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section8 - EE221A Section 8 1 Administrivia Midterms still...

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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 ) = ˆ t t 0 sin( t - τ ) u ( τ ) 3 Liebniz integration ∂z ˆ b ( z ) a ( z ) f ( x, z ) dx = ∂b ∂z f ( b, z ) - ∂a ∂z f ( a, z ) + ˆ b ( z ) a ( z ) ∂f ∂z dx 4 Solving linear time-varying ODE’s Suppose you have the linear time varying scalar ordinary differential equation ˙ x ( t ) = a ( t ) x ( t ) + g ( t ) , x ( t 0 ) = x 0 Claim. This equation is solved by ψ ( t ) = x 0 φ ( t, t 0 ) + ˆ t t 0 φ ( t, s ) g ( s ) ds, where φ ( t, s ) = exp ´ t s a ( τ ) . Proof. First consider the initial condition, ψ ( t 0 ) = x 0 φ ( t 0 , t 0 ) + ˆ t 0 t 0 φ ( t, τ ) g ( τ ) = x 0 exp ˆ t 0 t 0 a ( τ ) = x 0 1
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5 State transition matrix 2 Now, take the derivative: d dt ψ ( t ) = x 0 d dt φ ( t, t 0 ) + d dt ˆ t t 0 φ ( t, τ ) g ( τ ) = x 0 d dt exp ˆ t t 0 a ( τ ) + φ ( t, t ) g ( t ) + ˆ t t 0 g ( τ ) d dt φ ( t, τ ) = x 0 exp ˆ t t 0 a ( τ ) d dt ˆ t t 0 a ( τ ) + g ( t ) + ˆ t t 0 g ( τ ) d dt exp ˆ t s a ( σ )
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