Application of dynamics inversion techniques to the force control of electrodynamic actuators used for active vibration absorption

Journal of Physics: Conference Series PAPER • OPEN ACCESS Application of dynamics inversion techniques to the force control of electrodynamic actuators used for active vibration absorption To cite this article: José Ramírez-Senent et al 2024 J. Phys.: Conf. Ser. 2647 172004 View the article online for updates and enhancements. You may also like Robust saturated control of humaninduced floor vibrations via a proof-mass actuator I M Díaz and P Reynolds - Downscaling of proof mass electrodynamic actuators for decentralized velocity feedback control on a panel Paolo Gardonio and Cristóbal González Díaz - Response-dependent velocity feedback control for mitigation of human-induced floor vibrations D Nyawako and P Reynolds - This content was downloaded from IP address 150.214.182.230 on 20/11/2024 at 10:04 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd XII International Conference on Structural Dynamics Journal of Physics: Conference Series 2647 (2024) 172004 IOP Publishing doi:10.1088/1742-6596/2647/17/172004 1 Application of dynamics inversion techniques to the force control of electrodynamic actuators used for active vibration absorption José Ramírez-Senent1, Iván M Díaz1, Jaime H García-Palacios2 and Javier F Jiménez Alonso3 1 Department of Continuum Mechanics and Theory of Structures. ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, C/Profesor Aranguren 3, 29040, Madrid, Spain 2 Department of Hydraulics, Energy and Environmental Engineering, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, C/Profesor Aranguren 3, 28040, Madrid, Spain 3 Department of Continuum Mechanics and Structural Analysis, ETS de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Seville, Spain [email protected] Abstract. The active vibration absorber has been successfully used, both in laboratory and actual facilities, to mitigate the human-induced vibrations taking place in lively, low-damping structures such as long-span floors or pedestrian footbridges. Nonetheless, the dynamic behavior of the electrodynamic proof-mass actuators employed for this purpose may negatively affect the overall vibration attenuation features due to the interaction with the motion of the structure under optimization. In this paper, the utilization of dynamic inversion techniques to improve the force control of electrodynamic proof-mass actuators is presented. The main idea behind this approach is to find an approximate inverse model of the actuator dynamics which, upon implementation on a real-time controller, leads to an approximate cancellation of actuator dynamics, making it behave closer to an ideal one. The selected way of proceeding also accounts for the actuatorstructure interaction phenomenon. This approach may be employed to improve the performance of any force-based vibration control algorithm. Herein, the dynamic inversion techniques are applied to the well-known Direct Velocity Feedback algorithm which aims at emulating the behavior of a dashpot connected between the control point and the ground in its simplest version. The goodness of the proposed procedure has been assessed both by experimental tests performed over a full-scale composite material pedestrian footbridge existing at the School of Civil Engineering of the Technical University of Madrid. The test results show that the dynamic inversion approach improves the force tracking of the proof-mass actuator to a great extent, therefore yielding better attenuation results that those achieved with the classical Direct Velocity Feedback scheme. 