I
October EE5307
EXAM 11, 2007
Name (Print): ___________________________ (Last) (First) I.D.: ___________________________ Solve ALL THREE problems. Time: 1 hr. 30 min. Maximum Score: 36 points. Problem 1 (a) Set up the state-variable description for the following circuit with input u, output y and state variables x1 and x2. (8 points) R1
x1
u(t)
R2
C L Current x2
y(t)
(b) Can x1 be deduced from a measurement of u and x2 over finite time? If not, when is this impossible? (3 points) Solution:
Problem 2 Consider the following system with input u and output y:
u
1 s k
x1
4
y
-1
1 s 2
x2
(a) The relation between u and y can be expressed as a 1 y a 2 y b 0 b1 u b 2 u y u Determine the coefficients as and bs in terms the of parameter k.
(6 points)
(b) Suppose k=0. Let u ( t ) 1, t 5, and let x 1 (5) 2 , x 2 (5) 0. Calculate the value of x 2 (3). (6 points) Solution:
y 1 1x 3u into an equivalent description z Azz bzu ( A z is a diagonal matrix) y cT z d z u z using the transformation x Vz where the (1,1) and (2,2) elements of V are chosen to be unity. (8 points) (a) Calculate V, A z , b z , c T and dz. z (b) Let the input u be identically zero. Determine all points on the unit circle in the figure below such that if the system starts from these points, the system trajectory will converge to the origin. (5 points)
x2
Problem 3 Convert the following system with input u and output y 0 4 0 x x u 1 0 1
1
-1 -1 Solution
1
x1