examples_2009_09_21_01

examples_2009_09_21_01 - 2-1 Linear Algebra Review S Lall...

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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
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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 .
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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 0 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
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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
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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 )
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2 - 6 Linear Algebra Review S. Lall, Stanford 2009.09.21.01 signal and interference power in wireless system n transmitter/receiver pairs transmitter j transmits to receiver j (and, inadvertently, to the other receivers) p j is power of j th transmitter s i is receiver signal power of i th receiver z i is receiver interference power of i th receiver G ij is path gain from transmitter j to receiver i we have s = Ap and z = Bp where a ij = braceleftBigg G ii for i = j 0 otherwise b ij = braceleftBigg 0 for i = j G ij otherwise A is diagonal; B has zero diagonal we’d like A ‘large’, B ‘small’
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