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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.
˙
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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
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Cost: w (x , u , t ) = 2 u T Ru + 1 x T Qx
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Final Cost: W (x ) = 1 x T QT x
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Optimal costtogo: J (x , t ) = 1 x T V (t )x + a(t )
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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
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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.
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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.
 Spring '14
 NickJones
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