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- 1. ISA Transactions 51 (2012) 50–64 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Fault tolerant synchronization of chaotic heavy symmetric gyroscope systems versus external disturbances via Lyapunov rule-based fuzzy control Faezeh Farivara,∗ , Mahdi Aliyari Shoorehdelib a Department of Mechatronics Engineering, Science and Research Branch, Islamic Azad University, P.O. Box: 14515/775, Tehran, Iran b Faculty of Electrical Engineering, Department of Mechatronics Engineering, K. N. Toosi University of Technology, Tehran, Iran a r t i c l e i n f o Article history: Received 12 January 2011 Received in revised form 25 June 2011 Accepted 27 July 2011 Available online 24 August 2011 Keywords: Fault tolerant Synchronization Chaotic gyroscope Nonlinear control Lyapunov rule-based fuzzy control a b s t r a c t In this paper, fault tolerant synchronization of chaotic gyroscope systems versus external disturbances via Lyapunov rule-based fuzzy control is investigated. Taking the general nature of faults in the slave system into account, a new synchronization scheme, namely, fault tolerant synchronization, is proposed, by which the synchronization can be achieved no matter whether the faults and disturbances occur or not. By making use of a slave observer and a Lyapunov rule-based fuzzy control, fault tolerant synchronization can be achieved. Two techniques are considered as control methods: classic Lyapunov-based control and Lyapunov rule-based fuzzy control. On the basis of Lyapunov stability theory and fuzzy rules, the nonlinear controller and some generic sufficient conditions for global asymptotic synchronization are obtained. The fuzzy rules are directly constructed subject to a common Lyapunov function such that the error dynamics of two identical chaotic motions of symmetric gyros satisfy stability in the Lyapunov sense. Two proposed methods are compared. The Lyapunov rule-based fuzzy control can compensate for the actuator faults and disturbances occurring in the slave system. Numerical simulation results demonstrate the validity and feasibility of the proposed method for fault tolerant synchronization. © 2011 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Since the synchronization of chaotic dynamical systems was observed by Pecora and Carroll [1] in 1990, chaos synchronization has become a topic of great interest. Chaos synchronization has many potential applications in laser physics, chemical reactors, secure communication, biomedicine, and so on [2–4]. The aim of chaos synchronization is to make two systems oscillate in a synchronized manner. Given a chaotic system considered as the master system, and another identical system considered as the slave system, the dynamical behaviors of these two systems may be identical after a transient time when the slave system is driven by a control input. But despite the amount of theoretical and experimental results already obtained, chaos synchronization seems a difficult task, particularly if it is considered that [5]: (1) due to the sensitive dependence of chaos on the initial conditions, it is almost impossible to reproduce the same starting conditions; (2) in matching the master and slave systems exactly, even infinitesimal parametric variations of any model will eventually result in the divergence of orbits starting nearby each other; and (3) parametric ∗ Corresponding author. Tel.: +98 9121463845; fax: +98 21 88788281. E-mail addresses: F.Farivar@srbiau.ac.ir, Faezeh_Farivar84@yahoo.com (F. Farivar), Aliyari@eetd.kntu.ac.i (M.A. Shoorehdeli). differences between chaotic systems yield different attractors. Many methods of chaos synchronization have been presented [6– 8]. On the other hand, the dynamics of a gyro presents a very in- teresting nonlinear problem in classical mechanics. The gyro has attributes of great utility to navigational, aeronautical and space engineering [9]. Gyros for sensing angular motion are used in airplane automatic pilots, rocket vehicle launch/guidance, space vehicle attitude systems, ship’s gyrocompasses and submarine in- ertial auto-navigators. The concept of chaotic motion in a gyro was first presented in 1981 by Leipnik and Newton [10], show- ing the existence of two strange attractors. In the past few years, gyros have been found with rich phenomena which are of bene- fit for the understanding of gyro systems. Different kinds of gyros with linear/nonlinear damping are investigated for predicting the dynamic responses, such as periodic and chaotic motions [11–13]. Some methods have been presented for synchronizing two identical/non-identical nonlinear gyro systems, such as active control [14], fuzzy sliding mode control [15], neural sliding mode control [16,17] etc. Fault diagnosis and isolation (FDI) and fault tolerant control (FTC) have been active research topics during the past two decades, with a view to increasing the safety and reliability of dynami- cal systems [18,19]. In the literature, faults normally occur at two places: the actuators and sensors. Actuator faults are faults that act on the system, resulting in deviation of the process variables. 0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2011.07.002
