{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern