Nonlinear Dyn DOI 10.1007/s11071-010-9868-2
O R I G I N A L PA P E R
Self-tuning fuzzy logic control of crane structures against earthquake induced vibration Ahmet Sagirli · C. Oktay Azeloglu · Rahmi Guclu · Hakan Yazici
Received: 30 April 2010 / Accepted: 6 October 2010 © Springer Science+Business Media B.V. 2010
Abstract Self-tuning fuzzy logic controllers (STFLC) for the active control of Marmara Kocaeli Earthquake excited crane structures are studied in this paper. Vibration control using intelligent controllers, such as fuzzy logic has attracted the attention of structural control engineers during the last few years, because fuzzy logic can handle, uncertainties and heuristic knowledge and even non-linearities effectively and easily. The improved seismic control performance can be achieved by converting a simply designed static gain into a real time variable dynamic gain through a self-tuning mechanism. A self-tuning fuzzy logic controller is designed to reduce the vibrations of the crane structure. The simulated system has a five degrees-offreedom and modeled system was simulated against the ground motion of the Marmara Kocaeli Earthquake (Mw = 7.4) in Turkey on August 17, 1999. At the end of the study, the time history of the crane bridge and portal legs displacements, accelerations, and frequency responses of the both uncontrolled and controlled cases are presented. Additionally, the performance of the designed STFLC is also compared with a PD controller. Simulations of an earthquake excited bridge and portal legs are performed to prove the validity of proposed control strategy.
A. Sagirli · C.O. Azeloglu · R. Guclu () · H. Yazici Faculty of Mechanical Engineering, Yildiz Technical University, Besiktas, Istanbul, Turkey e-mail:
[email protected]
Keywords Self-tuning fuzzy logic control · Crane structure · Earthquake-induced vibration
1 Introduction In the recent years, earthquakes have caused much more loss of life and financial damage compared to those in previous centuries. Cranes are also affected by seismic movements. A collapsed crane in an earthquake is shown in Fig. 1. Cranes damaged in earthquakes also cause loss of life together with economical losses. Moreover, cranes damaged on strategical points such as harbors and railways cause failure in logistic activities. Therefore, it is necessary to investigate the behavior of cranes during an earthquake to take suitable design precautions to prevent possible damages in the earthquake and to enable stability toward an earthquake with active-passive controllers. Abdel-Rahman et al. [2] examined the crane control strategies in their study; and consequently presented the related studies to be conducted in the future. Kobayashi et al. [3] observed the dynamic behaviors of container cranes under seismic effects. In the study, especially the contact problem between wheels and rails during the earthquake is the focus, a model of 1/8 ratio upon the wheel tray connection of crane is composed; dynamic effects on the system are then observed by applying actual earthquake data on an earthquake tray. Otani et al. [4] observed vertical vibrations that are
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Fig. 1 A collapsed crane in earthquake [1]
formed with the effect of an earthquake on overhead cranes by composing a model of 1/8 ratio of an overhead crane to observe dynamic effects on the system by applying actual earthquake data on an earthquake tray. Soderberg and Jordan [5] observed the dynamic behavior of jumbo container cranes and put forward suggestions for the design to reduce damages of an earthquake and to prevent collapse. In this study, self-tuning fuzzy logic controllers (STFLC) have been implemented to the seismic responses of a crane system in the Kocaeli earthquake.
