bv_cvxbook_extra_exercises

the disturbance v is random with e v 0 e vv t 2 i

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Unformatted text preview: given by √ w2 + h2 F2 , 2w which gives us the second constraint: √ w2 + h2 F2 ≤ σπ (R2 − r2 ). 2w We also impose limits wmin ≤ w ≤ wmax and hmin ≤ h ≤ hmax on the width and the height of the structure, and limits 1.1r ≤ R ≤ Rmax on the outer radius. In summary, we obtain the following problem: √ 2π (R2 − r2 ) w2 + h2 √ w2 + h2 F1 ≤ σπ (R2 − r2 ) subject to 2h √ w2 + h2 F2 ≤ σπ (R2 − r2 ) 2w wmin ≤ w ≤ wmax minimize hmin ≤ h ≤ hmax 1.1r ≤ R ≤ Rmax R > 0, r > 0, w > 0, h > 0. The variables are R, r, w, h. Formulate this as a geometric programming problem. 14.5 Optimizing the inertia matrix of a 2D mass distribution. An object has density ρ(z ) at the point z = (x, y ) ∈ R2 , over some region R ⊂ R2 . Its mass m ∈ R and center of gravity c ∈ R2 are given by 1 ρ(z ) dxdy, c= m= ρ(z )z dxdy, mR R and its inertia matrix M ∈ R2×2 is M= R ρ(z )(z − c)(z − c)T dxdy. (You do not need to know the mechanics interp...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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