1. Introduction The current trend to build lightweight, low-damping structures such as footbridges or long-span floors has led to an increased vibrational response when these elements are subjected to human actions. XII International Conference on Structural Dynamics Journal of Physics: Conference Series 2647 (2024) 172004 IOP Publishing doi:10.1088/1742-6596/2647/17/172004 2 Usually, the vibrations induced in this type of structures due to pedestrian loads do not represent a threat to their integrity; however, the Vibration Serviceability Limit State (VSLS) may not be fulfilled. A common approach to mitigate human-induced vibrations consists in installing passive tuned vibration absorbers (TVAs) [1]; nonetheless, when the properties of the structure vary over time, detuning issues may arise, limiting to a great extent the vibration mitigation performance of the TVAs. Usual measures to tackle these drawbacks comprise the use of multi-TVAs [2] or semi-active TVAs [3]. The active vibration absorber (AVA) has been satisfactorily used, both in laboratory and in actual environments, to minimize human-induced vibrations in scenarios which, in comparison to the previously mentioned approaches, require higher vibration attenuations or damping out the effects of multi-frequency loads. The main elements of an AVA system are the actuators, which exert the control forces on the structure, the sensors (usually accelerometers), which provide the control system with the structural kinematic information and the real-time controller, which calculates the control forces making use of the information provided by the sensors and the vibration control algorithm (VCA) in use. There are two main categories of VCAs [4]. The first one corresponds to the direct output feedback controllers. This approach usually consists in employing an actuator and a sensor located at the same point of the structure (collocated pair). The control force is obtained by multiplying the reading of the sensor by a gain that needs to be tuned. The main advantage of this family of VCAs is that an accurate model of the structure is not usually needed, and good results can be achieved by using a small number of actuators and sensors. The most widespread VCA within this category is the Direct Velocity Feedback (DVF) algorithm [4], which aims at reproducing a pure damping element between the actuator-sensor installation point and the ground. For this purpose, the accelerometer reading is integrated with respect to time to yield an estimate of the structure velocity and later on, multiplied by a gain (which acts as the damping coefficient) to yield the control force applied to the structure. An important advantage of the DVF is that it is unconditionally stable when the actuators and sensors dynamics are neglected; however, when these hypotheses cannot be assumed, there exists a limit value for the control gain over which the system becomes unstable [5]. Many efforts have been undertaken to improve the stability margin of the DVF. Some remarkable examples are the Compensated Acceleration Feedback Control [6], the Inner Control Loop Approach [7] or the Integral Resonant Control [8]. The model-based controllers constitute the other category of VCAs. In this case, an accurate model of the structure is employed to obtain estimates of the states to be controlled. A gain-matrix is applied to the controlled states to calculate the control forces to be reproduced by the actuators. Some representative examples of model-based controllers are the Pole Placement method [9], the Linear Quadratic Regulator and the Linear Quadratic Gaussian Control [10] and the 𝐻∞ and 𝐻2 methods [11]. Regardless of the VCA at hand, the common denominator to all of them is that their output is a control force to be exerted by the actuator on the structure under control. Due to the fact that the actuators do not behave ideally, i.e.: they exhibit their own dynamics, the performance of the VCAs may be seriously affected when the actuator dynamics interact with those of the structure under control. In this paper, the utilization of dynamic inversion (DI) techniques to improve the force control of electrodynamic proof-mass actuators (PMAs) usually employed in AVAs, for structures subjected to human-induced vibrations, is presented. The main idea behind this approach is to find an approximate inverse model of the actuator dynamics which, upon implementation on the real-time controller, leads to an approximate cancellation of the actuator dynamics, making it behave closer to an ideal one [12]. The proposed way of proceeding also accounts for the actuator-structure