisation technique and adaptive mechanism. Then existence conditions are given in terms of linear matrix inequality (LMI) for different integral sliding surface construction methods. The estimate parameters including the unknown weighted matrix , the approximation error , the unknown fault information , , , the unknown upper bounds of disturbance , are updated through adaptive laws. What's more, an integral SMC (ISMC) law is synthesised to guarantee the sliding dynamics stability and maintain the sliding mode thereafter. Assume can be factorised as (7)where and both have rank . The designed ISM controller in this paper consists of splitting the control variable into two parts (8)where (9)In (9), the term , where and , is responsible for disturbance attenuation and dealing with mismatched uncertainties, while is a discontinuous control action designed to compensate actuator faults and reject the non-linearity terms, forcing the system state on a suitably designed sliding manifold which will be shown in (11). Let be a positive parameter, which will be introduced in Lemma 3. In particular, is the estimate of reciprocal of , i.e. and (10)where is the smallest eigenvalue of matrix , the parameters , , , and are the estimate values of the weight matrix , the upper bound of reconstruction error , the fault effect factor , the upper bound of stuck fault and the upper bound of disturbances , respectively. is any positive scalar. To have a complete definition, assume that if . The integral switching surface is defined by the set (11)and two construction methods of integral switching surface are made as follows: Construction method I (12)Construction method II (13)where is the design freedom, in (15) is the estimate matrix of unknown actuator effectiveness level , which is updated through the following projection algorithms (14)where , is the th column of input matrix , is the th row of gain matrix and is an the adjusted parameter. The term achieves the property that , and hence the reaching phase is eliminated [22, 36]. Different from the design of a typical sliding mode controller, the ISM design is somewhat different: it can be shown [22] that the sliding motion associated with (12) and (13) is always nominally governed by and irrespective of the choice of , respectively. Recently, an approach has been suggested for which attempts to ameliorate the effects of unmatched uncertainty [36]. In this paper (15)is suggested. Notice that this choice of has the property that (16) Remark 3.Construction methods I in (12) and II in (13) about integral sliding surfaces are novel. On the one hand, in the traditional ISMC literature [36], there is a common design condition that , i.e. the sliding surface belongs to a null space of dimension . Once the actuators outage occurs, it will destroy the sliding modes. To tackle the problem, is designed in this paper. In this way of construction, outage of certain actuators occurs in a null space of dimension without affecting the sliding mode. On the other hand, the estimate of actuator effectiveness factor is used in (13) of construction method II, which takes fully use of fault information, in other words, it derives less conservativeness and better robustness better robustness than construction method I when designing the stability condition in Theorem 1. The existence conditions of sliding dynamics on the given integral sliding surface (11) and (12) will be shown in the following Theorem 1, and the performance index is no larger than in the presence of actuator faults, unmatched uncertainties and external disturbances. Theorem 1.An asymptotically stable sliding mode dynamics exists from the initial time and the performance bound is no larger than , if there exist positive scalar , , positive matrix , , and matrix , , such that for integral sliding surface (12) the following inequality (17)holds and for integral sliding surface (12) the following inequality (18)holds for all , , where , . Proof.The sliding dynamics stability analysis on the integral sliding surface (12) and (13) is similar, thus, for simplicity, only the stability analysis process for integral sliding surface (13) is shown here. To analyse the sliding motion associated with the surface in (13) and in (15) in the presence of faults, compute the time derivative of equation (13) (19)Substituting (1) into (19) gives (20)Since , the addition of column to matrix does not change its rank, i.e. , which will result in an infinite number of solutions to the equivalent control. Let is the Moore–Penrose inverse of . Then, the equivalent control [22, 23] is obtained as (21)Substituting (21) into (1) and using the fact that , the sliding dynamics are given by (22)Using as defined in (16), in (9) and further simplifying equation (22) gives (23)By Lemma 2, if there exists a positive-definite matrix such that the following inequality (24)holds, where , , then the sliding dynamics quadratical stability is guaranteed and the performance index is no larger than . Applying Schur's