Boundary control of a flexible crane system in two‐dimensional space
2017; Institution of Engineering and Technology; Volume: 11; Issue: 14 Linguagem: Inglês
10.1049/iet-cta.2016.1446
ISSN1751-8652
Autores Tópico(s)Contact Mechanics and Variational Inequalities
ResumoIET Control Theory & ApplicationsVolume 11, Issue 14 p. 2187-2194 Research ArticleFree Access Boundary control of a flexible crane system in two-dimensional space Shuang Zhang, Corresponding Author Shuang Zhang zhangshuang.ac@gmail.com School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083 People's Republic of China Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, University of Science and Technology Beijing, Beijing, 100083 People's Republic of ChinaSearch for more papers by this authorXiuyu He, Xiuyu He School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083 People's Republic of China Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, University of Science and Technology Beijing, Beijing, 100083 People's Republic of ChinaSearch for more papers by this author Shuang Zhang, Corresponding Author Shuang Zhang zhangshuang.ac@gmail.com School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083 People's Republic of China Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, University of Science and Technology Beijing, Beijing, 100083 People's Republic of ChinaSearch for more papers by this authorXiuyu He, Xiuyu He School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, 100083 People's Republic of China Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, University of Science and Technology Beijing, Beijing, 100083 People's Republic of ChinaSearch for more papers by this author First published: 05 July 2017 https://doi.org/10.1049/iet-cta.2016.1446Citations: 2AboutSectionsPDF 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 A flexible crane system with vibrating and varying cable is investigated in two-dimensional space. Two partial differential equations and four ordinary differential equations derived by the Hamilton's principle are used to describe the dynamics of the flexible crane system. The dynamic model of the crane system considers the variation of the tension of the cable. Boundary control design is given to suppress vibrations of the flexible crane system. The Lyapunov's direct method is employed to prove the uniform ultimate boundedness of the states of the cable system. The effectiveness and performance of the proposed control schemes are depicted via numerical simulations. Nomenclature L total length of the cable l (t) time varying length of the cable, m mass of the payload the uniform mass per unit length of the cable v (t) moving velocity of the cable, (upward) T (x, t) tension of the cable w (x, t) transverse displacement of the cable at position x for time t w (l (t), t) boundary of the cable in the transverse direction z (x, t) longitudinal displacement of the cable at position x for time t z (l (t), t) boundary of the cable in the longitudinal direction EA axial stiffness of the cable boundary disturbance on the tip payload in the transverse direction boundary disturbance on the tip payload in the longitudinal direction boundary control force applied on the tip payload in the transverse direction boundary control force applied on the tip payload in the longitudinal direction distributed disturbance along the cable in the transverse direction distributed disturbance along the cable in the longitudinal direction 1 Introduction Crane systems are widely used in modern society, such as the construction industries, manufacturing industries, harbour engineering, offshore engineering, and automotive facilities. The control performance for the crane system is pretty important when it is implemented for the accurate positioning and transporting tasks. The crane system actually can be modelled as a flexible string with a payload attached to its free end [1, 2]. The flexible systems have many advantages such as lightweight, better mobility, lower cost, and less energy consumption. However, a fast movement as well as the external disturbances will produce the swing of payload and the vibration of flexible structures, which lead the performance and accuracy of system to deteriorate. The excessive swing and vibration bring down the mechanical properties of system, give rise to inferior quality, result in premature fatigue failure, and impose restrictions on the utility of system. Hence, there is a necessary to suppress the vibration of flexible systems. However, dealing with flexible systems is difficult since they are modelled as the distributed parameter systems with infinite-dimensional state space [3–7]. Therefore, some control methods for finite-dimensional systems [8–11], such as neural network control [12, 13] and model predictive control [14], cannot be directly used for control of distributed parameter systems. In recent years, increasing attentions have been given on the boundary control on flexible systems, such as robotic manipulators [15, 