Improving hybrid simulation through Kalman filter

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Chapter 3 Delay Compensation for Fast hybrid simulation

Introduction

The conventional hybrid simulation method is unable to simulate the response of structure or structural component with rate-dependant behaviour, such as structures equipped with viscoelastic dampers or base isolation. Energy dissipation characteristics of these devices are highly sensitive to the rate of loading and evaluating these devices using the conventional hybrid simulation method will lead to inaccurate result.
The development of the fast (and real-time) hybrid simulation method [28-30] addresses the shortcoming by replicating the rate of loading the structure will experience in real earthquake. Dynamic interactions between the servo-valve, the actuator piston, and the specimen in fast hybrid simulation result in actuator delay. Actuator delay is the difference in time between the sampling rate of the actuator servo-controller and the instant the actuator physically realises the desired displacement. The time needed for computation and data communication add further delay into the system. Thus assuming all instruments have been properly calibrated, it is almost impossible to achieve the desired displacement within a restricted time limit, such as the sampling rate of the servo-controller, in a fast or real-time hybrid simulation since actuator delay cannot be completely eliminated [31].
Actuator delay in a hybrid simulation system introduces additional energy into the system, analogous to the effect of negative damping [32]. If the rate of energy dissipation, from viscous damping and hysteretic action, is less than the rate of energy addition resulting from response delay, the response of the specimen will grow without bound and the experiment result will be meaningless. Therefore, it is imperative to properly compensate the effect of actuator delay.

