Sareban: Evaluation of Three Common Algorithms for Structure Active Control

Transcription

1 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, Evalation of Three Common Algorithms for Strctre Active Control Mohammad Sareban Department of Civil Engineering Shahrood Branch, Islamic Azad University Shahrood, Iran Abstract Recently active strctre controllers were considered to deal with the impact of earthqake forces and the reslt of the investigations provided mltiple algorithms to calclate force control and many different ways to apply these forces on the strctre. In this stdy, the efficiency and effectiveness of three methods (linear qadratic reglator, fzzy logic and pole assigning) are investigated. In addition, three bildings with different height classes with an active tned mass damper (ATMD) on the top floor are considered to compare the active control methods. Examples with known mass and stiffness and with variable mass are considered. The reslts show that all three control methods sed for the ATMD device redce the strctral response. The fzzy control method, cased a sharp decline in relative displacement of bilding floors p to 80%. Bt in and pole allocation procedres the applied force is limited. The best performance of fzzy control is for high-rise bildings. The three different methods of control are stable in different masses and even nder a random change of floor masses, their effectiveness can be trsted. Keywords-active control; configrable active mass damper; linear qadratic optimal control; fzzy control; pole allocation control; ncertainty. I. INTRODUCTION Controlling wind or seismic indced vibrations can be performed by passive, active, semi-active or combined control systems. Each of these control systems ses varios instrments to redce seismic responses. As semi-active systems are reliable as passive systems and adaptive as active systems, they have attracted the attention of many researchers in the strctral control field. Active control systems, in contrast to passive systems, reqire an external energy for their operation. Active control systems can be mainly divided into force applying mechanisms and algorithms to calclate the control force. Althogh redcing damage cased by large loads is the main target in strctral engineering, little attention has been given to directly control damage parameters. This is becase valid damage parameters involve varios variables while modern control theories, sch as, LQG, sliding mode control based on state space models, have the ability to involve state variables in the performance index. Many researchers have stdied the se of different control algorithms and tools in order to redce response and improve the behavior of strctres and achieved good reslts. In [1], athors provided an overview of modeling and controlling strctres. They raised advancements in the field of modeling and seismic strctral control and considered a variety of control methods inclding linear control, nonlinear control and intelligent control. In [2], athors investigated the application of an active system in vibration control of three and two-floor non-linear strctres. De to the nonlinear behavior of the strctre, researchers sed fzzy control to determine the control force. In [3], athors sed a fzzy control system and an active mass damper (ATMD) to shield bildings against earthqakes. II. MATERIALS AND METHODS A. Active Control of Strctres An active control system is made from an external power sorce or hydralic or electromechanical actator force that applies a predetermined force to the strctre. Sch forces can be sed to attract or waste energy. The fnction of this system is based on measres by sensitive receptors that have been installed in floors and are sent to the CPU for processing. Then, based on a predetermined control algorithm, the optimal controlling force is decided. The control process starts when the measred amplitde of oscillations is eqal to or greater than a preset limit. Sch an approach is considered to bear increased efficiency compared to passive control strategies [4]. 1) Damper with variable mass This system is a modified form of a damper with coordinated mass in which, an additional object with a driver is connected to the primary system. Ths, there is a renewed impets in system that complements the generated force by the coordinated mass and increase the damping force of the system. 2) Linear qadratic method reglator () 2 eqations are one of the most widely sed optimization algorithms for active control. The eqation of mlti-floor bilding s motion as a shear frame model with controller is as follows: * * * M x(t) C x(t) K x(t) ηx g (t) ΓU(t) After obtaining the soltion to (1), the eqation of motion is transferred to state-space. (1)