16], heat equations [17, 18], aircraft wings [19, 20], marine risers [21, 22], air-breathing hypersonic vehicles [23], spacecraft [24], and beams [25–27]. In [16], boundary control combined with a novel non-linear PDE observer is designed for a flexible manipulator so that the end effector can track a desired reference trajectory, and the vibration of flexible manipulator can be suppressed simultaneously. Stabilisation of a cascade of heat PDE-ODE system is discussed by using boundary control scheme in [17], where the sliding model control combined with the backstepping approach is proposed to handle the external disturbance. In [19], boundary control is designed for flexible wings of a robotic aircraft to regulate both wing twist and bending on the original coupled dynamics. For offshore engineering, both state feedback boundary control and output feedback boundary control are proposed to regulate the transverse vibration of a flexible marine riser with input saturation in [22]. In [23], flexible air-breathing hypersonic vehicles are described by hybrid PDEs-ODEs, and a novel non-linear composite control strategy is proposed to deal with the flexible effects. The authors propose a boundary output feedback control law for an Euler–Bernoulli beam equation under external boundary disturbance in [27], where the exponential stability of the closed-loop system is obtained. In the earlier age of the study for the flexible string system, the researchers pay more attention to the system which the length of string is fixed and the vibration and disturbance just come from one dimension [28, 29]. Modeling and control of moving systems have typically been studied [30–32]. Yang et al. have investigated the axially moving string [33] in which the length between two rolls is steadfast but the string is allowed to move in the horizontal direction. Nguyen et al. have studied the asymptotic stabilisation of a non-linear axially moving string by adaptive boundary control [34]. Other similar topics have also been investigated from different respects [35–37]. Additional, Nguyen et al. have investigated the suppression of vibrations which come from more than one dimension, for example two dimensions, for string system [38, 39]. However, these works mainly deal with the control problem for moving systems with a constant length and transport speed. The varying length has a significant effect on the dynamic characteristics of a moving system. Practical applications for flexible systems with time-varying length in the engineering field include high-speed elevator, crane cables for dynamic positioning, and flexible manipulators with extensible arm. Park et al. have studied the vibration control of axially moving system with variable speed, variable tension, and variable length [40]. Zhu et al. have investigated the varying length of flexible string and beam [41–44] which are extremely different from their time-invariant counterparts that the length of string or beam is a constant. However, the consideration of the coupling effect between the transverse and longitudinal displacements, which would lead to a more precise dynamic model, has been ignored in the above papers. In this paper, we concentrate our interest on the modelling and control problem for a crane system with varying length and varying speed. In addition, the cable is subjected to both transverse and longitudinal vibrations under external distributed disturbances (as shown in Fig. 1). After considering the varying length and varying speed, the crane system is described by non-linear and coupled PDEs, which would make the system model quite different with the linear model. Along with the increase of vibration's dimension, the model of the flexible string system gets more close to the real world, but it is followed by that the complexity of the modelling and control design for the string system. On the basis of the previous works from others and us, we build a two-dimensional model for the flexible crane system. Then, boundary control laws are proposed to suppress vibrations of the crane system. Finally, based on the Lyapunov's direct method [45], the uniformly ultimately boundedness of the states of the cable system is proved. Fig. 1Open in figure viewerPowerPoint Schematic of a crane system under two-dimensional vibration 2 Problem formulation and preliminaries 2.1 Dynamics of the non-linear flexible crane system The schematic of the moving cable is shown in Fig. 1. The cable is subject to vibrations which come from two directions, the transverse and the longitudinal, with distributed disturbances along the cable and . The top of the cable can be treated as fixed end, and a tip payload that mass is m loads at the bottom accompanied with boundary disturbances and . The cable moves at the speed of . For the purposes of suppressing two-dimensional vibrations, two control forces and are applied at the bottom boundary. For clarity, notations , , , and are used throughout this paper. Motivated by the authors in [38, 43], the