Prior research

Existing delay compensation methods were largely based on predicting the response of the specimen using known information of the specimen in previous time steps. Horiuchi et al. [32] proposed utilising simple polynomial prediction using Lagrangean extrapolation. Using computed displacements at current and several integration time steps before, the method predicts required actuator displacement several integration time steps ahead to account for the delay.
An improvement of the displacement extrapolation method was proposed using linear extrapolation of acceleration over the expected time delay, which gave a third-order displacement prediction [33]. The improved method increased the stability limit both with respect to stiffness and mass of the specimen.
Darby et al. [34] showed that constant time delay used in the polynomial extrapolation method may not give satisfactory result if there was a large variation in the magnitudes of delay. Actuator delay is a function of the overall system stiffness thus its magnitude can vary significantly when the stiffness changes, for example due to specimen’s yielding. They proposed a method to estimate the delay on-the-fly during testing. The estimated delay was assumed proportional to the difference between the actual position of the specimen and the target displacement value specified in the test coordinator at every time step. Parameters of the delay estimation equation must be tuned for rapid convergence with relatively low oscillations.
Carrion and Spencer [35] proposed a model based prediction that considered physical characteristics of the system by incorporating known information about the specimen and the loading regime. These included the mass, damping, and external excitations to the specimen, as well as an estimate of the restoring force. Compared to the polynomial extrapolation method, the model based prediction method accommodated longer prediction time frame either due to larger time delay or higher natural frequencies.
Ahmadizadeh et al. [36] presented a different technique to estimate and compensate actuator delay. In their method time delay was estimated using polynomial fit to a few desired and measured displacement data points. Displacement command was modified to account for actuator delay using the same kinematic equations formulated in the Newmark direct integration method assuming constant acceleration variation over the expected time delay. Meanwhile, the measured restoring force was also compensated using polynomial fitting technique by seeking the time at which the desired displacement was achieved.
More complex methods regarding actuator delay compensation have also been developed using control and system identification theories. Conceptually, the compensation methods in this category utilised additional outer loop controller between the test coordinator and the closed-loop control system in the dual-loop implementation of the hybrid simulation method illustrated in Figure 2-9. Before being sent to the actuator servo-controller, the additional outer loop modified the target displacement computed by test coordinator to compensate for the anticipated delay. Zhao et al. [23] employed a first-order phase lead compensator to adjust the target displacement and the generated restoring force to account for a total time delay as large as 81 milliseconds (ms). Carrion et al. [37] adopted a feedforward-feedback compensation procedure as the additional outer-loop controller. Chen [24] used the inverse of a simplified model of servo-hydraulic system using first order discrete transfer function to compensate for actuator delay, known as the Inverse Compensation method.
Due to inherent nonlinearity in the servo-hydraulic system as well as the specimen, the performances of the delay compensation methods may become less effective when delay magnitude varies due to the nonlinearities. In this case it is preferable to use compensators with adaptive gains that seamlessly adjust their parameters in response to changes in the system dynamics. Wallace et al. [38] utilised adaptive parameters to minimise the error in estimating the magnitude of time delay, analogous to the method proposed by Darby et al. [34], and to control the magnitude of forward prediction. These adaptive parameters accommodate possible large variations in dynamic responses of the test specimen.
Chen and Ricles [39] incorporated adaptive parameters based on proportional and integral gains applied to the Tracking Indicator (TI) procedure [40] to improve the Inverse Compensation method [24] by minimising the effect of delay estimation error.
Bonnet et al. [41] adopted a versatile outer loop controller adaptable to changes in the system dynamics called Minimal Control Synthesis with modified demand (MCSmd). In the MCSmd approach, the input to the inner loop PID servo-controller is the sum of the compensated desired displacement from numerical integration and the actuator displacement, multiplied by adaptive gains. The adaptive gains aim to minimise the difference between the desired displacement from numerical integration and the actuator displacement.
Chae et al. [42] proposed an adaptive outer loop controller in time domain. The method was identified as the adaptive time series compensator (ATS), which minimise the sum of square difference between the compensated target displacement and the estimate of compensated target displacement over a certain period. The ATS method utilised adaptive coefficients from the Taylor series expansion determined using standard least square regression method.
Other delay compensation methods that have been identified included the virtual coupling method by Christenson et al. [43] and the ‘hockey-stick’ method by Elkhoraibi and Mosalam .The virtual coupling method provided an attractive feature from an inherent trade-off between performance and stability depending on the restoring force of the specimen, and was suitable for testing specimens with large dynamic range and highly nonlinear behaviour. The ‘hockey-stick’ model was particularly of interest because the delay compensation procedure was based on observed relationship between displacement actuation errors and actuator velocity demands, without the need of displacement prediction or system identification.
Additionally, Liu et al. [45] presented an effort to integrate the methods by Darby et al. [34], Horiuchi et al. [33], and Carrion et al. [37]. The integrated method achieved better displacement tracking compared to the performance of each constituting method considered separately.
The chapter presents the development of an error compensation technique that does not require actuator displacement prediction or system identification. Instead of improving actuator tracking by predicting the actuator displacement to minimise the effect of delay, the proposed technique simply compensates the actuator displacement errors after they occurred. The compensation is performed by introducing additional damping in the numerical integration and exploiting the properties of the errors, using measurements that are readily present or collected during a typical hybrid simulation.

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Systematic displacement control errors and their effect on energy content