2 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, r g Z t AZ t B U t B x (t) In procedre the below energy integral shold be minimal: t 1 T T J (Z QZ U RU)dt 0 If the matrix Q is large, strctral response will be less and control force increases. If R is large, the response of the strctre is high and control force decrease. The control force can be obtained from (4). As can be seen, the control force is linked to the strctral response. If the strctre response is large, more control force is applied respectively. The ontrol force U(t) can be obtained from the following eqation: 1 T U t GZ t R B PZ t (4) In which, G is the gain matrix and P is a matrix obtained from nonlinear Rikati eqation. By combining (2) and (4) we have: 1 T Z t AZ t B ( R B PZ t ) B f(t) (5) r * ( 1) T Considering A A B R B P we have: * Z t A Z t Brf(t) (6) To solve in state space we need the following eqation: y EZ t Lf(t) (7) 3) Fzzy control The fzzy control algorithm is based on fzzy logic and ses intition and knowledge of experts instead of differential eqations to explain the behavior of a system [5] and in this case to determine the control force. There are many ways to generate fzzy rles that inclde logical reasoning, and learning tests and simlations of the sample. So the control sing this method is natrally robst and yet simple. To get answers to the eqation (1), the eqation of motion is transmitted to state-space. r g Z t AZ t B U t B x (t) To condct a fzzy control, (8) is written as follows: Z t AZ t B (B U t B F(t)) (9) s F t B U t B F(t) r (2) (3) (8) f (10) r To solve in state space we also need the following eqation: * y EZ t LF t (11) f Fzzy control can be incorporated in a closed loop control system similar to conventional controllers. Fzzy rle defines the relationship between fzzy inpt and otpt based on a Mamdani model. The law contains two parts: introdction and reslts made of IF-THEN statements. Each rle is as follows: i R ifx A and X B then Y C (12) 1 i 2 i i Where i is nmber of control laws, X 1 and X 2 are introdction variables and Y is reslt variable. Also A i, B i and C i are lingistic vales of fzzy variables. In this stdy, the fzzy controller design directly from the nmerical and clear strctre data model. Controller acts based on two inpt variables (velocity and displacement of the top floor), each having five trianglar membership fnction (Figre 1a) and an otpt variable (the active control) with seven trianglar membership fnction (Figre 1b). The abbreviations of inpt and otpt membership fnctions for fzzy variables to define the phase space are: LP=big and positive; P=positive; Z=nll; N=negative; LN=large negative (for inpt variables), and PL=positive and large, PM=moderate positive; PS=Positive and small; ZR=nll; NL=large negative; NM=high negative; NS=negative and small (variable otpt). Details of inference rles sed in this stdy are shown in Table I. TABLE I. Σφάλμα! Τα αντικείμενα δεν μπορούν να δημιουργηθούν από την επεξεργασία κωδικών πεδίων.(displacement) FUZZY SYSTEM RULE BASE Σφάλμα! Τα αντικείμενα δεν μπορούν να δημιουργηθούν από την επεξεργασία κωδικών πεδίων.(velocity) LN N Z P LP LP NS NS NM NL NL P NS NM NM NM NL Z PS ZR ZR ZR NS N PL PM PM PM PS LN PL PL PM PS PS Fzzy controller ses the Mamdani method for making non-fzzy to change the lingistic vales to nmerical and clear vales. Lingistic otpt vales between -1 and 1 mst be mltiplied by a factor to prodce active control force. (b) (a) Fig. 1. (a) Membership fnctions of inpt variables (Displacement and Velocity), (b) Membership fnctions of the otpt variable (Active control force) 4) Pole allocation method Unlike algorithms, this algorithm solves strctres inversely to encorage them to the desired response. This means that the strctre's primary poles are determined from the beginning of the algorithm and then the system with respect to dampers increase the damping ratio of strctres and by solving the main determinant of algorithm determines the new poles of strctre so that systems moves toward increasing stability. Therefore new active strctre poles define the

