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MIT16_30F10_rec08

# MIT16_30F10_rec08 - 16.30/31 Fall 2010 Recitation 8 Brandon...

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16.30/31, Fall 2010 Recitation # 8 Brandon Luders November 1, 2010 In this recitation, we revisit LQR to perform a bit more analysis, then use it in part of the design of a DOFB controller. 1 LQR Revisited Suppose you are asked to solve the LQR problem specified by the following four matrices: 2 1 0 1 0 0 q , R = [ 1 ] , A = B , Q = = 0 0 , 2 2 1 0 q where q 1 > 0 and q 2 > 0. Note that this is equivalent to HW 5, Problem 3, part (a), with ρ = 1 and M = 1 kg, but with a different choice of Q . Thus we’re dealing with a double integrator with state x = [ x v ], where x is position/displacement and v is velocity, and input u corresponding to the force applied. First, let’s check our assumptions on the LQR problem: 1. The matrix Q must be positive semidefinite, i.e. Q 0 . 2 2 0, this will always be satisfied. 2 1 Since q 0 and q 2. The matrix R must be positive definite, i.e. R 0 . Since 1 > 0, this is satisfied. 3. The solution P to the algebraic Riccati equation is always symmetric, such that P T = P . We will use this below. 4. If ( A, B, C z ) is stabilizable and detectable, then the correct solution of the algebraic Riccati equation is the unique solution P for which P 0 . If ( A, B, C z ) is also observable, then P 0 . We have that 0 1 M c = [ B AB ] = 1 0 ; this is full rank, so the system is controllable. If we choose C z = I 2 , it is immediately clear that the system is also observable. Thus, when solving for P below, we can use the fact that P 0. We can represent P symbolically as a b P = , b c where a , b , and c are scalar quantities to be found.

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We now plug everything into the algebraic Riccati equation to solve for P : 0 = A T P + PA + Q P BR 1 B T P � � � � 0 0 a b a b 0 1 q 1 2 0 = + + 2 1 0 b c b c 0 0 0 q 2 � � a b 0 1 a b b c 1 1 [ 0 1 ] b c 0 0 0 a q 1 2 0 = + + 2 a b 0 b 0 q 2 � � � � a b 0 0 a b b c 0 1 b c � � q 1 2 a a b 0 0 = 2 a 2 b + q 2 b c b c q 2 a b 2 bc = a 1 2 b + q 2 2 bc c 2 q 1 2 b 2 a bc = 2 2 .
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