Control and Dynamics

How state-space models, feedback, estimation, and optimal control turn dynamical systems into systems that can be steered, stabilized, and observed.
Modified

April 26, 2026

Keywords

control, state-space, feedback, observability, LQR, MPC

1 Why This Module Matters

Dynamics asks:

  • what trajectory follows from a law of change
  • where the equilibria are
  • whether nearby trajectories stay close

Control asks one layer more:

  • what input should we apply
  • what state can we infer from measurements
  • what feedback stabilizes the system
  • what objective are we optimizing over trajectories

So this module is where the site turns dynamics you analyze into dynamics you can shape.

It is the bridge from ODEs and numerical simulation into feedback, estimation, optimal control, and learning-based systems work.

Prerequisites ODEs and Dynamical Systems should come first. Numerical Methods matters because real controllers and simulations live on sampled updates, not only exact flows. Linear Algebra is the language of state-space models, reachability, and feedback.

Unlocks Feedback design, reachability and observability, LQR, Kalman filtering, MPC, robotics, systems thinking

Research Use Reading papers or courses on control, estimation, dynamical systems, model-based RL, robotics, and continuous-time system design

2 First Pass Through This Module

The intended first-pass spine for this module is:

  1. State-Space Models, Inputs, and Outputs
  2. Controllability, Reachability, and Observability
  3. Feedback, Stability, and Pole Placement
  4. Linear Quadratic Regulation and Riccati Intuition
  5. Estimation, Kalman Filtering, and the Separation Principle
  6. Model Predictive Control and Constraint Handling
  7. Learning-Based Control, System Identification, and RL Bridges

The module now has a complete seven-page first-pass spine. The pages move from state-space modeling to reachability and observability, then into feedback design, LQR, Kalman filtering, MPC, and finally the bridge into learning-based control and RL-flavored sequential decision-making.

3 How To Use This Module

A good first pass through the live pages is:

  1. start with State-Space Models, Inputs, and Outputs
  2. continue to Controllability, Reachability, and Observability
  3. continue to Feedback, Stability, and Pole Placement
  4. continue to Linear Quadratic Regulation and Riccati Intuition
  5. continue to Estimation, Kalman Filtering, and the Separation Principle
  6. continue to Model Predictive Control and Constraint Handling
  7. continue to Learning-Based Control, System Identification, and RL Bridges
  8. pair those pages with Discretization, Time-Stepping, and the Bridge to Control
  9. use Linear Systems, Matrix Exponentials, and Modes as the continuous-time systems companion
  10. use Time-Stepping for ODEs and Stability to keep the continuous versus sampled distinction clear
  11. return to Optimization once the route reaches LQR or MPC

The design goal is to make the state-input-output viewpoint feel natural before introducing heavier control theorems.

4 Core Concepts

5 After This First Pass

Natural next directions after the completed spine are:

  • Applications > Control and Dynamics
  • stochastic control and dynamic programming
  • robust or adaptive control
  • deeper reinforcement learning theory

6 Applications

6.1 Robotics And Autonomous Systems

Control turns motion models into systems that can track, stabilize, and reject disturbances.

6.2 Estimation And Sensing

The same state-space language also supports observers, filtering, and hidden-state estimation.

6.3 ML And Sequential Decision-Making

Modern ML keeps rediscovering control language through model-based RL, system identification, differentiable simulators, and continuous-time generative viewpoints.

7 Go Deeper By Topic

The strongest adjacent live pages right now are:

8 Optional Deeper Reading After First Pass

The strongest current references connected to this module are:

9 Sources and Further Reading

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