Least squares parameter estimation methods for material decomposition with energy discriminating detectors

Abstract In this paper, offline and online parameter estimation methods for hydraulic systems based on stochastic gradient descent are presented. In contrast to conventional approaches, the proposed methods can estimate any parameter in mathematical models based on multi-step prediction error. These advantages are achieved by calculating the gradient of the multi-step error against the estimated parameters using Lagrange multipliers and the calculus of variations, and by forming differentiable models of hydraulic systems. In experiments on a physical hydraulic system, the proposed methods with three different gradient decent methods (normal gradient descent, Nesterov’s Accelerated Gradient (NAG), and Adam) are compared with conventional least squares. In the offline experiment, the proposed method with NAG achieves estimation error about 95% lower than that of least squares. In online estimation, the proposed method with NAG produces predictive models with about 20% lower error than that of the offline method. These results suggest the proposed method is a practical alternative to more conventional parameter estimation methods.

Download Full-text

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

Complexity ◽

10.1155/2019/4861708 ◽

2019 ◽

Vol 2019 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Zhengqing Fu ◽

Lanlan Guo

Keyword(s):

Parameter Estimation ◽

Least Squares ◽

Nonlinear Least Squares ◽

Real Data ◽

Estimation Methods ◽

Linear Parameter ◽

Nonlinear Parameters ◽

Marquardt Algorithm ◽

Tikhonov Regularisation ◽

Parameter Estimation Methods

This paper considers the classical separable nonlinear least squares problem. Such problems can be expressed as a linear combination of nonlinear functions, and both linear and nonlinear parameters are to be estimated. Among the existing results, ill-conditioned problems are less often considered. Hence, this paper focuses on an algorithm for ill-conditioned problems. In the proposed linear parameter estimation process, the sensitivity of the model to disturbance is reduced using Tikhonov regularisation. The Levenberg–Marquardt algorithm is used to estimate the nonlinear parameters. The Jacobian matrix required by LM is calculated by the Golub and Pereyra, Kaufman, and Ruano methods. Combining the nonlinear and linear parameter estimation methods, three estimation models are obtained and the feasibility and stability of the model estimation are demonstrated. The model is validated by simulation data and real data. The experimental results also illustrate the feasibility and stability of the model.

Download Full-text

A Hierarchical Approach for Joint Parameter and State Estimation of a Bilinear System with Autoregressive Noise

Mathematics ◽

10.3390/math7040356 ◽

2019 ◽

Vol 7 (4) ◽

pp. 356 ◽

Cited By ~ 18

Author(s):

Xiao Zhang ◽

Feng Ding ◽

Ling Xu ◽

Ahmed Alsaedi ◽

Tasawar Hayat

Keyword(s):

Parameter Estimation ◽

Least Squares ◽

Bilinear System ◽

Generalized Least Squares ◽

Estimation Methods ◽

Estimation Accuracy ◽

Data Filtering ◽

State Estimator ◽

Product Term ◽

Parameter Estimation Methods

This paper is concerned with the joint state and parameter estimation methods for a bilinear system in the state space form, which is disturbed by additive noise. In order to overcome the difficulty that the model contains the product term of the system input and states, we make use of the hierarchical identification principle to present new methods for estimating the system parameters and states interactively. The unknown states are first estimated via a bilinear state estimator on the basis of the Kalman filtering algorithm. Then, a state estimator-based recursive generalized least squares (RGLS) algorithm is formulated according to the least squares principle. To improve the parameter estimation accuracy, we introduce the data filtering technique to derive a data filtering-based two-stage RGLS algorithm. The simulation example indicates the efficiency of the proposed algorithms.

Download Full-text