6.003 Lab 3

Nada Amin

                     namin@mit.edu

May 5, 2005

Basic Problems

(a)

Given an unstable system H(s) which satisfies

\frac{dy(t)}{dt} = 2y(t) + x(t)

I determine the closed-loop system function Q(s) as a function of K

Q(s) = \frac{H(s)}{1 + G(s)H(s)}

when G(s) is a simple proportional feedback: G(s) = K.

From the differential equation of H(s), I find that

H(s) = \frac{1}{s − 2}

H(s) has one pole, at s = 2. Since the system is unstable, the ROC must be ℜe{s} > 2 so as not to include ℜe{s} = 0.

Now,

Q(s) = \frac{1}{s − 2} \frac{1}{1 + \frac{K}{s−2}}
Q(s) = \frac{1}{s − 2 + K}

(b)

Q(s) → ∞ when s − 2 + k = 0, thus the pole of the closed-loop occurs at s = 2 − k.