complement lemma, linear matrix inequality (20) can be equivalent to thus inequality (24) holds with . Therefore, it is guaranteed that the sliding dynamics exists and performance in no larger than from the every beginning. □ Remark 4.Though there is in LMI (18), the projection algorithm (14) ensures that belongs to convex set , so LMI (18) are solvable by using LMI tool in Matlab software. Remark 5.It should be mentioned that the control law is involved into the stability analysis of sliding mode dynamics, which is different from the traditional sliding mode FTC strategy [42, 43]. That is, the synthesis and analysis using ISM method is totally integrated. To some extent, the analysis of the closed-loop system is less complex and less conservative than the works in [42, 43], which requires a 'synthesis-followed-by analysis' procedure. Moreover, the ISM ensured sliding is achieved for all time, thus guarantee more robustness to actuator faults than the traditional SMC strategy [42, 43]. The following matrices decomposed form and adaptive laws are given before the main results are introduced Remark 6.Since is the basis function vector being chosen as the commonly used Gaussian function so that the basis vector is positive. From (25), it is easy to see that for any initial condition , the solution holds for . Without loss of generality, one further assumes that throughout this paper. Thus, one can obtain . Then, the following adaptive law is given to estimate unknown positive constant (25)Furthermore (26)where is designed in (10), , , , , , , , and are bounded initial values of , , , , and , respectively. The constants , , , , and are the positive design parameters. Let (27)Since , , , , , , are unknown parameters, one can rewritten the error systems as (28)In the next step, an important lemma we proposed in [42, 43] will be given for design purposes. Lemma 3.For the matrix full-rank factorisation (7), there exists a positive constant , such that the following inequality (29)holds for all . Proof.For space limitation, it is omitted here. □ Theorem 2.Consider the closed-loop FTC systems described by (1) subjected to actuator faults (2). Suppose LMI (18) is feasible, and Assumptions 1–4 are valid. Then the trajectory of the closed-loop system can be driven into the integral sliding manifold by employing the ISMC law in (9) and (10) and the parameter adaptive laws (25) and (26). Proof.To analyse the reachability, consider the following candidate Lyapunov function (30)where .Taking the time derivative of along the closed-loop system (1), it yields (31)Using the estimate error as defined in (27), and further simplifying (31) gives (32)Recalling the property , the above inequality (32) becomes The following inequality is true according to The following inequalities will be true according to Assumption (2) (33)Combining (31) with (33), it can be obtained that Taking (9) into account, one can get that (34)Substituting (5) and (27) into (34), it yields (35)Further, according to (6), the following inequality is true (36)Combining (36) with (35), it yields (37)From (27), it can be shown from (37) that (38)In view of adaptive mechanism (25), it is clear that holds for for any initial condition . With the help of Lemma 3, it follows that where is defined in (10), now it is readily obtained from (38) that (39)Taking (10) into account, one has Moreover, combining the adaptive laws (26) with the facts (28), the time derivative of the Lyapunov candidate function defined in (30) is (40)Further, from the adaptive laws given in (25), it gets that Thus, it can be shown from (40) that From above inequality, one gets is a non-increasing function of time. Define , thus the following is true that so . Moreover, exits as , and taking the integration manipulation on both sides of (40) results in (41)where . Further, as , the above inequality (41) becomes Thus , which means is a uniformly continuous function. Thus the system trajectories will remain on the integral sliding surface if applying Barbalat lemma to (3). This completes the proof. □ Remark 7.If the initial values are chosen sufficiently large, they can compensate the potential disturbances and faults in the systems,and the sliding function will stay in the small convergence set even after a fault occurs. However, the actuator faults are totally unknown beforehand, the proper initial values of adaptive gains are difficult to select. In this situation, although the sliding function may move out of the small convergence set once actuator faults or disturbances occur, the state trajectory will be pulled back to the small convergence set in finite time. 4 Simulation results In this subsection, a model of a B747-100/200 aircraft in [47] is considered here by adding a mismatched uncertain term. The system can be written in terms of the following parameters In our example, only the pitch rate (radians per second), true airspeed (metres per second), angle of attack (radians) and pitch angle (radians) are considered, and the actuators