- 2. F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 51 The result is that the command (control) signal sent to this device has no effect. Sensor faults are faults that act on the sensors that measure the system variables, and do not directly affect the pro- cess. The source of these faults could be wear and tear of the sen- sor leading to inaccurate readings, or a total failure of the sensor. Sensor faults could also arise from drift/poor calibration/extreme process conditions. These faults will affect the process if the output measurements are used to generate the input control signal [20]. FTC is an area of research that aims to increase applicability by specifically designing control algorithms capable of maintaining stability and performance despite the occurrence of faults. FTC structures have also been studied where a model-matching strat- egy is used to design linear [21] and nonlinear [22] controllers, etc. [23,24]. Observer-based methods are commonly used as a basis for FDI schemes. Residual generation approaches, using linear ob- servers, have been widely used, where the difference between the system output and observer output is processed to form so-called residuals. Ideally, these will be zero during fault-free operation but will give a specific response when a certain fault occurs. Residual generation techniques have been demonstrated in [25,26]. But when a fault happens in chaotic systems, the conventional methods for synchronizing the chaotic systems are invalid. In order to increase the safety and the reliability of synchronization systems when a fault arises, fault tolerant synchronization is essential to guarantee the synchronization of chaotic systems. The objective of fault tolerant synchronization (FTS) is to design a fault tolerant controller to achieve the synchronization no matter whether a fault occurs or not. To the best of the authors’ knowledge, the problem of FTS is still an open problem and seldom researched [27,28]. Various approaches for achieving chaos synchronization using fuzzy systems have been proposed [29]. Fuzzy set theory was first presented by Zadeh [30], while fuzzy logic control (FLC) schemes have been widely developed for almost 40 years, and have been successfully deployed in many applications [31]. Furthermore, adaptive fuzzy controllers have been used to control and synchro- nize chaotic systems [32,33]. Until now, there has been no investigation on the subject of fault tolerant synchronization of chaotic gyroscope systems. Two notable papers are studied, namely [27,28]. Ref. [27] considers Lur’e systems using time-delay feedback control. By making use of an observer-based fault estimator and a modified time-delay feedback controller, the fault tolerant master–slave synchroniza- tion is formulated so as to allow discussion of the global asymp- totic stability of the error system and the bound of the energy gain from fault to state, and fault estimation error vectors. Some delay-dependent criteria are derived in order to analyze the syn- chronization error system, and on the basis of the analysis re- sults, a sufficient condition for the existence of such a master–slave synchronization scheme and a solution for the controller and fault- estimator gain matrices are obtained in terms of linear matrix inequalities (LMI). Also, a Chua circuit is used to illustrate the ef- fectiveness of the proposed method. Ref. [28] presents an offline and data-based procedure for fault tolerant synchronization. In Ref. [28], to investigate the stability of the error system and facilitate the design of the fuzzy sampled- data controller, a Takagi–Sugeno (TS) fuzzy model is employed to represent the chaotic system dynamics. A parameter-dependent Lyapunov–Krasovksii functional and a relaxed stabilization tech- nique are considered and the stability conditions based on the lin- ear matrix inequality (LMI) are obtained in order to achieve fault tolerant synchronization of the chaotic systems. Motivation and overview of this study. Fault tolerant control (FTC) has been an active research topic during the past two decades with a view to increasing the safety and reliability of dynamical systems. FTC is an area of research that aims to increase applicability by specifically designing control al- gorithms capable of maintaining stability and performance despite the occurrence of faults. On the other hand, dynamic chaos is a very interesting nonlin- ear effect which has been intensively studied during the last three decades. Chaotic phenomena can be found in many scientific and engineering fields such as biological systems, electronic circuits, power converters, chemical systems, and so on. A chaotic system has complex dynamical behaviors that possess some special fea- tures, such as excessive sensitivity to initial conditions, broad spec- tra of the Fourier transform, bounded and fractal properties of the motion in the phase space, etc. Chaos in control systems and con- trolling chaos in dynamical systems have both attracted great in- terest in recent years. But when a fault happens in chaotic systems, the conventional methods for synchronizing the chaotic systems are almost invalid. In order to increase the safety and the reliability of synchroniza- tion systems when a fault happens, fault tolerant synchronization (FTS) is essential to guarantee the synchronization of chaotic sys- tems. The objective of fault tolerant synchronization is to design a fault tolerant controller to achieve the synchronization no mat- ter whether a fault arises or not. FTS is capable of maintaining sta- bility and performance despite the occurrence of faults in chaotic systems. Moreover, since the gyro has been utilized to describe modes in navigational, aeronautical or space engineering, the chaos syn- chronization procedure for chaotic gyroscope systems is important and this study may have practical applications in the future. Therefore, FTS of chaotic gyroscope systems versus external disturbances via Lyapunov rule-based fuzzy control is investigated in this study. Since fuzzy control schemes have been successfully deployed in many applications, we choose this control method for this study. In this study, on the basis of Lyapunov stability theory and fuzzy rules, the nonlinear controller and some generic sufficient conditions for global asymptotic synchronization are obtained. The fuzzy rules are directly constructed subject to a common Lyapunov function such that the error