This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 1 q , R = [ 1 ] , A = B , Q = = , 2 2 1 q where q 1 > 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 . 2 2 ≥ 0, this will always be satisfied. 2 1 Since q ≥ and q 2. The matrix R must be positive definite, i.e. R . 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 . If ( A, B, C z ) is also observable, then P . We have that 1 M c = [ B AB ] = 1 ; 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. 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 a b a b 1 q 1 2 = + + 2 1 b c b c q 2 a b 1 a b − b c 1 1 [ 0 1 ] b c a q 1 2 = + + 2 a b b q 2 a b a b − b c 1 b c q 1 2 a a b =...
View
Full
Document
This note was uploaded on 02/03/2012 for the course AERO 16.30 taught by Professor Ericferon during the Fall '04 term at MIT.
 Fall '04
 EricFeron

Click to edit the document details