Fourier Series-Based Approximation of Time-Varying Parameters in Ordinary Differential Equations

TL;DR

This work proposes a novel approximation method inspired by the Fourier series to estimate time-varying parameters (TVPs) in deterministic dynamical systems modeled with ordinary differential equations and demonstrates the capabilities of the proposed approach in estimating periodic parameters.

Abstract

Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic time-varying parameters of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.

Figures (12)

• Figure 1: Illustration of the EnKF two-step updating scheme. In the model prediction step, the ensemble members (represented as black circles, with sample variance in gray) at time $j$ are propagated forward (shown with black dashed arrows) to time $j+1$ by solving the model in \ref{['Eq:Aug_Dynamics']}. In the observation update, the ensemble predictions (green circles) are corrected (green arrows) using the observed system data at time $j+1$. The process continues using the corrected sample at time $j+1$ (black circles) until all available data are assimilated.
• Figure 2: Fourier series approximations of the sinusoidal function $h(t) = 2\sin(t) - 0.5\cos(2t/3)$ using different values of $Q$, which dictates the number of terms used in the approximation. In each plot, the Fourier series approximation is shown in dashed blue, while the true function is shown in solid black. The integrals in \ref{['Eq:Fourier_acoeff']} and \ref{['Eq:Fourier_bcoeff']} were computed numerically using adaptive quadrature via the integral function in MATLAB Shampine2008.
• Figure 3: Position and velocity data generated from the mass-spring system in \ref{['Eq:MS_system']} with sinusoidal forcing parameter $\theta(t) = 2\sin(t) - 0.5\cos(2t/3)$.
• Figure 4: EnKF time series estimates of position, velocity, and the seven unknown coefficients in the TVP approximation model \ref{['Eq:TV_form2']} of the sinusoidal forcing parameter $\theta(t) = 2\sin(t) - 0.5\cos(2t/3)$ in \ref{['Eq:MS_system']} when $M=3$, using the data in Figure \ref{['Fig:MS_Data']} and assuming a known period for $\theta(t)$. On each plot, the solid red line denotes the EnKF sample mean and the dashed red lines show $\pm2$ standard deviations around the mean. The plot for each coefficient, $c_0, \dots, c_6$, also shows the resulting histogram of the posterior sample at time $t=60$.
• Figure 5: Resulting Fourier series-based approximations of the sinusoidal forcing parameter $\theta(t) = 2\sin(t) - 0.5\cos(2t/3)$ in \ref{['Eq:MS_system']} for different values of $M$, using the data in Figure \ref{['Fig:MS_Data']} and assuming a known period for $\theta(t)$. On each plot, the solid red line denotes the TVP approximation $\bar{\theta}_{M}(t)$ computed using the posterior mean coefficient values in Table \ref{['Tab:KnownPeriod_Results']} and the dashed black line shows the true underlying $\theta(t)$.