Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 7058–7063 Multi-Modal Active Vibration Control of a Multi-Modal Active Vibration Control of a Multi-Modal VibrationFootbridge Control of a LightweightActive Stress-Ribbon Lightweight Stress-Ribbon Footbridge Lightweight Stress-Ribbon Footbridge Based on Subspace Identification Based on Subspace Identification Based on Subspace Identification ∗ ∗∗ ∗
Xiaohan Liu Thomas Schauer Arndt Goldack ∗∗ ∗ ∗ Arndt Goldack ∗ Xiaohan Liu ∗∗∗ Thomas Mike Schauer Schlaich∗∗ ∗∗ ∗ Xiaohan Liu Thomas Mike Schauer Schlaich ∗∗∗ Arndt Goldack Mike Schlaich ∗ Chair of Conceptual and Structural Design, Technische Universit¨ at ∗ ∗ Chair of Conceptual and Structural Design, Technische Universit¨ at Berlin, Germany (e-mail:
[email protected]). ∗ Conceptual and Structural Design, Technische ∗∗Chair of Berlin, Germany (e-mail:
[email protected]). Control Systems Group, Technische Universit¨ at Berlin, Universit¨ Germanyat ∗∗ (e-mail:
[email protected]). ∗∗ Control Berlin, SystemsGermany Group, Technische Universit¨ at Berlin, Germany ∗∗ Control Systems Group, Technische Universit¨ at Berlin, Germany Abstract: Experimental identification of the modal state-space model for Active Vibration Abstract: Experimental of the modal state-space model for Active Vibration Control (AVC) is proposedidentification based on Subspace Identification Method (SIM). A stress-ribbon Abstract: Experimental identification of the modal state-space model for Active Control (AVC) is proposed based on Subspace Identification Method (SIM). A stress-ribbon bridge built at the Technische Universit¨at Berlin was taken as research object. A Vibration real-time Control (AVC) is based on Subspace Identification (SIM). bridge at with theproposed Technische Universit¨ at Berlin wasinertial taken Method as research object. A control built system pneumatic muscle actuators and sensors was set Aupstress-ribbon forreal-time AVC of bridge built at with the purpose Technische Universit¨ afeedback t Berlinand wasinertial taken as research real-time control system pneumatic muscle sensors wasaobject. set up A for AVC of the bridge. For the of AVC, theactuators control design requires modal state-space control system with pneumatic muscle inertial sensors wasa set up for AVCand of the bridge. Fordescribes the purpose of AVC, theactuators feedback and control requires modal state-space model, which the multi-modal characteristics of design resonance modes of the bridge the bridge. For the purpose of AVC, the feedback control design requires a modal state-space model, which describes the multi-modal characteristics of resonance modes of the bridge and the input/output relationship of the controlled system. Analytically, a state-space model of model, which thefrom multi-modal characteristics resonance the bridge the bridge input/output ofEuler-Lagrange the controlled system.ofAnalytically, a state-space modeland of candescribes be relationship derived mechanism. Howevermodes this isofcomputationally the input/output relationship of the controlled system. Analytically, a state-space model of the bridgeand canthe be determination derived from of Euler-Lagrange However Comparably, this is computationally expensive damping ratiosmechanism. is also inconvenient. SIM offers the bridge can be derived from Euler-Lagrange mechanism. However this is computationally expensive and the determination of damping ratios is also inconvenient. Comparably, SIM offers an attractive alternative due to simple and general parametrisation for Multiple-Input Multipleexpensive andalternative thesystems. determination of damping ratios parametrisation ismodel also inconvenient. Comparably, SIM offers an attractive due to identified simple and general forcan Multiple-Input MultipleOutput (MIMO) The state-space from SIM easily be transformed an attractive alternative due to simple and general parametrisation for Multiple-Input MultipleOutput (MIMO)modal systems. The identified model from SIM can easily be transformed into a reduced state-space modelstate-space for the AVC controller design. The goodness of the Output (MIMO) systems. Theand identified state-space model from SIM can easily be transformed into a reduced modal state-space model for the modal AVC controller design. The of the identified state-space model the truncated state-space model wasgoodness investigated in into a reduced modal state-space model for the modal AVC controller goodness of the identified state-space model and and the truncated state-space modelThe was investigated in the stabilization diagram of SIM with model fit indexes. The design. identified modal parameters identified state-space and and the with truncated model was investigated in the stabilization diagram of were SIM model fit indexes. The identified modal parameters of the footbridge frommodel SIM compared withmodal that ofstate-space previous free-vibration tests and the the stabilization diagram of SIM and with model fit indexes. The identified modal parameters of the footbridge SIM were compared withModal