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Decentralised fault‐tolerant finite‐time control for a class of interconnected non‐linear systems

2015; Institution of Engineering and Technology; Volume: 9; Issue: 16 Linguagem: Inglês

10.1049/iet-cta.2014.1264

1751-8652

Changchun Hua, Yafeng Li, Hongbin Wang, Xinping Guan,

Stability and Control of Uncertain Systems

IET Control Theory & ApplicationsVolume 9, Issue 16 p. 2331-2339 Research ArticlesFree Access Decentralised fault-tolerant finite-time control for a class of interconnected non-linear systems Changchun Hua, Corresponding Author Changchun Hua cch@ysu.edu.cn Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004 People's Republic of ChinaSearch for more papers by this authorYafeng Li, Yafeng Li Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004 People's Republic of ChinaSearch for more papers by this authorHongbin Wang, Hongbin Wang Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004 People's Republic of ChinaSearch for more papers by this authorXinping Guan, Xinping Guan Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004 People's Republic of ChinaSearch for more papers by this author Changchun Hua, Corresponding Author Changchun Hua cch@ysu.edu.cn Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004 People's Republic of ChinaSearch for more papers by this authorYafeng Li, Yafeng Li Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004 People's Republic of ChinaSearch for more papers by this authorHongbin Wang, Hongbin Wang Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004 People's Republic of ChinaSearch for more papers by this authorXinping Guan, Xinping Guan Institute of Electrical Engineering, Yanshan University, Qinhuangdao City, 066004 People's Republic of ChinaSearch for more papers by this author First published: 01 October 2015 https://doi.org/10.1049/iet-cta.2014.1264Citations: 21AboutSectionsPDF 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 investigates decentralised fault-tolerant finite-time stabilisation problem for a class of interconnected non-linear systems with gain fault in each actuator. Each subsystem of the interconnected system is with lower-triangular structure. In the procedure of controller design, unknown faults of the system are eliminated via robust control technology. Then, based on finite-time Lyapunov stability theorem and recursive design algorithm, decentralised state feedback controller is proposed to guarantee global finite-time stabilisation of the interconnected system. Finally, two simulation examples are given to verify the effectiveness of the proposed design method. 1 Introduction Finite-time stabilisation control is a kind of time optimal control. In contrast to the commonly used notion of asymptotic stability, its main characteristic is to render state of the system to equilibrium in finite time and keep them there thereafter. Besides, there are also some other nice features such as faster convergence rates, higher accuracies [1], better disturbance rejection properties [2, 3] and so on. Due to these so good features above, finite-time stabilisation control has attracted more and more attention. Similarly, finite-time control for high-order non-linear system has also drawn an increasing attention in recent years. Many kinds of controller design methods are proposed to guarantee finite-time stabilisation of the non-linear system, such as [4–7]. In [4], the global finite-time stabilisation problem was studied for a class of uncertain non-linear systems. A recursive design algorithm was developed for the construction of a Höder continuous, global finite-time stabiliser as well as a c1 positive definite and proper Lyapunov function. The work [5] investigated the problem of global finite-time stabilisation in probability for a class of stochastic non-linear systems. Based on homogeneous domination approach and stochastic finite-time stability theorem, it was proved that the designed controller can render the closed-loop system to equilibrium in finite time and stay there thereafter. In [6], a class of non-linear systems dominated by a lower-triangular model with a time-varying gain were considered and state feedback finite-time stabilisation controller was proposed with gains being tuned online to guarantee finite-time stabilisation of the non-linear system. Interconnected systems can be found in practical engineering, such as power networks, transportation, aerospace, economics and management. In the interconnected systems, each subsystem includes state of the other subsystems, which makes the decentralised controller design more difficult and challenging, see [8–14]. The work [11] investigated adaptive decentralised fuzzy output feedback fault-tolerant control problem for a class of non-linear large-scale systems in strict-feedback form. In [12], the adaptive laws and output feedback controllers were developed for non-linear interconnected time-delay systems via backstepping technology. In [13], the adaptive fuzzy output feedback control problem was investigated for a class of non-linear time-delay systems with unknown control direction. It is well known