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continuous_time_lqr_2009_11_18_01_2up

continuous_time_lqr_2009_11_18_01_2up - 17 1...

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Unformatted text preview: 17 - 1 Continuous-Time LQR S. Lall, Stanford 2009.11.18.01 17. Continuous-Time LQR • Continuous-time LQR problem formulation • Dynamic programming solution • Hamilton-Jacobi equation • Riccati differential equation • Steady-state regulator • Infinite-horizon regulator 17 - 2 Continuous-Time LQR S. Lall, Stanford 2009.11.18.01 The Key Points of This Section • LQR problem in continuous-time has an integral cost function • solved via dynamic programming • solution is Riccati differential equation • controller is linear state feedback u ( t ) = K t x ( t ) • steady-state/infinite horizon solution gives constant controller 17 - 3 Continuous-Time LQR S. Lall, Stanford 2009.11.18.01 The Continuous-Time LQR Problem continuous-time system ˙ x ( t ) = Ax ( t ) + Bu ( t ) x (0) = x problem: choose u : [0 , T ] → R m to minimize J = Z T x ( τ ) T Qx ( τ ) + u ( τ ) T Ru ( τ ) dτ + x ( T ) T Q f x ( T ) • T is the time-horizon • Q ≥ , Q f ≥ , R > are state cost , final state cost and input cost matrices an infinite dimensional problem: trajectory u is the variable 17 - 4 Continuous-Time LQR S. Lall, StanfordS....
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