that ofVelocity previousFeedback free-vibration tests and the analytical model. from Implementation of a Delayed Control (DMVFC) of the on footbridge from SIM were compared withModal that carried ofVelocity previous free-vibration tests and the analytical model. Implementation of a Delayed Control (DMVFC) based the obtained modal state-space model was out Feedback for the full-scale lightweight analytical model. Implementation of a Delayed Modal Velocity Feedback Control (DMVFC) based on the obtained modal state-space model was carried out for the full-scale lightweight bridge. The effectiveness of the proposed experimental identification method was shown in case based on the obtained modal modelexperiments. was carried out formethod the full-scale lightweight bridge. The effectiveness of the proposed control experimental identification was shown in case of vertical resonance modes in state-space vibration bridge. The effectiveness of the proposed experimental identification method was shown in case of vertical resonance modes in vibration control experiments. of vertical resonance modes in vibration control Control) experiments. © 2017, IFAC (International Federation of Automatic Hosting by Elsevier Ltd. All rights reserved. Keywords: Active Vibration Control; Subspace Identification Method; Lightweight footbridges Keywords: Active Vibration Control; Subspace Identification Method; Lightweight footbridges Keywords: Active Vibration Control; Subspace Identification Method; Lightweight footbridges 1. INTRODUCTION technique where the system model is identified without 1. INTRODUCTION technique where the theinternal system physical model ismechanism. identified without trying to model Among 1. INTRODUCTION technique where theinternal system model ismechanism. identified without trying to model the physical Among the main advantages of such a method are numerical Recent lightweight footbridges, such as the stress-ribbon trying to model the internal physical mechanism. Among main advantages of such a method numerical robustness and efficiency, the ability to dealarewith MIMO Recent such the stress-ribbon bridge inlightweight the lab of footbridges, the TU Berlin, areashighly flexible and the the main advantages of such a method are numerical Recent lightweight footbridges, such as the stress-ribbon robustness and efficiency, the ability to deal with MIMO problems in a straightforward way and its ease of use bridge in the lab of the TU Berlin, are highly flexible and commonly prone to pedestrian induced vibrations. The robustness and efficiency, the ability to deal with MIMO problems insimplicity a straightforward way and The its ease of use bridge in the lab of the TU Berlin, are highly flexible and due to its (Viberg (1995)). method is commonly prone to pedestrian induced vibrations. The stress-ribbon bridge at TU Berlin is supported by very problems in a straightforward way and its ease of use commonly tomade pedestrian induced vibrations. The due to its simplicity (1995)). The state-space method is particularly attractive (Viberg for high-order MIMO stress-ribbon bridge at TU is supported by plasvery thin stress prone ribbons of Berlin carbon-fibre-reinforced due to identification. its simplicity (Viberg (1995)). The state-space method is stress-ribbon bridge at TU Berlin is etsupported by plasvery particularly attractiveThe for subspace high-order MIMO model identification method thin stress ribbons of Bleicher carbon-fibre-reinforced tics (Schlaich et al. made (2007)). al. (2011a,b,c,d) particularly attractive for high-order MIMO state-space model identification. The subspace identification method thin stress made of Bleicher carbon-fibre-reinforced plas- has been extensively studied. A good overview and an tics (Schlaich et al. Vibration (2007)). al. (2011a,b,c,d) showed thatribbons Active Controlet(AVC) using pneumodel identification. The subspace identification method has been studied. A presented good overview and an tics (Schlaich et al. (2007)). Bleicher et al. (2011a,b,c,d) account of extensively performance have been by Overschee showed that Active Vibration Control (AVC) using pneumatic muscles could overcome vibration problems of the has been extensively studied. A good overview and an showed that Active Vibration Control (AVC) using pneuaccount of performance have been presented by Overschee et al. (1996) and by Verhaegen (1994). Output-only modal matic vibration problems of the account of performance have been presented by Overschee bridge muscles withoutcould usingovercome tuned mass dampers. State-space matic could vibration of the et al. (1996) andonbySIM Verhaegen (1994). Output-only analysis based was widely applied in recentmodal years bridge without usingovercome tuned mass dampers. State-space modelsmuscles characterising the dynamics of the problems physical system et al. (1996) andonby Verhaegen (1994). Output-only modal analysis based SIM was widely applied in recentof years bridge without using tuned mass dampers. State-space for experimental identification of modal