dynamics of two identical chaotic motions of symmetric gyros satisfy stability in the Lyapunov sense. The goal of this paper is to synchronize two chaotic heavy sym- metric gyroscope systems versus actuator faults and external dis- turbances. To achieve this goal, Lyapunov rule-based fuzzy control is applied as the active fault tolerant control. Also, an observer is designed, assuming that the nonlinear term satisfies a Lipschitz condition. In addition, the results of this paper may be extended to synchronize many classes of nonlinear chaotic systems. The aforementioned Refs. [27,28] have a different viewpoint on the subject of FTS to ours. In our work, this subject is considered as an online procedure and the fault occurs in the slave system. We attempt to design a convenient and robust controller. Also, the controller is easy to implement. On the other hand, since the gyro has been utilized to describe modes in navigational, aeronautical and space engineering, the FTS procedure for chaotic gyroscope systems is important and this study may have practical applications in the future. This paper is organized as follows. In Section 2, the dynamics of a heavy symmetric gyroscope system and the chaos synchro- nization problem are explained. In Section 3.1, fault tolerant control and synchronization are introduced. The scheme of fault tolerant synchronization is presented in Section 3.2. In Section 3.3, classic Lyapunov-based control is designed for chaos synchroniza- tion. Then, Lyapunov rule-based fuzzy control is designed for chaos synchronization in 3.4. In Section 3.5, various forms of slave system as regards actuator faults and external disturbances are presented. An appropriate observer is designed, and also residuals are gener- ated in this section. In Section 3.6, the stability of FTS via Lyapunov
- 3. 52 F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 rule-based control is presented. A theorem concerning the state es- timator, actuator faults and external disturbances is presented and proven to verify the stability of the system under these conditions. Finally, simulations are presented in Section 4, to show the effec- tiveness of the proposed FTS methods for gyroscope systems versus disturbances. At the end, the paper is concluded in Section 5. 2. System description and the synchronization problem The dynamics of a heavy symmetric gyroscope with linear- plus-cubic damping of angle θ mounted on a vibrating base can be described in terms of Euler’s angles θ, φ and ψ. The vibration dynamics of the base can be described in terms of the multiple harmonic motion ∑n k=1 Ak sin ωkt. sin x. Let the state variables be x = [θ ˙θ ˙φ]T ; then the dynamic equations can be obtained as [11] ˙x1 = x2 ˙x2 = − (βφ − βψ cos x1)(Bψ − βφ cos x1) I2 1 sin3 x1 − c1 I1 x2 + Mg l I1 sin x1 − Mg A I1 sin ωt sin x1 ˙x3 = −2x2x3 cos x1 sin x1 + βψ I1 sin x1 x2 (1) where I1 and I3 are the polar and equatorial moments of inertia of the typical gyroscope, Mg is the gravity force, and l is the distance between the center of gravity and O. It is clear that coordinates θ and φ are cyclic, which provides the conjugate momenta. The momentum integrals are βφ and βψ . In order to simplify the following procedure, two nonlinear functions are defined as follows: p(x1, x2) = − (βφ − βψ cos x1)(Bψ − βφ cos x1) I2 1 sin3 x1 − c1 I1 x2 + Mg l I1 sin x1 − Mg A I1 sin ωt sin x1 (2) q(x1, x2, x3) = −2x2x3 cos x1 sin x1 + βψ I1 sin x1 x2. (3) The gyroscope system, Eq. (1), performs complex dynamics and has been extensively studied by Ge [11], with specific values set as follows: βφ = 2, βψ = 5, l = 0.25, I1 = 1, Mg = 4, c1 = 0.5, ω = 2, A = 50. In the numeric simulational, the dynamic behavior gyroscope system, Eq. (1), exhibits a chaotic motion as shown in Fig. 3(a) with initial conditions (x1, x2, x3) = (1, −1, 1). Consider two coupled, chaotic heavy symmetric gyro systems, where the master and slave systems are denoted by x and y, respectively. The master system is shown in Eq. (1) and the slave system is presented as follows: ˙y1 = y2 ˙y2 = p(y1, y2) + u1(t) ˙y3 = q(y1, y2, y3) + u2(t) (4) where u1, u2 ∈ R are the control inputs attached to the nonlinear gyroscope system. The control problem considered in this study is that of different initial conditions of the master and slave systems to be synchro- nized by designing appropriate input controls. We define synchronization errors between the master and slave systems as follows: e1 = y1 − x1 e2 = y2 − x2 e3 = y3 − x3. (5) The synchronization error dynamics can be obtained as follows: ˙e1(t) = e2(t) ˙e2(t) = p(y1, y2) − p(x1, x2) + u1(t) ˙e3(t) = q(y1, y2, y3) − q(x1, x2, x3) + u2(t). (6) The objective of the current control problem is to design the appropriate control inputs u1(t) and u2(t) such that for any initial conditions of the master and slave systems, the synchronization errors converge to zero such that the resulting synchronization error vector satisfies Eq. (7): lim t→∞ ‖y(t) − x(t)‖ → 0 (7) where ‖·‖ is the Euclidean norm of a vector. 