hw1_sol - Optimal Control Winter 2009 Solutions to Homework...

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Unformatted text preview: Optimal Control Winter 2009 Solutions to Homework 1 Shaunak D. Bopardikar bshaunakeumail.ucsb.edu Problem 2 Let G : X —r X TAX , H denote the trace function and Y :2 XTAX. Using the property that the derivative of the composition is the composition of the derivatives. we have [3(§;GJ (1,0)] WM (we )aTDii ()G(mo) —]V. Now 6(tmce(Y)) _ A1 [ BY (3:0)] — I. shpuicfl Muj be wrii-iw SO %[X'FAX](V) = )Z’TAV + VAX. [(?x (x Ax“ (@100 Thus. using the chain rule, 6(tmce(XTAX)) _ ”T "" u .. [Ti’i/L X AV+VAX. h M 4a.. incurs a”; Problem 3 We have 6 _c./ am xiii/i: a _( —[ A (V): VA: 66X“ —m G. gamma By the linearity property of the derivative operator, we have (A +29") vw (MEN cum-u}. R k. x {finmi’nt— Problem 4 Let XTl denote the solution to the Ricatti equation at the n—th iteration. Then the Newton iteration can be written as %[R(X)](V) = AW + VA + X'RV + VRX. filmxonxw) = film/emu) - 7%th A‘Xnn + XnHA + XnRXflH + XnHRX = A‘Xn + XnA + 2XnRXn A (Am + XnA + XfiRXn + Q), => A‘XW + XMIA + XHRXHH + Xvi“ RXH = XnRXn — Q- Let Qn := XnRXn — Q. Note that QR is symmetric. Thus the above equation becomes (A + R-Xn)‘Xn+l + Xn+1(A + RXn) : Qua V- which is a Lyapunov equation in Xfl+1 since at each iteration, X“ is known. ...
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