System identification of buildings equipped with closed-loop control devices

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1 System identification of bildings eqipped with closed-loop control devices Akira Mita a, Masako Kamibayashi b a Keio University, Hiyoshi, Kohok-k, Yokohama , Japan b East Japan Railway Company (Former Gradate Stdent, Keio University) ABSTRACT The prpose of this stdy is to provide a system identification tool to obtain dynamic strctral properties of bildings when closed-loop control devices are in operation so that we will be able to detect possible damages or changes in the bilding strctres withot sspending the control devices. The difficlty associated with closed-loop systems, where noise, inpt and otpt signals are correlated, can be resolved sing the otpt over-sampling approach. Using the approach, we were able to sccessflly obtain the open-loop properties of the bilding strctres even when the control device is operated. Until now, it has been a common practice to temporarily sspend the closed-loop control circits to measre the properties of the bilding withot the inflence of the control device. The control device is sed as an exciter for the bilding strctre with no feedback to the device. However, the tre dynamic properties of the bilding when sbject to control forces generated by the control devices that are operated as closed-loop systems may be different from those neglecting the control force. Ths, the otpt over-sampling approach was employed to overcome these difficlties. The employed approach was indeed able to estimate the properties of the bilding when the AMD, that is a typical vibration control device, is in operation nder the condition that the control system can hold its control signal for the sampling period T. Keywords: system identification, closed-loop, strctral control, damage detection 1. INTRODUCTION Vibration control devices for bilding strctres have been stdied for many years to mitigate the possible damage de to strong winds or/and earthqakes. It is essential, however, to know the exact dynamic properties of a bilding to design any control device with excellent performance. It is frther desired to obtain the dynamic behavior when the device is in operation. They are sally estimated from inpt and otpt signals for the bilding sing the system identification techniqe. However, most system identification tools are only applicable to open-loop systems 1). They are not applicable to a system that has some feedback components sch as a bilding with control devices. In this sitation, feedback excitation de to the device always exists on the bilding. When the feedback components are inclded in the inpt signal, the conventional identification tools are no more effective. In this paper, as an example of the control device, an active mass damper (AMD) is selected. The AMD has a feedback closed-loop with respect to vibration response of a bilding. The existence of the closed-loop components in a system makes the online identification very difficlt. If we apply a method developed for open-loop systems directly to the closed-loop systems withot carefl consideration, the correlated noise, inpt and otpt signals wold reslt in erroneos estimation. The otpt over-sampling approach employed here is known as a system identification tool for closed-loop systems developed for other engineering fields. Applicability of the approach was extensively examined by simlations and experiments for or target, bilding strctres. From or extensive stdy, it will be shown that the otpt over-sampling approach is indeed able to estimate the dynamic properties of the bilding when the AMD is in operation nder the condition that the control system can hold the control signals for the sampling period T. It is fond that, however, when the observation noise is higher than a threshold vale, the method might fail. Ths, by keeping the observation noise lower than the threshold vale, identification of bildings eqipped with closed-loop devices becomes possible. mita@sd.keio.ac.jp; phone & fax Keio University

2 2. SYSTEM IDENTIFICATION FOR CLOSED-LOOP SYSTEMS 2.1. Definition of the problem System identification for the closed-loop systems is difficlt de to correlation between noise and control signals. Their existence reslts in bias components in the estimation as well as lack of identifiability conditions 2),3). Several methods have been proposed to keep away from this difficlty 4). However, most methods reqire artificial noise or other inpts to achieve the process. The prpose of this stdy is to identify the dynamic properties of a bilding from the inpt and otpt signals when the control device is in operation withot adding any other signals. The state space representation of a bilding with an AMD is expressed by the eqations as x y k = Ax = Cx k + B + Bv k+1 k k k (1) where x, y,, v are the state vector, the otpt vector, the control force and the external force, respectively. The corresponding block diagram is shown in Fig. 1. v B B x A C y based AMD on A c, B c, C c Figre 1: Typical closed-loop system for a bilding with AMD Over-sampling approach An attractive approach 5) has been proposed recently. The approach introdces open-loop sbsystems intentionally by over-sampling the otpt signals. The mechanism is explained below. The feedback system with sampling dration T is presented in Fig. 2. If it is possible to sample p data instead of jst one dring the period T, the system can be represented by the block diagram as shown in Fig. 3 where =T/p. In addition to this condition, if the control force can be constant for the period T, the system can be modeled by the SIMO (single inpt mltiple otpts) model as depicted in Fig. 4. From this figre, it is nderstood that only one sbsystem becomes a closed-loop system while other sbsystems are open-loop systems. Ths, this over-sampling approach enables s to identify the closed-loop system by artificially adding open-loop sbsystems. (b) AMD (mt ) v(mt ) (a) strctre (T-model) e(mt ) y(mt ) controller plant Figre 2: Conventional closed-loop system.. (b) AMD controller (mt ) v( k ) (a) strctre (-model) plant e( k ) y( k ) T = p Figre 3: Closed-loop system with otpt over-sampling.

