Sliding-window Least-squares Research Articles

Vehicle platform attitude estimation method based on adaptive Kalman filter and sliding window least squares

Precision instrument measurement on an unstable platform is a difficult engineering problem, and the commonly used method is to compensate the instrument measurement results through platform attitude estimation. This paper derives a three-dimensional attitude estimation method based on inertial sensors including gyroscope and inclinometer. In order to deal with the inertial sensor noise, low-pass filter, Kalman filter, adaptive Kalman filter (AKF) and sliding window least squares (SWLS) are chosen to test the filtering performance from the frequency domain perspective. Practical filtering experiments indicate that AKF achieves the best filtering performance for gyroscope, while SWLS has the best filtering performance for inclinometer. Using AKF and SWLS to deal with inertial sensor outputs, the attitude estimation of the vehicle platform is realized. The proposed method is verified on vehicle-mounted electro-optical measurement equipment.

Fast algorithm for least‐squares based image prediction

A new computationally efficient algorithm for two-dimensional sliding-window least-squares prediction is presented in this study. The fast algorithm is based on a recursive update of the Cholesky decomposition. Compared with the state-of-the-art algorithm, the proposed algorithm reduces the computational complexity from O(D3) to O(D2h), where D is the predictor order and h is the height of the prediction patch. The computational improvement is made at the stage of solving the normal equations for which an update algorithm for the Cholesky decomposition of the covariance matrix is proposed. It is shown that a large part of the Cholesky decomposition at location n can be efficiently calculated by performing orthonormal updates on the Cholesky decomposition at n − 1. The computational improvement is made without requiring additional storage space. Extensive experiments using causal and non-causal predictors of varying shapes and sizes have confirmed that the proposed algorithm is consistently faster than the state-of-the-art algorithm and produces identical prediction images. The efficiency of the proposed algorithm is shown to be affected by the order in which pixels are sampled, thus an ordering procedure is proposed to minimise the number of numerical operations.

Moving-Horizon State Estimation for Nonlinear Systems Using Neural Networks

Moving-horizon (MH) state estimation is addressed for nonlinear discrete-time systems affected by bounded noises acting on system and measurement equations by minimizing a sliding-window least-squares cost function. Such a problem is solved by searching for suboptimal solutions for which a certain error is allowed in the minimization of the cost function. Nonlinear parameterized approximating functions such as feedforward neural networks are employed for the purpose of design. Thanks to the offline optimization of the parameters, the resulting MH estimation scheme requires a reduced online computational effort. Simulation results are presented to show the effectiveness of the proposed approach in comparison with other estimation techniques.

Dynamics- and Laser-Based Boundary Effect Evaluation Method for Damage Inspection of One- and Two-Dimensional Structures

This paper presents new damage indicators and current capabilities of a dynamics-based Boundary Effect Evaluation Method (BEEM) for damage inspection of large one-and two-dimensional structures. Damage introduces new boundaries to a structure, and influences of boundaries on steady-state high-frequency dynamic response are spatially localized effects. The BEEM is a signal processing method that takes advantage of these localized effects in performing area-by-area extraction of damage-induced boundary effects from steady-state Operational Deflection Shapes (ODSs) to reveal damage locations. Steady-state ODSs of a structure can be measured using a scanning laser vibrometer or any other full-field measurement tool, and the BEEM decomposes an ODS into central and boundary solutions by using a sliding-window least-squares data-fitting technique. Numerical and experimental results show that boundary solutions are excellent damage indicators because of Gibb’s phenomenon, and the central solutions can be used to clearly identify actual boundary conditions. Except for experimental ODSs of the damaged structure the method requires no model or historical data for comparison. Experimental results of many one-and two-dimensional structures validate the high sensitivity and accuracy of BEEM for detection and estimation of multiple small defects in structures.

Identification and applications of boundary effects in beams

Presented here is a signal decomposition method for extracting boundary effects from an operational deflection shape (ODS) of a structure under harmonic excitation. It decomposes an ODS into central and boundary solutions using a sliding-window least-squares curve-fitting technique, and the boundary solutions can be used to reveal damage locations, and the central solutions can be used to identify boundary conditions. Except an experimental ODS the method requires no model or historical data for comparison. Exact mode shapes and ODSs of beams with damage are obtained by spectral element analysis. Boundary and central solutions caused by different boundary conditions, different loading conditions, and different damage with or without noise are simulated and characterized. Numerical results show that Gibbs' phenomenon caused by the use of continuously differential functions to fit ODSs with discontinuous first-, second-, and/or third-order derivatives actually makes boundary solutions excellent damage indicators. Several experiments are performed using a scanning laser vibrometer for sensing and a lead zirconate titanate (PZT) patch for actuation. The experimental results confirm the feasibility and accuracy of this boundary effect detection method in detecting multiple defects and identifying boundary conditions of beams.

Damage detection of beams using operational deflection shapes

This work is an in-depth study of a boundary effect detection (BED) method for pinpointing locations of small damages in beams using operational deflection shapes (ODSs) measured by a scanning laser vibrometer. The BED method requires no model or historical data for locating structural damage. It works by decomposing a measured ODS into central and boundary-layer solutions using a sliding-window least-squares curve-fitting technique. For high-order ODSs of an intact beam, boundary-layer solutions are non-zero only at structural boundaries. For a damaged beam, its boundary-layer solutions are non-zero at the original boundaries and damage locations because damage introduces new boundaries. At a damage location, the boundary-layer solution of slope changes sign, and the boundary-layer solution of displacement peaks up or dimples down. The theoretical background is shown in detail. Noise and different types of damage are simulated to show how they affect damage locating curves. Experiments are performed on several different beams with different types of damage, including surface slots, edge slots, surface holes, internal holes, and fatigue cracks. Experimental results show that this damage detection method is sensitive and reliable for locating small damages in beams.

LOCATING STRUCTURAL DAMAGE BY DETECTING BOUNDARY EFFECTS

This paper presents a method of pinpointing structural damage locations using operational deflection shapes (ODSs) measured by a scanning laser vibrometer. The method assumes the form C1cosβx +C2sin βx+C3coshβx +C4sinh βx for an ODS to match with its experimental data using a sliding-window least-squares method to determine the four coefficients Ci. It is shown thatC1 represents the central solution of displacement, C3the boundary-layer solution of displacement caused by boundary constraints, C2β the central solution of slope, andC4β the boundary-layer solution of slope. Moreover,C1 *C3is proportional to the difference of kinetic and elastic energy densities,C3 −C1is proportional to the curvature, andC4 −C2is proportional to the spatial derivative of the curvature. Because cracks on a structure introduce new boundaries to the structure and these coefficients change significantly at boundaries, crack locations are clearly revealed by the peaks on the C1*C3curve, sign change of C4, peaks on the fitting error curve, and sudden changes of C3−C1andC4 −C2. Moreover, numerical and experimental results show that each of these coefficients requires a different length of data points for curve fitting in order to smooth the curve. Based on this different data length requirement and the peaks and sign change of these coefficients we derive a boundary effect detection (BED) method for finding damage locations. A non-linear sliding-window least-squares fitting technique is also derived for estimating the extent of structural damage. Numerical and experimental studies on beams with different damages have been performed to prove the accuracy and reliability of the BED method.