Adding timeshifted input trajectories to opty.

[chris-konrad]

Motivation

For a dynamic system $\dot{x}(t) = f(x(t)) + u(t-\tau)$, one may want to estimate the unknown input delay $\tau$. Currently, opty does not support timeshift symbols and hence can't explicitly estimate this delay (See csu-hmc#224).

Method

A potential way to enable opty to find the desired $\tau$ directly is outlined as follows (credits to @moorepants)

• Substitute $u(t-\tau) = u'(t)$ and consider $u'(t)$ an unknown trajectory and $\tau$ an unknown free parameter.
• Consider $u(t)$ a known trajectory.
• Create instance constraints $c_i(u'[i], \tau) = u'[i] - u[i-\frac{\tau}{\Delta t}]$ for all collocation nodes enforcing that.

Calculating the constrain Jacobian

The timeshift constraints have to be added to the constraint jacobian. For the constraints $c_i$ as defined above, the Jacobian is

• for midpoint integration:
  $G_i = [\frac{\partial c_i}{\partial u[i]}, \frac{\partial c_i}{\partial \tau}] = [1, \frac{u'[i-\frac{\tau}{\Delta t}+1] - u'[i-\frac{\tau}{\Delta t}-1]}{2\Delta t} \cdot \frac{1}{\Delta t}]$

• for backwards euler integration:
  $G_i = [\frac{\partial c_i}{\partial u[i]}, \frac{\partial c_i}{\partial \tau}] = [1, \frac{u'[i-\frac{\tau}{\Delta t}] - u'[i-\frac{\tau}{\Delta t}-1]}{\Delta t} \cdot \frac{1}{\Delta t}]$

Status

Development is ongoing in branch input-time-shifts. The current code does not run without exceptions and the added functions are untested.

[chris-konrad]

TODO List

• Add a function to identify and substitute trajectories with a timeshift (16cc611, 89b7db3)
• Verify that timeshift trajectories are given in the correct format (6f79482)
• Require unshifted input to be a known trajectory (11ab07d)
• Generate instance constraints (89b7db3)
• Enable _find_closest_free_index() to create the indices of constraints with timeshift (b52a96f, 1e9a710, 11ab07d)
• Enable _instance_constraints_func() to handle timeshift constraints. (b52a96f, 1e9a710, 11ab07d)
• Enable _instance_constraints_jacobian_indices() to handle timeshift constraints (6f79482).
• Enable jacobian_indices() to find the dynamic indices of timeshift constraints
• Test if constraints_jacobian() needs to be adressed
• Test code and unittests