parameters civil models characterising the dynamics of the physical system are needed for the purpose of AVC design. Bleicher et al. analysis based on SIM was widely applied in recent years for experimental identification of modal parameters of civil models characterising the dynamics of the physical system engineering structures, see e.g. (Peeters et al. (2016)). are needed for the purpose of AVC design. Bleicher et al. (2011a) analytically derived a state-space model of the for experimental identification of modal parameters of civil are needed for purpose design. Bleicher al. engineering see e.g. (Peeters al. (2016)). For AVC instructures, the frequency domain, an etapplication of (2011a) analytically derivedof aAVC state-space model ofetthis the bridge via the the Euler-Lagrange formalism. However engineering structures, see based e.g. (Peeters etapplication al. (2016)). (2011a) analytically derived a state-space model of the For AVC in the frequency domain, an of experimental identification on SIM can be found in bridge viaisthe Euler-Lagrange formalism. this For AVC in the frequency domain, an application of approach computationally expensive andHowever cumbersome bridge viaisthe Euler-Lagrange formalism. this experimental (Nestorovic etidentification al. (2006)). based on SIM can be found in approach computationally expensive andHowever cumbersome with increasing number of degrees of freedom. For poorly experimentaletidentification approach is computationally expensive and cumbersome (Nestorovic al. (2006)). based on SIM can be found in with increasing number degrees of For poorly understood systems, the of derivation of freedom. a state-space model (Nestorovic Subspace identification of a modal state-space model is et al. (2006)). with number of degrees of For poorly understood systems, thephysics derivation of freedom. a state-space model Subspace from increasing first principles of is almost impossible. identification of amulti-modal modal state-space is proposed in this paper for AVC in model the time understood systems, of thephysics derivation of a state-space Subspace identification of a modal state-space model is from first principles is almost impossible.model proposed in this paper for multi-modal AVC in the time domain. State-space modelling of the bridge dynamics for An alternative way of by experimental proposed from first principles of building physics ismodels almostisimpossible. in this paper for multi-modal AVC in the time modelling in of Section the bridge dynamics for AVC willState-space be shortly described 2. Section 3 will An alternative way ofHere, building models by experimental system identification. the aim is toisestimate dynamic domain. domain. State-space modelling of Section the bridge dynamics for An alternative way of building models is by experimental AVC will be shortly described in 2. Section 3 will introduce the proposed identification method based on system the aim is toand estimate dynamic models identification. directly fromHere, observed input output data. AVC will be shortly described in Section 2. Section 3 will introduce the proposed identification method based on system identification. Here, the aim is to estimate dynamic SIM. Identification experiments on the bridge and model models directly from observed input and output data. Subspace identification is the name for a general class of introduce the proposed identification method based on models directly from observed input and output data. SIM. Identification experiments on the bridge and model validation of the obtained models will be described in Subspace identification is the name a general class of SIM. system identification methods; it is afor black-box modelling Identification experiments on the and model Subspace identification is the name a general class of validation of the obtained models willbridge be described in system identification methods; it is afor black-box modelling validation of the obtained models will be described in system identification methods; it is a black-box modelling
Copyright © 2017 IFAC 7329 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 7329 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 7329Control. 10.1016/j.ifacol.2017.08.1352
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Fig. 1. The stress-ribbon bridge with AVC at TU Berlin. detail in Section 4. In Section 5, the obtained state-space model from the experimental identification will be used in the multi-modal AVC of the lightweight footbridge. Finally, the control efficiency of AVC based on the identified model will be presented. 