Actuator delay is an example of systematic displacement actuation error due to its reproducible recurrence pattern [1]. Beside actuator delay, systematic displacement actuation error happens if the actuator exhibits constant undershoot or overshoot errors, in which the actuator moves in phase with the command displacement signal but exceeds or falls behind at peak values, mainly due to the dynamics between the actuator and the servo-controller [34]. In a fast to real time hybrid simulation, the two mechanisms of displacement actuation errors occur simultaneously and are difficult to distinguish between one another. However, both mechanisms affect the energy balance in the system and can thus be compensated using a single algorithm [1, 46].
The change in mechanical energy content in a structural system is the sum of the work done by the inertial, dissipative and elastic restoring components. This balances with the work done by the external forces in lieu of any errors. This can be expressed mathematically in Equation 3-1, which is simply the governing equation of motion integrated with respect to displacements. The following relationship assumes the structural system is initially at rest and the system is elastic.
The third term in Equation 3-1 represents the strain energy content. In a hybrid simulation, this quantity depends on the measured restoring force from the test specimen. Any displacement actuation errors will result in erroneous displacements and hence erroneous measured restoring forces. This in turn distorts the energy balance in the system described by Equation 3-1. The perceived energy content is expressed mathematically as.
In Equation 3-2 and 3-3, Rm is the restoring forces vector measured from the specimen, Rd is the restoring forces vector at the same time step had the desired displacements been correctly applied, and Rer is the vector of restoring force errors due to the displacement actuation errors. Thus, the energy error across a time-step, Eer, is equal to Equation 3-4 is a general form of the formula proposed by other researchers [47-49] which does not treat numerical and experimental substructures in a hybrid simulation separately.
In a hybrid simulation, it is common for the numerical integration algorithm to calculate the time history response using the measured restoring forces and the computed (target) displacements. This leads to an interesting conundrum as the command and measured displacements are not necessarily the same due to the presence of experimental errors. Consequently, the measured restoring force is not a corresponding quantity to the commanded displacement. Consider a linear-elastic structure evaluated using hybrid simulation, Figure 3-1 shows a perceived force-deformation relationship of a linear-elastic structure where consistent displacement actuation errors are present. The behaviour of the test structure as observed by the integration algorithm can be thought as a vertical departure of the restoring forces from the true linear elastic response. The departure occurs as the result of time difference between when the displacement is commanded and when the structure reaches the target displacement. In the case of consistent time delay depicted in Figure 3-1a, the measured forces lag the command displacement, which leads to a perceived counter-clockwise hysteretic response. This behaviour adds energy into the test specimen, represented by the shaded area inside the plot. Conversely, Figure 3-1b depicts the perceived response under consistent time lead. In this case, the perceived clockwise hysteretic response represents additional energy dissipation from the system [47]. Time delay errors are more common in fast hybrid simulations due to the dynamic interactions between the specimen and the test apparatus (servo-controller, servo-valve and the hydraulic actuator).
The energy addition behaviour of time delay error as illustrated in Figure 3-1 can also be illustrated in Figure 3-2, which is a graphical representation of Equation 3-4 for the case of time delay error. Figure 3-2 shows a time histories of computed displacement, restoring force error due to time delay error, as well as the increments of the computed displacement in a normalised scale. The restoring force error is in phase with the incremental displacement, which is always 90 out-of-phase with the computed displacement, but with opposite sign. The cumulative energy error, defined as the integration of restoring force error with respect to the incremental displacement (Equation 3-4) therefore will always be negative. Moving Eer to the right-hand side in Equation 3-2 is equivalent to introducing additional input energy into the structure.

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Computation of energy error

The amount of energy error across a time step during a hybrid simulation is approximately equal to the quadrilateral area between the true and perceived force-displacement response over the displacement increment magnitude [46]. This energy error can be estimated using the trapezoidal rule, at time step.

Modification in energy computation

Due to the difference between the commanded incremental displacements (∆uc) and the measured incremental displacements (∆um), the computations of energy error described in Equation 3-6 to 3-8 are modified. Consider a linear elastic structure with a force-deformation relationship as shown in Figure 3-3.
The energy error as computed by Equation 3-5 is highlighted by the bold area in Figure 3-3a. While the energy error as computed by Equation 3-6 is the difference in areas between the bold and dotted trapezoid.
The energy error calculated using Equation 3-6 is an approximate of the true energy error represented in Figure 3-3a. Despite the inexact nature, it is a more practical measure as it does not require knowledge of the instantaneous stiffness of the system.
It is further proposed that displacements are integrated with respect to restoring forces to compute energy error instead of restoring forces with respect to displacements. Adopting this modification, Equation 3-6 to 3-8 are now expressed.
This modification in calculating energy error can be perceived as using complementary quadrilateral area and is analogous with the actuator tracking indicator method [40]. A graphical explanation of Equation 3-9 to 3-11 is depicted in Figure 3-4.

