Efficient total least squares state and parameter estimation for differentially flat systems

This paper proposes an efficient framework for the total least squares (TLS) estimation of differentially flat system states and parameters. Classical ordinary least squares (OLS) estimation assumes: (i) that only the dependent (i.e., output) signals are noisy, and that (ii) the independent (i.e., input) variables are known. In contrast, TLS estimation assumes both the input and output signals to be noisy. Solving TLS problems can be computationally expensive, particularly for nonlinear problems. This challenge arises because the input trajectory must be estimated in a TLS problem, rather than treated as given. This paper addresses this challenge for differentially flat systems by utilizing a pseudospectral expansion to express the input, state, and output trajectories in terms of a flat output trajectory. This transforms the TLS problem into an unconstrained nonlinear programming (NLP) problem with a small number of optimization variables. We demonstrate this framework for an example involving estimating the states and parameters of a second-order nonlinear flat system. Our approach reduces the number of optimization variables from 1503 to 33, while achieving state and parameter estimation errors below 5% and 7%, respectively.