Equivalent Linear Model Research Articles

Observer design for a nonlinear diffusion system based on the Kirchhoff transformation

This paper deals with the state estimation for a nonlinear diffusion system. An observer that reconstructs the whole state, from the available measurements, is proposed based on an equivalent linear diffusion model obtained using the Kirchhoff tangent transformation. This bijective mapping allows to apply the available and powerful state estimation theory of linear distributed parameter systems and simplifies the observer design. Hence, an observer can be designed for the obtained equivalent linear diffusion system and by using the Kirchhoff transformation, the whole state of the original nonlinear diffusion system is recovered. The observability analysis of the nonlinear diffusion system and the convergence of the proposed observer are also investigated based on the equivalent linear diffusion system. The effectiveness of the proposed observer is shown, through numerical simulation runs, in the case of a heated steel rod by considering both an uniformly distributed and a punctual boundary sensing.

Floor response spectra for beyond design basis seismic demand

Vulnerability assessment of Structures Systems and Components (SSCs) of nuclear facilities for earthquake ground motion exceeding the design basis ground motion has become a key issue to ensure safety in case of highly improbable but possible extreme earthquake event beyond design basis. In order to assess seismic safety of SSCs located at different floors of the building, floor response spectra (FRS) for such beyond design basis seismic demand, taking into account possible nonlinear behavior of structure, is required. Currently used methods (e.g. IAEA Safety Series 28) do not capture softening of structure at increased demand. A performance based approach would be ideal under such situations. FRS for higher seismic demand may be developed through nonlinear time history analysis of the structure. However, this approach is complicated and cumbersome for structural configurations such as that of NPPs and requires deep insight of solution algorithm and large resources for numerical solution. In this work a simplified methodology for FRS generation for higher level of seismic demand using linear time history analysis of an equivalent linear structural model is proposed. Nonlinear behavior of structure and associated damping are estimated from Nonlinear Static Pushover Analysis. Through this approach realistic estimation of stiffness degradation of the structure and increase in damping at different levels of seismic demands can be made. Subsequently, linear time history analysis is performed on degraded structure, accounting for stiffness reduction and hysteretic damping to obtain the floor time histories which are then converted to FRS. Application of this method is shown through case studies.

Neural-Based Compensation of Nonlinearities in an Airplane Longitudinal Model with Dynamic-Inversion Control.

The inversion design approach is a very useful tool for the complex multiple-input-multiple-output nonlinear systems to implement the decoupling control goal, such as the airplane model and spacecraft model. In this work, the flight control law is proposed using the neural-based inversion design method associated with the nonlinear compensation for a general longitudinal model of the airplane. First, the nonlinear mathematic model is converted to the equivalent linear model based on the feedback linearization theory. Then, the flight control law integrated with this inversion model is developed to stabilize the nonlinear system and relieve the coupling effect. Afterwards, the inversion control combined with the neural network and nonlinear portion is presented to improve the transient performance and attenuate the uncertain effects on both external disturbances and model errors. Finally, the simulation results demonstrate the effectiveness of this controller.

An equivalent linear model for shear pin fractures and its experimental verification

For bridges in seismically active areas, lower horizontal bearing stiffnesses are usually designed to reduce structural seismic responses and protect the piers and foundations against earthquake damage. But lower horizontal bearing stiffnesses are not preferred under normal service conditions. In the seismic designs of Chinese bridges, shear pins are extensively integrated into bearings to resist horizontal force and restrict the relative longitudinal displacement between the superstructure and the substructure under normal service conditions. Shear pins are designed to be sheared off in a certain earthquake to activate the seismic isolation mechanism. For lack of a proper constitutive model for shear pin fractures, effects of shear pin fractures have not been considered in current bridge seismic isolation designs yet. In order to establish a constitutive model of seismic isolation bearings considering shear pin fracture effects, shear fracture tests of shear pins were conducted and an equivalent linear model was established. Ultimate deformation and equivalent stiffness of the equivalent linear model are approximately proportional to the diameter of the shear pin, and load capacity is approximately proportional to the square of the diameter of the shear pin. A series of shaking table tests were conducted, and then test results were compared with numerical results calculated with the equivalent linear model. It was found the equivalent linear model was accurate enough and suggested to simulate the effects of shear pin fractures in bridge seismic designs.

