Single pendulum on a cart - Identification results #4

Analytical model with all physical parameters = 1
image

State data X was generated by a numerical simulation of an analytically derived model. State derivatives dX (dot X) were computed from state data X using spectral differentiation. Other nonlinear functions were constructed from these signals.

The function library A (regression matrix) signals A_col (regressors) look as follows:
sim_data

Correlation matrix of the function library:
theta_corr

The goal of regression is to find sparse right-hand side solution (x) to a left-hand side guess term (b). There's only one regression hyperparameter - the threshold. For each LHS guess term, the regression is repeated many times for different threshold values. Nonzero, unique solutions are then saved for future comparison and modelling.

After trying all the possible LHS guess terms and solving the regression problems, we're left with a set of implicit equations that are candidates for equations describing the system dynamics. If two particular terms are in the real dynamics, each one of those should generate a solution if used as a LHS guess. We can use this fact to look at all the models and pick the models which contain the same active terms.

Each model equation is described by a vector of parameters and the corresponding term. The identified implicit equations are below, each row is a separate model. The labels on the left are in the form "{LHS term guess | model index}:
implicit_eqns_raw

For each model, I calculate an activation vector, that describes which parameters are non-zero, or which terms were identified as active in the model. Non-zero elements of the original solution vector become 1, others stay 0:
active_terms

We can then compare different models by calculating the L1 distance between their activation vectors. An activation distanceof 0 means that the models contain exactly the same terms, although their actual parameters will be almost always different, because the equations are implicit. An activation distance of 1 means the models differ by one term, and so on...

Matrix of activation distances between identified implicit equations:
activation_dist

The labels for each distance matrix element contain two parts, the left one is the LHS guess and the right one is the model index. Activation distances on the diagonal are all 0, because each model is identical to itself. The matrix is symmetrical, so we can focus only on the lower triangular part. We can for example see that model 8 (with a LHS guess of dX4) contains the same parameters as model 5. These models are therefore good candidates for equations describing the implicit dynamics.

Implicit models, normalized so that the smallest parameter is 1 or -1, so that the model parameters can be directly compared:
implicit_eqns

We can see that the implicit equations 5, 8 and 17 are nearly identical. These equations accurately describe the implicit form of the state equation for dX4. Note that it could also be used for calculating dX3, the simulation model was however defined so that dX3 = u, and the input signal u was removed from the function library, because it was perfectly correlated to dX3 and therefore would make the regression ill-conditioned.

Here are the results if input signal u is included in the function library:
Correlation matrix:
theta_corr_with_u
Activation distance matrix:
activation_dist_poorcond
Implicit solutions:
implicit_eqns_raw_poorcond
It seems that the correct implicit equation was also identified (models 20, 10, 7).
The positions of x-axis labels for larger trig functions might be confusing, I'll rotate them so they're vertical next time.