hw3 - 18 1 Dierentiability and the calculus of variations y...

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7. Consider the calculus of variations problem Φ ( z )= ± t f t 0 φ ( z ( t ) , ˙ z ( t ) ,t ) dt where t 0 , t f are Fxed, z ( t 0 ) and z ( t f ) are free except for the following mixed end-point constraint z ( t 0 )+2 z ( t f )=1 . Obtain the correct transversality conditions on any stationary curve ¯ z for this objective.
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2.3 Exercises 29 1. In this problem you will derive the Algebraic Ricatti Equation (ARE) of the LQR problem using constrained Fnite dimensional optimization. a) Given a stable LTI system without input ˙ x ( t )= Ax ( t ) ,x (0) = x o , it is desired to evaluate the performance index J := ± 0 x * ( t ) Mx ( t ) dt. Show that J is given by J = x * o Px o , where P solves A * P + PA = - M. b) Now given an LTI system with input ˙ x ( t Ax ( t )+ Bu ( t ) (0) = x o , it is desired to Fnd the optimal static state feedback law, i.e. u ( t Kx ( t ) (where K is a constant matrix), such that the following per- formance index is minimized J := ± 0 x * ( t ) Qx ( t u * ( t ) Ru ( t ) dt, x (0) = x o . Using the answer to the Frst part above, recast this problem as a
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This note was uploaded on 03/23/2009 for the course ME 254 taught by Professor Bamieh during the Winter '09 term at UCSB.

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hw3 - 18 1 Dierentiability and the calculus of variations y...

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