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stoch_lqr

stoch_lqr - EE363 Winter 2008-09 Lecture 5 Linear Quadratic...

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Unformatted text preview: EE363 Winter 2008-09 Lecture 5 Linear Quadratic Stochastic Control • linear-quadratic stochastic control problem • solution via dynamic programming 5–1 Linear stochastic system • linear dynamical system, over finite time horizon: x t +1 = Ax t + Bu t + w t , t = 0 ,...,N- 1 • w t is the process noise or disturbance at time t • w t are IID with E w t = 0 , E w t w T t = W • x is independent of w t , with E x = 0 , E x x T = X Linear Quadratic Stochastic Control 5–2 Control policies • state-feedback control: u t = φ t ( x t ) , t = 0 ,...,N- 1 • φ t : R n → R m called the control policy at time t • roughly speaking: we choose input after knowing the current state, but before knowing the disturbance • closed-loop system is x t +1 = Ax t + Bφ t ( x t ) + w t , t = 0 ,...,N- 1 • x ,...,x N , u ,...,u N- 1 are random Linear Quadratic Stochastic Control 5–3 Stochastic control problem • objective: J = E parenleftBigg N- 1 summationdisplay t =0 ( x T t Qx t + u T t Ru t ) + x T N Q f x N parenrightBigg with Q , Q f ≥ , R > • J depends (in complex way) on control policies φ ,...,φ N- 1 • linear-quadratic stochastic control problem : choose control policies φ ,...,φ N- 1 to minimize J (‘linear’ refers to the state dynamics; ‘quadratic’ to the objective) • an infinite dimensional problem: variables are...
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stoch_lqr - EE363 Winter 2008-09 Lecture 5 Linear Quadratic...

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