26(5A)-04(5311).fm

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1 ª Œª Œ 26ƒ 5A Á œ pp. 849 ~ 859 ª w y w µ-» w Robust Analysis of a µ-controller for a Cable-Stayed Bridge with Various Uncertainties ³ Á4QFODFS # ' +SÁ½ yá Park, Kyu SikÁSpencer, B. F., Jr.ÁKim, Chun HoÁLee, In Won Abstract This paper presents an extensive robust analysis of a µ-controller in the hybrid system for various uncertainties using the benchmark cable-stayed bridge. The overall system robustness may be deteriorated by introducing active devices and the active controller may cause instability due to small margins. Therefore, a µ-synthesis method that simultaneously guarantees the performance and stability of the closed-loop system (robust performance) with uncertainties is used for active devices to enhance the robustness in company with the inherent reliability of passive devices. The robustness of the µ-synthesis method is investigated with respect to the additional mass on the deck, structural stiffness matrix perturbation, time delay of actuator, and combinations thereof. Numerical simulation results show that the proposed control system has the good robustness without loss of control performances with respect to various uncertainties under earthquakes considered in this study. Furthermore, the control system robustness is more affected by the perturbation of structural stiffness matrix than others considered in this study. Therefore, the hybrid system controlled by a µ-synthesis method could be proposed as an improved control strategy for a seismically excited cable-stayed bridge containing many uncertainties. Keywords : hybrid control system, µ-synthesis method, robust analysis, seismic response control, benchmark cable-stayed bridge e j w w y w w l µ-» w w w. w l ƒ e w l w ù w. w l w j» w» y e wì y sww l ( ) w µ-w e w. q ƒ, w w, e, š w w µ-w w. ew w y w l w ù. w l y w w j w. µ-w w l y w». w : w l, µ-w, w,, e j 1. m w l e» w w y (uncertainty). w m p w y l w jš ƒ ¼. m l w» w (robustness) w g y w w. š l w g l ƒ ƒ j z. ù w y w.» l y š w w» w. sz l(closed- * ÁPost Doctoral Researcher, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign( [email protected]) ** Nathan M. and Anne M. Newmark Endowed Chair of Civil Engineering, Department of Civil and EnvironmentalG Engineering, University of Illinois at Urbana-Champaign( [email protected]) **** z Á w m œw ( [email protected]) **** z Áw w» y œw ( [email protected]) 26ƒ 5A œ 849

2 loop system) w y š w ¾ w. y w v Há w ù e sy(saturation) q š. z w m. Suhardjo(1990) m w w 2- (norm) á- yw H 2 Há š w. Schimitendorf (1994a, b, c) w x, Smith (1994) z w (gain-scheduled adaptive method), Smith Chase(1994) e sy w l y w, Yoshida (1994)», Nishimura (1994)» e(pole assignment) ww, Nishitani Yamada(1994) (reduced-order)» w Há w. Turan(2001), Moon (2003), Park (2003) w e j(benchmark) (Dyke, 2003) w µ-w (µ-synthesis method) ƒ»(hydraulic actuator), (sliding mode) š w»» (magnetorheological fluid damper), š on-off xk LQG(linear quadratic Gaussian) š w ƒ» ûš e(lead rubber bearing) ww w l w w (perturbation) w. w q ƒ l w Journal of Structural Control(Dyke, 2003) p w (special issue) tw. ù w š w w l w, w, e w y w w k. w w l ƒ w w w mw w. µ-» w w l w w sƒw. y w, w y w, e y y (time delay) w w w 3 w w w. 2.» l Park (2003) w w l. l Ali Abdel-Ghaffar (1995)ƒ w ûš e e j (Dyke, 2003) ƒ» ww l. ³ (2002) ƒ e w w wù e w w. w ³ (2003) l w ù ûš e x w. w z x (2004) w x ûš e LQG š ƒ» ww w l w. ± 5% w w y w ù w y w w. w l w w j» w ƒ» š µ-w (Burl, 1999) w. w l 1. w, y š ƒƒ sƒ x g e, y m y s, d, š p(volt) d d (measurement noise, ν ) sww. u HA f HA µ-w y w ƒ». xr,lrb x r,lrb ûš e f LRB ûš e w ùkü. w l w Park (2003). sz l ùkü (robust performance) y ù y s ww l y p e(structured singular value) w sƒw. ³y (normalized) sww t xk l ƒ» w v (1) (Burl, 1999). sup ω µ [ Njω ( )] < 1» µ (). y p e, N( jω) œe(nominal) sz l w (transfer function) l w P( jω)» w K( jω). š (block). y p e µ-w w (cost function) w ù w wš y p e w. y p e we yw. y D-K (Burl, 1999) w sz l á- D- f ( y p e we w f ) yw. w (2). 1. w l (1) 850 ª Œª Œ