3. Fault tolerant synchronization of chaotic gyroscopes versus disturbances 3.1. Fault tolerant control and synchronization As previous mentioned, fault diagnosis and isolation (FDI), and fault tolerant control (FTC) have been active research topics during the past two decades with a view to increasing the safety and reliability of dynamical systems [18,19]. Faults normally occur in two places: the actuators and the sensors. Actuator faults are faults that act on the system, resulting in the deviation of the process variables. The result is that the command (control) signal sent to this device has no effect. Sensor faults are faults that act on the sensors that measure the system variables, and they do not directly affect the process. The source of these faults could be wear and tear of the sensor leading to inaccurate readings, or a total failure of the sensor. Sensor faults could also arise from drift/poor calibration/extreme process conditions. These faults will affect the process if the output measurements are used to generate the input control signal [20]. FTC is an area of research that aims to increase applicability by specifically designing control algorithms capable of maintaining stability and performance despite the occurrence of faults. When faults happen in chaotic systems, the conventional methods for synchronizing the chaotic systems are invalid. In order to increase the safety and the reliability of synchronization systems when a fault happens, fault tolerant synchronization (FTS) is essential to guarantee the synchronization of chaotic systems. The objective of fault tolerant synchronization (FTS) is to design a fault tolerant controller to achieve the synchronization no matter whether a fault occurs or not. To the best of the authors’ knowledge, the problem of FTS is still an open problem and seldom researched [27,28]. In this study, the actuator faults are considered as intermittent faults and ramp faults. These occur in the slave system. Also, the external disturbances happen in the slave system, as shown in Fig. 1. The proposed method is the active fault tolerant synchronization. The disturbances are assumed to be bounded as follows: |d(t)| ≤ γ , where γ is a positive constant value. Also, the actuator faults assumed to be bounded as follows: |fa(t)| ≤ β, where β is a positive constant value which is the maximum power of actuator faults happening in the slave system. The actuator faults and external disturbances considered in this study are shown in the simulation section. 3.2. Description of fault tolerant synchronization of chaotic gyro- scopes versus disturbances via classic Lyapunov-based control and Lyapunov rule-based fuzzy control The scheme of fault tolerant synchronization of chaotic gyro- scope systems versus disturbances via Lyapunov rule-based fuzzy control is shown in Fig. 1. First, Lyapunov-based control (classic nonlinear control) is designed for chaos synchronization of chaotic gyroscope systems in Section 3.3. Second, Lyapunov rule-based
- 4. F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 53 Fig. 1. The scheme of fault tolerant synchronization of chaotic gyroscopes versus disturbances. fuzzy control (classic intelligent control) is designed for chaos syn- chronization of chaotic gyroscope systems in Section 3.4. Then, it is assumed that the states of slave systems are not available. An ap- propriate observer is designed for the linear part of the slave sys- tem in Section 3.5. Therefore, Lyapunov rule-based fuzzy control is designed for fault tolerant synchronization of the master and slave systems, considering the actuator faults, disturbances, and a state estimator. A theorem and its proof for the stability of fault tolerant synchronization of chaotic gyroscope systems under the aforemen- tioned conditions are presented at the end of Section 3.6. 3.3. Classic Lyapunov-based control for synchronization of chaotic gyroscopes In this section, classic nonlinear control based on Lyapunov sta- bility theory is designed for chaos synchronization of chaotic gyro- scope systems. To solve the chaos synchronization problem presented in Eq. (6), define a Lyapunov function as follows: V = 1 2 e2 1 + e2 2 + e2 3 . (8) We differentiate Eq. (8) with respect to time: ˙V = e1 ˙e1 + e2 ˙e2 + e3 ˙e3. (9) We substitute Eq. (6) into Eq. (9): ˙V = e2 (e1 + ˙e2) A + e3 ˙e3 B . (10) The corresponding requirement for Lyapunov stability is [34] ˙V < 0. (11) If A < 0 and B < 0, then the Lyapunov stability conditions will be satisfied. The following cases will satisfy all the stability conditions. This discussion is separately presented for A and B terms. Case 1 for A: If e2 > 0, then e1 + ˙e2 < 0. (12) Substituting Eq. (6) into Eq. (12), we obtain u1(t) < −e1 − p(y1, y2) + p(x1, x2). (13) A solution of Eq. (13) is considered: u∗ 1−A1 (t) = −e1 − p(y1, y2) + p(x1, x2) − λ (14) where λ is a positive constant value. Case 2 for A: If e2 < 0, then e1 + ˙e2 > 0. (15) Substituting Eq. (6) into Eq. (15), we obtain u1(t) > −e1 − p(y1, y2) + p(x1, x2). (16) A solution of Eq. (16) is considered: u∗ 1−A2 (t) = −e1 − p(y1, y2) + p(x1, x2) + λ (17) where λ is a positive constant value. Case 3 for A: For e1 > 0 and e2 ∈ zero, e2 (e1 + ˙e2) = −η |e2| (18) where η is a positive constant value. Substituting Eq. (6) into Eq. (18), we simplify as follows: u1(t) < −ηsgn(e2) − p(y1, y2) + p(x1, x2). (19) A solution of Eq. (19) is considered: u∗ 1−A3 (t) = −ηsgn(e2) − p(y1, y2) + p(x1, x2) − λ (20) where λ is a positive constant value. Case 4 for A: For e1 < 0 and e2 ∈ zero, e2 (e1 + ˙e2) = −η |e2| (21) where η is a positive constant value. Substituting Eq. (6) into Eq. (21), we simplify as follows: u1(t) > −ηsgn(e2) − p(y1, y2) + p(x1, x2). (22) A solution of Eq. (22) is considered: u∗ 1−A4 (t) = −ηsgn(e2) − p(y1, y2) + p(x1, x2) + λ (23) where λ is a positive constant value. If the system satisfies the conditions of Cases 1–4 then A < 0 and the error state will be asymptotically driven to zero. Case 5 for B: If e3 > 0, then ˙e3 < 0. (24) Substituting Eq. (6) into Eq. (24), we obtain u2(t) < −q(y1, y2, y3) + q(x1, x2, x3). (25)