2 Mathematical model of the crane In this study, a self-tuning fuzzy logic controller is implemented to a five degrees-of-freedom gantry crane which is modeled using spring-mass-damper subsystems. Since the destructive effect of earthquakes is a result of horizontal vibrations, the degrees-of-freedom have been assumed to be occurring only in this direction. Control structure is defined as a mechanical system which is installed in a structure to reduce structural vibrations during loadings imposed by earthquakes. The control system can be divided into two parts: active control device and the control algorithm [6]. An important element of an active control strategy is the actuators. These are active control devices that attenuate disturbances at the corresponding subsystems or reduce the vibration on crane bridges when they are installed [7]. In this study, two actuators are used to suppress earthquake induced vibrations. The first actuator is installed between the bridge and the portal legs and the second one is placed between portal legs and the ground. STFLC is used as a
Fig. 2 Physical model of crane for earthquake motion
control algorithm for both control devices. It supplies control voltage directly to suppress magnitude of undesirable earthquake vibrations. The crane system is shown in Fig. 2. The mathematical model includes the following assumptions: • The motion of the crane is modeled as a planar motion. The direction of the ground motion is on this plane and the motion of all masses is also in this plane. • The degrees-of-freedom have been assumed to be occurring only in the horizontal direction. • All springs and dampers are considered as acting only in a horizontal direction. • In this model, ground, portal legs, crane bridge, trolley, and payload are considered as point masses m1 , m2 , m3 , m4 , m5 , respectively. • The cable is considered as massless and rigid. • In the model, portal legs are fixed on the ground. • Two actuators are used to suppress earthquake induced vibrations. The first actuator is installed between the bridge and the portal legs and the second one is placed between portal legs and the ground. The equation of motion of the system is [M]x¨ + [C]x˙ + [K]x = Fd + Fu
(1)
where x = [x1 x2 x3 x4 θ ]T , Fd = [−(c1 x˙0 + k1 x0 ) 0 0 0 0]T and Fu = [−Fu1 (Fu1 −Fu2 ) Fu2 0 0]T . Fd is the force induced by an earthquake. Fu1 and Fu2 are the control forces produced by linear motors. The equations of motion can be derived using the Lagrangian
Self-tuning fuzzy logic control of crane structures against earthquake induced vibration
equation: d ∂Ek ∂Ek ∂Ep ∂ED + + = Qi − dt ∂ q˙i ∂qi ∂qi ∂ q˙i (i = 1, . . . , 5)
(2)
where Ek is system kinetic energy, Ep is system potential energy, ED is system damping energy, qi is a generalized coordinate, and Qi is external force. Finally, the following equations of motion are obtained: m1 x¨1 + (k1 + k2 )x1 − k2 x2 + (c1 + c2 )x˙1 − c2 x˙2 = −(c1 x˙0 + k1 x0 ) − Fu1 ,
(3)
m2 x¨2 + 2k2 x2 − k2 x1 − k2 x3 + 2c2 x˙2 − c2 x˙1 − c2 x˙3 = Fu1 − Fu2 ,
(4)
m3 x¨3 + (k2 + k3 )x3 − k2 x2 − k3 x4 + (c2 + c3 )x˙3 − c2 x˙2 − c3 x˙4 = Fu2 ,
(5)
(m4 + m5 )x¨4 + k3 x4 − k3 x3 + c3 x˙4 − c3 x˙3 + m5 Lθ¨ cos θ − m5 Lθ˙ 2 sin θ = 0, 2¨
m5 L θ + m5 Lx¨4 cos θ + m5 gL sin θ = 0
(6) (7)
where m1 , m2 , m3 , m4 , m5 denote the mass of the ground, portal legs, bridge, trolley, and payload, respectively; k1 , 2k2 , k3 denote the stiffness of the ground, portal legs, and wheels of trolley, respectively; c1 , 2c2 , c3 denote the damping of the ground, portal legs, and wheels of trolley, respectively; L is the rope length and g is the gravitational acceleration. x0 is the earthquake-induced ground motion disturbance imposing on the crane structure. All springs and dampers are acting in horizontal direction. The system parameters of a real gantry crane are presented in the Appendix.