that sensor or component faults frequently occur and actuators may undergo complete failures or partial loss of effectiveness faults during operation, which may cause system performance deterioration and even lead to instability. In order to guarantee stabilisation of the system and improve the system performance, many effective approaches have been developed, see [15–18]. The study [19] focused on a class of multi-input and multi-output non-linear systems in strict-feedback form and developed an adaptive fuzzy control method for accommodating actuator faults. In [16], robust fault-tolerant control against time-varying actuator faults and saturation was investigated. A set of invariance conditions were developed and applied to fault-tolerant control problem against the actuator effectiveness loss for linear systems subject to system uncertainties and actuator saturation. In this paper, we consider the problem of decentralised fault-tolerant finite-time stabilisation for a class of non-linear interconnected systems, of which each subsystem is with lower-triangular structure. Based on finite-time Lyapunov stability theory and recursive design algorithm, state feedback stabilisation controllers are designed to guarantee finite-time stabilisation of the interconnected system. Robust control technology is applied to deal with the faults that possibly occur during the operation. Finally, two simulation examples are given to verify the effectiveness of the proposed designed controller. The rest of the paper is organised as follows. In Section 2, the system descriptions and some necessary assumptions are provided. Moreover, the Lyapunov theory for finite-time stabilisation of continuous autonomous systems and another three useful lemmas are given. In Section 3, the main result of this paper is presented, in which a non-Lipschitz continuous state feedback control laws is explicitly constructed for a family of non-linear interconnected systems. Based on the finite-time stabilisation theory, it is proved that the designed controller can guarantee finite-time stabilisation of the system. Then, two simulation examples verify the effectiveness of the proposed method in Section 4. Finally, the paper is concluded in Section 5. 2 System formulation and preliminaries In this paper, we consider a class of large-scale non-linear systems with the i th subsystem described by (1)where 1 ≤ i ≤ n, 1 ≤ j ≤ ni, xij(t) ∈ R and yi(t) ∈ R are state variable and output of the i th subsystem, respectively. ui(t) ∈ R is the input of the i th subsystem. is the vector of state of the i th subsystem. is the vector of outputs. gij (·) and fij (·) are uncertain non-linear functions satisfying Assumptions 1 and 2, respectively. δij is used to describe the unknown fault, which satisfies Assumption 3. Assumption 1.The non-linear function gij (·) satisfies the following condition (2)where is a known c1 function. Assumption 2.The non-linear function fij (·) satisfies the following inequalities (3)in which for k = 1, 2, …, n, φijk(yk) ≥ 0 is a known c1 function and αijk ≥ (2N − 1)/(2N + 1), N = max {n1, n2, …, nn} . Assumption 3.δij represent the fault that occurs during the operation of the system, which is an unknown bounded scalar with the known sign. Without losing generality, we assume δij > 0, and its upper bound and lower bound are known positive scalars and , respectively. Remark 1.Assumption 1 can be found in the work [4], which is used to deal with the non-linear term only including the i th subsystem's state variables. In this paper, we consider the interconnection of outputs. In order to design decentralised controller to render state variables and outputs to zero, we introduce Assumption 2. Assumption 3 shows that the precise value of parameters δij need not be obtained, but its sign and bound need be available, which is a common assumption in the existing literature. Remark 2.The unknown parameter can be viewed as the gain fault of actuator of the i th subsystem, which is also called linear fault in [15]. In the existing literature, with some appropriate assumptions for the faults, many effective fault-tolerant methods have been proposed to guarantee stabilisation of the system, such as [15, 16, 19]. In this paper, we assume that upper and lower bounds of the gain fault are known and the fault-tolerant controller is designed by robust technology to guarantee stabilisation of the interconnected system. To study finite-time stabilisation, we introduce some basic concepts and lemmas as the basis for the design of our decentralised finite-time controller. We also recall the theorem for finite-time stability of autonomous systems, which was discussed previously in [4, 20, 21]. Definition 1 (Hong et al. [20]; Haddad et al. [21]).Consider a system (4)where f : U0 × R+ → Rn is continuous with respect to x on an open neighbourhood U0 of the origin x = 0. The equilibrium x = 0 of the system is (locally) finite-time stable if it is Lyapunov stable and finite-time convergent in a neighbourhood U ⊆ U0 of the origin. By ‘finite-time convergence’, we mean that, for any initial condition x (t0) = x0 ∈ U at any given initial time t0, there