N is the process noise or disturbance at time are IID with 0 is independent of with 0 Linear Quadratic Stochastic Control 52 brPage 3br Control policies statefeedback control 0 N called the control policy at time roughly speaking we choo ID: 26157 Download Pdf

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N is the process noise or disturbance at time are IID with 0 is independent of with 0 Linear Quadratic Stochastic Control 52 brPage 3br Control policies statefeedback control 0 N called the control policy at time roughly speaking we choo

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EE363 Winter 2008-09 Lecture 5 Linear Quadratic Stochastic Control linear-quadratic stochastic control problem solution via dynamic programming 5–1

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Linear stochastic system linear dynamical system, over ﬁnite time horizon: +1 Ax Bu , t = 0 , . . . , N is the process noise or disturbance at time are IID with = 0 is independent of , with = 0 Linear Quadratic Stochastic Control 5–2

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Control policies state-feedback control: = 0 , . . . , N called the control policy at time roughly speaking: we choose input after knowing the current state,

but before knowing the disturbance closed-loop system is +1 Ax B ) + , t = 0 , . . . , N , . . . , x , u , . . . , u are random Linear Quadratic Stochastic Control 5–3

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Stochastic control problem objective: =0 Qx Ru with R > depends (in complex way) on control policies , . . . , linear-quadratic stochastic control problem : choose control policies , . . . , to minimize (‘linear’ refers to the state dynamics; ‘quadratic’ to the objective an inﬁnite dimensional problem: variables are functions , . . . , Linear Quadratic Stochastic Control 5–4

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Solution via

dynamic programming let be optimal value of objective, from on, starting at ) = min ,..., Qx Ru subject to +1 Ax Bu we have ) = (expectation over Linear Quadratic Stochastic Control 5–5

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can be found by backward recursion: for , . . . , ) = Qz + min Rv +1 Az Bv expectation is over we do not know where we will land, when we take optimal policies have form ) = argmin Rv +1 Ax Bv Linear Quadratic Stochastic Control 5–6

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Explicit form let’s show (via recursion) value functions are quadratic, with f orm ) = , t = 0 , . . . , N, with = 0 now assume that +1 ) = +1 +1

Linear Quadratic Stochastic Control 5–7

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Bellman recursion is ) = Qz + min Rv (( Az Bv +1 Az Bv ) + +1 Qz Tr WP +1 ) + +1 min Rv + ( Az Bv +1 Az Bv we use +1 ) = Tr WP +1 same recursion as deterministic LQR, with added constant optimal policy is linear state feedback: ) = +1 +1 (same form as in deterministic LQR) Linear Quadratic Stochastic Control 5–8

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plugging in optimal gives ) = , with +1 +1 +1 +1 +1 Tr WP +1 ﬁrst recursion same as for deterministic LQR second term is just a running sum we conclude that are same as in deterministic LQR strangely,

optimal policy is same as LQR, and independent of Linear Quadratic Stochastic Control 5–9

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optimal cost is Tr XP ) + Tr XP ) + =1 Tr WP interpretation: is optimal cost of deterministic LQR, with = 0 Tr XP is average optimal LQR cost, with = 0 Tr WP is average optimal LQR cost, for = 0 = 0 Linear Quadratic Stochastic Control 5–10

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Inﬁnite horizon choose policies to minimize average stage cost = lim =0 Qx Ru optimal average stage cost is Tr WP ss where ss satisﬁes the ARE ss ss ss ss ss optimal average stage cost doesn’t depend on Linear Quadratic

Stochastic Control 5–11

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(an) optimal policy is constant linear state feedback ss where ss ss ss ss is steady-state LQR feedback gain doesn’t depend on Linear Quadratic Stochastic Control 5–12

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Example system with = 5 states, = 2 inputs, horizon = 30 chosen randomly; scaled so max = 1 = 10 ∼N (0 , X = 10 ∼N (0 , W = 0 Linear Quadratic Stochastic Control 5–13

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Sample trajectories sample trace of and 10 15 20 25 30 −4 −2 10 15 20 25 30 −1 blue: optimal stochastic control, red: no control ( = 0 Linear Quadratic

Stochastic Control 5–14

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Cost histogram cost histogram for 1000 simulations 100 200 300 400 500 600 700 100 200 100 200 300 400 500 600 700 100 200 100 200 300 400 500 600 700 100 200 100 200 300 400 500 600 700 100 200 pre ol nc Linear Quadratic Stochastic Control 5–15

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Comparisons we compared optimal stochastic control ( = 224 ) with ‘prescient’ control decide input sequence with full knowledge of future disturbanc es , . . . , u computed assuming all are known pre = 137 ‘open-loop’ control , . . . , u depend only on , . . . , u computed assuming = 0 ol =

423 no control = 0 nc = 442 Linear Quadratic Stochastic Control 5–16

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