Optimal Quadratic Control Chapter 2 W&S Calculus of Extrema
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Transcript of Optimal Quadratic Control Chapter 2 W&S Calculus of Extrema
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Optimal Quadratic ControlChapter 2 W&S
Calculus of ExtremaDr. Hatem ElaydiEE Department, Islamic University of GazaFeb 18, 2012
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Dr. H Elaydi, EELE 6302, IUG 2
Unconstrained Extrema
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Dr. H Elaydi, EELE 6302, IUG 3
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Dr. H Elaydi, EELE 6302, IUG 4
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Dr. H Elaydi, EELE 6302, IUG 5
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Dr. H Elaydi, EELE 6302, IUG 6
For multivariate functions
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Dr. H Elaydi, EELE 6302, IUG 7
Example 2.1-1
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Dr. H Elaydi, EELE 6302, IUG 8
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Dr. H Elaydi, EELE 6302, IUG 9
Back substitution & Lagrange Multiplier
Extrema of functions with equality constraints
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Dr. H Elaydi, EELE 6302, IUG 10
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Dr. H Elaydi, EELE 6302, IUG 11
A tin manufacturer wants to maximize the volume of a certain run of cans subject to that the area of tin used be a given constant. If a fixed metal thickness is assumed, a volume of thin constraint implies that the cross-sectional area is constrained.
Example 2.2-2
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Dr. H Elaydi, EELE 6302, IUG 12
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Dr. H Elaydi, EELE 6302, IUG 13
Lagrange Multiplier Approach
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Dr. H Elaydi, EELE 6302, IUG 14
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Dr. H Elaydi, EELE 6302, IUG 15
Example 2.2-2 Using Lagrange
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Dr. H Elaydi, EELE 6302, IUG 16
Solution
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Dr. H Elaydi, EELE 6302, IUG 17
Non-scalar case
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Dr. H Elaydi, EELE 6302, IUG 18
Lagrange Multiplier: Multivariate
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Dr. H Elaydi, EELE 6302, IUG 19
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Dr. H Elaydi, EELE 6302, IUG 20
1st variation
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Dr. H Elaydi, EELE 6302, IUG 21
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Dr. H Elaydi, EELE 6302, IUG 22
Hessian
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Dr. H Elaydi, EELE 6302, IUG 23
Example 2.2-4
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Dr. H Elaydi, EELE 6302, IUG 24
Solution
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Dr. H Elaydi, EELE 6302, IUG 25
Sol…
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Dr. H Elaydi, EELE 6302, IUG 26
Example 2.2-5
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Dr. H Elaydi, EELE 6302, IUG 27
Solution
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Dr. H Elaydi, EELE 6302, IUG 28
Example 2.2-6
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Dr. H Elaydi, EELE 6302, IUG 29
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Dr. H Elaydi, EELE 6302, IUG 30
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Dr. H Elaydi, EELE 6302, IUG 31
Nonlinear Programming
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Dr. H Elaydi, EELE 6302, IUG 32
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Dr. H Elaydi, EELE 6302, IUG 33
Solution
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Dr. H Elaydi, EELE 6302, IUG 34
Sol…
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Dr. H Elaydi, EELE 6302, IUG 35
2.1, 2.3, 2.7, 2.8
Homework