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Research Papers

Adjoint- and Hybrid-Based Hessians for Optimization Problems in System Identification

[+] Author and Article Information
Souransu Nandi

Department of Mechanical and
Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: souransu@buffalo.edu

Tarunraj Singh

Professor
Department of Mechanical and
Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: tsingh@buffalo.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received June 30, 2017; final manuscript received April 19, 2018; published online May 22, 2018. Assoc. Editor: Jongeun Choi.

J. Dyn. Sys., Meas., Control 140(10), 101011 (May 22, 2018) (14 pages) Paper No: DS-17-1332; doi: 10.1115/1.4040072 History: Received June 30, 2017; Revised April 19, 2018

An adjoint sensitivity-based approach to determine the gradient and Hessian of cost functions for system identification of dynamical systems is presented. The motivation is the development of a computationally efficient approach relative to the direct differentiation (DD) technique and which overcomes the challenges of the step-size selection in finite difference (FD) approaches. An optimization framework is used to determine the parameters of a dynamical system which minimizes a summation of a scalar cost function evaluated at the discrete measurement instants. The discrete time measurements result in discontinuities in the Lagrange multipliers. Two approaches labeled as the Adjoint and the Hybrid are developed for the calculation of the gradient and Hessian for gradient-based optimization algorithms. The proposed approach is illustrated on the Lorenz 63 model where part of the initial conditions and model parameters are estimated using synthetic data. Examples of identifying model parameters of light curves of type 1a supernovae and a two-tank dynamic model using publicly available data are also included.

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Figures

Grahic Jump Location
Fig. 1

Forward difference error analysis for the gradient of J

Grahic Jump Location
Fig. 2

Visualization of the second derivative tensors where a∈ℝn, b∈ℝp, and c∈ℝq: (a)(d2a/dbdc) and (b) (∂2a/∂b∂c)

Grahic Jump Location
Fig. 3

Convergence of different algorithms

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Fig. 4

Illustration of system dynamics for parameters estimated using the hybrid method for the example problems: (a) Lorenz-63 model, (b) Supernova 1999dq luminosity, and (c) two-tank model

Grahic Jump Location
Fig. 5

Convergence for different algorithms in supernova system identification: (a) Supernova 1999dq, (b) supernova 1998aq, and (c) supernova 1990n

Grahic Jump Location
Fig. 6

Convergence of different algorithms

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