2. STATE-SPACE MODELLING OF THE BRIDGE The dynamics of the stress-ribbon bridge shown in Fig. 1 is very complex because of its prestressed geometry as well as inherent nonlinearities. In general, the bridge dynamics can be described with a nonlinear MIMO state-space model based on physical principles. After linearisation, a linear state-space model of the bridge dynamics is given by X˙ = AX + BU, Y = CX + DU (1) T U = [u1 u2 u3 ] (2) T T (3) Y = [y1 y2 y3 y4 ] = [q1 q˙1 q4 q˙4 ] where X is the state vector, U is the input vector and Y is the output vector. A is the system matrix, B is the input matrix, C is the output matrix and D is the direct feedthrough matrix. As shown in Fig. 2, the output vector Y consists of the dynamic displacements q1 and q4 as well as the corresponding velocities q˙1 and q˙4 . The input vector U contains the forces introduced by the pneumatic muscle actuator (PMA) pairs as shown in Fig. 1 and Fig. 2 to control the first three vertical modes. Analytically, a nonlinear MIMO state-space model was derived based on Euler-Lagrange formalism from the simplified rigid body model of the footbridge (Bleicher et al. (2011a)) shown in Fig. 2. After linearisation, the 14th order linear state-space model [A, B, C, D] was obtained with a state vector X = [q1 q˙1 q2 q˙2 · · · q7 q˙7 ]T in nodal form. Alternatively, system identification methods can be applied to directly identify a linearized state-
¯ B, ¯ C, ¯ D] ¯ space model. The identified state-space model [A, ¯ describes the input/output (I/O) with a state vector X relationship. For a model-based controller design, the linear state-space model obtained analytically or from experimental system identification can be transformed into a modal statespace representation and reduced using modal truncation techniques (Gawronski (2004)). The reduced modal statespace model can be written as (4) X˙ r = Ar Xr + Br U, Y = Cr Xr + Dr U (5) Xr = [xrh x˙ rh . . . xr1 x˙ r1 ]T where the modal state vector Xr contains the first h ≥ 3 vertical bridge modes. 3. SYSTEM IDENTIFICATION PROCEDURE The aim of the proposed system identification method is to ¯ B, ¯ C, ¯ D] ¯ that is obtain an estimated state-space model [A, used to derive a reduced state-space model [Ar , Br , Cr , Dr ] in modal form while avoiding an analytical system modelling approach. 3.1 Subspace identification method The measurements for practical applications are usually sampled-data, so the system to be identified from the input-output measurements is expressed in a general form of a discrete-time linear state-space model: (6) Xk+1 = ΦXk + GUk + wk (7) Yk = CXk + DUk + vk where Xk ∈ Rn , Uk ∈ R3 , Yk ∈ R4 , n = 2, 4, 6, . . . is the system order and k is the sample index. The process noise wk and the measurement noise vk are white noise vector sequences with zero mean. The identification is first of all
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Fig. 2. Simplified physical model of the bridge with sensors and actuators. to determine the system matrices Φ ∈ Rn×n , G ∈ Rn×3 , C ∈ R4×n and D ∈ R4×3 .
At first, measured input and output data are organised into block Hankel matrices. Based on the unbiased combined deterministic-stochastic identification algorithm presented in (Overschee et al. (1996)), the singular values are then accordingly determined by singular value decomposition (SVD) for user-defined system orders n from the weighted oblique projection. The subspace algorithm ’N4SID’ is used for the identification of the combined deterministic-stochastic system. Furthermore, the ¯ C¯ can be computed in a least identified matrices Φ, ¯ and squares sense, and also the estimated matrices G ¯ are solved from the solution of a minimization criteD rion by least squares. Finally, the identified discrete-time ¯ G, ¯ C, ¯ D] ¯ will be transformed linear state-space model [Φ, ¯ B, ¯ C, ¯ D] ¯ correto a continuous-time state-space model [A, sponding to the user-defined system order n. 3.2 Modal state-space model transformation As a black-box model structure is assumed in the subspace-based method, i.e. the parameterization of the model is not specified before identification, the identified ¯ B, ¯ C, ¯ D] ¯ will not be the same as those in the matrices [A, physically derived models, such as [A, B, C, D]. Even when the system orders of both models are the same, that is to say n = 14. If a transformation matrix is chosen as a modal ma¯ B, ¯ C, ¯ D] ¯ can trix T ∈ Rn×n , the identified matrices [A, be also transformed into the modal state-space model ¯m , C¯m , D ¯ m ], which contains maximum n/2 conju[A¯m , B gate complex modes of the bridge system. 3.3 Stabilization diagram of SIM A typical challenge in system identification is the determination of the model order n. When trying to get an appropriate estimation of the state-space model from real measurement datas, it is quite common to over-specify the model order considerably; here we also try from the data to fit high-order models that contain much more modes than real modes of bridge dynamics. Afterwards, interpreting a so-called stabilization diagram helps to separate the true physical modes (stable poles) from the spurious numerical ones. The spurious numerical poles will not stabilise at all during the process of increasing the system order and can be sorted out of the modal parameter data set (Peeters et al. (2004)). The model order n, which contains no spurious numerical poles between the stable ones in the stabilization diagram, can be chosen as an appropriate system order.
