# pendulum¶

probnum.problems.zoo.filtsmooth.pendulum(measurement_variance=0.1024, timespan=(0.0, 4.0), step=0.0075, initrv=None)[source]

Filtering/smoothing setup for a (noisy) pendulum.

A non-linear, discretized state space model for a pendulum with unknown forces acting on the dynamics, modeled as Gaussian noise. See e.g. Särkkä, 2013 1 for more details.

$\begin{split}\begin{pmatrix} x_1(t_n) \\ x_2(t_n) \end{pmatrix} &= \begin{pmatrix} x_1(t_{n-1}) + x_2(t_{n-1}) \cdot h \\ x_2(t_{n-1}) - g \sin(x_1(t_{n-1})) \cdot h \end{pmatrix} + q_n \\ y_n &\sim \sin(x_1(t_n)) + r_n\end{split}$

for some step size $$h$$ and Gaussian process noise $$q_n \sim \mathcal{N}(0, Q)$$ with

$\begin{split}Q = \begin{pmatrix} \frac{h^3}{3} & \frac{h^2}{2} \\ \frac{h^2}{2} & h \end{pmatrix}\end{split}$

$$g$$ denotes the gravitational constant and $$r_n \sim \mathcal{N}(0, R)$$ is Gaussian mesurement noise with some covariance $$R$$.

Parameters
Returns

• regression_problemRegressionProblem object with time points and noisy observations.

• statespace_components – Dictionary containing

• dynamics model

• measurement model

• initial random variable

References

1

Särkkä, Simo. Bayesian Filtering and Smoothing. Cambridge University Press, 2013.