- 5. 54 F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 A solution of Eq. (25) is considered: u∗ 2−B1 (t) = −q(y1, y2, y3) + q(x1, x2, x3) − λ (26) where λ is a positive constant value. Case 6 for B: If e3 < 0, then ˙e3 > 0. (27) Substituting Eq. (6) into Eq. (24), we obtain u2(t) > −q(y1, y2, y3) + q(x1, x2, x3). (28) A solution of Eq. (28) is considered: u∗ 2−B2 (t) = −q(y1, y2, y3) + q(x1, x2, x3) + λ (29) where λ is a positive constant value. If the system satisfies the conditions of Cases 5 and 6 then B < 0 and the error state will be asymptotically driven to zero. Therefore, the Lyapunov-based control is as follows: u1(t) = −e1 − p(y1, y2) + p(x1, x2) − λ, e2 > 0 −e1 − p(y1, y2) + p(x1, x2) + λ, e2 < 0 −ηsgn(e2) − p(y1, y2) + p(x1, x2) − λ, e1 > 0, e2 ∈ zero −ηsgn(e2) − p(y1, y2) + p(x1, x2) + λ, e1 < 0, e2 ∈ zero (30) u2(t) = −q(y1, y2, y3) + q(x1, x2, x3) − λ, e3 > 0 −q(y1, y2, y3) + q(x1, x2, x3) + λ, e3 < 0 (31) where λ and η are positive constant values. 3.4. Lyapunov rule-based fuzzy control for synchronization of chaotic gyroscopes A set of the fuzzy linguistic rules based on expert knowledge are applied to design the control law for fuzzy logic control. To overcome the trial-and-error tuning of the membership functions and rule base, the fuzzy rules are directly defined such that the error dynamics satisfies stability in the Lyapunov sense. The basic fuzzy logic system is composed of five function blocks as follows [35]. (1) A rule base contains a number of fuzzy if–then rules. (2) A database defines the membership functions for the fuzzy sets used in the fuzzy rules. (3) A decision-making unit performs the inference operations on the rules. (4) A fuzzification interface transforms the crisp inputs into degrees of match with a linguistic value. (5) A defuzzification interface transforms the fuzzy results for the inference into a crisp output. The fuzzy rule base consists of a collection of fuzzy if–then rules expressed in the form of if a is A then b is B, where a and b denote linguistic variables, and A and B represent linguistic values that are characterized by membership functions. All of the fuzzy rules can be used to construct the fuzzy associated memory. In this study, the two FLC are designed as follows. The signal (e1, e2) in Eq. (6) is the antecedent part of the proposed FLC1 for designing the control input u1 that will be used in the consequent part of the proposed FLC1: u1 = FLC1(e1, e2). (32) The signal e3 in Eq. (6) is the antecedent part of the proposed FLC2 for designing the control input u2 that will be used in the consequent part of the proposed FLC2: u2 = FLC2(e3) (33) where the FLCs accomplish the objective of stabilizing the error dynamics Eq. (6). The ith if–then rule of the fuzzy rule base of the FLC1 is of the following form: Rule i: if e1 is X1 and e2 is X2 then u1−Li ≡ f1−i(e1, e2) (34) Table 1 Rule table for FLC1. Rule Antecedent Consequent e1 e2 u1−Li 1 P P u1−L1 = −e1 − p(y1, y2) + p(x1, x2) − λ 2 P Z u1−L2 = −ηsgn(e2)−p(y1, y2)+p(x1, x2)−λ 3 P N u1−L3 = −e1 − p(y1, y2) + p(x1, x2) + λ 4 Z P u1−L4 = −e1 − p(y1, y2) + p(x1, x2) − λ 5 Z Z u1−L5 = zero 6 Z N u1−L6 = −e1 − p(y1, y2) + p(x1, x2) + λ 7 N P u1−L7 = −e1 − p(y1, y2) + p(x1, x2) − λ 8 N Z u1−L8 = −ηsgn(e2)−p(y1, y2)+p(x1, x2)+λ 9 N N u1−L9 = −e1 − p(y1, y2) + p(x1, x2) + λ Table 2 Rule table for FLC2. Rule Antecedent Consequent e3 u2−Li 1 P u2−L1 = −q(y1, y2, y3)+q(x1, x2, x3)−λ 2 Z u2−L2 = zero 3 N u2−L3 = −q(y1, y2, y3)+q(x1, x2, x3)+λ where X1 and X2 are the input fuzzy sets, and u1−Li is the out- put which is the analytical function f1−i(.) of the input variables (e1, e2). For given input values of the process variables, their de- grees of membership µxi, i = 1, 2, . . . , n, called rule-antecedent weights, are calculated. The centroid defuzzifier evaluates the out- put of all rules as follows: u1 = n∑ i=1,µ̸=0 µ1−i.u1−Li n∑ i=1,µ̸=0 µ1−i , µ1−i = µX1(e1)µX2(e2). (35) u2 = n∑ i=1,µ̸=0 µX (e3).u2−Li n∑ i=1,µ̸=0 µX (e3) . (36) u2−Li is the output which is the analytical function f2−i(.) of the input variable (e3). Its degrees of membership µx, called rule- antecedent weights, are calculated to give the input values of the process variables. Table 1 lists the fuzzy rule base in which the input variables in the antecedent parts of the rules are e1 and e2, and the output variable in the consequent is u1−Li. Table 2 lists the fuzzy rule base in which the input variable in the antecedent parts of the rules is e3, and the output variable in the consequent is u2−Li. We use P, Z and N as input fuzzy sets representing ‘positive’, ‘zero’ and ‘negative’, respectively. The Gaussian membership function is considered. The combination of the two input variables (e1, e2) forms n = 9 heuristic rules in Table 1 and each rule belongs to one of the three fuzzy sets P, Z and N. The input variable (e3) forms n = 3 heuristic rules in Table 2 and each rule belongs to one of the three fuzzy sets P, Z and N. The rules in Tables 1 and 2 are read as, taking rule 1 in Table 1 as an example, ‘rule 1: if input 1 e1 is P and input 2 e2 is P, then the output is u1−L1’. If the system satisfies the conditions of Cases 1–4 then A < 0 and the error state will be asymptotically driven to zero. In order to achieve this result, we will design u1−L by using the rules in Table 1 for FLC1. According to the stability analysis method [34], only those fuzzy subsystems that correspond to each rule need to be considered. From Table 1, it can be seen that (e1, e2) belong to fuzzy sets P, Z and N. The heuristic rules in Table 1 are divided into five parts in order to discuss the stability of each subsystem. For rules 1, 4 and 7 in Table 1, the error state e2 is positive, and
