Direct Acceleration Feedback Control of Shake Tables with Force. Stabilization

Transcription

1 Direct Acceleration Feedback Control of Shake Tables with Force Stabilization Matthew Stehman [1] and Narutoshi Nakata [2] [1] Graduate Student, Department of Civil Engineering, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA, [2] Assistant Professor, Department of Civil Engineering, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA, ABSTRACT This study presents a new strategy for shake table control that uses direct acceleration feedback without need for displacement feedback. To ensure stability against table drift, force feedback is incorporated. The proposed control strategy was experimentally validated using the shake table at the Johns Hopkins University. Experimental results showed that the proposed control strategy produced more accurate acceleration tracking than conventional displacement-controlled strategies. This paper provides the control architecture, details of the controller design and experimental results. Furthermore, the impact of input errors in shake table testing on the structural response is also discussed. Keywords: Shake tables, Acceleration control, Dynamic tests, Earthquake engineering, Error analysis 1

2 1. Introduction The role of experimental studies is of considerable significance in earthquake engineering. In particular, shake table testing is the primary experimental means for the performance assessment of structures subjected to earthquake ground motions. Since the development of the first handpowered shake table in the late 19 th century [Severn, 2011], a number of shake table facilities have been constructed in many parts of the world (US, Japan, Germany, Italy, China, etc.), and shake tables have been employed in a variety of earthquake engineering studies (e.g. [Jacobsen, 1930; Filiatrault et al., 2001; Elwood and Moehle, 2003]). It is not an overstatement to say that the development of earthquake engineering today would not be possible without the outcomes, advancements, and achievements in shake table testing. While shake table testing is by far the most preferred experimental method among others in earthquake engineering, it has yet a number of challenges. Some of the challenges are due to practical matters and economical constraints. For example, though full-scale shake table tests with multi-directional loading might be desirable, such tests are rarely possible because of limited access to experimental facilities and lack of funds. Such practical and economical challenges are often beyond the engineer s control. However, some of the other challenges in shake table testing are technical issues that should be addressed by engineers. Technical challenges in shake table testing range from control issues to post-processing of test results: reproduction of desired table accelerations; shake table nonlinearities; control of multiple actuators for multi-dimensional loading and asynchronous ground motions; instrumentations to capture structural responses of interest; boundary conditions such as soil-structure interaction 2

3 and interaction with surrounding members that are omitted in shake table testing; difficulties to incorporate substructure techniques; extrapolation of scale effects; interpretation, implication, and generalization of test results; and so on. All of the above cannot be addressed by a single study. This study focuses on the challenge of reproducing desired table acceleration in shake table testing. The challenge in acceleration control of shake tables is mainly due to the inherent dynamics of the control system (i.e., servo hydraulic actuators) and its interaction with the test structures, often referred to as control-structure interaction [Dyke et al., 1995]. In general, the hydraulic actuators used in shake tables are displacement-controlled with proportional-integral-differential (PID) controllers; where reference displacements are determined a priori by integrating the acceleration time history and removing drifting components. In some cases, velocity and acceleration feedback are added to the displacement control (e.g., three-variable controller by MTS [Tagawa and Kajiwara, 2007; Nowak et al., 2000]). Iterative approaches are often incorporated in commercial shake table controllers to compensate for the dynamics of the control system by modifying the reference displacement outside the servo control loops [Spencer and Yang, 1998]. Displacement control in shake tables provides reasonable performance in the low frequency range if evaluated in the frequency domain (e.g., performance spectrum often used in manufacturer specifications). However, displacement control generally produces poor acceleration tracking in the time domain and does not provide adequate repeatability in generated accelerations [Nakata, 2010]. Efforts to improve acceleration control accuracy of shake tables over the conventional displacement control strategies can be found in literature. Stoten and Gomez [2001] proposed 3

