{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

practice-midterm2

# practice-midterm2 - (c Compute a value for the initial...

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

P RACTICE M ID - TERM E XAM L INEAR S YSTEMS Jo˜ao P. Hespanha Please explain all you answers. 1. Consider the nonlinear systems ¨ y + ˙ y + y = u 2 1 . (a) Compute a state-space representation for the system with input u and output y (b) Linearize the system around the solution y ( t ) = 0, u ( t ) = 1, t 0. 2. Consider the system ˙ x = ( A bk ) x + bu , y = cx , where A = bracketleftbigg 0 1 0 0 bracketrightbigg , b = bracketleftbigg 0 1 bracketrightbigg , k = bracketleftbig k 1 k 2 bracketrightbig , c = bracketleftbig 0 1 bracketrightbig where k 1 and k 2 are constant scalars. (a) Compute the system’s transfer function. (b) Determine values for k 1 and k 2 such that the transfer function is equal to s s 2 + s + 1 Hint: Your answer to (a) should appear as a function of the constants k 1 and k 2 . 3. Consider the matrix A = 2 1 0 0 2 0 0 0 3 (a) Compute the characteristic polynomial of A . Is

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.

Unformatted text preview: . (c) Compute a value for the initial state x ( ) : = ± x 1 ( ) x 2 ( ) x 3 ( ) ² ′ such that the solution to ˙ x = Ax , y = ± 1 1 1 ² x is equal to y ( t ) = te − 2 t , ∀ t ≥ . 1 Hint: To solve (c), start by writing y ( t ) as a function of x 1 ( ) , x 2 ( ) , x 3 ( ) and then determine values for these constants to get the desired output. 4. Compute a matrix A ( t ) such that Φ ( t , t ) = 1 e t 2 − t 2 2 − 1 e t 2 − t 2 2 is the state transition matrix of the homogeneous ODE ˙ x = A ( t ) x 2...
View Full 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