---
title: 'Control theory: from feedback to verifiable dynamics'
url: https://doc.liz6.com/en/theory/02-control-theory
locale: en
area: theory
tags:
- Theory
- Control theory
date: 2026-09-10
modified: 2026-09-12
description: 'Eleven articles build a calculable foundation for control theory: nine core chapters move from feedback and thermal modeling to digital implementation and autoscaling; two advanced chapters introduce state estimation and optimal control. Information theory and physical equipment are not prerequisites.'
---

# Control theory: from feedback to verifiable dynamics

Eleven articles build a calculable foundation for control theory: nine core chapters move from feedback and thermal modeling to digital implementation and autoscaling; two advanced chapters introduce state estimation and optimal control. Information theory and physical equipment are not prerequisites.

After the core route, you should be able to identify feedback signals, write a simple dynamic model, separate stability from speed and accuracy, explain PID and sampling delay, and recognize actuator limits and model uncertainty. The advanced chapters open a path to state-space and MPC material without replacing a full specialist course.

## Core route: 01–09

1. [Feedback and control](/en/theory/02-control-theory/01-feedback-and-control)
2. [Dynamic models](/en/theory/02-control-theory/02-dynamic-models)
3. [Response and stability](/en/theory/02-control-theory/03-response-and-stability)
4. [PID control](/en/theory/02-control-theory/04-pid-control)
5. [Transfer functions and poles](/en/theory/02-control-theory/05-transfer-functions-and-poles)
6. [Frequency response and stability margins](/en/theory/02-control-theory/06-frequency-response-and-margins)
7. [Sampling and digital control](/en/theory/02-control-theory/07-sampling-and-delay)
8. [Constraints and robustness](/en/theory/02-control-theory/08-constraints-and-robustness)
9. [Control in computing systems](/en/theory/02-control-theory/09-control-in-computing-systems)

## Advanced route: 10–11

10. [Advanced: state feedback and observers](/en/theory/02-control-theory/10-state-feedback-and-observers)
11. [Advanced: optimal control and MPC](/en/theory/02-control-theory/11-optimal-and-predictive-control)

## Choose a route

- First encounter: 01 → 02 → 03 → 04. Work through one thermal loop.
- Hardware stability: 02 → 03 → 05 → 06 → 08, then revisit op-amps and power circuits.
- Software control and autoscaling: 01 → 02 → 04 → 07 → 08 → 09; use 05–06 for frequency-domain questions.
- Robotics and advanced control: study 02, 03, 05, 07, then 10 → 11 with linear algebra.

## Prerequisites appear when needed

01 begins with algebra. 02–04 introduce derivatives, integrals and exponentials; 05–06 add complex numbers, Laplace transforms and logarithmic axes; 07 uses difference equations; 10–11 need matrices, eigenvalues and quadratic forms. Check quantities, units and assumptions before calculating. The thermal baseline is consistently 2000 J/K, 20 W/K and 1000 W; normalized examples explicitly declare new parameters.

## Use experiments to test a prediction

Predict first, vary one parameter, compare output with action, then try the end-of-chapter checks. Each chapter has two expandable reasoning answers. Equations, examples, diagrams and summaries remain readable without interaction. Physical simulation time does not depend on rendering frame rate.

Models identify initial conditions, disturbances, constraints and axis units. A finite trace is evidence about a model, not proof of stability under every condition. Analytical solutions and independent numerical calculations check the teaching models; real plants need identification and measurement.

## Notation

| Symbols | Meaning |
| --- | --- |
| $r,e,y,y_m$ | Reference, error, actual output, measurement |
| $u_c,u$ | Unclipped request, applied action |
| $P(s),C(s),L(s)$ | Plant, controller, loop transfer |
| $S(s),T(s)$ | Sensitivity, complementary sensitivity |
| $x,\hat x$ | State, estimated state |
| $T_s,h$ | Physical sample period, numerical integration step |

In state space, $A,B,C$ are matrices. Thermal capacitance is $C_{\mathrm{th}}$. Temperature differences use K and readings use °C; increments have the same numerical size but different roles.

## References and continuation

Scope is informed by the [Caltech control course](https://murray.cds.caltech.edu/CDS_110/ChE_105,_Spring_2024) and its Feedback Systems textbook entry. Reproducible calculations use [python-control](https://python-control.readthedocs.io/en/stable/linear.html); implementation references accompany each chapter. Teaching examples are independently constructed with explicit simplifications.

Continue with system identification, nonlinear stability, robust synthesis or stochastic estimation. The [queueing theory series](/en/theory/03-queueing-theory) explains arrivals, service and waiting; the autoscaling chapter focuses on workload balance and dynamic adjustment.