# CDS 110b: Linear Quadratic Optimal Control

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This Wednesday 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.

## Frequently Asked Questions

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