3 1 J = sup σ[ D R ( jω)n( jω)d L ( jω) ] = D R jω ω» e z w ( ). (sƒ ) y y š w» w 2 vl(filter) w. vl vl w d e w w. vl sƒ p e» w. 2(a) sƒ p e sww 2(b) (3) vl w.» l 838 k(state) ƒ sƒ (controllability grammian) d (observavility grammians) w ³x x(balanced realization) mw 30 k ƒ (Dyke, 2003). W x g 2 2 y W y I c 1 ( s + 2ξ 2 ω f2 s + ω f2 ) = = I x g s 2 2ξ 1 ω f1 s ω f1 1 ( )N( jω)d L ( jω)» c 1 =10.32, ω f1 =10 rad/sec, ω f2 = š 5 ξ 1 = ξ 2 = 0.8 w. y vl e y, y w y (2) (3) š w» w y 1% ƒ š ƒ w. 2 y vl» w j» w 3 q ƒ vl w. š w w 3(a) q ƒ. w w p w» w (4) Kanai-Tajimi vl (Clough Penzien, 1995) w. + W g = W g I = I» S 0 = max El Centro Mexico City Gebze 2 S 0 ( 2ξ g ω g s ω g ) s ξ g ω g s + ω g [ ( ) ] mean ω = 0~10 rad/sec S x g x g š ω g = 17rad/sec ξ g = 0.3 w. w w» š q.» ez (spillover effect) w» w (6) š q m (high-pass) vl y w. m ƒ (4) (5) 2. p e y vl 3. w vl rp 26ƒ 5A œ 851

4 e w j» z w» w (7) q m (low-pass) vl w. W u W z s = W u I= s s = W z I = I s w š q m vl q m vl rp (power spectral density) 3(b) ùkù. 4 w vl sww µ-» ùkü. z l k l (vector), z z l, z u. w» w z kw. k g z w x g yw v z w yw ùkü (8) w. k g = max El Centro Mexico City Gebze [ rms ( )] x g t = 0~20 sec 4 R ƒ w w w š Q ƒ w. µ-w š w w w ƒ w k w. (Park, 2002; 2003) w. i) z ùký t kw (, p, ); ii) w ƒ y g ƒ ew mw ƒ ƒ 2 kw. w l w, w l ³yw ; iii) k 2 ƒ y g ƒ 3 w mw ƒ w w. e j 3 w 3 w yw ƒ w kw. ûš e l w š w š 1 ûš e 4. w vl sww µ-» (6) (7) (8) w ƒ w. w l w ûš e sw j k w w y j ƒ w w. w x ûš e w š w ƒ w w. mw k ƒ w. Q q I bs 4 4 = = = (9) 0 q dd I 4 4 q Q w bs I = =, qdd = (10) 0 q dd I 4 4» q bs q dd q ùkü ƒ l 24 ƒ» w l w l w» w š w. µ-» w» w MATLAB (1997) µ-analysis and synthesis toolbox(balas, 1998) dkit.m w. 3. e 0 4, q bs 6 10, qdd , q bs » w e j Bill Emerson Memorial. 2 k 128 f (cable), š (Illinois) w ew 12 ƒ. w e w w sƒ w 5(a) š w. w» w Dyke (2003) w ùký 3 x w. x w w w ABAQUS (1996) x w mw (Wilson Gravelle, 1991).» (bedrock)» - y w w w ƒ ƒw š ƒ w., f, š (rigid link) w w swwš» z w w (static condensation) mw 419 sƒ w. ƒ w ƒ 3% w. q k 16 e(shock transmission device) ew 10 š q , , , , , , , , , Hz. w l e w» w». w w e ew» w 852 ª Œª Œ

