Optimal Control and Trajectory Planning

A bridge page showing why many systems problems are not only about stability, but about choosing trajectories and control actions that trade off effort, accuracy, time, and safety.
Modified

April 26, 2026

Keywords

optimal control, trajectory planning, LQR, MPC, cost

1 Application Snapshot

Many real systems do not ask only:

can I stabilize this system?

They ask:

  • can I reach the target smoothly?
  • can I do it without wasting too much energy?
  • can I avoid overshoot, actuator stress, or unsafe states?
  • can I plan a whole path, not just react locally?

That is the shift from bare stabilization to optimal control and trajectory planning.

This page is the shortest bridge from feedback language into the idea that a controller is often chosen by optimizing a cost over time.

2 Problem Setting

Suppose the system evolves as

\[ x_{t+1} = f(x_t, u_t) \]

or, in continuous time,

\[ \dot{x}(t) = f(x(t), u(t)). \]

A controller is no longer judged only by whether it keeps the system stable.

It is also judged by a performance objective, such as

\[ \text{cost} = \sum_{t=0}^{T-1} \ell(x_t, u_t) + \phi(x_T) \]

or, in continuous time,

\[ \text{cost} = \int_0^T \ell(x(t), u(t))\,dt + \phi(x(T)). \]

Here:

  • \(\ell\) is a running cost
  • \(\phi\) is a terminal cost
  • \(T\) is a horizon

The planner or controller must now choose actions that respect both:

  • the system dynamics
  • the cost tradeoffs

3 Why This Math Appears

This language reuses several math layers already on the site:

  • Control and Dynamics: feedback stabilizes the system, but does not by itself specify the best tradeoff among behaviors
  • Optimization: objectives, constraints, and tradeoffs become part of the control design itself
  • Stochastic Control and Dynamic Programming: finite-horizon and infinite-horizon planning are sequential optimization problems
  • Numerical Methods: many trajectory-planning problems are solved approximately, repeatedly, and under discretization
  • ODEs and Dynamical Systems: the state evolution is still the load-bearing object underneath the optimization

So optimal control is not “control plus a bonus objective.” It is the point where dynamics and optimization become one problem.

4 Math Objects In Use

  • state \(x_t\)
  • control input \(u_t\)
  • dynamics law \(f\)
  • running cost \(\ell(x_t,u_t)\)
  • terminal cost \(\phi(x_T)\)
  • horizon \(T\)
  • reference trajectory or target set
  • sometimes constraints on state or input

In the simplest linear-quadratic setting, these objects become

\[ x_{t+1} = Ax_t + Bu_t \]

and

\[ \ell(x_t,u_t) = x_t^\top Q x_t + u_t^\top R u_t. \]

That is the first clean place where readers usually meet LQR.

5 A Small Worked Walkthrough

Imagine a drone that must move from one altitude to another.

A stabilizing controller could simply try to make the altitude error go to zero.

But several trajectories might all be stable while behaving very differently:

  • one reaches the target fast but uses too much thrust
  • one uses little thrust but takes too long
  • one overshoots and oscillates before settling

Now define a cost like

\[ \sum_{t=0}^{T-1} \left( q(h_t-h_{\mathrm{ref}})^2 + r u_t^2 \right). \]

This cost says:

  • penalize being far from the target altitude
  • also penalize aggressive thrust commands

The controller now has to balance two goals:

  • accuracy
  • effort

That tradeoff is the systems-level meaning of optimal control.

Trajectory planning adds one more layer:

  • the target may not be a single equilibrium
  • it may be a whole path through state space

Then the question becomes not only “how do I stabilize?” but “which state sequence should I follow, and how should I generate the controls that realize it?”

6 Implementation or Computation Note

Three practical branches appear quickly:

  1. Quadratic regulation When the model is linear and the cost is quadratic, LQR is the clean first-pass answer.

  2. Constraints and receding horizons When state or input limits matter, planning often becomes MPC-like.

  3. Uncertainty and partial observability When the state is estimated rather than measured directly, planning and estimation must interact.

Use these pages as the strongest follow-on support:

7 Failure Modes

  • thinking every stable controller is equally good
  • writing a cost that ignores actuator effort or physical limits
  • confusing trajectory planning with point stabilization
  • assuming an optimal open-loop plan remains good after disturbances appear
  • forgetting that many “planning” problems are actually solve-estimate-replan loops

8 Paper Bridge

9 Sources and Further Reading

  • 16.323 / Principles of Optimal Control - First pass - official MIT lecture-note hub for optimal control. Checked 2026-04-25.
  • 16.323 syllabus - First pass - useful compact framing of what optimal control is trying to solve. Checked 2026-04-25.
  • Topic 18: Linear Quadratic Regulator - Second pass - a clean official MIT entry into quadratic cost tradeoffs. Checked 2026-04-25.
  • AA203 / Optimal and Learning-Based Control - Second pass - official Stanford course anchor for planning and optimal control. Checked 2026-04-25.
  • EE364b course information - Bridge to constrained planning - official Stanford convex-optimization course info with strong optimization-to-control relevance. Checked 2026-04-25.
  • EE364b lectures - Bridge to constrained planning - useful once you want optimization-backed receding-horizon and constrained-control context. Checked 2026-04-25.
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