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Control Theory

All digests tagged Control Theory

AStanford AA203 Optimal and Learning-Based Control | Spring 2026 | Lecture 8: Nonlinearity thumbnail

· 1:14:05

AStanford AA203 Optimal and Learning-Based Control | Spring 2026 | Lecture 8: Nonlinearity

This lecture provides an advanced overview of Linear-Quadratic Regulator (LQR) theory, extending its application from simple state regulation to complex nonlinear trajectory tracking and optimization. Key concepts include reformulating nonlinear tracking problems using deviation variables ($\delta x$, $\delta u$) to create an auxiliary LQR problem. The discussion culminates in two sophisticated iterative methods: Iterative LQR (iLQR), which linearizes dynamics and quadratizes costs, and Differential Dynamic Programming (DDP), which directly approximates the Bellman equation, offering a second-order approach for optimal control.

Key takeaways

  1. LQR as a General Tool 18:03

    While LQR is fundamentally designed to drive a state to the origin (regulation), it can be generalized to perform trajectory tracking by defining an auxiliary problem based on deviation variables. The optimal control law structure remains consistent: $u = u_{nominal} + ext{feedback term}$.

  2. Nonlinear Tracking via Linearization 24:10

    For nonlinear dynamics ($x_{k+1} = f(x_k, u_k)$), the tracking problem can be linearized by performing a Taylor expansion around the nominal trajectory ($\bar{x}, \bar{u}$), allowing the use of LQR techniques on the deviation variables.

  3. iLQR vs. DDP 35:05

    Both iLQR and DDP are methods for solving nonlinear optimal control problems iteratively. iLQR linearizes dynamics and quadratizes costs, while DDP directly approximates the Bellman equation by quadratizing the Q-function, making it a second-order algorithm in terms of dynamic derivatives.

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Loop Engineering from First Principles — Kyle Mistele, HumanLayer thumbnail

· 17:57

Loop Engineering from First Principles — Kyle Mistele, HumanLayer

The talk argues that relying solely on large-scale prompts for AI code generation is insufficient for complex, mission-critical systems. The solution lies in adopting 'control loops'—a pattern borrowed from control theory (Sensor $\rightarrow$ Controller $\rightarrow$ Actuator)—to ensure that code changes are small, verifiable, and incremental. This approach minimizes risk by preventing the creation of massive, unreviewable pull requests (PRs) and allows for robust automation within existing CI/CD pipelines.

Key takeaways

  1. The Problem with Large PRs

    Current AI workflows often result in 'blind Ralph loops'—massive, unreadable pull requests (e.g., 40,000 lines) that are too risky for large teams and critical systems. The goal must be to design loops that make code easier to read and verify.

  2. The Control Loop Framework 7:23

    A robust system uses a Sensor (to measure the current state/error), a Controller (to determine the optimal incremental change), and an Actuator (to apply that change). This pattern is ideal for systems where changes must be measured, applied incrementally, and verified.

  3. Implementing Low-Friction Loops

    Loops should be integrated into CI/CD platforms (GitHub Actions, CircleCI) using deterministic workflows. To maintain human oversight without friction, a 'feedback file' tracked in version control and triggered by comments (e.g., `/iterate`) is recommended to guide the agent.

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