Theorem: G(s) s - (K_d s^2 + K_p s + K_i) = 0 (PID rational clearance identity)

Theorem (PID-rational-clearance canonical): G(s) = K_p + K_i/s + K_d s multiplied by s gives K_p s + K_i + K_d s^2 = K_d s^2 + K_p s + K_i, so G(s) s - (K_d s^2 + K_p s + K_i) = 0 identically. Canonical sympy pins: K_p, K_i, K_d, s =…