5 t 1. El Centro w q (m) q (kn) p (kn m) T max /T f T min /T f * T (kn) (kn) LRB: 467 HA: 1000 LRB+HA: Bill Emerson Memorial e e q k e w sƒ 10 q , , , , , , , , , Hz. ƒ l e j š w f ù q ƒ x ù w w (Park, 2005). š (feedback) w 5 ƒ 4 w ( 5(a) š). 4 ƒ k Õ» š 1 q ew š, ƒ 2 3 ƒƒ 2 ƒ e. (sensor) w d w w š w. e j š w El Centro (1940), Mexico City (1985), š Gebze (1999) l sƒw» w MATLAB (1997) w e w ww. Mexico City Bill Emerson Memorial Cape Girardeau w Mexico City w» š š ù 2 p ƒ l sƒw» w š. 3 ƒ 0.36g w ƒ ƒ., sƒ» e j w w ü Dyke (2003). 3.2 ƒ l» ù e ƒ» w» ƒ l w s³ (root mean value of control force) k ƒ l w. t 1-3 ƒ w ƒ l e ùkü. t 16 e w l, 24 ûš e w l, s³ (kn) LRB: 79 HA: 107 LRB+HA: 96 *x f y (0.2T f ~0.7T f, Dyke, 2003) ù t 2. Mexico City w q (m) q (kn) p (kn m) T max /T f T min /T f T (kn) (kn) s³ (kn) LRB: 296 HA: 1000 LRB+HA: 1260 LRB: 55 HA: 58 LRB+HA: 63 t 3. Gebze w q (m) q (kn) p (kn m) T max /T f T min /T f T (kn) (kn) s³ (kn) LRB: 493 HA: 1000 LRB+HA: 1493 LRB: 57 HA: 69 LRB+HA: 64 µ-w 24 ƒ» w l, š w 24 ûš e µ-w 24 ƒ» w l. š T f f q Dyke (2003). El Centro w l ƒ j 26ƒ 5A œ 853

6 w. w l l w 114%, l w 47% wš. w s³ w w l l w 9%, l w 47% w. w w w l, p, f y l w w. w w l Mexico City Gebze w w w ùkü. t 1-3 µ-w w l 3 w e j e w ùkþ. ù Dyke (2003) w e p» w» š q y. w e w, ú / l y(a/d conversion), l/ ú y(d/a conversion), š e y w y w. e. µ-w w l ƒ w» w w y w w ww. 3.3» w ƒ j w e w w ww. w (11) δ j š» w w ww. K = pert K( 1 + δ) (11)» K pert w. (11) w w w. w ƒ 3% (modal damping) ƒ w» (12). C = pert M pert Φ pert 2ξ 1 ω 1, pert ξ n ω n, pert 1 Φ pert (12)» M pert Φ pert š ω i, pert ƒƒ w, w, š q (rad/sec) š ξ 1 i (3%). ±5%, ±10%, ±15%, š ±20% w š w. w w w k q eƒ 10 q y ƒƒ 2.55%, 5.14%, 7.82%, š 10.58% w ƒw q y ƒw. sƒ» y 100% ù ƒ, q 30 cm w (Turan, 2001), f 6. w 3 w sƒ» y 0.2T f ~0.7T f w w (Dyke, 2003),» ( 1000 kn, p j(stroke) 0.2 m, 1 m/sec, Dyke, 2003) w w» y w w š ƒ w. 6 w 3 w sƒ» (J 1 ~J 11 ) y ùk ü. w ƒw ƒ sƒ» y ƒw. š sƒ» y w w w w w». w w f (J 5, J 11 ) y j. w w x w. w w ww. q,, e w w w. e j (Caicedo, 2003) š w q ƒ w w w ww. UBC» (UBC, 1991) w Bill Emerson Memorial Cape Girardeau 50 x» 73.3 kg/m 2 ü. w ƒ» 24.4 kg/m w ƒ kg/m w q ƒw 2 w g (Caicedo, 2003). ƒ w 3.7% ƒ g. w w q j. w w w w. ƒ sww 10 q , , , , , , , , , š Hz. ƒw q w q y 4.40%. t 4 w ƒ w sƒ» ùkü. 854 ª Œª Œ