4 adaptive control using the minimal control synthesis (MCS) algorithm for shake tables. The MCS algorithm improves the control of shake tables when using the first order strategy of displacement control in the low-medium frequency range. Kuehn et al. [1999] developed a feedback/feed-forward control strategy based on receding horizon control (RHC). Experimental results showed that the RHC based controller had better phase characteristics in the acceleration transfer function than a feedback control using the linear quadratic regulator control. Trombetti and Conte [2002] employed a method that combines displacement feedback, velocity feedforward and differential pressure feedback control in a test-analysis comparison study. Their sensitivity analysis showed the effectiveness of the velocity feed-forward term in the magnitude characteristics of the displacement transfer function. Nakata [2010] developed an acceleration trajectory tracking control strategy that combines acceleration feed-forward, displacement feedback, command shaping, and a Kalman filter for the displacement measurement. Experimental results showed superior performance and repeatability of the acceleration trajectory tracking control over the conventional displacement control. While those methods improve control accuracy of shake tables to some extent, acceleration tracking in a wide range of frequencies, particularly in high frequency range, is still a challenging problem. This paper presents a new approach to reproduce accurate reference acceleration for shake tables. The proposed acceleration control strategy employs direct acceleration feedback without utilizing displacement feedback. While the idea of acceleration feedback is mentioned in literature [Stoten and Gomez, 2001], it has not been thoroughly investigated due to the issue of table drift; table position cannot be inferred by acceleration measurement. The proposed method overcomes this challenge by adding a force feedback loop while maintaining high-fidelity 4

5 acceleration control in both low and high frequency ranges. Following the hardware requirements as well as sample controller designs for the proposed acceleration control strategy, experimental validations conducted on the Johns Hopkins University uni-axial shake table are presented. Furthermore, the performance of the proposed control strategy is discussed in comparison with the conventional displacement-controlled strategy. Finally, the impact of erroneous input ground motions in shake table tests on the structural response is presented. 2. Direct Acceleration Feedback Control of Shake Tables This paper presents a new approach for acceleration tracking of shake tables that adopts direct acceleration feedback control without displacement feedback. As mentioned in the previous section, due to issues of table drift, implementation of acceleration feedback for shake tables is fairly limited. In this study, force feedback control is incorporated into the actuator control system to provide stability for preventing table drift. 2.1 Control Architecture The goal of the control system in shake table testing is to reproduce reference accelerations. While the acceleration control strategy proposed herein can be applied to multi-axis shake tables, for illustrative purposes, only single axis shake tables are considered in this study. Figure 1 shows the block diagram of the proposed acceleration control strategy. The control system consists of two parallel feedback loops: an acceleration control loop and a force control loop. In the acceleration control loop, the reference acceleration, a r, is first prefiltered by the pre-gain, P a, to obtain the modified reference acceleration, â r. This technique is used to compensate the dynamics of the closed loop controller. This modified reference 5

6 acceleration is then sent to the acceleration feedback control loop. The acceleration feedback controller, C a, generates the valve command, u a, from the acceleration error between the modified reference and the measured accelerations. In the same way, the reference acceleration is forwarded to the force control loop that contains the force pre-gain, P f, and the force feedback controller, C f. The reference force, f r, is the converted signal from the reference acceleration and is sent to the force feedback control loop. The force feedback controller, C f, generates the valve command, u f, from the force error between the reference and the measured forces. The total valve command is the sum of the acceleration and force valve commands. As shown in the block diagram, acceleration and force measurements of the table are used as feedback in the servo control loops; in the figure, transfer functions, H au, and H fu, denote the open-loop dynamics of the shake table from the valve command to the measured acceleration and force, respectively. It should be noted that the control strategy proposed here does not utilize displacement measurement, and thus it is fundamentally different from the conventional shake table controllers that are relying on displacement feedback. a r Acc. Pre-Gain â P r a Force Pre-Gain + Acc. FB. Controller C a Force FB. Controller Shake Table/Structure Dynamics u a H au a m u P f f r + C f u f H fu f m FIGURE 1 Block diagram of proposed acceleration control strategy. 6