kinetic energy of the cable system can be presented by (1)where the notation denotes . The potential energy of the cable system due to the tension can be obtained from (2) can be obtained as [46] (3)where g is the acceleration of gravity. The virtual work done by spatiotemporally varying distributed disturbance along the cable and and the time-varying boundary disturbance on the tip payload and is given by (4)The virtual work done by control forces and can be achieved by (5)Therefore the total virtual work can be accounted as follows: (6)Hence, according to the Hamilton's principle, we achieve the governing equations as (7) (8) , and the boundary conditions as (9) (10) (11) (12)where (13) (14) 2.2 Assumptions Assumption 1.We assume that , , , and are bounded, respectively. In other words, there exist constants , , , , such that , , and , , . Assumption 2.In this paper, we consider that which means the cable moves upward. In addition, the range of the acceleration is considered to be such that the potential energy is positive definite. Assumption 3.For the flexible crane system and from the expression of the distributed tension (3), we assume that , and are bounded by known constant lower and upper bounds as follows: (15) (16) (17) 3 Control design The control objectives are to suppress the longitudinal and transverse vibrations of the cable system governed by (7)–(12). Meanwhile, the length of the cable is varying and distributed disturbances intersperse along the cable and lumped disturbances impact on payload. The Lyapunov's direct method is used to analyse the stability of the closed-loop system and construct the boundary control laws and at the bottom boundary of the cable. We choose Lyapunov candidate function as (18)where (19) (20) (21)where and are two positive constants, is designed based on the mechanical energy and , called the energy term, the auxiliary term is related to the payload, and the crossing term is designed to facilitate the stability analysis. , and are auxiliary functions defined as (22) (23) (24)The boundary control laws are designed as follows: (25) (26)where and are positive control gains. Theorem 1.Consider the crane system defined by (7)–(12), with boundary control laws (25) and (26), the states of the controlled system and are bounded stable. Proof.Using the inequality [47–49] which explains the relationship between the longitudinal and transverse deflections of the flexible crane cable, we have (27)where is a positive constant. Let the positive satisfying and , we have (28)where (29)On the other hand, we have (30)where (31)To sum up, we obtain (32) For , we have (33)Apparently, and . If we set (34)we will obtain (35)Let (36) (37)Obviously, we have (38)Therefore, we obtain (39)The time derivative of is given as (40)where – are positive constants.Let (41) (42) (43) (44) (45) (46) (47)In addition, the following inequalities should be satisfied by choosing the designed parameters: (48) (49) (50) (51) (52)Hence, we obtain (53)where (54)Multiplying (53) by yields (55)Integrating the above inequality, we obtain (56)which demonstrates is bounded. Utilising Wirtinger's inequality (or Poincaŕe) [50] and (19), we have (57) (58)Appropriately rearranging the terms of the aforementioned inequalities, we obtain w (x, t) and z (x, t) are uniformly bounded as follows: (59) (60) .Furthermore, we obtain (61) (62) . □ Remark 1.Only bounded stability can be ensured due to the term . When the upper bounds of the distributed disturbances are large, the control performance would be affected. However, a better control performance can also be obtained by tuning the designed parameters. From inequalities (41)–(47), by adjusting the designed parameters, such as increasing and will bring a larger . Then, the value of will increase, which will produce a better vibration suppression performance. However, increasing in the control gains would bring a high gain control scheme. Therefore, in practical applications, the designed parameters should be adjusted carefully for achieving suitable transient performance and control action. Remark 2.The signals used in the boundary controls (25) and (26) can be measured by sensors or obtained by the backward difference algorithm. v (t) can be sensed by a velocimeter, and can be sensed by a laser displacement sensor, , , , and with only one time differentiating with respect to time can be calculated with a backward difference algorithm. 