Optimizing on-time arrival probability and percentile travel time for elementary path finding in time-dependent transportation networks: Linear mixed integer programming reformulations

Aiming to provide a generic modeling framework for finding reliable paths in dynamic and stochastic transportation networks, this paper addresses a class of two-stage routing models through reformulation of two commonly used travel time reliability measures, namely on-time arrival probability and percentile travel time, which are much more complex to model in comparison to expected utility criteria. A sample-based representation is adopted to allow time-dependent link travel time data to be spatially and temporally correlated. A number of novel reformulation methods are introduced to establish equivalent linear integer programming models that can be easily solved. A Lagrangian decomposition approach is further developed to dualize the non-anticipatory coupling constraints across different samples and then decompose the relaxed model into a series of computationally efficient time-dependent least cost path sub-problems. Numerical experiments are implemented to demonstrate the solution quality and computational performance of the proposed approaches.

Site response analyses using downhole arrays at various seismic hazard levels of Singapore

Local site conditions can significantly influence the characteristics of seismic ground motions. In this study, site response analyses using one-dimensional linear elastic (LE), equivalent-linear (EQL) and nonlinear (NL) approaches are performed at different seismic hazard levels of Singapore. Two seismic stations, namely, the KAP and BES stations located at soft soil sites, are selected from the national network of Singapore. Firstly, site response estimates using the LE, EQL (SHAKE04) and NL (DEEPSOIL) approaches are compared with the borehole recordings. Results show favorable matches between the predictions and the observations at the KAP site, while under-predictions are observed for all the three site effect approaches at the BES site. Secondly, the applicability of the LE, EQL and NL models is examined at different hazard levels of Singapore. It is found that for the hazard level at a return period of 475 years, the computed maximum strain (γmax) is 0.06% and then the EQL model can provide accurate site response predictions. However, for the hazard level at a return period of 2475 years, the calculated γmax is larger than 2%, resulting in notable differences in the predictions of different site response models. This study highlights the importance of site effects in seismic hazard analysis of Singapore.

A Modeling Methodology for Robust Stability Analysis of Nonlinear Electrical Power Systems Under Parameter Uncertainties

This paper develops a modeling method for robust stability analysis of nonlinear electrical power systems over a range of operating points and under parameter uncertainties. Standard methods can guarantee stability under nominal conditions, but do not take into account any uncertainties of the model. In this study, stability is assessed by using structured singular value (SSV) analysis, also known as $\mu$ analysis. This method provides a measure of stability robustness of linear systems against all considered sources of structured uncertainties. The aim of this study is to apply the SSV method for robust small-signal analysis of nonlinear systems over a range of operating points and parameter variations. To that end, a modeling methodology is developed to represent any such system with an equivalent linear model that contains all system variability, in addition to being suitable for $\mu$ analysis. The method employs symbolic linearization around an arbitrary operating point. Furthermore, in order to reduce conservativeness in the stability assessment of the nonlinear system, the approach takes into account dependences of operating points on parameter variations. The methodology is verified through $\mu$ analysis of the equivalent linear model of a 4-kW permanent magnet machine drive, which successfully predicts the destabilizing torque over a range of different operating points and under parameter variations. Further, the predictions from $\mu$ analysis are validated against experimental results.

Piezoelectric energy harvesting computer controlled test bench

In this paper a new computer controlled (C.C.) laboratory test bench is presented. The patented test bench is made up of a C.C. road traffic simulator, C.C. electronic hardware involved in automating measurements, and test bench control software interface programmed in LabVIEW™. Our research is focused on characterizing electronic energy harvesting piezoelectric-based elements in road traffic environments to extract (or "harvest") maximum power. In mechanical to electrical energy conversion, mechanical impacts or vibrational behavior are commonly used, and several major problems need to be solved to perform optimal harvesting systems including, but no limited to, primary energy source modeling, energy conversion, and energy storage. It is described a novel C.C. test bench that obtains, in an accurate and automatized process, a generalized linear equivalent electrical model of piezoelectric elements and piezoelectric based energy store harvesting circuits in order to scale energy generation with multiple devices integrated in different topologies.

