Fast stable parameter estimation for linear dynamical systems

Dynamical systems describe changes in processes that arise naturally from their underlying physical principles, such as the laws of motion or the conservation of mass, energy or momentum. These models facilitate a causal explanation for the drivers and impediments of the processes. Extracting these governing equations from data is a central challenge in many diverse areas of science and engineering. A methodology for estimating the solution; and the parameters of linear dynamical systems from incomplete and noisy observations of the processes is introduced. Building on the parameter cascading approach, where a linear combination of basis functions approximates the implicitly defined solution of the dynamical system. The systems’ parameters are estimated so that this approximating solution adheres to the data. By taking advantage of the linearity of the system, the parameter cascading estimation procedure is simplified, and by developing a new iterative scheme, fast and stable computation is achieved. An illustrative example obtains a linear differential equation that represents real data from biomechanics. A comparison of the proposed approach with popular methods for estimating the parameters of linear dynamical systems, namely, the non-linear least-squares approach, simulated annealing, parameter cascading and smooth functional tempering reveals a considerable reduction in computation and an improved bias and sampling variance.