CDS 110b: Optimal Control

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Course Home

L7-2: Sensitivity

L8-1: Robust Stability

L9-1: Robust Perf

Schedule

This lecture provides an overview of optimal control theory. Beginning with a review of optimization, we introduce the notion of Lagrange multipliers and provide a summary of the Pontryagin's maximum principle.

Lecture Outline

1. Introduction: two degree of freedom design and trajectory generation
2. Review of optimization: necessary conditions for extrema, with and without constraints
3. Optimal control: Pontryagin Maximum Principle
4. Examples: bang-bang control and Caltech ducted fan (if time)

Lecture Materials

• Lecture Presentation
• Lecture notes on optimal control
• Homework 1

References and Further Reading

• Notes on Pontryagin's Maximum Principle (courtesy of Doug MacMynowski) - this comes from a book on dynamic programming (DP) and uses a slightly different notation than we used in class.

Frequently Asked Questions

Q: What do you mean by penalizing something, from Q>=0 "penalizes" state error?

According to the form of the quadratic cost function J, there are three quadratic terms such as $x^{T}Qx$, $u^{T}Ru$, and $X(T)^{T}P_{1}X(T)$. When $Q\geq 0$ and if Q is relative big, the value of x will have bigger contribution to the value of J. In order to keep J small, x must be relatively small. So selecting a big Q can keep x in small value regions. This is what the "penalizing" means.

So in the optimal control design, the relative values of Q, R, and $P_{1}$ represent how important X, U, and X(T) are in the designer's concerns.

Zhipu Jin,13 Jan 03