are elevator defection, total thrust and horizontal stabiliser. The initial values of above states are set . Define the fuzzy membership functions as follows (42)The fuzzy basis functions can be obtained as follows (43)In the simulation, choose as and define as Then from calculation, it gets the corresponding Using sliding surface construction method I, the solution matrices and of LMI (7) are as follows Using sliding surface construction method II, the solution matrices and of linear matrix inequality 18 are as follows the performance index are , , respectively, and the corresponding integral sliding surface parameter is For simulation, some initial values of estimate parameters and adjust gains are given as follows: , , , , , , , , , , , , , . Then the disturbance is chosen as A severe fault scenario is considered in the simulation. At , the first actuator suffers from a partial loss of effectiveness fault, and the third actuator has stuck at in the meantime. The non-linear vector part of ISM law 9 is replaced by a continuous approximation in order to reduce the chattering phenomenon. Fig. 1 shows the comparisons results of the pitch rate, true airspeed, angle of attack and pitch angle using construction methods I and II, respectively. Furthermore integral sliding surface comparisons responses are given in Fig. 2. It can be observed that the asymptotically stability can be guaranteed using both of construction methods. However, better robustness against actuator faults using construction method II is obtained than that using construction method I. Fig. 3 reflects the time histories of the estimates of when using the second construction method. From Figs. 4-7–8, it is clear that the estimates , , , , and are convergent, and they are not necessary to converge to their exact values in our method. A conclusion can be made from the simulation results that the closed-loop uncertain non-linear system is quadratically stable in the presence of actuator outage faults. Fig. 1Open in figure viewerPowerPoint Response comparisons of pitch rate, true airspeed, angle of attack and pitch angle using construction methods I and II, respectively Fig. 2Open in figure viewerPowerPoint Response comparisons of integral sliding surface using construction methods I and II, respectively Fig. 3Open in figure viewerPowerPoint Estimations of unknown parameter using construction method II Fig. 4Open in figure viewerPowerPoint Estimations comparisons of unknown parameter using construction methods I and II, respectively Fig. 5Open in figure viewerPowerPoint Estimations comparisons of the upper bound of disturbance using construction methods I and II, respectively Fig. 6Open in figure viewerPowerPoint Estimations comparisons of unknown parameter using construction methods I and II, respectively Fig. 7Open in figure viewerPowerPoint Estimations comparisons of unknown actuators factors using construction methods I and II, respectively Fig. 8Open in figure viewerPowerPoint Estimations comparisons of weight vector using construction methods I and II, respectively 5 Conclusion In this paper, an ISMC-based FTC scheme for a class of non-linear systems subject to actuator faults including outage and external disturbances has been proposed. Integrating the matrix factorisation technique and adaptive laws, two construction methods of integral sliding surfaces have been given. One is fault free, the other is taking fault information into account. From the existence conditions of sliding dynamics, a less conservativeness and better robustness can be obtained using the second construction method. Furthermore, without the need for an FDI mechanism, the gain of the non-linear unit vector term of integral SMC law is designed to compensate for non-linearities and actuator faults even certain total actuator failures. Finally, a model of B747-100/200 aircraft is used to demonstrate the effectiveness of the proposed design method. Though an ISMC FTC problem has been partly resolved for a class of non-linear systems, the unmatched non-linearities is another interesting problem, which deserves the future research interest. 6 Acknowledgments This work of L. Hao and D. Ye was supported in part by National Natural Science Foundation of China (grant nos. 61503055, 61273155, 61322312), China Postdoctoral Science Foundation (no. 2015M571291), a Foundation for the Author of National Excellent Doctoral Dissertation of P.R. China (grant no. 201157), the Fok Ying Tung Education Foundation (grant no. 141060), the Fundamental Research Funds for the Central Universities (grant no. N140405006) Educational Commission of Liaoning Province of China (no. L2014277), the Scientific Research Foundation of Dalian Ocean University (no. HDYJ201412). Also, the work of J.H. Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10005201). 7 References 1Li H. Liu H., and Gao H., et al.: 'Reliable fuzzy control for active suspension systems with actuator delay and fault', IEEE Trans. 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