# CDS 110b: Optimal Control

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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

• Excerpt from LS95 on optimal control - This excerpt is from Lewis and Syrmos, 1995 and gives a derivation of the necessary conditions for optimaliity. A few pages have been left out from the middle that contained some additional examples (which you can find in similar books in the library, if you are interested). 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.

Q: In the example on Bang-Bang control discussed in the lecture, how is the control law for $u$ obtained?

Pontryagin's Maximum Principle says that $u$ has to be chosen to minimise the Hamiltonian $H(x,u,\lambda )$ for given values of $x$ and $\lambda$. In the example, $H=1+({\lambda }^{T}A)x+({\lambda }^{T}B)u$. At first glance, it seems that the more negative $u$ is the more $H$ will be minimised. And since the most negative value of $u$ allowed is $-1$, $u=-1$. However, the co-efficient of $u$ may be of either sign. Therefore, the sign of $u$ has to be chosen such that the sign of the term $({\lambda }^{T}B)u$ is negative. That's how we come up with $u=-sign({\lambda }^{T}B)$.

Shaunak Sen, 12 Jan 06

Q: Notation question for you: In the Lecture notes from Wednesday, I'm assuming that $T$ is the final time and $T$ (superscript T) is a transpose operation. Am I correct in my assumption?

Yes, you are correct.

Jeremy Gillula, 07 Jan 05

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