lqr_2009_11_11_02

lqr_2009_11_11_02 - 15 - 1 The Linear Quadratic Regulator...

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Unformatted text preview: 15 - 1 The Linear Quadratic Regulator S. Lall, Stanford 2009.11.11.02 15. The Linear Quadratic Regulator Regulation and the least squares formulation of regulation The LQR problem formulation Constrained optimization formulation Dynamic programming example: path optimization Solving the Hamilton-Jacobi equation The Riccati recursion Summary of LQR solution via DP Example: force on mass The steady-state regulator Time-varying systems and tracking problems Infinite-horizon problems The Algebraic Riccati equation 15 - 2 The Linear Quadratic Regulator S. Lall, Stanford 2009.11.11.02 The Key Points of This Section idea of regulation; keep the output small, using as little input as possible multi-objective problem: allows trade-off to be made between input effort and regu- lation can be formulated as a large least squares problem instead, solve it via dynamic programming solution is Riccati recursion ; much faster to compute controller is linear state feedback u ( t ) = K t x ( t ) we often use the steady-state solution; to find it, solve the Algebraic Riccati Equation 15 - 3 The Linear Quadratic Regulator S. Lall, Stanford 2009.11.11.02 Regulation usual discrete-time system x ( t + 1) = Ax ( t ) + Bu ( t ) x (0) = x y ( t ) = Cx ( t ) multiobjective problem regulation: keep y ( t ) small on t = 0 ,...,N 1 ; wed like to keep small J out = N 1 summationdisplay t =0 bardbl y ( t ) bardbl 2 using low input effort ; wed like to keep small J in = N 1 summationdisplay t =0 bardbl u ( t ) bardbl 2 15 - 4 The Linear Quadratic Regulator S. Lall, Stanford 2009.11.11.02 least-squares formulation as before we have y (0) y (1) y (2) . . . y ( N 1) = CB CAB CB . . . . . . CA N 2 B CA N 3 B ... CB u (0) u (1) u (2) . . . u ( N 1) + C CA CA 2 . . . CA N 1 x (0) = Lu seq + Mx multiobjective least squares problem: J out ( u seq ) + J in ( u seq ) = bardbl Lu seq + Mx bardbl 2 + bardbl u seq bardbl 2 = vextenddouble vextenddouble vextenddouble vextenddouble bracketleftbigg L I bracketrightbigg u seq + bracketleftbigg Mx bracketrightbiggvextenddouble vextenddouble vextenddouble vextenddouble 2 least-squares solution is open-loop ; does not use measurements of x ( t ) on t = 0 ,...,N 1 15 - 5 The Linear Quadratic Regulator S. Lall, Stanford 2009.11.11.02 cost function J ( u seq ) = J out ( u seq ) + J in ( u seq ) = N 1 summationdisplay t =0 bardbl y ( t ) bardbl 2 + bardbl u ( t ) bardbl 2 = N 1 summationdisplay t =0 x ( t ) T C T Cx ( t ) + u ( t ) T u ( t ) well use the slightly more general cost function J ( u seq ) = N 1 summationdisplay t =0 parenleftBig x ( t ) T Qx ( t ) + u ( t ) T Ru ( t ) parenrightBig + x ( N ) T...
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lqr_2009_11_11_02 - 15 - 1 The Linear Quadratic Regulator...

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