State and Parameter Estimation of Partially Observed Linear Ordinary Differential Equations with Deterministic Optimal Control

Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give unsatisfactory results because the inverse problem can be ill-posed, even when the differential equation is linear. Following recent approaches that use approximate solutions of the ODE model, we propose a new method that converts parameter estimation into an optimal control problem: our objective is to determine a control and a parameter that are as close as possible to the data. We derive then a criterion that makes a balance between discrepancy with data and with the model, and we minimize it by using optimization in functions spaces: our approach is related to the so-called Deterministic Kalman Filtering, but different from the usual statistical Kalman filtering. e show the root-$n$ consistency and asymptotic normality of the estimators for the parameter and for the states. Experiments in a toy model and in a real case shows that our approach is generally more accurate and more reliable than Nonlinear Least Squares and Generalized Smoothing, even in misspecified cases.Comment: 45 pages, 1 figur

Retrospective Cost Methods for Combined State and Parameter Estimation

This dissertation is principally concerned with the combined state and parameter estimation problem, where the goal is to estimate the state of a discrete-time, linear time-invariant system with structured uncertainty in the system dynamics. First, we prove necessary and sufficient conditions for the identifiability of unknown parameters within a state-space realization. Next, we evaluate the performance of classical techniques for solving the combined state and parameter estimation problem. We then formulate and test the retrospective cost parameter estimation algorithm under the assumption that the initial states are known. Two variants of the retrospective cost parameter estimation and smoothing algorithm are formulated and tested in the case where the initial states are unknown. Finally, the retrospective cost Kalman filter algorithm is formulated and tested for state estimation despite uncertain noise covariances and potentially nonzero-mean sensor and process noise.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138515/1/mingray_1.pd

Difference Algebra and System Identification

The framework of differential algebra, especially Ritt’s algorithm, has turned out to be a useful tool when analyzing the identifiability of certain nonlinear continuous-time model structures. This framework provides conceptually interesting means to analyze complex nonlinear model structures via the much simpler linear regression models. One difficulty when working with continuous-time signals is dealing with white noise in nonlinear systems. In this paper, difference algebraic techniques, which mimic the differential algebraic techniques, are presented. Besides making it possible to analyze discrete-time model structures, this opens up the possibility of dealing with noise. Unfortunately, the corresponding discrete-time identifiability results are not as conclusive as in continuous time. In addition, an alternative elimination scheme to Ritt’s algorithm will be formalized and the resulting algorithm is analyzed when applied to a special form of the nfir model structure