Linear equivalent models at the maximum power point based on variable weather parameters for photovoltaic cell

In order to sweep completely the obstacle to the whole linearization of photovoltaic (PV) system with non-linear PV cell, in this paper, the voltage-current characteristic of PV cell at the maximum power point (MPP) is linearized and two linear equivalent models including Thevenin equivalent model and Norton equivalent model are proposed. On the basis of this work, the whole linearization of PV system is workable and reasonable, and then the conventional linear theories or laws can be used to study PV system conveniently. Meanwhile, in this work, the direct relationships between three linear model parameters and variable weather parameters (VWP) were found, which ensures the strong adaptation of these proposed models to the varying weather conditions. Finally, some simulation experiments verify that these proposed models are feasible and available in practical application, illustrate that the characteristics of three linear model parameters are influenced by varying weather conditions and unaffected by varying load, and show that PV system using the proposed Thevenin equivalent model has the same maximum power point tacking (MPPT) stead-state performance and similar MPPT transient-state performance with the conventional four-parameter model under fast varying weather conditions.

Comparison of Equivalent Linear and Nonlinear Site Response Analysis Results and Model to Estimate Maximum Shear Strain

We compared the results of equivalent linear (ELA) and nonlinear site response analyses (NLA) and found that the differences between the values of the peak ground acceleration ( PGA), peak ground velocity ( PGV), Arias intensity ( I a), significant duration ( D5–75), and response spectrum for periods between 0.025 s and 2 s predicted by each method are non-negligible for maximum shear strain values predicted by ELA ( γ max, ELA) greater than 0.04% to 1.0%. As γ max, ELA increases, ELA in general predict smaller shear strain and D5–75 values, and larger PGA, PGV, I a, mean period, and response spectral values for periods less than 0.1 s and periods near the natural site period than NLA. To help researchers and practitioners decide when to use ELA and/or NLA, we developed a model to estimate γ max, ELA before conducting a site response analysis.

Seismic microzoning from synthetic ground motion earthquake scenarios parameters: The case study of the city of Catania (Italy)

The city of Catania (Italy) in the South-Eastern Sicily has been affected in past times by several destroying earthquakes with high values of estimated magnitude. The seismogenic area to the south of Volcano Etna, known as Iblean Area, is placed between the African and the Euro-Asiatic plates on the west of the Ibleo-Maltese escarpment, to the south of the Graben of the Sicilian channel and on the east of the overlapping front of Gela. Basing on the seismic history of Catania, the following earthquake scenarios have been considered: the “Val di Noto” earthquake of January 11, 1693 (with intensity X-XI on MCS scale, magnitude MW=7.41 and epicentral distance of about 13km); the “Etna” earthquake of February 20, 1818 (with intensity IX on MCS scale, magnitude MW=6.23 and epicentral distance of about 10km). The soil response analysis at the surface, in terms of time history and response spectra, has been obtained by 1-D equivalent linear models for about 1200 borings location available in the data-bank of the central area of Catania of about 50km2, using deterministic design scenario earthquakes as input at the conventional bedrock.Seismic microzoning maps of the city of Catania have been obtained in terms of different peak ground acceleration at the surface and in terms of amplification ratios for given values of frequency.

Comparing Populations in Data Involving Spatial Information

Observations corresponding to spatial units are commonly studied. If we want to see whether a continuous variable has the same distribution in a group of populations, different methods can be used according to the characteristics of the data. It could occur that observations in geographical data are related because they correspond to the same spatial unit, in which case we can use a repeated measures model. Whether or not repeated measures are involved, parametric and non-parametric methods are available. We analyze how repeated measures can be seen as a linear model and their relationship. We illustrate all these methods using data concerning economical activity in five sectors in specific regions in Mexico, where we want to see if all sectors are equally relevant. We also show through simulated data how by not selecting an adequate model we can obtain wrong inferences. In data involving spatial units, the independence assumption associated with a one-factor ANOVA could be violated when a variable changes spatially so that there are similar values between neighbors. Then, an equivalent linear model involving that spatial information could be used. We use a geographically weighted regression and illustrate the method through data concerning income in Mexico. We also show how the lack of independence is solved through the spatial model and perform a post hoc analysis