interaction (ASI) phenomenon [5]; thus, making the behavior of the actuator independent from the flexibility of the structure onto which it is mounted. The control system is completed with a parallel three-variable-controller (TVC) which accounts for the unavoidable modelling errors arising in the DI process. This approach may be employed to improve the performance of any force-based vibration control algorithm; in this work, the DI techniques are applied to the DVF algorithm leading to the Direct Velocity Feedback with DI (DVFDI) algorithm. The goodness of the proposed procedure has been assessed by means of experimental tests performed over a full-scale composite material pedestrian footbridge existing at the School of Civil Engineering of the Technical University of Madrid. XII International Conference on Structural Dynamics Journal of Physics: Conference Series 2647 (2024) 172004 IOP Publishing doi:10.1088/1742-6596/2647/17/172004 3 The remainder of this paper is organized as follows. Section 2 briefly describes the structure over which the AVAs have been implemented. Section 3 explains the DI suggested procedure along with the implemented control architecture. Section 4 presents the hardware architecture and the experimental results obtained with the proposed methodology. In addition, a comparison with the outcomes obtained with the classical DVF approach is offered. Finally, section 5 summarizes the conclusions drawn from the presented research. 2. Structure description The structure employed to perform the AVA tests is a simply supported glass-fiber reinforced plastic footbridge with a length of 10 m, a width of 1.5 m and a linear mass density of only 80 kg/m (figure 1). It has been designed so as to meet the static requirements at Serviceability and Ultimate Limit States. Nonetheless, the VSLS is not fulfilled, on purpose, by the bare structure, as it has been engineered to comply with VSLS through the integration of inertial vibration controllers [13]. Figure 2 shows the results obtained in an experimental modal analysis (EMA), undertaken for modelling and simulation purposes, which have been reproduced here to complete the description of the target structure. The values of the first three modal frequencies are approximately of 6.5 Hz (first vertical bending mode), 9.3 Hz (first lateral-torsional mode; mostly lateral) and 11.9 Hz (second lateral-torsional mode; mostly torsional), respectively. The EMA has been carried out by exciting the footbridge with the actuator-sensor pair installed at its midspan, that is, with the same configuration as that used for the actual experiments. Figure 1. Footbridge photograph: general view and detail of supports. (a) (b) Figure 2. Footbridge accelerance at its midspan. (a) Magnitude. (b) Phase. XII International Conference on Structural Dynamics Journal of Physics: Conference Series 2647 (2024) 172004 IOP Publishing doi:10.1088/1742-6596/2647/17/172004 4 3. Control system The control system implemented in this research is dealt with in detail in this section. Figure 3 presents the block diagram corresponding to the architecture under study. In this figure, 𝐹𝑡ℎ represents the total force transmitted by the pedestrian to the footbridge, the rest of the blocks and symbols appearing in the figure will be explained in the next subsections. Figure 3. Control system block diagram. 3.1. Dynamics inversion algorithm In order to calculate the voltage which achieves theoretical PMA dynamics cancellation (𝑢𝑐,𝐷𝐼 in figure 3), a particularization of the Structure Algorithm to linear systems in which some of the system inputs are known, as described in [12] has been followed. The first step in this process is to work out a statespace representation of the PMA system. This model takes the following form [12]: {𝑧󰇗𝑎 𝑧󰇘𝑎 𝑑𝑖𝑎𝑑𝑡 ⁄}= [ 0 1 0 −𝑘𝑎 𝑚𝑎−𝑐𝑎 𝑚𝑎𝐾𝑒 𝑚𝑎 0 −𝐾𝑒 𝐿𝑐−𝑅𝑐 𝐿𝑐 ] {𝑧𝑎 𝑧󰇗𝑎 𝑖𝑎}+ [ 0 0 0 𝑘𝑎 𝑚𝑎𝑐𝑎 𝑚𝑎0 0𝐾𝑒 𝐿𝑐𝐾𝑎𝑣 𝐿𝑐 ] {𝑧𝑠1 𝑧󰇗𝑠1 𝑢𝑐}, (1) 𝐹𝑡𝑎=[𝑘𝑎𝑐𝑎−𝐾𝑒]{𝑧𝑎 𝑧󰇗𝑎 𝑖𝑎}+[−𝑘𝑎−𝑐𝑎0]{𝑧𝑠1 𝑧󰇗𝑠1 𝑢𝑐}. (2) where 𝑧𝑎 and 𝑧𝑠1 are the displacement of the proof-mass and of the point of the structure where the PMA was installed (overdot notation has been employed to denote derivatives