The corresponding state-space model with order n is the estimated model used for further analysis. 3.4 Model reduction A model with a large number of degrees of freedom causes numerical difficulties in dynamic analysis and control design. Thus, a reduced order system solves the problem if it acquires the essential properties of the high-order model. Among many reduction techniques, the modal truncation method gives results close to the optimal one (Gawronski (2004)). The states of the estimated model ¯m , C¯m , D ¯ m ] can be written as [A¯m , B (8) Xm = [Xt Xr ]T where Xr is the vector of the retained states and Xt is a vector of truncated states. If there are h retained modes out of a total number of n/2 modes, then Xr is a vector of 2h states, and Xt is a vector of 2(n/2 − h) states. Let the ¯m , C¯m , D ¯ m ] from Section modal state quadruples [A¯m , B 3.2 be partitioned accordingly: At 0 ¯m = Bt , C¯m = [Ct Cr ] . (9) A¯m = , B 0 Ar Br A reduced model is obtained by deleting the first 2(n/2−h) ¯m , the first 2(n/2 − h) columns of A¯m rows of A¯m and B ¯ and Cm . Formally, this operation can be written as follows: ¯m , Cr = C¯m LT , Dr = D ¯m Ar = LA¯m LT , Br = LB (10) with L = [02h×2(n/2−h) I2h ]T . Considering the frequency range of pedestrian excitation and the response spectrum of the bridge, the first three vertical resonance modes are considered to be controlled by AVC, therefore h was set as 3, and L = [06×2(n/2−3) I6 ]T . The truncated model for the first three modes can be written as (11) X˙ r = Ar Xr + Br U, Y = Cr Xr + Dr U 0 1 Ar = diag(Am3 , Am2 , Am1 ), Ami = (12) −ωi 2 −2ζi ωi where ωi = 2πfi and ζi are the natural angular frequencies and damping ratios of the footbridge. 4. IDENTIFICATION EXPERIMENTS ON THE STRESS-RIBBON BRIDGE For the identification experiments and vibration control experiments, the observer and controller were implemented on a PC running the Ubuntu operating system. The real-time code was generated with Simulink Embedded Coder using the Linux ERT target 1 . Three actuator 1
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PMA pairs excite the bridge successively. Because of the persistently exciting characteristics of MLPR signals, the dominant vertical modes of the bridge could be properly excited. Under open-loop conditions of the control system in the laboratory, the input/output data for a duration of 300 s were collected in the identification experiment and used for SIM. 2000
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Fig. 4. Flow chart of the Matlab/Simulink implementation. 4.1 Identification experiment A flow chart of the Matlab/Simulink implementation is shown in Fig. 4. In the identification experiments the two-way switch connects to the multi-level pseudo random (MLPR) signals. The internally feedback-controlled PMAs (see Bleicher et al. (2011a)) excite the stress-ribbon bridge at the handrail level, and the modal states of the bridge are estimated by a modal state observer together with an adaptive filter (Band-limited Multiple Fourier Linear Combiner (BMFLC) estimator) from the acceleration measurements and PMA forces. The detailed setup and the applied observer design using two accelerometer sensors can be found in (Liu et al. (2016)). The input vector [u1 u2 u3 ]T contains the generated input forces from the three PMA pairs at three positions on the bridge. The dynamic displacements q1 and q4 as well as the corresponding velocities q˙1 and q˙4 are estimated from two accelerometers (denoted as S01 and S02) are taken to build up the output vector. Multi-level pseudo random signals (Godfrey (1993)) are generated in Matlab/Simulink and applied as reference force signals to the controlled PMA pairs (see Fig. 5). The three