7 The closed-loop transfer functions from the reference acceleration to the measured acceleration and force can be expressed as: H am a r = a C m = a P a + C f P f H au (1) a r 1+ C a H au + C f H fu H fm a r = f C m = a P a + C f P f H fu (2) a r 1+ C a H au + C f H fu It can be seen that all of the controller terms, namely P a, P f,c a and C f, affect the closed-loop acceleration and force transfer functions. In this study, those controller terms are designed based on the above transfer functions: acceleration tracking performance is evaluated from the acceleration transfer function while stability is judged from the force transfer function. Therefore, those controller terms need to be designed for each test setup. 2.2 Hardware Requirements The acceleration control strategy developed in this study also has certain requirements in terms of hardware. The first requirement of the proposed control strategy is a restoring force member between the shake table and the fixed base. The restoring force member such as springs provide forces proportional to the absolute position of the table, allowing force from the restoring member to serve as a reference to the table position. To measure the force from the member, a force transducer such as a load cell has to be installed in series with the actuator rod and table platform. Figure 2 shows a schematic of a uni-axial shake table test setup to illustrate the requirements for the proposed combined acceleration control strategy. The spring attached between the shake table and the fixed base provides the restoring force that is proportional to the position of the 7

8 table. It should be mentioned that under dynamic loading the measurement from the load cell includes inertial forces of the table and test structure; base shear from the test structure; and restoring force from the spring. For acceleration feedback, accelerometers have to be mounted on the shake table. It should be also clarified that because the displacement signal is not used in the proposed control strategy, linear variable differential transducers (LVDTs) are not required in the test setup. FIGURE 2 Schematic of a uni-axial shake table setup for the proposed acceleration control strategy. 8

9 3. Experimental Setup To demonstrate an application of the proposed acceleration control strategy, shake table tests are conducted using the shake table in the Smart Structures and Hybrid Testing (SSHT) laboratory at the Johns Hopkins University. Figure 3a shows the uni-axial shake table with a three-story test structure. The shake table has a 1.2m x 1.2m (110kg) aluminum platform driven by a Shore Western hydraulic actuator (Model: 911D). The actuator has a dynamic load capacity of 27kN and a maximum stroke limit of ± 7.6cm. An MTS 252 series dynamic servo valve is used to control the fluid flow through the actuator chambers. The table platform is capable of moving a maximum payload of 0.5 ton at an acceleration of 3.8 g. To meet the hardware requirements discussed in Section 2.2, an accelerometer, a 22.5kN dynamic load cell and four compression springs connecting the table platform to the floor were installed. Together the springs act as a single kn/m restoring member; 2 of the springs are shown in Figure 3b. The overall spring stiffness was chosen such that static drifting of the table resulted in reasonable force levels measured through the load cell. The test structure is 2m tall with individual floor masses of 60kg and a total mass (including support connection) of approximately 225kg. Because the total mass of the structure is more than double the mass of the table platform, the structure has a high influence on the dynamics of the actuator than the table platform. Thus, the test setup here is subjected to a high influence of control-structure interaction, and is used to demonstrate the capabilities of the proposed control strategy for such challenging conditions. Dynamics characteristics of the three-story structure are listed in Table I. 9

10 This paper will appear in Journal of Earthquake Engineering Taylor & Francis. FIGURE 3 The shake table in the SSHT lab at Johns Hopkins: (a) shake table with three-story structure; (b) view of restoring springs. Table I. Dynamic characteristics of three-story test structure. Vibration Mode 1 Vibration Mode 2 Vibration Mode 3 Natural Frequency, Hz Damping Ratio, %

