Active vibration control of a light and flexible stress ribbon footbridge using pneumatic muscles

Transcription

1 Proceedings of the 18th World Congress he International Federation of Autoatic Control lano (Italy) August 8 - Septeber, 011 Active vibration control of a light and flexible stress ribbon footbridge using pneuatic uscles A. Bleicher*,. Schauer**, M. Valtin**, J. Raisch**, M. Schlaich* Berlin Institute of echnology (U Berlin), Berlin, Gerany * Chair of Conceptual and Structural Design, Departent of Civil and Structural Engineering ** Control Systes Group, Departent of Electrical Engineering and Coputer Science Abstract: his paper describes the developent of an active vibration control syste for a light and flexible stress ribbon footbridge. he 13 span Carbon Fiber Reinforced Plastics (CFRP) stress ribbon bridge was built in the laboratory of the Departent of Civil and Structural Engineering, Berlin Institute of echnology. Its lightness and flexibility result in high vibration sensitivity. o reduce pedestrianinduced vibrations, very light pneuatic uscle actuators are placed at handrail level introducing control forces. First, a reduced discretized analytical odel is derived for the stress ribbon bridge. o verify the analytical prediction, experients without feedback control are conducted. Based on this odel, a velocity feedback control strategy is designed to actively control first ode vibrations. o handle the nonlinearities of the uscle actuator a subsidiary nonlinear force controller is ipleented based on exact linearisation ethods. he stability of the entire closed-loop syste with actuator saturation is investigated by the Popov Criterion. Control perforance is verified by experients. It is deonstrated that handrail introduced forces can efficiently control the first ode response. Keywords: active vibration control; light-weight structure; stress ribbon bridge; pedestrian induced vibration; pneuatic uscle actuator; velocity feedback control; exact linearization 1. INRODUCION Stress ribbon bridges are aong the ost elegant and lightest bridges. Due to their static and dynaic characteristics, they have been ainly designed for pedestrian traffic rather than for road or rail traffic (Strasky (005), Baus et al. (008)). he suspension cable and the bridge deck are cobined into one load-bearing eleent, which is anchored in the abutents. Usually the ribbons are ade of steel cables or steel plates. o show the potential of high-strength Carbon Fiber Reinforced Plastics (CFRP), a 13 span stress ribbon bridge with CFRP ribbons was built in the laboratory of the Departent of Civil and Structural Engineering, U Berlin by Schlaich et al. (007). his coposite aterial allows considerably saller cross sections. he bridge s tensile force under dead and live load is carried by only six ribbons with a cross section of 1.1 x 50 each. he cobination of low extensional stiffness using CFRP for the ribbons and a light-weight bridge deck leads to considerable dynaic responses caused by pedestrian loads. In order to reduce the vibration apli- tudes a ulti-odal active vibration control (AVC) syste was developed (Bleicher (011)). In particular it is necessary for extreely light structures, where the syste properties like ass and stiffness becoe tie-variant by changing pedestrian traffic. In this paper, the results of the AVC syste to reduce first ode responses are presented. herefore, control forces are introduced on handrail level by two very light pneuatic uscle actuators (PMA) placed in idspan (Fig. 1, Fig. ). Fig.. Equipent to control the force of the PMA F Fig. 1. Concept of active vibration control for the first ode /11/$ IFAC / I