4.2 Stabilization diagram and modal parameters of the bridge The identified discrete-time linear state-space models were transformed to continuous-time state-space models with system orders n from 2 to 80. According to Section 3.2, the continuous-time state-space models can be transformed into modal state-space models using the transformation matrix T ∈ Rn×n . Modal state-space models with model orders of n = 2h include modal information in terms of natural frequencies fi , damping ratios ζi and mode shape vectors Vi (i = 1, 2, ..., h). These parameters can be used for model verification. The stabilization diagram of the identified natural frequencies (below 10 Hz) from the system matrix A¯m is shown in Fig. 6. The natural frequencies f1 =1.33 Hz, f2 =2.52 Hz, f3 =4.04 Hz, f4 =5.57 Hz and f5 =6.88 Hz are quite stable in the whole process in terms of system orders from 4 to 80. In order to estimate the natural frequencies and modal damping ratios of the bridge without AVC, individual free-vibration tests were made in the first three vertical modes. As shown in Tab. 1, the first two vertical natural frequencies identified from SIM match good with that obtained from the analytical modelling approach and from individual free-vibration tests on the bridge. There is a small deviation of 7% for the third natural frequency. The identified modal damping
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An appropriate state-space model with a model order n should be chosen for AVC design. A model which includes false modes resulting from spurious numerical poles in frequency range of interest should be excluded. Following this argumentation, from our identification experiment a model order n = 10 was chosen. The resulting model describes the first five vertical modes of the bridge system. 4.3 Model validation As written in Section 3.4, the previously identified and chosen modal state-space model with a system order n of 10 was truncated to a reduced state-space model describing only the first three vertical modes of the bridge. Model reduction by modal truncation of stable modes always produces a stable reduced state-space model, since the poles of the reduced model are a subset of the poles of the full-order model. Reduction errors comes from the question how good the retained states Xr = LXm will represent the dynamic characteristics of the system response. Therefore model validation of the reduced model and the identified model are performed by inspecting model fit indexes. The velocities at midspan and one-eight span estimated from measurement are plotted together with the simulated model outputs using the identified model ¯m , C¯m , D ¯ m ] (system order 10) and the reduced [A¯m , B
In order to inspect the goodness of the identified and the truncated models, model fit indexes were investigated from the outputs estimated from acceleration measurements and the simulated outputs. The function goodnessOfFit of the System Identification Toolbox (The Mathworks, Inc., US) was used, the cost function was chosen as Normalized Mean Square Error (NMSE), which varies between -Inf (bad fit) to 1 (perfect fit). Model fit indexes regarding the outputs y1 , y2 , y3 and y4 are given in Tab. 2. A model fit of 82% for y2 and y4 is obtained with the identified model. The truncated modal state-space model also shows a reasonable model fit of 55% and 75%. Table 2. Model fit indexes of the identified model and the reduced model. State-space model Identified model Truncated modal model
y1 0.68 0.40
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5. MULTI-MODAL ACTIVE VIBRATION CONTROL The Delayed Modal Velocity Feedback Control (DMVFC) controller was applied to control the first three vertical resonance modes of the bridge. Based on the reduced modal state-space model from system identification of the footbridge in Section 4, a MIMO controller was designed and implemented in Matlab/Simulink. In the implementation, as shown in Fig. 4, the switch will be connected to the second circuit with the designed AVC controller. A detailed description about the controller design can be found in (Bleicher et al. (2011a,b,c,d)).
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ACKNOWLEDGEMENTS M1 with AVC M1 without AVC
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The first author would like to thank the financial support from the PhD Program of the Chinese Scholarship Council (CSC). The professional support of Prof. Achim Bleicher for this research is also sincerely appreciated.