3 Self-tuning fuzzy logic controller design The theory of fuzzy sets have been extensively used in variety of fields including control applications since its first inventions by Zadeh [8–15]. If any one of FLC’s tunable parameters (scaling factors, membership functions, and rules) changes when the controller is being used, FLC can be called as an adaptive controller. An adaptive FLC that fine tunes an already working controller by modifying either its membership functions or scaling factors or both of them is described
as a self-tuning FLC [16]. Since the output scaling factor of the existing FLC is adjusted on-line by the gain tuning mechanism, the proposed FLC is called the self-tuning FLC. Therefore, the self-tuning fuzzy logic controller is suitable choice for control algorithm. The superior qualities of this method include its simplicity, satisfactory performance, and its robust character. This study introduces an implementation of the selftuning fuzzy logic controller to crane systems using Matlab Simulink and Fuzzy Toolbox. STFLC for the crane system uses the errors (e = xr2 − x2 , e1 = xr3 − x3 ) in the portal legs and bridge motions, and their derivatives (de/dt = x˙r2 − x˙2 , de1 /dt = x˙r3 − x˙3 ) as the input variable while the control voltages (u) and (u1 ) are their outputs. Reference values (xr2 , x˙r2 ) and (xr3 , x˙r3 ) are considered to be zero. A block diagram of the proposed STFLC for crane system is shown in Fig. 3. Figure 3 shows that the output scaling factors of the controllers are modified by a self-tuning mechanism, shown by the dotted boundary. Figure 4 illustrates Gaussian membership functions (with equal base and 50% overlap with the neighboring) are used to achieve a good controller performance. All membership functions for controller inputs, error (e), and derivative of error (de) are defined on the common interval [−1, 1], while the membership functions for the gain updating factor (α) are defined on the common interval [0, 1] [16]. Where P, N, ZE, B, M, S, and V represent Positive, Negative, Zero, Big, Medium, Small, and Very, respectively. The membership functions for both scaled inputs (eN , deN , eN 1 , deN 1 ) and output (uN , uN 1 ) of the controller have been defined on the common interval [−1, 1]. The values of actual inputs and outputs (e, de, e1 , de1 , u, u1 ) are mapped onto [−1, 1] by the input and output scaling factors (Se, Sde, Su, and Se1 , Sde1 , Su1 ) in Fig. 3. The choice of suitable values of scaling factors is determined through a trial and error approach to achieve the best possible control performance. The values of scaling factors are presented in the Appendix. The relationship between the scaling factors and the input and output variables of the STFLC is defined as follows: eN = e.Se,
(8)
eN 1 = e1 .Se1 ,
(9)
deN = de.Sde,
(10)
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Fig. 3 Block diagram of the proposed self-tuning fuzzy logic controller
deN 1 = de1 .Sde1 ,
(11)
u = uN .(Su.α),
(12)
u1 = uN 1 .(Su1 .α).
(13)
It is observed that the effective gain of the self-tuning controllers are Su.α and Su1 .α, not simply Su and
Su1 . Therefore, the output gain of the proposed selftuning FLC does not remain fixed while the controller is in operation. It is modified in each sampling time using the gain tuning mechanism, depending on the trend of controlled crane system output. The reason behind this on-line gain variation is to make the controller respond according to the desired performance
Self-tuning fuzzy logic control of crane structures against earthquake induced vibration
Fig. 4 Membership functions of (a) e, de, u, and (b) gain updating factor (α) Table 1 (a) Fuzzy rules for computation of u, (b) fuzzy rules for computation of α (a)
(b)
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specifications. The limited number of IF-THEN rules using membership functions and fixed valued scaling factors are not sufficient to produce the required nonlinear controller output for the desired control performance. To eliminate this problem, the number of membership functions and rules can be increased, but it will cause a design complexity. In the proposed selftuning scheme, the controller output is generated by the continuous and nonlinear variation of α. The most important case of this method is that α depends on the instantaneous process states, not any process parameter. Therefore, the proposed self-tuning scheme is model independent [17]. The rule base for computing u and u1 is shown in Table 1(a). This is a common rule base designed with a two-dimensional phase plane in mind where the FLC drives the system into the so-called sliding mode. The controller output u is calculated using fuzzy rules of the form in Table 1(a): If e is NB and de is NB, then u is NB. And the gain updating factor (α) is calculated using fuzzy rules of the form in Table 1(b): If e is NB and de is NB, then α is VB. All the rules are written similarly using the Mamdani method to apply to fuzzification in Table 1. Overall control performance, using the rule base in Table 1(b) for computation of α. Table 1(b) is designed
in conjunction with rule base in Table 1(a) [17]. In this study, the centroid method is used for the defuzzification process.