2 lano (Italy) August 8 - Septeber, 011 Fig plate odel for set-up o design a odel-based velocity feedback controller an analytical odel is derived for the bridge in Section. he odel is also used to estiate odal states with a Kalan filter, which is described in Section 3. An analytical nonlinear odel of the actuator syste is developed in Section 4. Based on this odel an I/O linearization controller design is applied for the PMA in Section 5. As a result the actuator dynaics becoes a first order syste. he velocity feedback controller is designed in Section 6. In Section 7, experiental results of the cascaded active vibration control are shown.. ANALYICAL MODEL OF HE SRESS RIBBON BRIDGE Fro the distributed syste of the stress ribbon bridge, an analytical plane rigid body odel for ulti-variable control syste design is developed (cf. Fig. 3 and Bleicher (011)). In order to get good agreeent with experiental data for the odes to be controlled a plane odel with seven degrees of freedo q i is developed using Euler-Lagrange equations. he paraeters of the 8-plate odel are listed in able 1. able 1. Paraeters of the 8-plate odel L otal bridge length Plate length L 1.63 Plate thickness d 0.10 otal ass M 4336 kg Plate ass i 54 kg Mass oent of inertia I 10 kg Spring stiffness k N/ Pre-stressing force P N Handrail post length h 1.09 yyi he resulting seven nonlinear ordinary differential equations of nd order can be linearized at the equilibriu point in case of sall aplitudes guaranteed by active vibration control. herefore, the nonlinear differential equations are derived into a nonlinear state space description, where the force generated by the two PMA FM 4 AC1 is the input variable u: x f( x, u) (1) For the 8-plate odel, seven generalized coordinates and their first derivatives are selected as state vector x. x q1 q1 q7 q7 x1 x x13 x14 z1 z z6 z7 z z z z he linear state space description (nodal for) is obtained by deterining the Jacobian atrices (A, B) at the equilibriu point. Siulation and experients of free vibrations without control are conducted to verify the dynaic behaviour of the odel. In able the natural frequencies of the first three odes of the 8-plate odel and of the easured bridge set-up are listed. he frequencies agree well with the easured frequencies. able. Natural frequencies of the 8-plate odel and the Bridge set-up Mode 8-plate odel Bridge set-up 1. vertical syetric 1.33 Hz 1.35 Hz. vertical asyetric.54 Hz.76 Hz 3. vertical syetric 3.85 Hz 4.08 Hz o design a odel based controller for specific odes it is ore convenient to use the odal state space representation, Gawronski (004). his representation is obtained fro the already derived nodal state space representation by transforation using the odal atrix R. In the odal description all odes are decoupled. -1 x R x (3) 1 1 A R AR, B R B, C CR (4) Where x is the odal state vector, A is 141 the odal syste atrix, B is the odal input 1414 vector, C is the odal output atrix and the nodal 1414 output atrix C is the identity atrix. () 3. KALMAN FILER O ESIMAE MODAL SAES o estiate the odal states a Kalan filter is eployed, Lunze (006). By easuring only the vertical displaceent x 7 in idspan the syetrical odes can be observed. herefore, only the syetrical odes are extracted fro the vectors / atrices of Eq. (4): A diag A, A, A, A (5) B B B B B (6) 91