Comparing Populations in Data Involving Spatial Information

Observations corresponding to spatial units are commonly studied. If we want to see whether a continuous variable has the same distribution in a group of populations, different methods can be used according to the characteristics of the data. It could occur that observations in geographical data are related because they correspond to the same spatial unit, in which case we can use a repeated measures model. Whether or not repeated measures are involved, parametric and non-parametric methods are available. We analyze how repeated measures can be seen as a linear model and their relationship. We illustrate all these methods using data concerning economical activity in five sectors in specific regions in Mexico, where we want to see if all sectors are equally relevant. We also show through simulated data how by not selecting an adequate model we can obtain wrong inferences. In data involving spatial units, the independence assumption associated with a one-factor ANOVA could be violated when a variable changes spatially so that there are similar values between neighbors. Then, an equivalent linear model involving that spatial information could be used. We use a geographically weighted regression and illustrate the method through data concerning income in Mexico. We also show how the lack of independence is solved through the spatial model and perform a post hoc analysis

Erratum to: Obtaining the surface PGA from site response analyses based on globally recorded ground motions and matching with the codal values

Destructive moderate to major earthquakes across the globe have highlighted various possible forms and extent of damages. Subsoil properties play a vital role in controlling the actual damage scenario. Induced effects such as liquefaction and landslide are also the functions of surface ground shaking. In the present work, site response analyses based on equivalent linear model using SHAKE2000 are attempted to assess the surface scenario. In the absence of recorded data at the site under consideration, globally recorded ground motions from Pacific Earthquake Engineering Research database are considered. Site response analyses results show variation in amplification factor as high as 7.4. Post-filtering of the analyses results has been proposed in this work considering the site condition as well as the design requirements. Based on the filtering, the above value of amplification factor has been reduced to 2.5. This reduction will considerably affect the design values and subsequently the construction cost. In addition, the value of surface peak ground acceleration proposed in this work matches closely with the earlier published literature as well as the codal provisions.

Performance-Based Placement of Manufactured Viscoelastic Dampers for Design Response Spectrum

In this study, a viscoelastic damper (VED) is developed by using a VE material with low temperature dependence, and a performance-based placement-design procedure of VEDs is developed for finding the storywise distribution of VEDs in a building such that each peak interstory drift coincides with the prescribed value. The mechanical properties of the employed VED’s dependence on amplitude and frequency of the excitation as well as material temperature are taken into account and a mechanical nonlinear four-element model that comprises two dashpot elements and two spring elements is proposed for the VED. The developed performance-based design procedure utilizes equivalent linearization of the VED and the expanded complete quadratic combination (CQC) method, which involves modal analysis with complex eigenvalue analysis. An equivalent linear Voigt model of the VED is determined by the prescribed peak interstory drift and the fundamental natural period of the structure for which the VEDs are installed. Seismic response analyses are carried out for high-rise building models installed with the necessary number of wall-type VEDs, with the results demonstrating the effectiveness and validity of the proposed performance-based placement-design procedure.

Equivalent circuit model for the insulated core transformer

The design of the insulated core transformer (ICT) needs to consider the flux leakage effects. An equivalent linear circuit model is proposed based on the principle of duality. It is composed by two types of leakage inductances: conventional leakage between windings and special leakage introduced mainly by the insulation gaps. The values of leakage inductances depend on the dimensions of the core, gaps, or windings and the property of magnetic materials. The circuit allows for quantitatively evaluating influences of ICT internal parameters on its output properties. The winding self- and mutual inductance matrix is mathematically converted to derive the inductance formula. As an example, the leakage parameters of a six-stage two-dimensional (2D) ICT are calculated and analyzed.

