Take a linear system, hit it with a pure oscillation, and everything it does collapses to a single ratio of polynomials: H(s) = C(sI−A)⁻¹B + D. The poles are the denominator roots — the eigenvalues of A, the system's natural modes. The zeros are the numerator roots — the frequencies the system swallows. Down the center a small 2×2 system goes in, the map is formed, and the pole–zero picture plus the step response come out. The blue team builds and defends it; the red team tries to break it.
source Laplace-domain transfer functions — Heaviside's operational calculus (Electromagnetic Theory, 1893) matured into H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, 1945) — archive.org/details/dli.ernet.15701. No single canonical primary, so AMBER. Rendered, not quoted.
The transfer function is not looked up; it is built from the realization (A,B,C,D):
den(s) = det(sI − A) — its roots are the poles, and they are exactly the eigenvalues of A. num(s) = C·adj(sI − A)·B + D·det(sI − A) — its roots are the zeros. Stability is a pure statement about pole location: BIBO-stable iff every pole has negative real part.
For the current system, computed live:
| quantity | value |
|---|
This is the-state-space seen from the other side. There the system is A and B in the time domain; here it is poles and zeros in the frequency domain — the same object, one Laplace transform apart.
Feed the poles and zeros into a Cartesian or polar plot and you get the-nyquist and the-bode curves: the classical pictures are all drawn from H(s). Each sphere is the next one's premise.
The blue team's live check: recompute the poles as roots of det(sI − A), confirm they equal the eigenvalues of A, and confirm the reported stability label agrees with the actual pole locations. If red flips a pole across the axis, this badge turns red.
A minimal SISO system is four objects: A (2×2, the dynamics), B (input), C (output), and D (feedthrough). We build A in controllable-canonical form from two chosen poles, and set C to place one zero:
A = [[0, 1], [−p₁p₂, p₁+p₂]], B = [0; 1], C = [−K·z, K], D = 0. The eigenvalues of A are then exactly p₁, p₂ and the single zero is at z. Choose them below and the machine forms H(s) = C(sI − A)⁻¹B on the spot.
Poles set the natural modes; a pole in the right half-plane (Re>0) is an unstable mode.
Every number here is computed from (A,B,C,D): eigenvalues by the 2×2 characteristic equation, the step response by a scaled-and-squared matrix exponential e^{At}. Nothing is looked up.
What the machine produces, proven at boot on the default system (poles −1, −2, zero −3): the poles equal the eigenvalues of A to 1e−8; BIBO-stable because both poles have negative real part; the DC gain H(0) = 1.5 computed two independent ways (num(0)/den(0) and −C·A⁻¹B), and the step response's final value equals it to 1e−6 — the final value theorem, checked, not asserted.
The blue team's witness (left) confirms the pole/eigenvalue identity live; the red team (right) tries to make a right-half-plane pole read as stable.
Kalman (1963) proved H(s) captures only the controllable-and-observable subsystem — the minimal realization. Nonlinear or time-varying systems have no transfer function at all. "Stable transfer function" is not the same as "internally stable".
"Pole–zero cancellation is a harmless simplification." Cut. It deletes an internal mode from view, not from existence. Cancel an unstable pole against a zero and the tidy H(s) hides a machine that diverges.
"The poles are always the eigenvalues of A." Corrected. Only for a minimal (controllable + observable) realization. A cancellation drops that eigenvalue from the poles — the minimal order falls.
"DC gain is the step response's final value." Kept, corrected. Only when stable. The final value theorem requires every pole in the open left half-plane; push one to the RHP and the step diverges — H(0) is then just a number, not a limit.
The red team's move: flip one pole's real part into the right half-plane while the code keeps reporting the system "stable". The blue team's witness (window 7) recomputes stability from the actual pole locations and catches the lie.
Move a pole to Re>0 but keep the "stable" label. The witness compares the label against the poles it computes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.