>[!warning] >This content has not been peer reviewed. # Feedback control and stability in dynamical systems ## What it is (established result) Classical **control theory** studies how dynamical systems use **feedback** to regulate their behaviour. A controller compares a reference (setpoint) with the measured output, forms an **error signal**, and drives an actuator to reduce that error. For linear time-invariant systems, this is expressed in terms of transfer functions and **loop gain** \(L(s)\). Closing the loop changes the system's response and stability properties: - The **sensitivity function** \(S(s) = 1/(1 + L(s))\) measures how strongly disturbances and plant uncertainty affect the output. - The **complementary sensitivity** \(T(s) = L(s)/(1 + L(s))\) measures how well the system tracks the reference \([1,2]\). - Properly designed feedback can stabilise unstable plants, reject disturbances, and shape bandwidth, but is subject to trade-offs (e.g. Bode's integral constraints). These results are standard in modern control theory and widely applied in engineering, neuroscience, and cybernetics \([1–3]\). --- ## How RRT / RST uses it RST reads feedback control as the **operational form** of the identity loop: - The reference signal is the **requested identity** \(I\); the actual output is \(\Omega \cdot \mu\). - The error drives **reallocation of budget** in the Resource Triangle to keep key variables within tolerances despite noise. - Loop gain \(L\) plays the role of an effective **signal-to-noise ratio** \(\eta = \Omega/N\); increasing it moves the system into a higher-fidelity regime. - The sensitivity function \(S = 1/(1+L)\) mirrors the **noise fraction** of the budget, while the complementary sensitivity tracks the useful fraction. In this view, stable, homeostatic systems are those that maintain a **high-\(\eta\), high-\(\mu\)** regime via feedback, rather than relying on open-loop tuning. --- ## Links | Role | Link | |:---|:---| | Foundation mapping | [[foundation/Control and feedback/Control and feedback (RST)]] | | RST script | [[foundation/Control and feedback/Control and feedback - Code]] | | Substrate properties | [[Concrete properties of the substrate]] | --- ## References [1] K. Ogata, *Modern Control Engineering*, 5th ed., Prentice Hall, 2010. [2] K. J. Åström and R. M. Murray, *Feedback Systems: An Introduction for Scientists and Engineers*, Princeton University Press, 2008. [3] N. Wiener, *Cybernetics: Or Control and Communication in the Animal and the Machine*, 2nd ed., MIT Press, 1961.