4 Simulation The effectiveness of the proposed boundary control laws is illustrated by numerical simulations using the finite difference method. For simulation study, the external distributed disturbances are simulated with several sinusoids signals with different frequencies which can be used to model the ocean disturbances in offshore engineering [21, 51]. We consider the boundary disturbances on the tip payload and represented as the following equations: (63) (64)From (63) and (64), we have and . Hence, we set and . The time-varying distributed disturbances along the cable and are described as (65) (66)The initial conditions are and . We choose the parameters of the varying length cable system as shown in Table 1. Table 1. Parameters of the varying length string system Parameters Description Value L total length 10 m m mass of the payload 10 kg mass per unit length 1.67 kg/m tension EA axial stiffness 10 N As the purpose of comparison for effectiveness, we take two PD controllers into account which are described as (67) (68)where , , , and are the control gains. We consider moving velocity as ) m/s. Accordingly, the length of the crane system at time t is . Figs. 2 a and 4 a show the transverse and longitudinal free vibrations of the varying length cable under the external disturbances without control inputs, i.e. . They show that there are large displacements in both the transverse and longitudinal directions that the maximum of is 6.486 m and the maximum of is 6.898 m due to the external disturbances included boundary disturbances and distributed disturbances. With the rising of the cable, the vibrations of the cable gradually decrease when its length gets shorter. However, the decrement is very small and slow when there is no any control force. Fig. 2Open in figure viewerPowerPoint Transverse displacement of the cable at (a) without control, (b) with PD control, (c) with the proposed boundary control Fig. 3Open in figure viewerPowerPoint Transverse displacement of the cable at (a) without control, (b) with PD control, (c) with the proposed boundary control Fig. 4Open in figure viewerPowerPoint Longitudinal displacement of the cable at (a) without control, (b) with PD control, (c) with the proposed boundary control From Figs. 2 b and 4 b, it can be seen that the vibrations can be suppressed in a small range around zero by selecting the designed parameters as and . Besides, Figs. 2 c and 4 c indicate that with the proposed boundary control laws (25) and (26), the vibrations can be suppressed greatly and swiftly in a small neighbourhood of zero by choosing the control parameters as , , , and . As an additional demonstration for the effectiveness of the proposed control laws, we also provide the vibrations of and shown in Figs. 3 and 5. As the pictures shown, the proposed control laws likewise can suppress the vibration of the cable at . The PD control inputs and the proposed boundary control inputs are given in Fig. 6. In Fig. 6, for transverse displacement, the PD control inputs and the proposed boundary control inputs vary between and , and for longitudinal displacement, the PD control inputs and the proposed boundary control inputs vary between and . These two control inputs vary in the similar range. Fig. 5Open in figure viewerPowerPoint Longitudinal displacement of the cable at (a) without control, (b) with PD control, (c) with the proposed boundary control Fig. 6Open in figure viewerPowerPoint Control inputs (a) the PD control input and the boundary control input , (b) the PD control input and the boundary control input From the simulation results, we can conclude that both the proposed control and the PD control are effective in regulating the transverse displacement and the longitudinal displacement of the cable. However, obviously, compared with the PD control, the transverse displacement and the longitudinal displacement converge faster under the proposed boundary control, which means a better control performance can be obtained with the proposed boundary control. 5 Conclusion This paper has investigated the control design for a flexible crane system with varying length with the external disturbances. The dynamic characteristics of moving system with varying length described by two partial differential equations (PDEs) and four ordinary differential equations (ODEs) have been derived by the Hamilton's principle, where both the changes in the tension and length of the cable have been considered. As an additional achievement of the former step, the uniform ultimate boundedness of the cable system's states has been achieved with the proposed control laws generated from the the Lyapunov's direct method. In the simulation study, by appropriate choices of the control parameters, the simulation examples have demonstrated the effectiveness and performance of the proposed control schemes. In this paper, in the numerical simulation, it can be seen that the PD control is also effective in suppressing the vibrations of the flexible cable. The control parameters are selected based on the control performance by trials and errors. We were not able to provide the theoretical guidance for choosing parameters in this paper. How to choose parameters for the PD control is a very good and meaningful topic for future research, and we would like to address the problem in our future work. 6 Acknowledgments The authors thank the Editor-In-Chief, the Associate Editor, and the anonymous reviewers for their constructive comments which helped improve the quality and presentation of this paper. 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