- 6. F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 55 u1−L1 = u1−L4 = u1−L7 = u∗ 1−A1. For rules 3, 6 and 9 in Table 1, the error state e2 is negative, and u1−L3 = u1−L6 = u1−L9 = u∗ 1−A2. For rule 2 in Table 1, the error state e1 is positive and e2 is zero, and u1−L2 = u∗ 1−A3. For rule 8 in Table 1, the error state e1 is negative and e2 is zero, and u1−L8 = u∗ 1−A4. For rule 5 in Table 1, the error states e1 and e2 are zeros. This condition is included in the other rules, and we define u1−L5 = 0. If the system satisfies the conditions of Cases 5 and 6 then B < 0 and the error state will be asymptotically driven to zero. In order to achieve this result, we will design u2−L by using the rules of Table 1 for FLC2. According to the stability analysis method [34], only the fuzzy subsystems that correspond to each rule need to be considered. From Table 2, it can be seen that e3 belongs to fuzzy sets P, Z and N. The heuristic rules in Table 2 are divided into three parts in order to discuss the stability of each subsystem. For rules 1, the error state e3 is positive, and u2−L1 = u∗ 2−B1. For rules 3, the error state e3 is negative, and u2−L3 = u∗ 2−B2. For rule 2 in Table 2, the error state e3 is zero. This condition is included in the other rules, and we define u2−L2 = 0. Therefore, all of the rules in FLC1 and FLC2 can lead to Lyapunov stable subsystems under the same Lyapunov function, Eq. (8). Furthermore, the closed-loop rule-based system Eq. (6) is asymptotically stable for each derivative of the Lyapunov function that satisfies ˙V < 0 in Tables 1 and 2. That is, the error states guarantee convergence to zero and the system of two nonlinear gyros is synchronized. 3.5. Design of an observer and residuals Consider the slave system when the actuator faults and external disturbances are added to the system. It is represented as follows: ˙y1 = y2 ˙y2 = p(y1, y2) + u1(t) + Fa1 (t) + d1(t) = − c1 I1 y2 + p(y1, 0) + u1(t) + Fa1 (t) + d1(t) ˙y3 = q(y1, y2, y3) + u2(t) + Fa2 (t) + d2(t) Yout = C y1 y2 y3 T (37) where C = I3×3. Fa1 and Fa2 are actuator faults. d1 and d2 are external disturbances. The slave system is regarded as consisting of two parts: the linear and nonlinear parts. The linear parts of Eq. (30) are as follows: A = 0 1 0 0 − c1 I1 0 0 0 0 ; B = 0 1 1 ; C = 1 0 0 0 1 0 0 0 1 . (38) Notice that the pair (A, C) is observable. A Luenberger observer is designed for the linear part of Eq. (38): ˙ˆY = A ˆY + Bu + L(Yout − ˆYout) Yout = C ˆY (39) where ˆY = [ˆY1 ˆY2 ˆY3]T . The observer gain L is selected such that the eigenvalues of A0 = A − LC are located in the left half- plane. We define the estimated errors as follows: eo = Yout − ˆYout. (40) Hence the dynamics of observer error is ˙eo = AoCeo. (41) Since Ao is stable, for any given positive-definite matrix Qo, there exists a unique positive-definite Po such that the Lyapunov equation is established [36]: AT o Po + PoAo = −Qo. (42) In this study, L is calculated from pole-placement method. A residual is defined as follows: R(t) = Yout(t) − ˆYout(t). (43) From Fig. 4, the estimated synchronization error is defined as es1 = ˆY1 − x1 es2 = ˆY2 − x2 es3 = ˆY3 − x3. (44) Notice that Eq. (44) will be used for designing the input controls via Lyapunov rule-based fuzzy control. These are considered as input variables for FLC1 and FLC2. 3.6. Stability of fault tolerant synchronization of chaotic gyroscope systems In this section, a theorem is proven for the stability of fault tolerant synchronization of chaotic gyroscope systems, concerning the state estimator, actuator faults and external disturbances via Lyapunov rule-based fuzzy control. Theorem. Consider the master and the slave chaotic gyroscope sys- tems presented by Eqs. (1) and (37). Actuator faults and disturbances are occurring in the slave system. The states of the slave system are unavailable and the state estimator presented by Eq. (39) is utilized to observe the states of the slave system. The estimated synchroniza- tion errors are represented by Eq. (44). Then, FTS is achieved by two Lyapunov rule-based fuzzy controls as follows: u1 = FLC1(es1, es2) and u2 = FLC2(es3) where es1, es2, and es3 presented by Eq. (44) are the antecedent parts of the FLCs and the control inputs (u1 and u2) are the outputs of the FLCs. Proof. The two FLCs accomplish the objective of stabilizing the error dynamics of Eq. (36). The ith if–then rule of the fuzzy rule base of the FLC1 and FLC2 are of the following forms: Rule i: if es1 is X1 and es2 is X2 then u1−Li ≡ f1−i(es1, es2) (45) Rule i: if es3 is X1 then u2−Li ≡ f2−i(es3) (46) where Xi are the input fuzzy sets, u1−Li and u2−Li are the outputs which are the analytical functions f1−i(.) and f2−i(.) of the input variables (es1, es2) and (es3). In the ‘‘3.6.1: Consequent parts of the FLCs’’ subsection, u1−Li and u2−Li are obtained. For given input values of the process variables, their degrees of membership µxi, i = 1, 2, . . . , n, called rule-antecedent weights, are calculated. The centroid defuzzifier evaluates the output of all rules as follows: u1 = n∑ i=1,µ̸=0 µ1−i.u1−Li n∑ i=1,µ̸=0 µ1−i , µ1−i = µX1(es1)µX2(es2) (47) u2 = n∑ i=1,µ̸=0 µX (es3).u2−Li n∑ i=1,µ̸=0 µX (es3) . (48) Table 3 lists the fuzzy rule base in which the input variables in the antecedent parts of the rules are es1 and es2, and the output variable in the consequent is u1−Li. Table 4 lists the fuzzy rule base in which the input variable in the antecedent part of the
- 7. 56 F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 Fig. 2a. Actuator faults: intermittent faults. Fig. 2b. Actuator faults: ramp faults. Fig. 2c. External disturbances. rules is es3, and the output variable in the consequent is u2−Li. We use P, Z and N as input fuzzy sets representing ‘positive’, ‘zero’ and ‘negative’, respectively. The Gaussian membership function is considered. Notice that λ = γ +β+k, where k is a positive constant value. Therefore, all of the rules in the FLC1 and FLC2 can lead to Lyapunov stable subsystems under the same Lyapunov function as follows: V = 1 2 e2 s1 + e2 s2 + e2 s3 . (49) The closed-loop rule-based system is asymptotically stable for each derivative of the Lyapunov function that satisfies ˙V < 0 in Tables 3 and 4. Notice that according to the Lyapunov equation presented by Eq. (42), the estimated errors presented by Eq. (40) will converge to zero. Therefore, it is guaranteed that the error states converge to zero and fault tolerant synchronization is achieved. 3.6.1. Consequent parts of the FLCs In this subsection, u1−Li and u2−Li are obtained.
