State-Space Control, Controllability, Observability, and LQR/LQG: Kalman Rank Conditions, Pole Placement, Riccati Equations, Separation Principle, and LQG's Guaranteed Margin Loss

Deep End Era. Engineering with Sankalp · Oct 29, 2026 · 1h 4m

In classical control, an engineer shapes a single Bode plot or root locus connecting one sensor to one actuator. What happens when a high-speed magnetic-bearing rotor, a reusable rocket booster, or an aircraft flight controller has six coupled states, four actuators, and three noisy sensors—and a hidden unstable internal mode cancels out of the input-output transfer function? We build modern state-space control from first principles: Kalman's controllability and observability rank conditions, the PBH test, pole placement vs LQR and the Algebraic Riccati Equation, Kalman's duality and the…