CDS 110b: Optimal Control: Difference between revisions
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<p>According to the form of the quadratic cost function J, there are three quadratic terms such | <p>According to the form of the quadratic cost function J, there are three quadratic terms such | ||
as <math>x^T Q x</math>, <math>u^T R u</math>, and <math> | as <math>x^T Q x</math>, <math>u^T R u</math>, and <math>x(T)^T P_1 x(T)</math>. When <math>Q \geq 0</math> and if <math>Q</math> is relative big, the value of <math>x</math> will have bigger contribution to the value of <math>J</math>. In order to keep <math>J</math> small, <math>x</math> must be relatively small. So selecting a big <math>Q</math> can keep <math>x</math> in small value regions. This is what the "penalizing" means.</p> | ||
selecting a big Q can keep x in small value regions. This is what the "penalizing" means.</p> | |||
<p>So in the optimal control design, the relative values of Q, R, and <math>P_1</math> represent how important | <p>So in the optimal control design, the relative values of <math>Q</math>, <math>R</math>, and <math>P_1</math> represent how important <math>X</math>, <math>U</math>, and <math>X(T)</math> are in the designer's concerns.</p> | ||
X, U, and X(T) are in the designer's concerns.</p> | |||
<p>Zhipu Jin,13 Jan 03</p> | <p>Zhipu Jin,13 Jan 03</p> | ||
</blockquote> | </blockquote> | ||
Revision as of 21:08, 2 January 2006
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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
- Introduction: two degree of freedom design and trajectory generation
- Review of optimization: necessary conditions for extrema, with and without constraints
- Optimal control: Pontryagin Maximum Principle
- Examples: bang-bang control and Caltech ducted fan (if time)
Lecture Materials
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 , , and . When and if is relative big, the value of will have bigger contribution to the value of . In order to keep small, must be relatively small. So selecting a big can keep in small value regions. This is what the "penalizing" means.
So in the optimal control design, the relative values of , , and represent how important , , and are in the designer's concerns.
Zhipu Jin,13 Jan 03