3 lano (Italy) August 8 - Septeber, 011 C C C C C (7) A i, B i and 14 C i are the co- Where ponents of ode i. he dynaics of the estiator are given by xˆ ˆ ˆ A x B u Lys CsC x (8) ˆ y s y s x7 x8 q4 q 4 (9) where xˆ are the estiated syetric odal states. he easureent is y s ( x 8 is obtained by nuerical differentiation) and y ˆ s is the estiation of this easureent based on x ˆ. he atrix C s selects the estiated eleents ˆx 7 and ˆx 8 out of the vector C ˆ x. he convergence of x ˆ x is deterined by the observer gain L. his gain is found by iniizing a quadratic cost function. 4. NONLINEAR MODEL OF HE ACUAOR SYSEM he used PMA is developed and anufactured by the copany Festo, Festo (010). he pulling-only actuator consists of a flexible tube. By inflating the tube with copressed air it expands in radial direction and contracts in longitudinal direction. he force F produced by the PMA i {1, } depends on the pressure p inside the uscle and its contraction length s. Detailed inforation about odelling of pneuatic actuators including ass flow, pressure dynaics and force characteristics are given in Hildebrandt et al. (003) and Hildebrandt (009). Input of the actuator syste i is the reference position u vi of the proportional directional control valve. Output of the actuator syste is the generated force F. In Bleicher (011), the nonlinear actuator odel is given by,, s s p p f p s g p s uvi 1 (9) 0 1s s p 0 C10p, buvi, uvi 0 nr p p L 0 1s s p C10p, buvi, uvi 0 pv F hp, s (10) 3 p p0av c1c1pps cs c3s where p is the easurable syste state, s is a tievariant easurable syste variable, and q, b b is the flow rate function defined by Eq. (11). Further paraeters are listed in able 3. able 3. Paraeters of the actuator syste using a PMA type DMSP N and a valve type MYPE-5-1/4-40-B, Festo (010) Virtual piston area A v Valve paraeter C litre/(s bar) Specific gas constant R 87 J/kgK Gas teperature L 93 K Valve paraeter b 0.3 1p kg/s kg/s kg/s Polytrophic exponent n p 1.0 Abient pressure p N/ Supply pressure p v 6 p 0 Volue constant v litre Volue constant v litre/ Volue constant v litre/ Isentropic exponent 1.4 Air density kg/ 3 5. FORCE CONROL USING I/O LINEARISAION Exact I/O linearization is applied to render the nonlinear actuator dynaics linear for forces in the range 0 to 3000 N and for -1 u vi 1. he linearising and stabilising control law is given by 0 0,, Lh f p, s s af a uvi F, ref Lh g p s Lh g p s (1) Lh p s Ac s f p s (13),, f v 1p, 1, Lh p s Ac s g p s (14) g v p s c c p c s c s s (15) p where F is the easured force, F, ref is the reference force, Lh f and Lh g are the L i - derivatives, s is the derivative of the tie-variant easurable syste variable and a 0 is the control paraeter to set the tie constant of the resulting linear actuator dynaics which is given by a0 1 GP s. (16) 1 s a0 1 s he control paraeter is set to a Hz respectively 0.05 s, Bleicher (011). he controller behaves experientally as expected as long as the supply pressure is constant also under load changes and if the sapling rate of the digital controller realisation is high enough. qb b 1, qb b (subcritical flow) qb, b 1 b 1, qb b (supercritical flow) (11) 913

4 lano (Italy) August 8 - Septeber, VELOCIY FEEDBACK CONROL O REDUCE FIRS MODE VIBRAIONS he concept to control first ode vibrations is based on a cascade control structure with a subsidiary force control and a superior velocity feedback control. he control gain K for positive feedback (negative plant gain) of the observed odal velocity was found by applying the root locus ethod (Dorf (008)) assuing a non-saturated control signal (force) and negligible observer dynaics. Hence, closed-loop poles are defined as roots of the equation P1 1G s G s K 0 (17) where G is the transfer function of the first ode and the linear actuator dynaics given by Eq. (16). G P 1 is he transfer function of the first ode (input: total force FM FM1 FM produced by the two pneuatic uscles, output: odal velocity x M1 ) is derived using the Laplace transfor of the odal state-space odel: b 1s Gs (18) s 11s1 Paraeters of this transfer function are the natural frequency rad/s, the daping factor and b rad/(s N). he optial daping ratio ˆ 1, opt 0.3 of the closed-loop syste is achieved with the gain setting K 17,500 Ns/rad. Root locus analysis furtherore revealed that high gain velocity feedback cannot render the closed-loop syste unstable. Due to the special pole-zero constellation (a zero at the origin and 3 poles) the conjugate coplex poles will converge to 10 j for K (at least theoretically for this ideal linear syste odel). Very large gains will lead to an undesirable increase in the natural frequency and a decrease of the daping ratio. Up to now, no constraints in the control signal have been taken into account. Due to the nature of the chosen actuator, only liited unidirectional (positive) control forces can beapplied. hese forces push the bridge downwards. Siulations have shown that the practically applied gain should be doubled to obtain the above given optial daping ratio. he Popov Criterion (Khalil (001)) was successfully used to prove the stability of the closed-loop syste fored by a static nonlinear gain (saturated velocity feedback law) Kx1 for 0 Kx 1 FM ax FM, ref FM ax for Kx 1 FM ax (19) 0 for x 1 0 and the linear syste x () s G s G () s F () s (0) M1 P1 M, ref Having a axial force of 3,000 N for one PMA it follows that FM ax 6, 000 N. he results of this control strategy are shown and discussed in Section 7. Fig. 4. Control structure for experient 914

