>[!warning] >This content has not been peer reviewed. # Control and feedback — RST foundation Classical **control theory** studies how systems use **feedback** to regulate their behaviour: maintaining a setpoint, rejecting disturbances, and stabilising otherwise unstable dynamics. In RST, the **identity loop** \(I = \Omega \cdot \mu\) and homeostatic systems are naturally expressed in control-theoretic terms: they sense deviations, allocate budget, and act to keep key variables within tolerances. --- ## I. Classical side - **Feedback loop:** A controller compares a reference signal (setpoint) to the measured output and applies an action based on the **error**. - **Open-loop vs closed-loop gain:** In linear systems, the **loop gain** \(L(s)\) determines stability and response; closing the loop reshapes dynamics and can suppress disturbances. - **Sensitivity and robustness:** The **sensitivity function** \(S(s) = 1 / (1 + L(s))\) quantifies how strongly disturbances affect the output; small \(|S|\) over a bandwidth means good disturbance rejection and robust tracking \([1,2]\). --- ## II. Mapping to RRT / RST | Control concept | RRT / RST identity | Axiom / equation | |:---|:---|:---| | **Setpoint / reference** | Target identity / request \(I\) the system tries to maintain. | Resource Allocation Equation. | | **Error signal** | Deviation between current \(\Omega \cdot \mu\) and requested \(I\); drives reallocation of budget. | Identity loop; A2, A3. | | **Loop gain \(L\)** | Effective **signal-to-noise ratio** \(\eta = \Omega/N\) achieved by the controller. | \(\eta\) in \(\mu(\eta, n)\). | | **Sensitivity \(S = 1/(1+L)\)** | Fraction of disturbances that survive; complements the useful fraction \(\mu\). | Noise fraction; \(\mu^n + \nu^n = 1\). | | **Stability / homeostasis** | Regime where feedback keeps key observables in a high-\(\mu\) band despite noise. | Identity loop fixed point; RST field equation. | In this reading, a **controller** is a configuration of relations that continually adjusts \(\Omega\) to maintain \(I\) against variations in \(N\), trading complexity for robustness. High loop gain corresponds to high signal-to-noise \(\eta\) and thus high fidelity \(\mu\). --- ## Links - **Foundation index:** [[../Foundation index]] - **RRT axioms:** [[Relational Resolution Theory (RRT)]] - **Applications Roadmap:** [[../../Applications Roadmap]]