3 (mt ) ( z ) A( z ) M 1( z ) ( z ) B p B A B A ( z ) ( z ) M e 1( m ) p y ( ) p m e 1( m ) y 1 ( m) e 0 ( m ) y 0 ( m ) M } Open loop To feedback controller Figre 4: Single inpt mltiple inpt system. 3. SIMULATIONS 3.1. Model description A single degree of freedom system is considered to represent a bilding strctre as shown in Fig. 5. An AMD is attached to the top of the bilding. The AMD is driven by an actator whose feedback gains are given by k ( x x ) G ( x& x& ) ( t) = G (2) s d The AMD has linear feedback force with respect to its relative displacement and velocity. In fact, this control algorithm is eqivalent to the TMD(tned mass damper) with a spring and a dashpot. Ths, these gains were determined to emlate the optimm TMD 6). Table 1: Parameters sed for simlations parameters gains m s [kg] 2.54 G k [N/m] 548 m d [kg] G c [Ns/m] 2.16 f s [Hz] h s c s d y d x d m d y s x s m s c s k s y g Figre 5: Simlation model.

4 3.2. Closed-loop system identification The system identification was applied to the total system considering the random grond motion. The sampling freqency for driving the AMD was set at 500Hz. The MOESP algorithm 7) was sed for this identification. The identified two modes are smmarized in Table 2. The transfer fnction between the displacement of the bilding and the grond motion is plotted in Fig. 6 compared with the bilding withot the AMD. The effects of the optimized AMD are clearly nderstood from this figre. Table 2: Identified modal parameters. Mode nmber Freqency Damping ratio Uncontrolled controlled Figre 6: Transfer fnctions. To evalate the over-sampling method, the sampling freqency was set in the range of 1,000Hz to 5,000Hz for the otpt signals while keeping the sampling freqency for the control system at 500Hz. This choice reslts in p=2,..,10. Two excitation cases, free vibration and the forced vibration considering the random grond motion, were considered. For the forced vibration, time histories of the grond motion and the control signal are plotted in Fig. 7 for p=4. The MOESP algorithm was sed for identification. The singlar vales for an identification example are plotted in Fig. 8. Largest three singlar vales were sed for deriving the modal information. The identified modal properties are smmarized in Table 3. The transfer fnctions are plotted in Fig. 9. From these table and figres, it is clearly recognized that the direct system identification does not work and that the over-sampling algorithm works well for both free and forced vibration. The inpt signals sed for identification were the grond motion acceleration and the control force. The otpt signal was the acceleration response of the bilding. y(k) [V] Ue(mT) [V] Time [sec] Figre 7: Grond motion and control signal. (p=4)