with respect to time), 𝑖𝑎 is the current through PMA coil, 𝑢𝑐 is the control voltage applied to the actuator through its power amplifier and 𝐹𝑡𝑎 is the total force transmitted by the actuator to the structure. Furthermore, 𝑚𝑎, 𝑘𝑎 and 𝑐𝑎 are, respectively, the proof-mass value and its equivalent stiffness and damping coefficient, 𝐾𝑒 is the electromagnetic constant of the circuit, 𝑅𝑐 and 𝐿𝑐 represent circuit resistance and inductance and 𝐾𝑎𝑣 constitutes the gain of the power amplifier driving the actuator. Dynamics inversion Acc. sensor Three variable controller ++ 𝑧 𝑎 𝑧󰇗 𝑎 𝑎 𝑧 𝑠1 𝑧󰇗 𝑠1 𝑢𝑐,𝐷𝐼 𝑢𝑐, 𝑢𝑐 𝐹𝑡ℎ 𝐹𝑡𝑎 𝑧󰇘𝑎 𝑧󰇘𝑠1 𝑧󰇘𝑠1,𝑚 𝑧󰇘𝑎,𝑚 𝐹󰇗𝑡𝑎, Footbridge Power amplifier 𝑢𝑎 1 HP Filter 1 Acc. sensor 1 HP Filter 1 HP Filter 𝐾 𝐷𝐼 1 1 −1 𝑚𝑎 𝑧󰇗𝑎, 𝑧𝑎, 𝑧󰇘𝑎, Actuator XII International Conference on Structural Dynamics Journal of Physics: Conference Series 2647 (2024) 172004 IOP Publishing doi:10.1088/1742-6596/2647/17/172004 5 The PMA employed in this work has been an APS 400 commercial electrodynamic shaker with a frequency range from 0 to 200 Hz, a force capacity of 445 N and a stroke of 158 mm [14]. The PMA is driven by an APS 125 amplifier [15]. The values of the parameters of the state-space model represented by equations (1a) and (1b) have been estimated through an identification process in which the PMA was installed on a rigid floor [12] yielding the following values: 𝑚𝑎=30.52 kg, 𝑘𝑎=303.56 N/m, 𝑐 𝑎= 150.44 Ns/m, 𝑅𝑐= 2.45 Ω, 𝐿𝑐= 17.34 mH, 𝐾 𝑒= 19.11 Vs/m and 𝐾 𝑎𝑣= 18.00 V/V, where hats denote estimates. Once the state-space model is available, the DI procedure requires differentiating with respect to time the output equation (2) and substituting equation (1) into the differentiation result until an explicit relationship between the control order (𝑢𝑐 in this case) and the system output or its time-derivatives (𝐹󰇗𝑡𝑎 in this case) is obtained. Then, the control voltage required to yield the desired response can be factored out as shown in the following expression: 𝑢𝑐,𝐷𝐼 = 󰇗𝑡𝑎,𝑟−𝐂𝐀 𝐱−𝑘 𝑎𝑐 𝑎 𝑚 𝑎𝑧 𝑠1−(𝑐 𝑎 2 𝑚 𝑎−𝐾 𝑒 2 𝐿 𝑐−𝑘 𝑎)𝑧󰇗 𝑠1+𝑐 𝑎𝑧󰇘𝑠1,𝑚 −𝐾 𝑒𝐾𝑎𝑣 𝐿 𝑐 ⁄, (3) in which subindices DI have been added to the control voltage to denote that this is the voltage calculated by the DI algorithm, 𝐀  and 𝐂 are the estimates of the state and output matrices of the state-space model of the PMA, subindex 𝑟 has been appended to the force transmitted by the actuator to emphasize that it is the reference force to be developed by the PMA (calculated by the VCA) and subindex 𝑚 denotes measured quantities. Regarding the estimate of the state vector, 𝐱, the displacement and the velocity of the proof-mass of the actuator (𝑧 𝑎 and 𝑧󰇗 𝑎) have been obtained by high-pass filtering (via a second order Butterworth filter with a cutoff frequency of 0.1 Hz) and successively integrating the signal from an accelerometer, whereas the current through PMA coil, 𝑎, is taken directly from an available output in the power amplifier. The approximations for the displacement and velocity of the structure (𝑧 𝑠1 and 𝑧󰇗 𝑠1) at the PMA installation point (denoted by 1) have been obtained by means of a similar process as that followed for the proof-mass. It is worth noting that the inversion formula derived in equation (3): (i) may be used for any VCA which outputs a target actuator force and (ii) takes into account the ASI phenomenon [5] by means of the kinematic information of the structure at the PMA installation point which in turn constitutes the known inputs to the system. 3.2. Parallel Three Variable Controller So as to cope with the unavoidable modelling errors taking place in the derivation of the PMA DI, the control scheme proposed in this work is completed with a parallel TVC which provides simultaneous corrections for the displacement, velocity and acceleration of the proof-mass. The TVC action, 𝑢𝑐, , can be expressed according to: 𝑢𝑐, =𝐾𝑑(𝑧𝑎, −𝑧 𝑎)+𝐾𝑣(𝑧󰇗𝑎, −𝑧󰇗 𝑎)+𝐾𝑎(𝑧󰇘𝑎, −𝑧󰇘𝑎,𝑚), (4) in which 𝐾𝑑, 𝐾𝑣 and 𝐾𝑎 are the control gains for displacement, velocity and acceleration, respectively, and 𝑧𝑎, , 𝑧󰇗𝑎, and 𝑧󰇘𝑎, correspond to the displacement, velocity and acceleration references which have been calculated by filtering the reference force 