Optimal model-based state estimation in mechanical and structural systems

Abstract

In many applications, such as control, monitoring, assessment, and failure detection in mechanical and structural systems, the state of the system is sought. However, in almost all cases, the full state vector is not directly measured, and it must be reconstructed from sparse measurements contaminated by noise. If in addition, the excitations are not (or cannot be) measured, such as the case with wind loads, wave loading, and others, the problem becomes more challenging and standard deterministic methods are not applicable. This paper presents a model-based state estimation algorithm for online reconstruction of the complete dynamic response of a partially instrumented structure subject to realizations of random loads. The proposed algorithm operates on noise-contaminated measurements of dynamic response, a finite element model of the system, and a spectral density description of the random excitation and noise. The main contribution of the paper is the development of a second-order estimator that can be directly implemented as a modified version of the finite element model of the system while minimizing the state error estimate covariance. The proposed observer results in a suboptimal Kalman filter that can be implemented as a modified version of the original finite element model of the system with added dampers and collocated forces that are linear combinations of the measurements. The proposed method is successfully illustrated and compared with the standard Kalman filter in a mass-spring-damper system under various ideal and nonideal conditions. trajectory of a system subject to unmeasured disturbances based on a state-space model of the system and noise-contaminated measurements. The algorithm was initially proposed in its present form by R. E. Kalman in the 1960s, and it has since found widespread applicability in many engineering fields [10][11][12][13][14].The KF operates by using a feedback gain matrix that is optimal in the sense of balancing the effect of noise in the measurements used as feedback and the effect of unmeasured disturbances to minimize any linear function of the trace of the state error covariance matrix. In the case of structural dynamics and vibration monitoring, the disturbances correspond to unmeasured excitations, and the measurements typically consist of accelerations and/or strain time histories at discrete locations within the structure.The original derivation is given for linear systems in which measurement noise and disturbances are independent realizations Gaussian random processes and for which the model is known with accuracy. In real world application, these assumptions are seldom fully satisfied; however, experience in many applications has shown that the KF can also operate robustly in nonideal conditions. This will be illustrated by means of simulations in the example at the end of the paper. For the implementation of the KF in nonlinear applications, various modifications of the original algorithm known as extended KFs (EKFs) have been proposed [15,16]. Other methods fo...

Estimation of element‐by‐element demand‐to‐capacity ratios in instrumented SMRF buildings using measured seismic response

“…The matrices Φ w w ( ω ) and Φ v v ( ω ) are, respectively, the power spectral density of the process noise and measurement noise. A detailed derivation of the MBO along with some implementation examples can be found in Hernandez . In the derivation of the MBO, it is assumed that the system is linear; however, for the application of interest, it is possible that during the strong motion, some localized parts of the system behave in a nonlinear fashion.…”

“…A detailed derivation of the MBO along with some implementation examples can be found in Hernandez. 15,16 In the derivation of the MBO, it is assumed that the system is linear; however, for the application of interest, it is possible that during the strong motion, some localized parts of the system behave in a nonlinear fashion. One of the aspects to be investigated in this paper is the suitability of the MBO to deal with mild nonlinearity present in steel structures during potentially damaging earthquakes.…”

“…In the context of this application, partially instrumented means that not all stories are instrumented. The estimation is carried out using a model‐based observer (MBO) derived in Hernandez . The MBO combines a finite element model of the structure and response measurements to estimate the complete dynamic response of the structure at all degrees of freedom (DoF) of the model.…”

“…The estimation is carried out using a model-based observer (MBO) derived in Hernandez. 15,16 The MBO combines a finite element model of the structure and response measurements to estimate the complete dynamic response of the structure at all degrees of freedom (DoF) of the model. The MBO assumes that the unmeasured excitation and measurement noise are realizations of random processes with known power spectral densities.…”

Bayesian framework for simultaneous input/state estimation in structural and mechanical systems

“…Moreover, in the railway transportation, the estimation of the wheel–rail contact forces is crucial to assess rolling contact fatigue and damage detection on the track as well as to certify the vehicle design safety . In addition to the input identification, the state estimation has also received very much attention in the structural dynamics to predict stress and fatigue life, to identify damage, and to determine the response in inaccessible points …”