5 Table 3: Identified reslts for different inpts and sampling rates. Case Freqency Damping ratio Tre Free vibration (p=1) Free vibration (2 p 10) Random vibration (2 p 10) Figre 8: Singlar vales. tre P=1 P=2 Figre 9: Transfer fnctions. 4. EXPERIMENTS The simlation models had no noise inpts and the control force was assmed constant ntil the next sampling step. However, the realistic system shold have noise contamination. In addition, it is difficlt to keep the control signal to be constant de to the noise. Ths, it is or prpose to test the over-sampling algorithm for realistic systems sing a simple experiment system. The system is depicted in Fig. 10. A bilding consists of for colmns and a floor system made of alminm. The mass of the bilding is 2.54kg. This bilding is eqipped with an AMD whose mass is 0.123kg. This AMD is driven by a linear actator with its stroke of ±0.02m. The control system for the AMD is presented in Fig. 11. The identified transfer fnction of the total system is plotted in Fig. 12 compared with the transfer fnction for the system withot AMD. These transfer fnctions were obtained considering the forced vibration sing random inpts at the base.

6 Power Amplifier xd x s md & y& s m s w L e H A/D,D/A Board & Control Compter & y& g Shaking Table Figre 10: Experiment model. e w x d x, x& x& s d s Digital Controller Linear Motor Figre 11: Control system for AMD. Uncontrolled controlled Figre 12: Transfer fnctions identified from experiments.

7 The over-sampling approach was applied to the experiments to obtain the dynamic properties of the bilding when the AMD is in operation. In this case, the sampling freqency to drive the AMD was set at 500Hz. The sampling freqency for the otpt signal was 2,000Hz (p=4). The feedback gains were set to achieve the optimm TMD. Time histories for free and forced vibration are plotted in Figs. 13 and 14. A sample time history of the control force for the forced vibration is plotted in Fig. 15. Using the grond motion acceleration and the control force as the inpt signals and the acceleration response of the bilding as the otpt signal, the system identification sing the MOESP algorithm was condcted. The modal properties obtained here are slightly deviated from the tre vales. However, the error in the modal freqency is less than 1% so that for most prposes this algorithm may provide satisfactory reslts. The control signal shown in Fig. 15 has a spike noise at the end. This is de to electrical noise in the control circit. To improve the accracy of the identification, it is essential to minimize the spikes. Table 4: Identified reslts from experiments. Case Freqency [Hz] Damping ratio Tre Free vibration (p=4) Random vibration (p=4) Figre 13: Free vibration Figre 14: Random vibration Figre 15: Control signal.

8 5. CONCLUDING REMARKS It was shown that the over-sampling approach is a promising tool to obtain dynamic properties of a bilding when closed-loop control devices sch as AMDs are in operation. In practical sitations, the closed-loop control device has been sed as an exciter by moving the system as an open-loop control system so that the conventional system identification tools shold work fine. However, operation of the control devices may change the dynamic properties of the bilding so that there is a strong need to obtain the properties when the control device is in operation. The difficlty associated with the closed-loop systems, where noise, inpt and otpt signals are correlated, was removed by sing the otpt over-sampling approach. By introdcing the approach, we were able to sccessflly obtain the open-loop properties of the bilding strctres. However, the sccess of the over-sampling approach relies on keeping the noise level as low as possible. For example, the spike noise in the control signal shold be careflly avoided. REFERENCES 1. Doebling, W., Farrar, R., Prime, B. and Shevitz, W. : Damage Identification and Health Monitoring of Strctral and Mechanical Systemes from Changes in their Vibration Characteristics; a Literatre Review. Los Alamos National Laboratory Soderstrom, T. Gstavsson, I. and Ljng, L.:Identifiability Conditions for Linear Systems Operating in Closed Loop, Int. Jornal of Control, 21(2), pp , Wellstead, P.E. and Edmnds, J. M.: Least-sqares Identification of Closed-loop Systems; Int. Jornal of Control, 21(2), pp , Soderstrom, T.: System Identification, Englewood Cliffs, NJ; Printice-hall, Sn, L., Li, W., and Sano, A.: Over-Sampling approach to closed-loop identification; Proc. 36th IEEE Conf. Decision and Control, , Den Hartog, J.P.: Mechanical Vibration; 4th edn, McGraw-Hill, New York, Verhaegen, M. and Dewilde, P.: Sbspace model identification --- Part 1. The otpt-error state-space model identification class of algorithms, Jornal of Control, 56(5), pp , 1992.