This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 steadystate 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 steadystate 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 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 •...
View
Full
Document
This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.
 Fall '08
 BOYD,S

Click to edit the document details