文档介绍：Handout #5
Prof. J. P. How October 19, 2007
. TBD Due: October 26, 2007
Homework Assignment #5
1. A third order system with two inputs and two outputs has the familiar representation
x˙(t) = Ax(t) + Bu(t) (1)
y(t) = Cx(t) (2)
with G(s) = C(sI − A)−1B and
⎡⎤⎡⎤
−3 −1 1 1 0
1 1 −1 1
A = ⎢ 1 −5 −1 ⎥ B = ⎢ 1 1 ⎥ C =
2 ⎣⎦⎣⎦−2 2 2
2 −2 −4 0 1
Suppose that the eigenstructure of A is given to be:
T
• It has an eigenvalue at λ1 = −1 with right eigenvector v1 = 1 0 1 and left
T 1
eigenvector w
1 =
2 1 −1 1
T
• It has an eigenvalue at λ2 = −2 with right eigenvector v2 =
1 1 0 and left
T 1
eigenvector w
2 =
2 1 1 −1
T
• It has an eigenvalue at λ3 = −3 with right eigenvector v3 =
0 1 1 and left
T 1
eigenvector w
3 =
2 −1 1 1
It is a fact that the 2 × 2 transfer function matrix G(s) has the form:
1
G(s) = R (3)
s + 3
where R is a real constant 2 × 2 matrix (. independent of s).
Given this information, what can you conclude about the poles, zeroes, controllability,
observability, stabilizability, and detectability of the system. Quantify your answers
whenever appropriate.
2. For the system
⎡⎤⎡⎤⎡⎤⎡⎤
x˙ 1 x1 1
−1 0
⎢⎥= ⎣⎦⎢⎥+ ⎢⎥ u
⎣⎦ 1 −2 ⎣⎦⎣⎦
x˙ 2 x2 1
⎡ x ⎤
1
⎢⎥
y = −1 1 ⎣⎦− u
x2
1
Cite as: Jonathan How, course materials for