CDS 110b: Linear Quadratic Optimal Control

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CDS 110b Schedule Project FAQ Reading

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.

Course Materials

References and Further Reading

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