11 4. Experimental Investigation of the Proposed Acceleration Control Strategy The proposed acceleration feedback control strategy is experimentally investigated using the shake table with the three-story structure described in the previous section. First, the open loop dynamics of the shake table system are experimentally obtained and modeled using system identification techniques. Then, based on the system models, acceleration and force feedback controllers along with pre-gains are designed. An experimental investigation is performed using a series of earthquake ground motions as the reference to the shake table. This section presents details of the system models, controller design and experimental results Experimental Modeling of the Shake Table System Open loop dynamics of the shake table system are determined using experimental system identification techniques for the design of the controllers. Relationships of interest here are those from the valve command to measured acceleration and to measured force. Those open loop relationships are the primary plants, H au, and H fu, in the proposed acceleration feedback control strategy, as shown in Figure 1. Figure 4 shows the experimentally obtained open loop relationships: valve command to measured acceleration (Figure 4a) and valve command to measured force (Figure 4b). The valve to acceleration relationship has small magnitude at low frequencies. The magnitude begins to increase with frequency. Then a pole-zero pair appears around 13Hz, which corresponds to the second natural frequency of the structure, influence from the first natural frequency is not apparent. Around 25Hz, a peak is present due to vibrations of support connections after which the magnitude begins to decrease. Another pole-zero pair occurs around 38Hz, which is the third 11

12 natural frequency of the three-story structure. On the other hand, the valve to force relationship exhibits peaks and valleys within 5Hz. The first zero appears at 1.45 Hz, which corresponds to the natural frequency of the table with springs. The first peak and the second valley are a polezero pair around 2.5 Hz, which is close to the first natural frequency of the structure. Beyond 5 Hz, the general trend in the valve to force relationship is similar to the valve to acceleration relationship except for the level of influence of the third mode of the structure around 38 Hz. It can be clearly seen that the dynamics of the test structure influences the open loop dynamics of the shake table system, indicating a high level of control-structure interaction in the test setup. Analytical models of the relationships are obtained using curve fitting techniques and are also plotted in Figure 4. The valve to acceleration relationship and the valve to force relationship are captured by 8 th order and 9 th order rational polynomial functions, respectively. The mathematical representations of the analytical models are presented in Table II. As shown in Figure 4, the analytical models capture the experimental relationships with reasonable accuracy throughout the entire frequency range Experimental Analytical Magnitude 10 1 Magnitude (a) 10 2 Experimental Analytical Frequency (Hz) (b) Frequency (Hz) FIGURE 4 Open loop dynamics for shake table system: (a) valve command to measured acceleration; (b) valve command to measured force. 12

13 H au = H fu = Table II. Analytical representations of open loop shake table dynamics s s s s s s s s s s s s s s s s s s s s s s s s s s s s

14 4.2. Design of the Feedback Controllers and Pre-Gains A controller design of the proposed acceleration feedback control strategy is developed employing analytical models of the open loop dynamics of the shake table system. In this study, the acceleration feedback loop including pre-gain and feedback controller are designed to provide acceleration tracking of the shake table system while the force feedback loop is designed to provide stability to prevent table drift. Because acceleration tracking is the main goal of this study, the acceleration feedback controller, C a, is considered first. A loop shaping design methodology is employed for the design of the acceleration feedback controller. Loop shaping is a frequency-domain approach where the product of the controller and the plant, referred to as the loop transfer function, is formed to have desirable frequency characteristics [Doyle and Stein, 1981]. In this study, the acceleration feedback controller is designed to compensate the dynamics of the valve to acceleration relationship in the entire frequency range of interest. Figure 5a shows the frequency domain characteristics of the designed acceleration feedback controller along with the valve to acceleration relationship. The mathematical representation of the designed acceleration feedback controller is expressed as: C a = 30s s s s s s s s s s s s s (3) As seen in the figure, the acceleration feedback controller is essentially the reciprocal of the valve to acceleration relationship. 14

