examples_2009_09_21_01_2up

# examples_2009_09_21_01_2up - 2 1 Linear Algebra Review S...

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Unformatted text preview: 2 - 1 Linear Algebra Review S. Lall, Stanford 2009.09.21.01 2. Examples and Review • Linear equations • Application examples • Control interpretation • Example: forces on rigid body • Estimation Interpretation • Example: navigation • Block matrices • Block diagrams 2 - 2 Linear Algebra Review S. Lall, Stanford 2009.09.21.01 Linear equations some familiar equations: y 1 = a 11 x 1 + a 12 x 2 + · · · + a 1 n x n y 2 = a 21 x 2 + a 22 x 2 + · · · + a 2 n x n . . . y m = a m 1 x 2 + a m 2 x 2 + · · · + a mn x n write this as y = Ax , where y = y 1 y 2 . . . y m A = a 11 a 12 ... a 1 n a 21 a 22 ... a 2 n . . . . . . a m 1 a m 2 ... a mn x = x 1 x 2 . . . x n this defines a map from R n to R m ; this map is linear ; that is A ( x + y ) = Ax + Ay A ( λx ) = λAx for any x,y ∈ R n and any λ ∈ R . 2 - 3 Linear Algebra Review S. Lall, Stanford 2009.09.21.01 Engineering Examples final position/velocity of mass from applied forces • unit mass, with zero position/velocity at t = 0 , subject to force f ( t ) for ≤ t ≤ n • f ( t ) = x j for t in the interval [ j − 1 ,j ) . ( x is the sequence of applied forces, constant in each interval) • y 1 , y 2 are final position and velocity (i.e. at t = n ) f we have y = Ax • a 1 j gives influence of applied force during j − 1 ≤ t < j on final position • a 2 j gives influence of applied force during j − 1 ≤ t < j on final velocity 2 - 4 Linear Algebra Review S. Lall, Stanford 2009.09.21.01 heating system with multiple heating elements • x j is power of j th heating element • y i is change in steady-state temperature at location i • thermal transport via conduction sensor location heating element x 1 x 2 x 3 x 4 x 5 we have y = Ax • a ij gives influence of heater j at location i (in ◦ /W ) • j th col. of A gives pattern of steady-state temperature rise due to 1 W at heater j • i th row shows how heaters affect location i 2 - 5 Linear Algebra Review S. Lall, Stanford 2009.09.21.01 illumination with multiple lamps • n lamps illuminating m (small, flat) patches, no shadows • x j is power of j th lamp • y i is illumination level of patch i illumination y i lamp power ò ij r ij x j y = Ax , where • a ij = r − 2 ij cos θ ij • j th column of A shows illumination pattern resulting from lamp j (at 1 W ) 2 - 6 Linear Algebra Review S. Lall, Stanford 2009.09.21.01 signal and interference power in wireless system n transmitter/receiver pairs • transmitter...
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examples_2009_09_21_01_2up - 2 1 Linear Algebra Review S...

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