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Fig. 8. Vibration response with and without AVC (top: 1st resonance mode, bottom: 2nd resonance mode). In order to inspect the effectiveness of the identified statespace model and to evaluate the damping effect of the control system, the bridge was firstly excited in a resonance mode by the PMAs with harmonic reference force signals Fi = Vi sin(2πfi t) [kN]. After the resonance mode was enough excited, the switch in Fig. 3 was switched to actively control the motion of the bridge. For comparison, vibration responses of the footbridge with and without AVC are plotted in Fig. 8. For the first two resonance vibration modes, the modal velocities were effectively dampened within 2 seconds with AVC, while the free vibrations decrease very slowly because of small natural damping values. The AVC results show that the MIMO controller based on the estimated reduced modal statespace model works well. No spill-over effecte were observed in the AVC experiments. 6. CONCLUSIONS In this work, experimental system identification of a lightweight footbridge based on subspace identification methods was conducted in purpose of AVC. The effectiveness of the obtained reduced modal state-space model was verified in real AVC experiments. Subspace identification was firstly applied to identify and to further obtain a stable modal state-space model with an appropriate system order, which was verified and chosen regarding natural frequencies in the stabilization diagram of SIM. The obtained state-space model represents the MIMO relationship between the generated forces from the PMAs and the estimated nodal displacement and velocity outputs from accelerometer measurements. A reduced state-space model used for the controller design of AVC was then determined by modal truncation technique. The damping effect of the control system based on the identified statespace model showed its feasibility for multi-modal AVC of the full-scale footbridge. For a flexible vibrating structure, whose natural frequencies lie possibly very close together, the accuracy and stability of the described identification method based on subspace identification should be analysed and discussed in aspect of all modal characteristics, also including mode shapes.
A. Bleicher, M. Schlaich, Y. Fujino and T. Schauer. Modelbased design and experimental validation of active vibration control for a stress ribbon bridge using pneumatic muscle actuators. Engineering Structures, 33: 2237–47, 2011. A. Bleicher, M. Schlaich and T. Schauer. Multi-modal and multivariable active vibration control for a footbridge model-based design and experimental validation. Proceedings of the 8th International Conference on Structural Dynamics, 1525–32, Belgium, 2011. A. Bleicher, T. Schauer, M. Valtin, J. Raisch and M. Schlaich. Active vibration control of a light and flexible stress ribbon footbridge using pneumatic muscles. IFAC Proceedings Volumes , 44(1): 911–16, 2011, 18th IFAC World Congress. A. Bleicher. Aktive Schwingungskontrolle einer Spannbandbr¨ ucke mit pneumatischen Aktuatoren. Doctoral Dissertation, Technische Universit¨ at Berlin, 2011. B. Peeters, G. Roeck. Reference-based stochastic subspace identification for output-only modal analysis. Mechanical Systems and Signal Processing, 13(6):855–78, 1999. B. Peeters, P. Guillaume et al. Automotive and aerospace applications of the PolyMAX modal parameter estimation method. Proceedings of the 22th International Modal Analysis Conference, Dearborn, USA, 2004. K. Godfrey. Perturbation signals for system identification. Prentice Hall, 1993. M. Viberg. Subspace-based methods for the identification of linear time-invariant systems. Automatica, 31(12): 1835–51, 1995. M. Verhaegen. Identification of the deterministic part of MIMO state space models given in innovations form from input-output data. Automatica, 30(1): 61–74, 1994. M. Schlaich and A. Bleicher. Spannbandbr¨ ucke mit Kohlenstofffaser-Lamellen. Bautechnik, 84(5): 311–19, 2007. P.V. Overschee and B.D. Moor. Subspace Identification for Linear Systems: Theory - Implementation - Applications, Springer, US, 1996. T. Nestorovi´ c and U. Gabbert. Active control of a piezoelectric funnel-shaped structure based on subspace identification. Structural Control and Health Monitoring, 13: 1068–79, 2006. W.K. Gawronski. Advanced Structural Dynamics and Active Control of Structures, Springer, New York , 2004. X. Liu, T. Schauer, A. Goldack, A. Bleicher and M. Schlaich. Accelerometer-based estimation and modal velocity feedback vibration control of a stress-ribbon bridge with pneumatic muscles. Journal of Physics: Conference Series, 744, 2016.
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