15 The acceleration closed-loop transfer function from the choice of C a in Equation 3 is plotted in Figure 5b where the pre-gain P a is set to 1. While it contains certain desirable characteristics (i.e. flat plateau and decaying magnitude in higher frequencies), its magnitude is too low (approximately 0.36 between 1.5 and 10 Hz). To raise the magnitude to unity, the pre-gain is set to The acceleration control loop with the pre-gain of 2.75 is also plotted in Figure 5b. The figure shows that the selection of P a (=2.75) and C a provides desirable acceleration control performance as well as robustness. It should be noted that while the acceleration control loop in Figure 5b exhibits good performance, it does not ensure stability for table drift. To have stability against table drift, the force feedback controller is incorporated as discussed in section 2. However, the impact of the force feedback loop should be minimized to maintain the acceleration tracking performance. To meet the stability criterion, the force feedback controller is designed such that force closed-loop transfer function contains unit magnitude around 0Hz. To ensure small impact of the force feedback loop on the acceleration performance, the pre-gain for the force controller, P f, is set to 0, and the force feedback controller is designed to have sufficiently small magnitude at higher frequencies in the force closed-loop transfer function; the pre-gain of zero converts reference acceleration into a zero static reference force. To have a smoother transition from low to high frequencies in the force closed-loop transfer function, poles and zeros are placed in the force feedback controller to compensate the dynamics of the valve to force relationship. Figure 5c and 5d show the transfer function of the designed force feedback controller and corresponding force closed-loop transfer function, respectively. The designed force feedback controller is expressed as: 15

16 C f = s s s s s s s s s s s s s (4) As shown in Figure 5d, the force closed-loop transfer function has unit magnitude around 0Hz and smaller magnitude at higher frequencies while compensating the dynamics of the valve to force relationship. Finally, the overall performance and stability of the proposed acceleration control strategy using the choices of C a, P a, C f and P f are examined. Figure 6a shows the closed-loop acceleration to force relationship for stability assessment. The acceleration-to-force relationship has finite magnitude in the neighborhood of 0Hz, indicating the stability of the shake table system to prevent table drift. Then, the overall acceleration performance of the proposed control strategy with the incorporation of the force feedback loop is shown in Figure 6b. As shown in the figure, the designed force feedback loop hardly influences the overall acceleration transfer function. It can be seen from these figures that the designed acceleration control strategy provides acceleration control performance while gaining stability to prevent table drift. 16

17 Magnitude Magnitude 10 0 P a =1 P a =2.75 H au (a) C a Frequency (Hz) (b) Frequency (Hz) Magnitude 10 0 H fu C f Magnitude (c) Frequency (Hz) (d) Frequency (Hz) FIGURE 5 Controller design for the proposed acceleration control strategy: (a) acceleration feedback controller; (b) acceleration closed-loop transfer function; (c) force feedback controller; (d) force closedloop transfer function With Force Feedback Without Force Feedback Magnitude 10 2 Magnitude 10 1 (a) Frequency (Hz) (b) Frequency (Hz) FIGURE 6 Closed-loop transfer functions of shake table system using the proposed acceleration control strategy: (a) reference acceleration to measured force magnitude; (b) reference acceleration to measured acceleration magnitude. 17

18 4.3. Experimental Validation of the Proposed Acceleration Control Strategy The proposed acceleration control strategy with the designed feedback controllers and pre-gains is experimentally validated using a series of earthquake ground motions. Figure 7 (a and b) show the time histories of the reference acceleration and the measured table accelerations from the JMA record of the 1995 Kobe earthquake. For comparison, the measured table acceleration using a conventional displacement control strategy with PID controller is also plotted. Overall both acceleration and displacement control strategies show reasonable agreement with the reference acceleration (see Figure 7a). However, when inspected closely (see Figure 7b), the acceleration control strategy exhibited smaller amounts of discrepancy than the displacement control strategy. The two control strategies are also compared in the frequency domain as in Figure 7c. The figure reveals that while the performance of both acceleration and displacement control strategies are similar, the acceleration control strategy has less high-frequency errors. The measured force time history from the acceleration control strategy is shown in Figure 8. While the force varies during the intense part of the acceleration time history, it returns to zero afterwards. Because the force control loop regulates static forces, the table platform is stable and table drift is not observed. 18