3 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, system. However, changeable points in the algorithm shold be considered from first variations advance throgh sable reslts for passive strctral damper design. As a reslt, damping and stiffness vales in the system are introdced only where there is permission to change them. To get answers to the (1), the eqation of motion is transmitted to state-space. r g Z t AZ t B U t B x (t) Eigenvales of matrix A are as follows: 2 ρ 1 ; 1 (13) i i i i i j j (14) In (13), U(t) is as follows: U t GZ t (15) By combining (13) and (15), we have: Z t AZ t B ( GZ t ) B f(t) (16) Considering * A A BG * Z t A Z t Brf(t) r (13) is expressed as follows: (17) Eigenvales of matrix A * are as follows: * *2 ρi i i i 1 i j; j 1 (18) By increasing the amont ξ to ξ * and considering G nknown, G is acqired sing place command in Matlab software and ths the amont of control force is obtained. To solve the problem in state space we also need following eqation: y EZ t Lf(t) (19) B. Uncontrolled strctre Eqation of motion for mlti-floor bildings with shear frame model withot a controller is as follows: g M x(t) Cx(t) Kx(t) Mr x (t) (20) After obtaining the response of motion eqation (20), we transmit the eqation of motion to state-space. r g Z t AZ t B x (t) (21) In addition to solve in the state space eqation (21) is needed in addition to eqation (22). g y EZ t Lx (t) (22) III. NUMERICAL EXAMPLE Three bildings with different heights (low height of 5 floors, the average height of 10 floors and high height of 20 floors) are taken into accont. For the 5 and 10 floor bildings, the mass of each floor is 1*10 5 Kg and the stiffness of each floor is 5*10 7 N/m and a strctral damping ratio of ξ=0.05 is considered. Fig. 2. N floor bilding with shear frame and an ATMD in roof In these two cases, an ATMD is installed on the top floor (roof) to redce the strctral response (Figre 2). Its mass is eqal to m t =5*10 4 Kg. Since the first strctre freqency is crcial for the specification of ATMD, stiffness of ATMD is calclated from the following eqation: k m.w (23) 2 t t l Where w l is the first freqency strctres. ATMD damping ratio eqal to that taken into accont, since the damping ξ t =0.05 ATMD to be achieved as follows: ct 2ξ t.m t.w (24) l The only difference in the 20 floor bilding is the stiffness of floors which is considered eqal to 8*108 N/m. For each bilding, three control methods have been sed. The first control method employed is the linear qadratic reglator method (). In all cases R=1*10-4 and Q=αI, α=1*109 are considered in relation. The higher α increases control force. The reslts showed that for amonts larger than α=1*10 9 the controlling force does not increase and this is a disadvantage to this approach. The second control method employed is fzzy control. In this method the response of the strctre as displacement and velocity of top floor are sed as fzzy logic inpts. Control force otpt is between -1 and 1. It shold be mltiplied by a factor to reslt to a larger control force. This factor is considered eqal to 1*10 6 in all cases. The final method employed is pole assignment, with a strctral damping of ξ=0.05, increased to ξ=0.15 for the first two bildings and to ξ=0.07 for the 20 floor bilding. The ltimately gain matrix G and control force each time step is calclated with the relationship U (t)=- GZ (t). New damping ratios were calclated in a way that greatest redction occrs in floor displacements. Two earthqake records corresponding to Loma Prieta and Northridge earthqakes were sed for all states. These accelerograms can be seen in Figres 3 and 4..

4 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, The strctral response reslts of Northridge earthqake is frther explained in detail with tables and graphs. The impact of the Northridge earthqake is considered in Table II for three 5-floor bildings where the three methods were compared. Maximm ATMD controlling force in these three methods shows that with the fzzy control method a large control force can be prodced. A weaker control force is created with the other two methods. Also, the displacement of the controlled mode to ncontrolled in all floors and for all three control methods is nder one, which shows a decrease in displacement for all control methods. Becase of the large control force in fzzy method, the controlled to ncontrolled mode displacement ratio is very low and there is 50% to 70% redction in movement. This redction is greater on the pper floors. The redction in other two methods is abot 20%. peaks of graph p to 20% and this redction increased in frther time steps. This is a good reslt, althogh the initial peaks of graph redced slightly, high displacement redction in next time steps, cases less cyclic strctre fatige. floor 1 floor 5 Fig. 5. Time-history graph of controlled and non-controlled states of Mode 1 Fig. 3. Northridge earthqake accelerograms Fig. 4. Loma Prieta earthqake accelerograms TABLE II. DISPLACEMENT RATIO (UNCONTROLLED TO CONTROLLED STATE) AND MAXIMUM CONTROL FORCE IN THE 5 FLOOR BUILDING In Figre 6, the force control generated by the fzzy logic can be observed at different time steps. History shows that the force is very logical control force and corresponding actions earthqake is generated. The maximm control force is fond to be 5*104 KN. Time-history graph for the controlling force in method can be observed in Figre 6. Maximm experienced control force is 1*10 4 KN. In Figre 7, the maximm absolte displacement of varios floors for controlled and non-controlled states of a 5-floor bilding with method is presented. In this figre redced displacements for each floor are presented, the redction is larger for pper floors. According to Figre 8, the variation range of controlled state is less than that of the ncontrolled state on both floors which reflects the decreasing displacement in fzzy method for five floor bilding. The Fzzy control method redces the displacement of steps in same size and approximately the same decrease can be observed. In Figre 9, the control force generated by the fzzy logic control at different time steps can be observed. The history of control force shows that the force is very logical and control force is generated corresponding to applied earthqake. The maximm experienced control force is 5*104 KN. Stronger qake prodces greater control force. Control Algorithm Maximm control force (kn) Fzzy Pole allocation In Figre 5, controlled and non-controlled displacement time-history chart of first and fifth floors of a 5-floor bilding with the procedre can be observed. In both floors variation range of controlled state is less than that of the ncontrolled state. control method redced the initial Fig. 6. Time- history graph for control force in method of mode 1 In Figre 10, the maximm absolte displacement of different floors in controlled and ncontrolled states of mode 2

