Kalman · LQR

Plant

State is position and velocity, input is thrust, and the camera sees position only. Drag c slows the puck, and random force w (air currents, a bumpy table) pushes it around.

Zero-order hold over one sample Δt, computed from one matrix exponential:

LQR

Minimise the infinite-horizon quadratic cost of tracking error and thrust:

The optimal cost-to-go is x̃^⊤Px̃, where P solves the discrete algebraic Riccati equation (DARE):

The feedforward u_ff = m a_ref + c v_ref carries moving targets. Thrust is clipped at ±F_max per axis, which LQR does not know about.

Kalman filter

Predict with the model, then correct with the camera:

The prior covariance converges to the solution of the dual DARE, swapping A→A^⊤, B→C^⊤, Q→W, R→V:

A consistent filter has mean normalised innovation squared ν^⊤S⁻¹ν = 2 (two measured axes). Detune the model noise to see it drift away from 2.

Solvers and the continuous limit

The DARE is solved with the structure-preserving doubling algorithm. With G = BR⁻¹B^⊤, each step doubles the Riccati horizon:

Starting from A₀ = A, G₀ = G, H₀ = Q, H_j → P. For comparison, the continuous ARE is solved with the matrix sign function of the Hamiltonian:

As Δt → 0 the sampled gain K approaches K_c. Make Δt large and watch them part.