The 2012 Emilia earthquake (Italy): Geotechnical characterization and ground response analyses of the paleo-Reno river levees

In 2012 Northern Italy was hit by a seismic sequence with earthquakes of moderate local magnitudes and shallow hypocentral depths. After the main shocks of May 20, collapse of buildings and lifeline ruptures were widely observed in the epicentral area within a distance of about 15km, where large acceleration values with prevailing vertical component were recorded. Locally, at a greater distance, intense and spectacular liquefaction effects were observed. They mainly affected the earthen embankments of the old Reno river channel (paleochannel), and, to a lesser extent, their immediate surroundings. In these far-field areas, ground surface accelerations were significantly low and not consistent with the spreading and extent of the observed liquefaction effects. A detailed geotechnical survey was performed to identify depth, location and behaviour of liquefied soil layers and to analyse their influence on ground response and surface liquefaction effects. Results from in situ and laboratory tests, performed under static and dynamic loading conditions, are presented herein and a complete geotechnical model of the ancient levees and the underlying soil deposit is proposed for numerical site-specific seismic analyses. More attention was paid to analysing the influence of non-linear behaviour of interbedded liquefied sands on the expected surface ground motion. Implications and limits of the results obtained are discussed with, finally, proposals for adjustments to 1-D linear equivalent models operating in the frequency domain, for taking into account stiffness degradation of liquefiable layers at high strain levels.

Nonlinear 3D finite element analysis of soil–pile–structure interaction system subjected to horizontal earthquake excitation

The dynamic response of a seismic soil–pile–structure interaction (SSPSI) system is investigated in this paper by conducting nonlinear 3D finite element numerical simulations. Nonlinear behaviors such as non-reflecting boundary condition and soil–pile–structure interaction modeled by the penalty method have been taken into account. An equivalent linear model developed from the ground response analysis and the modified Drucker–Prager model are separately used for soil ground. A comparison of the two models shows that the equivalent linear soil model results in an underestimated acceleration response of the structure under this ground shaking and the soil behavior should be considered as a fully-nonlinear constitutive model in the design process of the SSPSI system. It was also observed that the dynamic response of the system is greatly affected by the nonlinearity of soil–pile interface and is not sensitive to the dilation angle of the soil. Furthermore, the effect of the presence of pile foundations on SSPSI response is also analyzed and discussed.

Equivalent linear elastic-viscous model of shape memory alloy for isolated structures

In this investigation, the equivalent linear elastic-viscous model of shape memory alloy (SMA) is established for seismic analysis of base-isolated structures using system identification method. The necessary key parameters to express the hysteresis loop of SMA are austenite stiffness, transformation strength, ductility ratio, and stiffness ratio. These parameters are considered in the modeling. This model is developed based on the American Association of State Highway and Transportation Officials (AASHTO) isolation guidelines. In order to validate the proposed model, the base-isolated benchmark building is analyzed by using proposed equivalent linear SMA model as well as the different non-linear SMA models. The evaluation criteria given in the benchmark problem and time variation of top-floor absolute accelerations and base-displacements are considered for comparing the linear and nonlinear models of SMA. An excellent agreement is achieved between proposed equivalent linear SMA model and its nonlinear models. The seismic code recommends that the equivalent linear model of the nonlinear system can be used for carry out the response spectrum analysis of base-isolated structures. Furthermore, the non-linear model requires computationally more time and effort, especially for larger degrees of freedom system. The proposed model may be useful to design engineers in order to over come the disadvantage of non-linear models.

Hybrid Robust Control Law with Disturbance Observer for High-Frequency Response Electro-Hydraulic Servo Loading System

Addressing the simulating issue of the helicopter-manipulating booster aerodynamic load with high-frequency dynamic load superimposed on a large static load, this paper studies the design of the robust controller for the electro-hydraulic loading system to realize the simulation of this kind of load. Firstly, the equivalent linear model of the electro-hydraulic loading system under assumed parameter uncertainty is established. Then, a hybrid control scheme is proposed for the loading system. This control scheme consists of a constant velocity feed-forward compensator, a robust inner loop compensator based on disturbance observer and a robust outer loop feedback controller. The constant velocity compensator eliminates most of the extraneous force at first, and then the double-loop cascade composition control strategy is employed to design the compensated system. The disturbance observer–based inner loop compensator further restrains the disturbances including the remaining extraneous force, and makes the actual plant tracking a nominal model approximately in a certain frequency range. The robust outer loop controller achieves the desired force-tracking performance, and guarantees system robustness in the high frequency region. The optimized low-pass filter Q(s) is designed by using the H∞ mixed sensitivity optimization method. The simulation results show that the proposed hybrid control scheme and controller can effectively suppress the extraneous force and improve the robustness of the electro-hydraulic loading system.