is a settling time 0< T < + ∞, such that every solution x (t ; t0 ; x0) of system (4) defined with x (t ; t0 ; x0) ∈ U /{0} for t ∈ [t0, T) satisfies that limt →T x (t ; t0 ; x0) = 0, and x (t ; t0 ; x0) = 0, ∀t > T. When U = Rn, the origin is a globally finite-time stable equilibrium. Lemma 1.Suppose that, for system (4), there is c1 function V (x) defined on a neighbourhood of the origin, and real numbers c > 0 and 0 < α < 1, such that V (x) is positive definite on ; , . Then, the origin of system (4) is locally finite-time stable. The settling time, depending on the initial state x (0) = x0, satisfies Tx(x0) ≤ V1−α(x0)/(c (1 − α)), for all x0 in some open neighbourhood of the origin. If and V (x) is also radially unbounded [i.e. V (x)→ + ∞ as ], the origin of system (4) is globally finite-time stable. Proof.The proof is straightforward by following that in [2], and thus omitted. □ Lemma 2 (Huang et al. [4]).For any real numbers xij, i = 1, …, n, j = 1, …, ni and 0 < b ≤ 1, the following inequality holds: (5)When b = p /q ≤ 1, where p > 0 and q > 0 are odd integers (6) Lemma 3 (Huang et al. [4]).Let c, d be positive real numbers and γ (x, y) > 0 a real-valued function. Then (7) 3 Finite-time controller design In this section, a fault-tolerant controller is provided to guarantee finite-time stabilisation of the state xij . First, we choose Lyapunov function for the i th subsystem as (8)where the parameter qij = (2ni + 3 −2j)/(2ni + 1), and the virtual c0 controllers are defined by (9)in which is the i th subsystem controller and are c1 functions which will be designed in the following. Next, through magnifying or shrinking inequality and the inductive method, the controller is designed. Step 1: The derivative of is (10)Under Assumptions 1, 2 and Lemma 3, one can obtain (11)where di = 4ni /(2ni + 1). , , are c1 functions. , j = 2, 3, …, ni, are positive constants. We choose the virtual controller as (12)where is a c1 function. Substituting (12) into (11), we obtain (13)Step 2: The derivative of Vi 2 = Vi 1 + Wi 2 is (14)where and we define for simpleness. Then, we can obtain (15)Using Propositions 1–4 (which can be found in the Appendix) and inequality (13), one can obtain (16)We choose the virtual controller as (17)where is a c1 function. Substituting (17) into (16), we obtain (18)Inductive step: Suppose, at step k − 1, the derivative of c1 function Vi (k −1) is as follows (19)Then, the derivative of c1 function Vik = Vi (k −1) + Wik is (20)Using Propositions 1–4 and inequality (19), we can obtain (21)The virtual controller is chosen as (22)where is a c1 function. Substituting (22) into (21), then (21) can be converted to (23)Step ni : Using the inductive argument above, we can conclude that there exists a non-Lipschitz state feedback control law of the form (24)with , such that (25)For the whole interconnected system, we choose the Lyapunov function as follows: (26)Combined with inequality (25), we can obtain the derivative of V satisfying the following inequality: (27)Based on (27), by two steps, we illustrate the controller which we design can guarantee global finite-time stability of the interconnected system (1). First: We know that if the initial value of the vector belongs to R0 ⊆ Rn1 +n2 +⋯+nn, which makes established for ∀t ≥ 0, (27) can be converted to the following form (28)where d = max {d1, d2, …, dn} . Using Proposition 5, Lemma 2 and (26), we can obtain (29)Combined with (28), we can obtain (30)By Lemma 1, the designed controller can guarantee the finite-time stability of the interconnected system (1). Second: From the inequality (27), although the initial value of the vector x0 does not belong to R0, we can still conclude that there must exist t1 > 0 such that for ∀t ≥ t1 the value of the vector belongs to R0 . Combined with the conclusion of the first step, we can obtain the main theorem of this paper. Theorem: For system (1) satisfying Assumptions 1–3, the fault-tolerant controller (24) can guarantee the global finite-time stabilisation of the interconnected non-linear system (1). Remark 3.Based on Lemma 1, when the initial value of the vector x0 belongs to R0, one has the settling time Tx(x0) ≤ V1−(d /2) (x0)/({1/2}(1 − {d /2})). If the initial value of the vector x0 does not belong to R0, we can obtain the settling time by two steps. First, we calculate the time t1, which satisfies the inequality and then the value of the state becomes Second, based on Lemma 1, we can obtain the settling time Remark 4.In this paper, the fault-tolerant global finite-time control is extended to a class of large-scale systems. Compared with the control of the single non-linear system, the non-linear interconnections render the control design complex and difficult. The appearance of the fault in the actuator of each subsystem may cause system performance deterioration and even lead to instability. A systematic method is proposed for the decentralised control design. By using robust control technology, the unknown fault is well dealt with. Based on the backstepping method, the decentralised fault-tolerant controller is designed to guarantee global finite-time stabilisation of the interconnected