  Dynamical systems describe the 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. But do they
describe the behaviour of the observed data? And how can we quantify the
models' parameters that cannot be measured directly? This paper addresses these
two questions by providing a methodology for estimating the solution; and the
parameters of linear dynamical systems from incomplete and noisy observations
of the processes.
  The proposed procedure builds 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 then estimated so
that this approximating solution adheres to the data. By taking advantage of
the linearity of the system, we have simplified the parameter cascading
estimation procedure, and by developing a new iterative scheme, we achieve fast
and stable computation.
  We illustrate our approach by obtaining a linear differential equation that
represents real data from biomechanics. Comparing our 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.
