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sol5 - EE 221 Linear Systems HW#5 Solutions Sam Burden 1...

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EE 221 : Linear Systems HW #5 Solutions — Sam Burden — Oct 15, 2010 1 Note these solutions are overly terse and leave out some details; in your solutions, you should strive for greater clarity and rigor. Exercise 1. Time invariance is straightforward to verify. Let T s be the time- s shift operator and L an input/output map with ( Lu )( t ) = y ( t ). Then: T s Lu ( t ) = Z t + s -∞ e - ( t + s - τ ) u ( τ ) = Z t -∞ e - ( t - σ ) u ( σ + s ) = LT s u ( t ) with σ = τ + s. The issue of whether/how y may be seen as the readout or state transition map of a dynamical system was discussed in class. One way to think about it: consider the linear system ˙ x = - x + u . We know this is a dynamical system, and it yields the state transition map s ( t, τ, x 0 , u ) = e - ( t - τ ) x 0 + Z t τ e - ( t - σ ) u ( σ ) dσ. Using the readout map ρ ( t, τ, x 0 , u ) = s ( t, τ, x 0 , u ), we see that y ( t ) = lim τ →-∞ ρ ( t, τ, x 0 , u ). Exercise 2. With x = ( r, θ, ˙ r, ˙ θ ), u = ( u 1 , u 2 ), and f ( x ) = ( ˙ r, ˙ θ, - k r 2 + r ˙ θ 2 + u 1 , - 2 ˙ r ˙ θ r + 1 r u 2 ), ˙ ξ | O D x f | O ξ + D u f | O μ = 0 0 1 0 0 0 0 1 2 k p 3 + ω 2 0 0 2 0 0 - 2 ω p 0
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