5 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, can be observed. Redced displacement for each floor is visible, while the redction in pper floors is more for the fzzy control method. The variation range of controlled state is less than that of the ncontrolled state which shows a redction in displacement for pole allocation method of the 5-floor bilding. Displacement redction in pole allocation control method in all steps is approximately the same as method. In Figre 12, the control force prodced by the pole allocation method in different time steps can be observed. The maximm experienced control force is 5*10 3 KN. In Figre 13, the maximm absolte displacement of different floors in controlled and ncontrolled state of mode 3 is shown. Redction of displacement in all floors is observed in this Figre. Redced displacement for pole allocation method in all floors is observed, while the redction in the pper floors is larger. 30%. Floor displacement redction in fzzy method is 25 to 40%. In pole allocation method this redction is 20 to 35%. Weak performance of fzzy method compared to previos state is becase stiff floors in a 10 floor bilding, in comparison to mass and stiffness of a 5-floor bilding remained nchanged and the freqency and period of strctre has changed. Fig. 9. Time-history graph for control force in fzzy method of mode 2 Fig. 7. The maximm absolte displacement in different floors for controlled and ncontrolled states of mode 1 floor 1 Fig. 10. Maximm absolte displacement in different floors in controlled and ncontrolled state of mode 2 floor 5 floor 1 Fig. 8. Time-history graph of controlled and ncontrolled state for mode 2 As shown in Table III, in the case of 4 to 6 with 10 floors the displacement of controlled state to ncontrolled state in all floors and for all three control methods is the same that shows a decrease in displacement for all control methods and also the maximm control force is abot the same in all three control methods. Bt control force generated by the method is a bit lower which case redction in displacement abot 10 to floor 5 Fig. 11. Time-history graph of controlled and ncontrolled state for mode 3

6 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, by the is slightly larger than that of pole allocation. Displacement redction is abot 13% for, 35 to 50%. For the fzzy method and 11% for the pole allocation method. Fig. 12. Time-history graph for control force in pole allocation method of mode 3 Fig. 14. Maximm absolte displacement of different floors in controlled and ncontrolled state of mode 4 Fig. 13. Maximm absolte displacement in different floors in controlled and ncontrolled state of mode 3 TABLE III. DISPLACEMENT RATIO (UNCONTROLLED TO CONTROLLED STATE) AND MAXIMUM CONTROL FORCE IN 10 FLOOR BUILDING. Control algorithm Maximm control force (kn) Floor 6 Floor 7 Floor 8 Floor Fzzy Pole allocation Fig. 15. Maximm absolte displacement of different floors in controlled and ncontrolled state of mode 5 In Figres 14-16, the redction of maximm floor displacements in ncontrolled and controlled state of a 10-floor bildings can be seen. In all three control methods for 10-floor bilding, displacement redction for all floors is seen that the redction in the pper floors is more. The fzzy method shows weak performance, becase of the increasing stiffness of floors in comparison to the 5-floor bilding. In Table IV, states 7 to 9 with 20 floors, maximm ATMD control force in three considered methods is shown that a large control force is created by the fzzy method and a weaker control force is created by the other two methods. The control force generated Fig. 16. Maximm absolte displacement of different floors in controlled and ncontrolled state of mode 6

