文档介绍：Handout #1
Prof. J. P. How September 7, 2007
. TBD Due: September 14, 2007
Homework Assignment #1
Goals: Refresh skills for Matlab and classical analysis.
1. (Root Locus Analysis) [FPE , page 159]. Suppose that you are to design a unity
gain feedback controller for a first order plant (see figure). A specific controller config­
uration has been selected (it is PI). The objective is to pick the control gains so that
the poles lie in the regions shown in the figure to the right.
Im(s)
4
θ1
θ 2
+ e(t) + Κα 2
r(t) ΣΣΚ s + α y(t)
_ + Re(s)
_4 _2
Κ
l _
s 2
Figure by MIT OpenCourseWare.
Figure 1: Problem # 1
(a) What values of ωn and ζ correspond to these shaded regions in the s-plane Cal­
culate these exactly (they will be nonlinear) and then approximate to something
more useful for design?
(b) Let Kα= α= . Find the values of K and KI so that the poles of the closed-
loop system lie in the shaded region.
(c) Prove that, no matter what the values of Kα and α, the controller provides suffi­
cient flexibility to place the poles anywhere in the s-plane.
2. (Dominant Pole Locations) [FPE (a),(c),(d), page 161]. Consider the second order
system
ω2
H(s) = n
2 2
(s/p + 1)(s + 2ζωns + ωn)
The unit step response is
−pt −σt
y(t) = 1 + Ae + Be cos(ωdt −θ)
where
−ω2 p
A = n B