lecture28 - Today's goals State space so far Definition of...

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Lecture 28 – Wednesday, Nov. 14 2.004 Fall ’07 Today’s goals State space so far – Definition of state variables – Writing the state equations – Solution of the state equations in the Laplace domain – Phase space and phase diagrams Today – Stability in state space – State feedback control
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Lecture 28 – Wednesday, Nov. 14 2.004 Fall ’07 State space overview From the Equation of Motion to the State—Space representation: m ¨ x ( t )+ b ˙ x ( t )+ kx ( t )= w ( t ) μ x ˙ x q ( t μ q 1 q 2 state ,y ( t ) ˙ x ( t )output ˙ q ( t μ ˙ q 1 ˙ q 2 = μ 01 k/m b/m ¶μ q 1 q 2 + μ 0 1 w ( t ); y ( t )=(0 1) μ q 1 q 2 cq . k 1 b 1 m 1 x 1 tower sway w A = μ b/m b = μ 0 1 Solution to the state equations: s ˆq ( s Aˆq ( s )+ b W ( s ) ( s )=( s I A ) 1 b W ( s ) . Y ( s cˆq ( s c ( s I A ) 1 b W ( s ) .
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Lecture 28 – Wednesday, Nov. 14 2.004 Fall ’07 State space solution to the uncompensated 2.004 Tower model ˆq ( s )=( s I A ) 1 b W ( s )= 1 s 2 +( b 1 /m 1 ) s k 1 /m 1 ) μ 1 /m 1 s/m 1 W ( s ) . From this result we can obtain transfer functions for position, velocity: for position choose c =(1 0) ,X ( s ) Y ( s c ( s I A ) 1 b W ( s ) X ( s ) W ( s ) = 1 /m 1 s 2 b 1 /m 1 ) s k 1 /m 1 ) . for velocity choose c =(0 1) ,V ( s ) Y ( s c ( s I A ) 1 b W ( s ) V ( s ) W ( s ) = s/m 1 s 2 b 1 /m 1 ) s k 1 /m 1 ) . position velocity phase diagram velocity position
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Lecture 28 – Wednesday, Nov. 14 2.004 Fall ’07 Poles are the eigenvalues of A | Stability Consider the eigevalue problem for the matrix A : A ξ = μξ , where the solutions for μ are the eigenvalues and ξ are the eigenvectors . To solve the eigenvalue problem, we set det ( μ I A )=0 . That is, the eigenvalues are the roots of the determinant of the matrix ( μ I A ).
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lecture28 - Today's goals State space so far Definition of...

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