Projection / capability. Model intervention, close a feedback loop, diagnose stability, use estimates, and detect hidden costs.
Question this mathematics answers. If we act on the system, how does the generator change and does the closed loop actually behave safely?
Equation family
State/input/output: ẋ=Ax+Bu, y=CxFeedback: u=−Kx ⇒ ẋ=(A−BK)xScalar: ẋ=(a−bk)xContinuous-time modal stability: Re(λ_i(A−BK))<0
Symbols. x = state; u = control input; y = measured output; A = open-loop generator; B = input map; C = measurement map; K = feedback gain; λ_i = closed-loop eigenvalues.
Assumptions. Linear time-invariant local model. Stability criterion refers to continuous-time exponential modes. Real systems may have saturation, delays, nonlinearities, model error, and constraints.
Where the inputs come from. A,B,C from modeling/system identification; K from controller design; measurements from sensors; estimates from an estimator when full state is not observed.
Short derivation / justification
Substitute u=−Kx into ẋ=Ax+Bu: ẋ=Ax−BKx=(A−BK)x. Each eigenmode evolves roughly like e^{λt}; negative real part decays, positive real part grows.
Worked example
Scalar a=0.4,b=1. With k=0.2, rate=+0.2 → growth. With k=0.6, rate=−0.2 → decay. The threshold k=0.4 is explained by the sign change, not by a green UI light.
Cadet or real-world application. Autonomous vehicles, robotics, industrial processes, physiological control models.
What to look for. State may not equal measurement; controller may act on an estimate; proxy improvement may hide omitted costs.
Common misinterpretation. “Stable means safe, good, or authorized.”
Break / domain boundary. Stability is a technical property of the modeled loop. Permission, mission value, and unmodeled consequences remain separate.