5 lano (Italy) August 8 - Septeber, RESULS OF EXPERIMEN For experient the control structure was ipleented in a real-tie environent with a sapling frequency of 100 Hz. herefore, a PC running a Linux operation syste with the real-tie extension RAI ( was used. he software code was generated and copiled fro Scilab / Scicos block diagras. Counication to devices was provided by the HAR oolbox ( he excitation signal for this experient was a sinusoidal force signal in the first ode so that the pneuatic uscle actuators work first as exciter. Afterwards the vibrations are actively daped by the actuator. In Fig. 5 the reference force F M 1/, ref and the easured force F M 1, act after excitation are plotted. It can be observed that the easured force follows the expected linear actuator dynaics F M 1/, est, P1 in good agreeent. In Fig. 6 tie histories of estiated odal velocity with velocity feedback control and without control are plotted. he control strategy efficiently reduces the first ode response. he uncontrolled response decays very slowly with logarithic decreent Λ 1,A he controlled response decays about 6 ties faster with Λ 1,A,AVC Fig. 7. Pedestrian induced accelerations w/o and with AVC 8. CONCLUSIONS Fig. 5. Reference force, actual force and estiated force by first order syste (P1) he control perforance under pedestrian excitation is shown in Fig. 7. One person is walking fro on side to the other side with a step frequency of 1.35 Hz that corresponds to the bridge s first natural frequency. Without control a axiu acceleration aplitude of 6 /s occurs. With control the acceleration aplitude can be reduced by 89 % to less than 0.6 /s. No significant spillover effects in higher unsyetrical odes could be observed. Active vibration control for a stress ribbon bridge with an extreely light pneuatic actuator was investigated. For this purpose, an analytical nonlinear bridge odel was developed. By experients it was confired that the linearized odel represents the nonlinear behavior of the stress ribbon in good agreeent for ulti-odal otions. By using exact linearization ethods the nonlinear dynaics of the pneuatic actuator were transfored into fast linear dynaic behavior. his step greatly siplifies the design for the higher level velocity feedback controller. he proposed cascaded control structure for active daping of the bridge itigates first ode responses efficiently. o verify the odel-based control designs full-scale experients were conducted. he used control strategy can be easily extended to ultivariable control, thus in further studies a ulti-variable control syste will be ipleented to control asyetric and higher syetric odes. In addition, the efficiency of the controller reducing forced and rando vibrations induced by a nuber of pedestrians will be investigated. Future analysis will also deal with the robustness of the controller under strongly varying asses by pedestrians. ACKNOWLEDGEMEN he authors would like to thank Festo AG & Co. KG., Esslingen, Gerany for the generous aterial support. Fig. 6. Modal velocity of the first ode w/o and with AVC 915

6 lano (Italy) August 8 - Septeber, 011 REFERENCES Baus, U., Schlaich, M. (008). Footbridges. Birkhäuser, Basel. Bleicher, A. (011). Multiodal Active Vibration Control of a Stress Ribbon Bridge using Pneuatic Muscle Actuators. Ph.D. hesis, Institut für Bauingenieurwesen, U Berlin (in Geran). Dorf, R., Bishop, R. (008). Modern Control Systes (11e). Prentice Hall. Festo AG & Co. KG (010). Product Catalogues. Esslingen, Gerany. Gawronski, W.K. (004). Advanced Structural Dynaics and Active Control of Structures. Springer, New York. Hildebrandt, A., Sawodny, O., Neuann, R., Hartann, A. (003). A Cascaded racking Control Concept for Pneuatic Muscle Actuators. European Control Conference (ECC), Cabridge. Hildebrandt, A. (009). Control and Design of Servopneuatic Drives. Ph.D. hesis, Institut für Systedynaik, Universität Stuttgart (in Geran). Khalil, H. (001). Nonlinear Systes (3e). Prentice Hall Lunze, J. (006). Regelungstechnik. Springer, Berlin (in Geran). Schlaich, M., Bleicher, A. (007). Spannbandbrücke aus Kohlenstofffaser-Laellen. Bautechnik, Vol. 84, Issue 5, pp Ernst & Sohn (in Geran). Strasky, J. (005). Stress Ribbon and Cable Supported Pedestrian Bridges. hoas elford, London. 916