L10-StochOptControl

2 please prove the above ive modeled my notation on

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Unformatted text preview: V (t )x where V (t ) is obtained by solving: ˙ −V = Q + AT V + VA − VBR −1 B T V : and −a = 1 Tr (FF T V ) with conditions V (T ) = QT and a(T ) = 0. ˙ 2 Please prove the above - I’ve modeled my notation on Todorov [1] so the proof presented there should be particularly straightforward. Nick Jones Elements of Stochastic Optimal Control: ICDNS Linear Quadratic Gaussian Control III We will discuss this equation system looking at the role of a and F on the optimal control and the total cost. Dynamics: dx = [Ax + Bu ]dt + Fd ξ 1 Cost: w (x , u , t ) = 2 u T Ru + 1 x T Qx 2 Final Cost: W (x ) = 1 x T QT x 2 Optimal cost-to-go: J (x , t ) = 1 x T V (t )x + a(t ) 2 Optimal controls: u (x , t ) = −R −1 B T V (t )x ˙ Dynamics of V: −V = Q + AT V + VA − VBR −1 B T V 1 Dynamics of a: −a = 2 Tr (FF T V ) ˙ Nick Jones Elements of Stochastic Optimal Control: ICDNS A more general class of system Dynamics: dx = (b (x , t ) + Bu )dt + d ξ Uncontrolled dynamics: arbitrary. Noise: Gaussian and uncorrelated (assumed independent of u ). Control: limited to be linear. 1 Cost: w (x , u , t ) = 2 u T Ru + Q (x , t ) Costs are quadratic in the control but are arbitrary otherwise. Final cost: W (x ) = φ(xT ) Constraint: ν = λBR −1 B The constraint has two notable features that we can observe in the case B = I and R , ν diagonal. First we cannot have control in dimension i if νii = 0. Second if my passive dynamics/noise is large in dimension i (νii large) then it’s cheap for me to push t...
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This document was uploaded on 03/01/2014 for the course EE 208 at Imperial College.

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