HW#13

Due Tuesday, March 9

0. Pick a lab partner to work with Your first lab is on its way!!!

1. Given the following state variable model:

a) Find the characteristic equation for A, , then verify the Cayley-Hamilton theorem by substituting A for s.

b) Use your answer to part a) to help you find the eigenvalues and eigenvectors of A (you may check your answer on Matlab)

c) Is the system asymptotically stable, marginally stable, or unstable?

d) Using any of the three methods, find the state transition matrix,

e) Use the equation, to find x(t) when w(t)=u(t) and identify the zero-input and zero-state portion of your solution. Plug x(t) into the output equation to find y(t).

f) Check your answers using Laplace Transforms

g) Use the property of Linearity and Time Invariance to solve,

when the input is w(t)=3u(t-4).

2. Consider the system,

a) Use the similarity transformation, x=Pz, (where P is the matrix of eigenvectors) to decouple the system as follows:

b) Solve the decoupled system in problem 2a) then find x(t) from x(t)=Pz(t). Does your answer agree with problem 1e)?

c) Assuming that the intial state is zero, draw a block diagram of the decoupled system in problem 2a) and identify which eigenvalues are: i) controllable ii) observable iii) stable

d) From your block diagram, find the transfer function H(s)=Y(s)/W(s). Are the poles of H(s) the same as the eigenvalues? Why not?