- 8. F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 57 (a) Master: X. (b) Slave: Y. (c) Estimated slave: Yhat. Fig. 3. Classic Lyapunov-based control: time responses of the master, slave and estimated slave systems, when disturbances and intermittent actuator faults occur. (a) Synchronization error: y − x. (b) Estimation error: y − yhat. (c) Error: yhat − x. Fig. 4. Classic Lyapunov-based control: synchronization error, estimation error and estimated synchronization error, when disturbances and intermittent actuator faults occur. Table 3 Rule table for FLC1. Rule Antecedent Consequent es1 es2 u1−Li 1 P P u1−L1 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) − λ 2 P Z u1−L2 = −ηsgn(es2)−p(ˆY1, ˆY2)+p(x1, x2)−λ 3 P N u1−L3 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) + λ 4 Z P u1−L4 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) − λ 5 Z Z u1−L5 = zero 6 Z N u1−L6 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) + λ 7 N P u1−L7 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) − λ 8 N Z u1−L8 = −ηsgn(es2)−p(ˆY1, ˆY2)+p(x1, x2)+λ 9 N N u1−L9 = −es1 − p(ˆY1, ˆY2) + p(x1, x2) + λ As presented by Eq. (44), the estimated errors are as follows: es1 = ˆY1 − x1 es2 = ˆY2 − x2 es3 = ˆY3 − x3. (50) Table 4 Rule table for FLC2. Rule Antecedent Consequent e3 u2−Li 1 P u2−L1 = −q(ˆY1, ˆY2, ˆY3)+q(x1, x2, x3)−λ 2 Z u2−L2 = zero 3 N u2−L3 = −q(ˆY1, ˆY2, ˆY3)+q(x1, x2, x3)+λ The dynamics of Eq. (50) is obtained from Eqs. (1) and (37): ˙es1(t) = es2(t) ˙es2(t) = p(ˆY1, ˆY2) − p(x1, x2) + u1(t) + Fa1 (t) + d1(t) ˙es3(t) = q(ˆY1, ˆY2, ˆY3) − q(x1, x2, x3) + u2(t) + Fa2 (t) + d2(t). (51) We define a Lyapunov function as follows: V = 1 2 e2 s1 + e2 s2 + e2 s3 . (52) We differentiate Eq. (52) with respect to time: ˙V = es1 ˙es1 + es2 ˙es2 + es3 ˙es3. (53)
- 9. 58 F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 Fig. 5. Classic Lyapunov-based control: control inputs obtained via Lyapunov rule-based fuzzy control, when disturbances and intermittent actuator faults occur. Fig. 6. Classic Lyapunov-based control: time responses of the master, slave and estimated slave systems, when disturbances and ramp actuator faults occur. We substitute Eq. (51) into Eq. (53): ˙V = es2 (es1 + ˙es2) A + es3 ˙es3 B . (54) The corresponding requirement for Lyapunov stability is [34] ˙V < 0. (55) If A < 0 and B < 0, then the Lyapunov stability will be satisfied. The following cases will satisfy all the stability conditions. This discussion is separately presented for A and B terms. Case 1 for A: If es2 > 0, then es1 + ˙es2 < 0. (56) Substituting Eq. (51) into Eq. (56), we obtain u1(t) < −es1 − p(ˆY1, ˆY2) + p(x1, x2) − Fa1 (t) − d1(t). (57) A solution of Eq. (57) is considered: u∗ 1−A1 (t) = −es1 − p(ˆY1, ˆY2) + p(x1, x2) − λ (58) where λ = γ + β + k and k is a positive constant value. Case 2 for A: If e2 < 0, then es1 + ˙es2 > 0. (59) Substituting Eq. (51) into Eq. (59), we obtain u1(t) > −es1 − p(ˆY1, ˆY2) + p(x1, x2) − Fa1 (t) − d1(t). (60) A solution of Eq. (60) is considered: u∗ 1−A2 (t) = −es1 − p(ˆY1, ˆY2) + p(x1, x2) + λ (61) where λ = γ + β + k and k is a positive constant value. Case 3 for A: For es1 > 0 and es2 ∈ zero,
- 10. F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 59 Fig. 7. Classic Lyapunov-based control: synchronization error, estimation error and estimated synchronization error, when disturbances and ramp actuator faults occur. Fig. 8. Classic Lyapunov-based control: control inputs obtained via Lyapunov rule-based fuzzy control, when disturbances and ramp actuator faults occur. es2 (es1 + ˙es2) = −η |es2| (62) where η is a positive constant value. Substituting Eq. (51) into Eq. (62), we simplify as follows: u1(t) < −ηsgn(es2) − p(ˆY1, ˆY2) + p(x1, x2) − Fa1 (t) − d1(t). (63) A solution of Eq. (63) is considered: u∗ 1−A3 (t) = −ηsgn(es2) − p(ˆY1, ˆY2) + p(x1, x2) − λ (64) where λ = γ + β + k and k is a positive constant value. Case 4 for A: For es1 < 0 and es2 ∈ zero, es2 (es1 + ˙es2) = −η |es2| (65) where η is a positive constant value. Substituting Eq. (51) into Eq. (65), we simplify as follows: u1(t) > −ηsgn(es2) − p(ˆY1, ˆY2) + p(x1, x2) − Fa1 (t) − d1(t). (66) A solution of Eq. (66) is considered: u∗ 1−A4 (t) = −ηsgn(e2) − p(ˆY1, ˆY2) + p(x1, x2) + λ (67) where λ = γ + β + k and k is a positive constant value. If the system satisfies the conditions of Cases 1–4 then A < 0 and the error state will be asymptotically driven to zero. Case 5 for B: If es3 > 0, then ˙es3 < 0. (68) Substituting Eq. (51) into Eq. (68), we obtain u2(t) < −q(ˆY1, ˆY2, ˆY3) + q(x1, x2, x3) − Fa2 (t) − d2(t). (69) A solution of Eq. (69) is considered: u∗ 2−B1 (t) = −q(ˆY1, ˆY2, ˆY3) + q(x1, x2, x3) − λ (70) where λ = γ + β + k and k is a positive constant value. Case 6 for B: If es3 < 0, then ˙es3 > 0. (71)
- 11. 60 F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 Fig. 9. Lyapunov rule-based fuzzy control: time responses of the master, slave and estimated slave systems, when disturbances and intermittent actuator faults occur. Fig. 10. Lyapunov rule-based fuzzy control: synchronization error, estimation error and estimated synchronization error, when disturbances and intermittent actuator faults occur.