Proposed error compensation algorithm utilising artificial viscous damping

The HSEM method alone does not provide a means of correcting the error during simulation. The proposed scheme complements the HSEM method to intuitively compensate for errors interactively during a simulation, at or near real-time. This is achieved by introducing a variable amount of artificial viscous damping proportional to the energy error at each time step.
The amount of energy dissipated by viscous damping mechanism is given by the second term on the left-hand side of Equation 3-1. Using the trapezoid rule, this integration can be approximated.
where EDi is the energy dissipated through viscous damping across time step . It is assumed that the viscous damping matrix C is constant throughout the simulation. For clarity, this constant viscous damping is called the initial viscous damping matrix. The variable viscous damping matrix, represented by a new variable Ccori, compensates for energy error Eeri across the same step and can thus be derived by substituting EDi with Eeri in Equation 3-12, which yields
The term ∫ (Ccoriu̇)Tdu in Equation 3-14 compensates for the energy error defined by Equation 3-4, which is inherent in the vector of restoring forces Rm. Two things to note:
The negative sign preceding Eeri in Equation 3-13 indicates that the amount of additional viscous damping to compensate the energy error across a time step is the opposite amount introduced into the system due to the energy error. For instance, if the energy error across a time step introduces negative damping into the structure, then the additional viscous damping Ccori is a positive definite matrix.
When an explicit numerical integration method such as the NEM method is used to solve the equation of motion, the velocity at the current time step ̇ can not be determined before solving the equation of motion. To avoid solving Equation 3-13
with iteration, the current velocity     ̇ in Equation 3-13 is replaced by the predictor velocity u�̇i which is expressed .
The proposed compensation method is readily extensible for MDOF system as will be shown in Chapter 4. Equation 3-13 can be computed for each experimental as well as interface DOF of in hybrid simulation with substructuring. The following points are important for multiple experimental substructures implementation:
Coordinate transformation, if the global DOF does not coincide with the actuator DOF. In this case, the transformation [T] between the global and actuator DOF is defined .
In real time hybrid simulation (RTHS), the highest natural frequency of the system can be quite high due to multiple DOFs involved, such that delay magnitude becomes critical since the proposed method has limited stability range. In Chapter 4, it will be shown that such limitation can be overcome by the addition of Kalman filter.

Table of Contents
Abstract 
Acknowledgement 
Table of Contents 
List of Figures 
List of Tables
1. Introduction 
1.1. General
1.2. Research needs
1.3. Research objectives
1.4. Organisation of thesis
2. Literature Review 
2.1. Introduction
2.2. The hybrid simulation method
2.3. Numerical errors
2.4. Systematic experimental errors
2.5. Random experimental errors
2.6. Implementation of the hybrid simulation method
2.7. Numerical integration method
2.8. Hybrid simulation with substructuring
2.9. Notable development in the hybrid simulation method
3. Delay Compensation for Fast hybrid simulation
3.1. Introduction
3.2. Prior research
3.3. Systematic displacement control errors and their effect on energy content
3.4. Computation of energy error
3.5. Numerical verification
3.6. Experimental validation
3.7. Stability
3.8. Summary
4. Improving hybrid simulation through Kalman filter 
4.1. Introduction
4.2. Kalman filter
4.3. Influence of DOF coupling
4.4. Increasing simulation stability through Kalman filter
4.5. Effectiveness of Kalman filter algorithm for inelastic systems
4.6. Summary
5. Multi-axial Actuator Control 
5.1. Introduction
5.2. Prior research
5.3. Quasi-static tests on RC walls
5.4. Hybrid simulations on RC column
5.5. Study on error accumulation
5.6. Summary
6. Displacement paths effect in hybrid simulations considering nonlinear responses 
6.1. Introduction
6.2. Test setup and specimen
6.3. Displacement tracking strategies
6.4. Loading regime
6.5. Experiment results and discussions
6.6. Summary
7. Hybrid testing on concrete wall 
7.1. Introduction
7.2. Methodology
7.3. Test setup
7.4. Instrumentations
7.5. Implementation problems during tests
7.6. Results and discussions on global responses
7.7. Results and discussions on local responses
7.8. Numerical validations
7.9. Summary
8. Conclusions 
8.1. Future work
References
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