19 Acceleration (g) Referecence Acceleration Control Displacement Control (a) Time (s) Fourier Amplitude 10 2 Acceleration (g) (b) Time (s) (c) Frequency (Hz) FIGURE 7 Results with the 1995 Kobe ground motion as the reference acceleration: (a) shake table acceleration tracking comparison; (b) close up view of table accelerations; (c) frequency domain comparison of table accelerations Force (N) Time (s) FIGURE 8 Measured table force from the acceleration control strategy using the Kobe reference acceleration. 19

20 Displacement (mm) Acceleration Control Displacement Control Time (s) FIGURE 9 Comparison of measured table displacements with the Kobe reference acceleration. In addition to the performance and stability assessment, the displacement time histories from both acceleration and displacement control strategies are compared in Figure 9. It is interesting to see from the figure that the table displacements are quite different even though acceleration time histories from both control strategies are similar (see Figure 7). This observation indicates that two unique displacement time histories can produce almost identical acceleration time histories. However, from the experimental results here, the actual displacement experienced during this earthquake cannot be fully identified. It may also be inferred that deducing the displacement from a given acceleration time history that is often done in conventional displacement control strategies may not result in the true ground displacements of the earthquake. Further performance evaluations are conducted using a series of earthquake ground motions. A summary of the performance evaluations is discussed using Root Mean Squared (RMS) percentage errors between reference and measured table accelerations. Table III presents RMS errors from both acceleration and displacement control strategies. All reference accelerations are 20

21 scaled to a peak ground acceleration of 0.5g. In all of the tests, acceleration control strategy produces less RMS errors than displacement control strategy. The average RMS error in acceleration control strategy is 9.12% whereas the average RMS error in displacement control strategy is 11.29%. Furthermore, the variation of the RMS errors in the acceleration control strategy is more consistent than the displacement control strategy where the RMS errors range from 9.63% to 13.62%. These test results prove that the acceleration control strategy developed in this study provides more accurate acceleration tracking than the conventional displacement control strategy, showing a 19.2% improvement. Table III. RMS errors between measured and reference shake table accelerations. Acceleration Control, % Displacement Control, % Chi Chi, Coalinga, Imperial Valley, Kobe, Landers, Loma Prieta, Morgan Hill, Northridge, Taiwan, Average

22 5. Impact of Input Acceleration Errors in Shake Table Tests on Structural Response Uncertainties are inherent in structural response under dynamic loading. Common sources of uncertainties are material properties, construction qualities, deterioration, excitation disturbances, environmental effects, etc. While some of the uncertainties are difficult to measure and often treated as stochastic processes, some can be quantitatively assessed. One of such uncertainties is the accuracy of the input ground motion in shake table testing. This section investigates the impact of acceleration errors in shake table control (difference between reference and measured accelerations of the shake table) and their affect on the measured structural response. The investigation is carried out using measured structural responses from the previous shake table tests. The response from an analytical model of the three-story structure is used as the reference for the experimental response. The top floor acceleration responses of the structure during the 1995 Kobe ground motion are shown in Figure 10. Figure 10a shows that while structural responses are mostly captured using inputs from both acceleration and displacement control strategies, differences can be seen in the magnitude of the responses; the structural response in the displacement-controlled test shows larger error in the magnitude. In addition to the difference in magnitude, different frequency contents can be observed in the close-up view (see Figure 10b); the structural response in displacement-controlled test exhibits more highly frequency contents that are not present in the reference. Above observations can be further verified through the frequency domain comparison in Figure 10c. Responses from both tests capture the first vibration mode well. However, higher frequency responses, in particular second and third vibration modes, are heavily amplified by the 22