- 12. F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 61 Fig. 11. Lyapunov rule-based fuzzy control: control inputs obtained via Lyapunov rule-based fuzzy control, when disturbances and intermittent actuator faults occur. Fig. 12. Lyapunov rule-based fuzzy control: time responses of the master, slave and estimated slave systems, when disturbances and ramp actuator faults occur. Substituting Eq. (51) into Eq. (71), we obtain u2(t) > −q(ˆY1, ˆY2, ˆY3) + q(x1, x2, x3) − Fa2 (t) − d2(t). (72) A solution of Eq. (72) is considered: u∗ 2−B2 (t) = −q(ˆY1, ˆY2, ˆY3) + q(x1, x2, x3) + λ (73) where λ = γ + β + k and k is a positive constant value. If the system satisfies the conditions of Cases 5, 6 then B < 0 and the error state will be asymptotically driven to zero. In Table 3, for rules 1, 4 and 7 in Table 1, the error state es2 is positive, and u1−L1 = u1−L4 = u1−L7 = u∗ 1−A1. For rules 3, 6 and 9 in Table 3, the error state es2 is negative, and u1−L3 = u1−L6 = u1−L9 = u∗ 1−A2. For rule 2 in Table 3, the error state es1 is positive and es2 is zero, and u1−L2 = u∗ 1−A3. For rule 8 in Table 3, the error state es1 is negative and es2 is zero, and u1−L8 = u∗ 1−A4. For rule 5 in Table 3, the error states es1 and es2 are zeros. This condition is included in the other rules, and we define u1−L5 = 0. In Table 4, for rules 1, the error state es3 is positive, and u2−L1 = u∗ 2−B1. For rules 3, the error state es3 is negative, and u2−L3 = u∗ 2−B2. For rule 2 in Table 4, the error state es3 is zero. This condition is included in the other rules, and we define u2−L2 = 0. 4. Simulation results In this section, numerical simulations are given to demonstrate the FTS of the chaotic gyros versus disturbances via Lyapunov- based control and Lyapunov rule-based fuzzy control. The utilized actuator faults and disturbances are introduced in Section 3.1. d1 is attached between 3 < t < 4 and 7 < t < 8. d2 is attached between 1 < t < 2 and 5 < t < 6. The actuator faults in this study are shown in Figs. 2a–2c. The parameters of the nonlinear chaotic gyroscope systems are specified in Section 2. The initial conditions of the master and slave systems are defined as follows: [x1(0)x2(0)x3(0)] = [1 − 1 1], [y1(0)y2(0)y3(0)] = [2 − 2 − 4]. The initial conditions of the observer are defined as follows: [yO1(0)yO2(0)yO3(0)] = [1.5 1 2]. 4.1. Simulation results for Lyapunov-based control Figs. 3–5 are related to Lyapunov-based control when intermit- tent actuator faults and disturbances occur. The time responses of the master, slave and estimated slave systems are shown in Fig. 3((a)–(c)). The synchronization error, estimation error and es- timated synchronization error are shown in Fig. 4((a)–(c)). The er- rors illustrated in Fig. 4 converge asymptotically to zero. Notice
- 13. 62 F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 Fig. 13. Lyapunov rule-based fuzzy control: synchronization error, estimation error and estimated synchronization error, when disturbances and ramp actuator faults occur. that the residuals are the estimation errors shown in Fig. 4. Fig. 5 shows the control inputs obtained via Lyapunov-based control. Figs. 6–8 are related to Lyapunov-based control when ramp actuator faults and disturbances occur. The time responses of the master, slave and estimated slave systems are shown in Fig. 6((a)–(c)). The synchronization error, estimation error and estimated synchronization error are shown in Fig. 7((a)–(c)). Obviously, the errors illustrated in Fig. 7 converge asymptotically to zero. Notice that the residuals are the estimation errors shown in Fig. 7. Fig. 8 shows the control inputs obtained via Lyapunov rule- based fuzzy control. As shown in these simulation results (Figs. 3–8), the Lyapunov- based control method is not robust. Also, it cannot be useful for fault tolerant synchronization of chaotic gyroscope systems. The synchronization errors (Figs. 4 and 7) are bounded, but they do not converge to zero. 4.2. Simulation results for Lyapunov rule-based fuzzy control Figs. 9–11 are related to Lyapunov rule-based fuzzy control when intermittent actuator faults and disturbances occur. The time responses of the master, slave and estimated slave sys- tems are shown in Fig. 9((a)–(c)). The synchronization error, es- timation error and estimated synchronization error are shown in Fig. 10((a)–(c)). The errors illustrated in Fig. 10 converge asymp- totically to zero. Notice that the residuals are the estimation er- rors shown in Fig. 10. Fig. 11 shows the control inputs obtained via Lyapunov rule-based fuzzy control. Figs. 12–14 are related to Lyapunov-based control when ramp actuator faults and disturbances occur. The time responses of the master, slave and estimated slave systems are shown in Fig. 12((a)–(c)). The synchronization error, estimation error and estimated synchronization error are shown in Fig. 13((a)–(c)). Obviously, the errors illustrated in Fig. 13 converge asymptotically to zero. Notice that the residuals are the estimation errors shown in Fig. 13. Fig. 14 shows the control inputs obtained via Lyapunov rule-based fuzzy control. The simulation results for FTS via Lyapunov rule-based fuzzy control show good performances and confirm that the master and slave systems achieve the synchronized states, when actuator faults and disturbances occur. From the simulation results, it is shown that the Lyapunov rule-based fuzzy control is better than the classic Lyapunov-based
- 14. F. Farivar, M.A. Shoorehdeli / ISA Transactions 51 (2012) 50–64 63 Fig. 14. Lyapunov rule-based fuzzy control: control inputs obtained via Lyapunov rule-based fuzzy control, when disturbances and ramp actuator faults occur. control. It is robust and more capable than the classic control for FTS. Also, these results demonstrate that by using Lyapunov rule- based fuzzy control, the synchronization error states are regulated to zero asymptotically. It is clear that the proposed method (Lyapunov rule-based fuzzy control) is capable of achieving FTS, when actuator faults and disturbances occur. 5. Conclusion The fault tolerant synchronization of chaotic gyroscope systems via Lyapunov rule-based fuzzy control has been addressed. Two techniques are compared: classic Lyapunov-based control and Lyapunov rule-based fuzzy control. In this study, a slave observer is designed to estimate the states of the slave system with respect to actuator faults and disturbances. 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