Feedback mode

Manual gain K Pole placement (Ackermann) LQR (Q/R weights)

Gain k₁ (position) 3.00 Gain k₂ (velocity) 2.00

Desired pole λ₁ -3.00 Desired pole λ₂ -3.00

Q — position cost 5.00 Q — velocity cost 1.00 R — control effort cost 1.00

Compare with poorly-tuned K

Live readouts

Gain K = [k₁, k₂]—

Eigenvalues of A−BK—

Controllability—

Settling time (2%)—

Stability verdict—

Position x₁—

Velocity x₂—

Control u = −Kx—

State: x = [position; velocity]. Plant: ẋ = Ax + Bu with A = [[0,1],[−2,−0.3]], B = [0,1] (mass 1, spring k=2, damper c=0.3). Full-state feedback u = −Kx replaces A with the closed-loop matrix A−BK, moving its eigenvalues. Manual: drag K directly. Pole placement: pick desired eigenvalues, K is solved by Ackermann's formula. LQR: pick Q (state-error cost) and R (control-effort cost); K minimizes J=∫(xᵀQx+uᵀRu)dt via a real Newton/Riccati iteration on the algebraic Riccati equation. Higher Q → faster, tighter convergence; higher R → gentler, cheaper control effort.