文档介绍：Handout #6
Prof. J. P. How October 26, 2007
. TBD Due: Nov 4, 2007
Homework Assignment #6
1. Consider the simple system
s − z
G(s) = o
(s + 3)(s + 4)
(a) Confirm that one possible state-space model is given by:
⎡⎤⎡⎤
−7 1 1
A = ⎣⎦, B = ⎣⎦, C = 1 0 , D = 0
−12 0 −zo
(b) Evaluate the controllability of this system as a function of the value of zo.
(c) Use pole placement techniques to derive the full-state feedback gains
⎡⎤
x1
u = − k1 k2 ⎣⎦
x2
necessary to place the closed-loop poles at the roots of:
2 2
s + 2ζωns + ωn = 0
(d) Use your analytic expressions for k1 and k2 to support the following two claims:
•“The system has to work harder and harder to achieve control as the control­
lability is reduced.”
•“To move the poles a long way takes large gains.”
2. Given the MIMO system,
⎡ 6s + 5 7 ⎤
⎢ s2 + 3s + 2 s + 2 ⎥
⎢⎥
G(s) = ⎢⎥
⎣ 5 6s + 17 ⎦
s + 2 s2 + 5s + 6
Develop a state space model using:
(a) The technique described at the bottom of page 10–5
(b) Gilbert’s realization method on page 10–6/7
Confirm that the two state space models give the same transfer function matrix.
1
Cite as: Jonathan How, course materials for Feedback Control Systems, Fall 2007. MIT OpenCourseWare
(), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].