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Unformatted text preview: EE221A Section 8 10/21/11 1 Administrivia Midterms still being graded.. HW5 is out, due next Thurs (Oct 27) GSI office hours Mon Oct 24th time change to 1 PM (still in 504 Cory) 2 Dynamical systems Exercise 1. Show that the following system is time invariant: x ( t ) = t t sin( t- ) u ( ) d 3 Liebniz integration z b ( z ) a ( z ) f ( x,z ) dx = b z f ( b,z )- a z f ( a,z ) + b ( z ) a ( z ) f z dx 4 Solving linear time-varying ODEs Suppose you have the linear time varying scalar ordinary differential equation x ( t ) = a ( t ) x ( t ) + g ( t ) , x ( t ) = x Claim. This equation is solved by ( t ) = x ( t,t ) + t t ( t,s ) g ( s ) ds, where ( t,s ) = exp t s a ( ) d . Proof. First consider the initial condition, ( t ) = x ( t ,t ) + t t ( t, ) g ( ) d = x exp t t a ( ) d = x 1 5 State transition matrix 2 Now, take the derivative: d dt ( t ) = x d dt ( t,t ) + d dt t...
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.
- Fall '10