systems theory//closed-loop system

A closed-loop system is a system whose actions change the world and whose sensors bring that change back into its next decision, and it is the organizing object of everything that senses, thinks and acts: a drone holding position in a gust, a plant keeping a reactor at temperature, a fleet manager sending robots where the orders are. Follow one gust through a drone. It tilts the frame half a degree; the inertial unit feels it a millisecond later; an estimator turns noisy readings into a best guess of attitude with its uncertainty; the controller asks the motors for a correcting torque; the propellers push, the drone moves, and the cycle starts again. When software and physics share one loop like this, the system is a cyber-physical system: an error of computation becomes a wrong movement in the world.

A closed-loop system is a system whose actions change the world and whose sensors bring that change back into its next decision, and it is the organizing object of everything that senses, thinks and acts: a drone holding position in a gust, a plant keeping a reactor at temperature, a fleet manager sending robots where the orders are. Follow one gust through a drone. It tilts the frame half a degree; the inertial unit feels it a millisecond later; an estimator turns noisy readings into a best guess of attitude with its uncertainty; the controller asks the motors for a correcting torque; the propellers push, the drone moves, and the cycle starts again. When software and physics share one loop like this, the system is a cyber-physical system: an error of computation becomes a wrong movement in the world.

Every such system passes through the same seven stages, standing on two foundations: the mathematics, which reappears in each stage in another disguise, and the platform (embedded system), which decides what fits in the time available.

1Model the worldstate-space model2Sensesensor3Estimate the statestate estimation4Learn what cannot be writtenML5Diagnose and predictfault diagnosis6Decidedecision theory7Actcontroller design

Three equations hold the whole map:

xk+1=f(xk,uk)+wk,yk=h(xk)+vk,uk=π(y0,…,yk).x_{k+1}=f(x_k,u_k)+w_k,\qquad y_k=h(x_k)+v_k,\qquad u_k=\pi(y_0,\dots,y_k).xk+1​=f(xk​,uk​)+wk​,yk​=h(xk​)+vk​,uk​=π(y0​,…,yk​).

The dynamics fff moves the state xxx, with www everything the model does not know; the sensors see h(x)h(x)h(x) through noise vvv; and the policy π\piπ, your code, may use only what has been measured so far. Modelling writes fff, understanding a sensor writes hhh and describes vvv, estimating reconstructs xxx from the yyy, learning extracts fff, hhh or π\piπ from data, diagnosing notices that fff or hhh changed, and controlling and deciding design π\piπ.

Everything is a loop, and a loop is as good as its weakest stage. A poor sensor, an uncompensated loop delay or a saturated actuator cannot be rescued by the best Kalman filter or LQR elsewhere: each technique improves one stretch, and the loop pays for the worst one.

It is wider than one control loop. Feedback control is the mechanics of one loop (error, gain, stability), and feedback loop the modelling primitive of a variable influencing itself; a closed-loop system is the whole chain, estimation, learning and decision included, usually as loops nested by time scale (hierarchical control): rotor current in microseconds, attitude in milliseconds, missions in seconds, maintenance in weeks.

A loop always asks what happens if it acts. A model that predicts what usually happens, fitted on passive data, can score 98 % in test and still be useless inside a loop, because the loop's own actions move the data (causal inference).

The business is a loop too, only slow. Measuring a fleet, estimating its health, deciding what to service and acting through a work order has the same stages; a monthly report is a sensor with 30 days of latency, and with that delay nothing that changes within a week can be corrected.

The recurring ideas of the loop, from weighted averaging to the error that moves instead of disappearing, are collected in engineering patterns; the decision-side reading of its timing is the OODA loop; the history of the idea is cybernetics.