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

• Excerpt from Lewis and Srymos - This excerpt is from Lewis and Syrmos, 1995 and gives a derivation of the necessary conditions for optimaliity. Other parts of the book can be searched via Google Books and purchased online.
• 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