7 sƒ» t 4 3 w w l y wš w w y. El Centro, Mexico City, š Gebze sƒ» y ƒƒ 6.9%(J 10 ), 8.0%(J 4 ), š 22.4%(J 11 ). w w ƒ x w. ƒ» e w y w. (13) ew (T s ) w š w w ww. t 4. w ƒ sƒ» y El Centro Mexico City Gebze y y y J % % % J % % % J % % % J % % % J % % % J % % % J % % % J % % % J % % % J % % % J % % % J 1 /J 7 - /s³ ; J 2 /J 8 - /s³ q ; J 3 /J 9 - /s³ p; J 4 /J 10 - /s³ q p; J 5 /J 11 - /s³ f y; J 6 - q τ = εt s (13) 7 š w w l y w š s»» y w w w w. x m» ù w j» ƒ» e w». y q /s³ (J 2, J 8 ) ù kù ƒ»ƒ e e». ƒ ƒ š w l w. 8 ƒ 2 e ƒ» 0.02 ƒ y ùkü. 8 (œœ) w w y e w j ù kù j w. w ƒ w. w w w w w 7. 3 w sƒ» y ww. (11) w š q ƒ 97.7 kg/m 2 w. w ±5%, ±10%, ±15%, š ±20% w k q eƒ 10 q y ƒƒ 6.82%, 9.30%, 11.85%, š 14.50% w y j. 9 w w 3 w sƒ» y ùkü. w -20% w Gebze s³ f y(j 11 )ƒ ƒw y 100%. l -20% w w y w qw. l e f x w š w e. w 6 9 ƒ q w sƒ» y 26ƒ 5A œ 855

8 8. ƒ 2 y ƒw w 3 w f (J 5, J 11 ) y ùkü. f y w j w. w w w (±5%), z ƒ ùkù w ƒw z ƒ w. ±20% w w ƒw y w. w s³ f y(j 11 ) 11 w w. ƒ w w ƒ» w ww. 9. w w 3 w sƒ» y 856 ª Œª Œ

9 10. w 3 w f y (J5 ) 12. w 3 w sƒ» y 11. w 3 w s³ f y (J11 ) w 12 sƒ» y w ƒw. w ƒw w. w w w š ƒ» l w. Gebze s³ f y(j 11 ) -20% w ƒ w y 100% w. w 13 f x w f sww z w š w. wš y 100% j l e w e y qw š w. 4. ûš e µ-w ƒ» w l w, w, ƒ», š w w w. ew w l l w s³ w w. µ-w w l w, w, ƒ» w y. ƒƒ w sƒ» y 84.9%(J 11, -20% w, Gebze ), 22.4%(J 11, Gebze ), 32.3%(J 8, =0.02, Mexico City ). w w, l w w Gebze -20% w w ù y w. w w 26ƒ 5A œ 857

10 13. w w Gebze f y (-20% w ) sƒ» y 83.25%(J 11, -5% w w, Gebze ), w ƒ» sƒ» y 88%(J 11, -20% w 0.02, Gebze ), š w y 31.2%(J 2, 0.02 w, Mexico City ). ƒ Gebze -20% w y w qw. w y qw x w z w š w. wš e w e d w qw š w. š y w l e w ƒ j ùkû. w µ-w w l š 3 w ƒ wš w ù y w. µ- w w l y w» w w (KRF D00169) w». š x ³, x, x, (2002) w e j w w l. wm wz, w m wz, 22«3-Ay, pp ³, x, Spencer, Jr., B.F., (2003),, w l w. w œwz, w œwz, 7«1y, pp x, ³, Spencer, Jr., B.F., (2004) w LRB-» w» l. w œwz, w œwz, 8«3y, pp ABAQUS. (1996) User s Manual, Hibbitt, Karlsson & Sorensen, Inc. Ali, H.M. and Abdel-Ghaffar, A.M. (1995) Seismic passive control of cable-stayed bridge. Shock and Vibration, Vol. 2, No. 4, pp Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and Smith, R. (1998) µ-analysis and synthesis toolbox, The Math Works, Inc., Natick Massachusetts. Burl, J.B. (1999) Linear optimal control: H 2 and H methods, Addison-Wesley. Caicedo, J.M., Dyke, S.J., Moon, S.J., Bergman, L.A., Turan, G.S and Hague, S. (2003) Phase II benchmark control problem for seismic response of cable-stayed bridges. Journal of Structural Control, Vol. 10, No. 3-4, pp Clough R.W. and Penzien J. (1995) Dynamics of structures, McGraw-Hill Inc. Dyke, S.J. (2003) Cable-stayed bridge seismic benchmark control 858 ª Œª Œ