system. Remark 5.The global finite-time control problem is investigated for interconnected systems in this paper. Based on finite-time Lyapunov stability theorem, the decentralised controller is successfully designed via the recursive design algorithm and the global finite-time stability of the closed-loop system is achieved. Assumption 2 is an essential condition to design the decentralised controller. In every inductive step, the interconnection term is separated into multiple functions with each one containing single output variable, then we can design virtual controller to eliminate the influence of related items. In the next section, the proposed method is applied to a coupled inverted pendulums system and an interconnected non-linear system to illustrate its effectiveness 4 Simulation example In this section, to illustrate the validity of the proposed controller, two examples are given. Example 1.We consider the following interconnected non-linear system (31)The bound of the fault parameters is chosen as Using Propositions 1–5, the other parameters can be determined by the design method as follows: (32)where The virtual controllers are constructed as (33)Further, the controller can be designed as (34)where The initial values are chosen as x11 (0) = 5, x12 (0) = −20, x21 (0) = −6, x22 (0) = 15 . The simulation results are shown in Figs. 1 and 2, from which we can see that the outputs y1, y2 and the state variables x12, x22 converge to zero in finite time. Example 2.In order to further prove the effectiveness of the proposed method, the simulation is performed on two identical inverted pendulums connected by a hard spring [10]. The model of parallel inverted pendulum system can be described by (35)where b1 and b2 are damping coefficients, and Fig. 1Open in figure viewerPowerPoint Response of outputs in Example 1 Fig. 2Open in figure viewerPowerPoint Response of states in Example 1 For simulation, parameters of the coupled inverted pendulums are chosen as follows: δ1 = δ2 = 1, m1 = m2 = 1 kg, l1 = l2 = 0.5 m, l0 = 1 m, g = 9.8 m/s2, b1 = b2 = 0.009, k = 30, A = 0.1, and the spring position a1 = a2 = 0.1. Let x11 = θ1, x21 = θ2, (35) can be converted to (36)where g12 = 5.88 sin x11 − 0.036x12, g22 = 5.88 sin x21 − 0.036x22, f12 = 4Fa1 cos(θ1 − β), f22 = 4Fa2 cos(θ2 − β). Further, we can obtain (37)and (38)Based on the design method, the controller can be designed as (39)where Ci 2 = 2.3 and and The initial values are chosen as x11 (0) = 0.7, x12 (0) = −0.1, x21 (0) = −0.3, x22 (0) = 0.1 . The simulation results are shown in Figs. 3 and 4, from which we can see that the outputs y1, y2 and the state variables x12, x22 converge to zero in finite time. Fig. 3Open in figure viewerPowerPoint Response of outputs in Example 2 Fig. 4Open in figure viewerPowerPoint Response of states in Example 2 5 Conclusion The problem of decentralised fault-tolerant finite-time stabilisation control is investigated in this paper for a class of interconnected non-linear systems, in which there exists unknown gain fault with each actuator. Each subsystem is with lower-triangular structure. In order to overcome the effect of unknown gain faults, the robust control technology is used. Based on the finite-time Lyapunov stability theory and recursive design algorithm, the Höder continuous, decentralised state feedback stabiliser is proposed to guarantee finite-time stabilisation of the non-linear interconnected system. Finally, two simulation examples have been given to verify the effectiveness of the proposed design technique. 7 Appendix Proposition 1.For 1 ≤ i ≤ n, 1 ≤ k ≤ ni, there are c1 functions such that (40) Proof.By Lemma 2 and (9), for j = 2, …, k, we can obtain (41)Using (2) and 0 < qik < ⋯ < qi 2 < qi 1 = 1, we have (42)where is a c1 function.Combined with Lemma 3, we can obtain The proof is completed. □ Proposition 2.For 1 ≤ i ≤ n, 1 ≤ k ≤ ni, there are c1 functions such that (43) Proof.By Lemma 3 and Assumption 2, one has where are c1 functions. is a positive constant to be designed. □ Proposition 3.For 1 ≤ i ≤ n, 1 ≤ k ≤ ni, the following inequality is established (44)where φij 1(k −j +1) (y1) ≥ 0, φij 2(k −j +1) (y2) ≥ 0, …, φijn (k −j +1) (yn) ≥ 0 are c1 functions. Proof.By (9) and from the definition of Wij, for 1 ≤ j ≤ k − 1, one has (45)Following the idea of (43)–(46) in [4], we know (46)where is a c1 function.Using (2), (3) and (41), (42), (46), we can obtain Combined with Lemma 3, the above inequality can be converted to (47)where are c1 functions.Next, due to (48)and from (47), one has (49)The proof is completed. □ Proposition 4.For 2 ≤ k ≤ ni, one has (50) Proof.By inequality (6) and Lemma 3, we can obtain where The proof is completed. □ Proposition 5. is a c1 function, positive definite and properly satisfying Proof.From (8), we can obtain The proof is completed. □ 6 References 1Hong Y. Xu Y., and Huang J.: ‘Finite-time control for robot manipulators’, Syst. Control Lett., 2002, 46, (4), pp. 231– 236 (doi: https://doi.org/10.1016/S0167-6911(02)00119-6) 2Bhat S.P., and Bernstein D.S.: ‘Finite-time stability of continuous autonomous systems’, SIAM J. 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