𝐹𝑡𝑎, (with a similar high-pass filter as those described previously), dividing by the estimated value of the proof-mass, 𝑚𝑎 and integrating successively with respect to time. 3.3. Direct Velocity Feedback with Dynamics Inversion The last step is to define the reference force to be reproduced by the actuator. In this work, the DI techniques have been applied to the well-known DVF algorithm leading to the DVFDI algorithm. Consequently, the time-derivative of the force to be reproduced by the PMA can be expressed as: XII International Conference on Structural Dynamics Journal of Physics: Conference Series 2647 (2024) 172004 IOP Publishing doi:10.1088/1742-6596/2647/17/172004 6 𝐹󰇗𝑡𝑎, =𝐾 𝐷𝐼d𝑧󰇗𝑠1 d𝑡 ≈𝐾 𝐷𝐼𝑧󰇘𝑠1,𝑚, (5) where 𝐾 𝐷𝐼 denotes the control gain. A high-pass, second order, Butterworth filter with a cutoff frequency of 1 Hz has been employed to attenuate the low-frequency components of 𝑧󰇘𝑠1,𝑚. 4. Experimental results 4.1. Experimental setup The hardware architectures employed in this work to deploy the control software is briefly explained in this subsection. Figure 4 illustrates the experimental setup. The real-time control program runs in a National Instruments (NI) cRIO 9066 controller. The board employed to read the accelerometers signals and the current signal output by the PMA power amplifier is a NI 9234. Furthermore, the board employed for analog output generation is a NI 9263. The accelerometer used to measure the structural accelerations at point 1 is a PCB 393B12. Finally, the acceleration of the proof-mass is acquired by means of a PCB 333B50 accelerometer. Figure 4. Experimental setup 4.2. Experimental results All the tests were carried out by a single pedestrian with a mass of 85 kg traversing the footbridge at a pace rate of 99 beats per minute. This rate was chosen in order to excite the first mode of the footbridge with the fourth harmonic of the walking load. Three different tests were performed: (i) uncontrolled structure (inactive AVA system), (ii) structure controlled with the DVF algorithm and (iii) structure controlled with the VFDI algorithm. The main results obtained in the tests are shown in figures 5, 6 and 7; these include the acceleration waveform at the PMA installation point, 𝑧󰇘𝑠1,𝑚, which has been filtered using a second order band-pass Butterworth filter with a low and a high cut-off frequency of 1 Hz and 20 Hz, respectively and the corresponding moving RMS (1 s RMS) calculated over 500 samples acquired at time intervals of 2 ms. As expected by the footbridge design, the uncontrolled structure test results depicted in figure 5 show high accelerations in excess of 0.5 m/s2 leading to a minimum degree of comfort according to [16]. The maximum values for the acceleration and the 1 s RMS reached in this test were of 1.27 m/s2 and 0.65 m/s2, respectively. Figure 6 shows the results obtained for the structure controlled with the classical DVF algorithm. A control gain of 140 Vs/m was employed for this test. This value was chosen performing a root-locus analysis and then refined experimentally. As it can be appreciated, the DVF VCA noticeably reduces XII International Conference on Structural Dynamics Journal of Physics: Conference Series 2647 (2024) 172004 IOP Publishing doi:10.1088/1742-6596/2647/17/172004 7 the maximum attained vibration levels which, in this case, were respectively of 0.36 m/s2 and 0.12 m/s2 for the acceleration and the 1 s RMS. Finally, figure 7 shows the vibration level accomplished when the VCA in use was the DVFDI. The control gain value employed was 22000 Ns/m and the values of the control parameters for the TVC were 𝐾𝑑=7 V/m, 𝐾𝑣=5 Vs/m and 𝐾𝑎=0 Vs2/m; approximate values for all these parameters were calculated by means of numerical simulations and, later on, improved experimentally. The obtained results are significantly better than those obtained for the DVF case, reaching maximum values of 0.21 m/s2 and 0.08 m/s2 for the acceleration and the 1 s RMS waveforms, respectively. These values represent corresponding relative reductions of 41.67 % and 33.33 % with respect to the DVF case. Figure 5. Walking test results. Uncontrolled structure. Figure 