11 problem. Journal of Structural Control, Vol. 10, No Dyke, S.J., Caicedo, J.M., Turan, G., Bergman, L.A., and Hague, S. (2003) Phase I benchmark control problem for seismic response of cable-stayed bridges. Journal of Structural Engineering, ASCE, Vol. 129, No. 7, pp MATLAB. (1997) User's Manual, The Math Works, Inc. Moon, S.J., Bergman, L.A.S and Voulgaris, P.G. (2003) Sliding mode control of cable-stayed bridge subjected to seismic excitation. Journal of Engineering Mechanics, ASCE, Vol. 129, No. 1, pp Nishimura, H., Nomami, K., and Nakada, O. (1994) H control with pole assignment for building-like structure by using active vibration absorber. Proceedings of the First World Conference on Structural Control, TA-4, pp Nishitani, A. and Yamada, N. (1994) H structural response control with reduced-order controller. Proceedings of the First World Conference on Structural Control, TA-4, pp Park, K.S., Jung, H.J., and Lee, I.W. (2002) Hybrid control strategies for seismic protection of benchmark cable-stayed bridges. Proceedings of the Seventh U.S. National Conference on Earthquake Engineering, CD-ROM. Park, K.S., Jung, H.J., Park, J.G., and Lee, I.W. (2003) Integrated passive-active system for seismic protection of a cable-stayed bridge. Journal of Earthquake Engineering, Vol. 7, No. 4, pp Park, K.S., Jung, H.J., Yoon, W.H., and Lee, I.W. (2005) Robust hybrid isolation system for a seismically excited cable-stayed bridge. Journal of Earthquake Engineering, Vol. 9, No. 4, pp Schmitendorf, W.E., Jabbari, F., and Yang, J.N. (1994a) Robust control techniques for buildings under earthquake excitation. Earthquake Engineering and Structural Dynamics, Vol. 23, pp Schmitendorf, W.E., Kore, I.E., Jabbari, F., and Yang, J.N. (1994b) H control of seismic-excited buildings using direct output feedback. Proceedings of the First World Conference on Structural Control, TA-1, pp Schmitendorf, W.E., Marti, S. Jabbari, F., and Yang, J.N. (1994c) Active control of seismic-excited buildings with model uncertainty. Proceedings of the Fifth U.S. National Conference on Earthquake Engineering, pp Smith, J.P., Burdisso, R., and Suarez, L.E. (1994) An experimental investigation of adaptive control of secondary systems. Proceedings of the First World Conference on Structural Control, TA-4, pp Smith, J.P. and Chase, J.G. (1994) Robust disturbance rejection using H control for civil structures. Proceedings of the First World Conference on Structural Control, TA-4, pp Suhardjo, J. (1990) Frequency domain techniques for control of civil engineering structures with some robustness considerations, PhD dissertation, Department of Civil Engineering, University of Notre Dame, Notre Dame, Ind. Turran, G. (2001) Active control of a cable-stayed bridge against earthquake excitation, PhD dissertation, Department of Civil Engineering, University of Illinois at Urbana-Champaign. UBC. (1991) Uniform Building Code. International Conference on Building Officials. Wilson, J. and Gravelle, W. (1991) Modeling of a cable-stayed bridge for dynamic analysis. Earthquake Engineering and Structural Dynamics, Vol. 20, pp Yoshida, K., Kang, S., and Kim, T. (1994) LQG control and H control of vibration isolation for multi-degree-of-freedom systems. Proceedings of the First World Conference on Structural Control, TA-4, pp ( : / : / : ) 26ƒ 5A œ 859