Tikhonov

Tikhonov

Normalized Tikhonov planner for harmonic-extension references.

Provides a stateful first-order filter TikhonovFilter and helpers for computing harmonic equilibria, reference rates, feedforward corrections, and Lyapunov diagnostics.

AbstractTikhonovFilter{T, V}

Abstract base type for Tikhonov reference filters.

JointTikhonovFilter(x0, v0; epsilon) -> JointTikhonovFilter

Filters both reference position qstar and reference velocity qstar_dot simultaneously:

epsilon * xdot = -x + qstar(t)
epsilon * vdot = -v + qstar_dot(t)

TikhonovFilter(x0; epsilon) -> TikhonovFilter

Stateful realization of the normalized Tikhonov planner

epsilon * xdot = -x + qstar(t),

where qstar(t) is an already-solved harmonic reference. The normalization is the form used by the singular-perturbation model: the boundary-layer rate is 1 / epsilon, independently of the spectrum of the harmonic system.

tikhonov_dissipation(error, epsilon)

Evaluate Vdot = -norm(error)^2 / epsilon for V(error) = 1/2 * norm(error)^2 and a stationary harmonic reference.

Return the instantaneous harmonic reference solving H * qstar = rhs.

H may be an AbstractMatrix or a pre-computed Factorization (e.g. factorize(H)). Passing a cached factorization avoids repeated factorization when solving for many right-hand sides in a loop.

tikhonov_feedforward_reference(reference, reference_rate, epsilon)

Return the lag-canceling input reference + epsilon * reference_rate for

epsilon * xdot = -x + input(t).

For an exact reference derivative, the error x - reference obeys the homogeneous equation epsilon * edot = -e.

tikhonov_reference_rate(H, rhs_rate)

Return qstar_dot by solving H * qstar_dot = rhs_rate. This identity applies when the sheaf topology is fixed and target motion changes only the right-hand side of the harmonic system.

H may be an AbstractMatrix or a pre-computed Factorization (e.g. factorize(H)). Passing a cached factorization avoids repeated factorization when computing reference rates in a loop.

tikhonov_step!(filter, q0, q1, dt)
tikhonov_step!(filter, reference_at, t, dt)

Advance the normalized Tikhonov planner exactly over one interval under a first-order hold between harmonic references q0 and q1. The callable form evaluates reference_at(t) and reference_at(t + dt).

Unlike an explicit ODE integrator, this update remains stable for every dt > 0 and epsilon > 0. As epsilon approaches zero, the returned state approaches q1 without a numerical singularity.