6. Walking test results. Structure controlled with the DVF algorithm. Figure 7. Walking test results. Structure controlled with the DVFDI algorithm. XII International Conference on Structural Dynamics Journal of Physics: Conference Series 2647 (2024) 172004 IOP Publishing doi:10.1088/1742-6596/2647/17/172004 8 5. Conclusions In this paper, the application of DI techniques to the force control of electrodynamic PMAs employed in the implementation of AVAs for structures subjected to human-induced vibrations has been explained. The DI approach has been applied to the well-known DVF VCA leading to the DVFDI algorithm. The assessment of the proposed control methodology has been performed by means of experimental tests carried over a composite footbridge built in the Structures Laboratory of the School of Civil Engineering of the Technical University of Madrid. This structure meets the static requirements at the Serviceability and Ultimate Limit States; however, it exhibits excessive vibrational response when subjected to human loads, since it has been engineered to fulfill VSLS through the installation of inertial vibration controllers. Firstly, the state-space representation of the PMA system has been derived. Once this model is available, the DI procedure implies differentiating with respect to time the output equation until an explicit relationship between the control voltage and the system output or its time-derivatives is obtained. For the case at hand, a relationship between the control voltage and the first time-derivative of the control force has been obtained. The DI formula implies estimating the state vector of the PMA system as well as the displacement, velocity and acceleration at the PMA installation point. The kinematic variables have been estimated by means of a filtering and successive integration process whereas the current of the actuator has been taken from an output of the power amplifier. The control system has been completed with a parallel TVC which accounts for the unavoidable modeling errors. The time-derivative of the reference force to be reproduced by the PMA according to the DVFDI algorithm can be simply calculated by multiplying the structure acceleration at the installation point by the control gain. The attained experimental results show that the DVFDI clearly outperforms the results obtained when using the classical DVF approach achieving relative reductions of 41.67 % and 33.33 % in the peak acceleration and in the 1s RMS respectively. This improvement in performance is due to the DI procedure which makes the actuator behavior closer to the ideal one. Acknowledgments The authors acknowledge Grant PID2021-127627OB-I00 (Transport Infrastructures subjected to dynamic loading: assessment techniques for the sustainability, intelligent maintenance and comfort) funded by Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación and 10.13039/501100011033 FEDER, European Union. References [1] Yang F, Sedaghati R and Esmailzadeh E 2001 Vibration suppression of structures using tuned mass damper technology: A state-of-the-art review J. Vib. Control 28 812-836. [2] Li Q, Fan J, Nie H, Li Q and Chen Y 2010 Crowd-induced random vibration of footbridge and vibration control using multiple tuned mass dampers J. Sound Vib. 329 4068-4092. [3] Moutinho C, Cunha Á, Caetano E and de Carvalho JM 2018 Vibration control of a slender footbridge using passive and semiactive tuned mass dampers Struct. Control Health Monit. 25 e2208. [4] Hudson MJ and Reynolds P 2012 Implementation considerations for active vibration control in the design of floor structures Eng. Struct. 44 334-358. [5] Díaz IM, Gallegos CA, Ramírez-Senent J and Renedo CMC 2021 Interaction phenomena to be accounted for human-induced vibration control of lightweight structures Front. Built Environ. 7 658529. [6] Díaz IM and Reynolds P 2010 Acceleration feedback control of human-induced floor vibrations Eng. Struct. 32 163-173. [7] Díaz IM, Pereira E, Hudson MJ and Reynolds P 2012 Enhancing active vibration control of pedestrian structures using inertial actuators with local feedback control Eng. Struct. 41 157– 166.