4 Earthquake excitation and the response of the crane Modeled crane subjected to the ground motion of the Marmara Kocaeli earthquake (Mw = 7.4) in Turkey in August 17, 1999, has been simulated. Earthquake ground motion is used as input to the crane system. Accelerations are recorded at the Kandilli Observatory and Earthquake Research Institute at the Kucukcekmece Nuclear Research Center in Istanbul, Turkey, during the August 17, 1999, main shock in Fig. 5 [18]. Figure 6 shows the time responses of the portal legs, bridge, and trolley displacements and accelerations, load sway angle, and angular acceleration, respectively, for both controlled and uncontrolled cases. The performance of the designed STFLC is also compared with a PD controller using the same control forces. The dashed line indicates the uncontrolled case, the bold line indicates the controlled case by designed STFLC, and the straight line indicates the case
A. Sagirli et al. Fig. 5 Marmara Kocaeli earthquake excitation input to the crane structure
by the PD controller. The PD controller and its parameters are given in the Appendix. It is well known that the maximum displacements are expected at the top of the crane during an earthquake. Displacements of the crane bridge and trolley are minimized successfully using the STFLC. As compared with the PD controlled case, displacements of the crane bridge are suppressed better. Figure 7 shows the time history of control forces. Figure 8 shows the frequency responses of the bridge displacements and accelerations, respectively, for both uncontrolled and STFLC and PD controlled cases. Since the system has five degrees-of-freedom, there are five resonance values at 0.22, 0.97, 3.7, 12.06, and 22.64 Hz. As expected, the lower curves belong to the controlled systems. When the response plots of the structural systems with uncontrolled and controlled cases are compared, a superior improvement in terms of magnitudes with STFLC has been observed (Fig. 8). The first mode is expected to be the most dangerous for crane structures during an earthquake [13, 15] and it is suppressed successfully using STFLC. As compared with the PD controlled case, frequency responses of bridge displacements and accelerations are suppressed better.
5 Conclusion In this study, STFLC has been implemented to suppress structural vibrations of an earthquake excited crane. The performance of the designed STFLC is demonstrated by simulations of a five degrees-offreedom crane subjected to Marmara Kocaeli earthquakes ground motion. STFLC is an adaptive controller and the major advantage of this method is that the self-tuning mechanism modifies the value of output gain of the FLC in each sampling time and adapts itself even if system parameters or disturbances change. The crane system is then subjected to Marmara Kocaeli earthquake vibrational effects, treated as disturbance. The simulation results exhibit that the implementation of STFLC shows a good response. Since the vibrations due to earthquake effects are absorbed, the study also shows the destructive effects of high accelerations which occur during the earthquake. The performance of the designed STFLC is also compared with a PD controller. As compared with the PD controller, responses of STFLC controlled simulation results are more successful. These effects cannot be ignored during the structural design of cranes. It is seen that this controller descends the effects of such accelerations substantially. It can be concluded that the con-
Self-tuning fuzzy logic control of crane structures against earthquake induced vibration
Fig. 6 Controlled and uncontrolled displacement and acceleration time responses of portal legs, bridge, trolley, and load sway angle
Fig. 7 Time history of control forces
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Fig. 8 Controlled and uncontrolled frequency responses of the crane bridge
troller used may affect the structural design of cranes drastically.
Acknowledgement We are grateful to acknowledge the support of KM Kumsan Crane Turkiye for providing the design data.
Appendix Parameters of the gantry crane Mass parameters m1 = 450000 kg m2 = 10800 kg m3 = 20800 kg m4 = 5000 kg m5 = 40000 kg
c2 = 2000000 Ns/m c3 = 2000000 Ns/m Length parameters L=5m Lportal = 8.3 m H = 10.9 m
Stiffness parameters k1 = 18050000 N/m k2 = 27850000 N/m k3 = 67000000 N/m Damping parameters c1 = 26170 Ns/m
STFLC input-output scaling factors for x2
STFLC input-output scaling factors for x3
PD controller parameters for x2
Se Sde Su
Se1 Sde1 Su1
Kp KD τd
40 1 8000000
30 1 6000000
100000000 8000 0.00008
PD controller parameters for x3 Kp1 KD1 τd1
PD controller A general control inputs u(t) and u1 (t) are obtained as follows: de(t) u(t) = Kp e(t) + τd dt and
de1 (t) u1 (t) = Kp1 e1 (t) + τd1 dt where Kp , Kp1 , τd and τd1 are the proportionality constants and derivative times, respectively.
10000000 5000 0.0005
Self-tuning fuzzy logic control of crane structures against earthquake induced vibration
Free body diagrams
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