7 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, TABLE IV. DISPLACEMENT RATIO (UNCONTROLLED TO CONTROLLED STATE) AND MAXIMUM CONTROL FORCE IN 20 FLOOR BUILDING. Control Algorithm Maximm Control Force Floor 6 Floor 7 Floor 8 Floor Pole allocation Fzzy Fig. 18. Maximm absolte displacement of different floors in controlled and ncontrolled state of mode 8 Figres show the redction of the maximm displacement of floors in ncontrolled and controlled state of a 20-floor bilding. In all three control methods for 20-floor bilding redced displacement for all floors can be seen that the redction in the pper floors is more. Meanwhile, the fzzy method shows the best performance. Tables V-VII 7 show the displacement of controlled state to ncontrolled state and the maximm control force in 5, 10 and 20 floor bildings nder Loma Prieta earthqake. Same reslts of Northridge earthqake can be seen for Loma Prieta. Fig. 19. Maximm absolte displacement of different floors in controlled and ncontrolled state of mode 9 Fig. 17. Maximm absolte displacement of different floors in controlled and ncontrolled state of mode 7 In Figres with changing mass of floors from 20% less than the original mass to 20% more than the original mass, its impact on redction of floor displacements in ncontrolled and controlled states with three methods is shown. The reslts show that all three methods of control are stable in different masses and random change of floor masses we can safely redce strctral displacement. In 5-floor bildings, as shown in Figre 20, it can be seen the displacement of strctres for varios masses decreased in all three control methods, and also increasing floor masses, the redction of strctral displacement is redced. Graphs of displacement variation in three control methods are nearly parallel to displacement variation graph of ncontrolled displacement. As can be seen fzzy control method has high performance in 5-floor bilding. According to Figre 21, in 10-floor bildings, the displacement of strctres for varios masses decreased in all three control methods and displacement variation with mass changes is very small. In method, same reslts are observed, and despite the redced displacement, the displacement variation graph is nchanged. In pole allocation method with mass increase, displacement is slightly redced.

8 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, The greatest effect of floor mass change in 10 floor bildings is in fzzy control method that the mass increase cases a strong increase in displacement. The increase from 80% of mass to 120% of the mass is abot 25%. So it is possible that with excess increase in floor masses, fzzy logic method in 10-floor bilding lose its efficiency. control method. This shows that the fzzy method has very high performance in high-rise bildings. Also fzzy method drastically redces the strctral response of high rise bilding. TABLE V. DISPLACEMENT RATIO (UNCONTROLLED TO CONTROLLED STATE) AND MAXIMUM CONTROL FORCE IN 5 FLOOR BUILDING Control Algorithm Fzzy Pole allocation Maximm control force (kn) TABLE VI. DISPLACEMENT RATIO (UNCONTROLLED TO CONTROLLED STATE) AND MAXIMUM CONTROL FORCE IN 10 FLOOR BUILDING. Fig. 20. Maximm displacement change of the top floor of 5-floor bildings to change floor masses. Control Algorithm Maximm control force (kn) Floor 6 Floor 7 Floor 8 Floor Fzzy Pole allocation TABLE VII. DISPLACEMENT RATIO (UNCONTROLLED TO CONTROLLED STATE) AND MAXIMUM CONTROL FORCE IN 20 FLOOR BUILDING. Fig. 21. Maximm displacement change of the top floor for 10-floor bildings to change of floor masses. Control algorithm Maximm Control Force (kn) Floor 6 Floor 7 Floor 8 Floor Fzzy Pole allocation According to Figre 22 in 20-floor bildings, displacement of strctres for varios masses decreased in all three control methods, in controlled and ncontrolled case and increasing mass increases displacement. This change is less in fzzy Fig. 22. Maximm displacement change of the top floor for 20-floor bildings to change of floor masses. IV. CONCLUSION All three control methods sed in this stdy (, fzzy and allocation pole) for ATMD redces the strctral response. Bt fzzy control method drastically redced floor displacement p to 80%. Frther, in and pole allocation methods limited control force was condcted. Therefore, the

9 Engineering, Technology & Applied Science Research Vol. 7, No. 3, 2017, fzzy control method provides the best reslts. However, if the controlling force is greater than a specific amont, rather than redcing floor displacement, increases them. It shold be noted that the fzzy control method performed best in the case of high-rise bildings. REFERENCES [1] S. Thenozhi, W. Y, Advances in modeling and vibration control of bilding strctres, Annal Reviews in Control, Vol. 37, No. 2, pp , 2013 [2] M. Al-Dawod, B. Samali, K. Kwok, F. Naghdy, Fzzy controller for seismically excited nonlinear bildings, Jornal of Engineering Mechanics, Vol. 130, No. 4, pp , 2004 [3] R. Gcl, H. Yazici, Vibration control of a strctre with ATMD against earthqake sing fzzy logic controllers, Jornal of Sond and Vibration, Vol. 318, No. 1, pp , 2008 [4] S. M. Nigdeli, M. H. Bodrogl, Active Tendon Control of Torsionally Irreglar Strctres nder Near Falt Grond Motion Excitation, Compter Aided Civil and Infrastrctre Engineering, Vol. 28, No. 9, pp , 2013 [5] T. Kobori, R. Minai, Analytical stdy on active seismic response control, Transactions, Architectral Institte of Japan, Vol. 66, No. 2, pp , 1960