The Johns Hopkins UniversityDepartment of Electrical and Computer Engineering

520.454 Control Systems Design Spring 2009

Problem Set #1

Due: Friday, February 13, 2009. In class.

Problems:

1. Consider the following system:

G(s) = 2s+ 1 +

(s+ 1)(s+ 2).

(a) Compute the unit step response.

(b) Analyze your result for [1, 1].2. Consider a system where

P (s) =1

s+ 1, C(s) =

kps+ kis

, F (s) = 1.

(a) Write down the characteristic polynomial for the system.

(b) Choose the coefficients kp and ki to make the system stable by placing both closed-looppoles at s = 2.

(c) Suppose that the feedback sensor is not perfect, but that instead is given by a loss-passfilter of the form:

F (s) =

s+ .

Determine the stability of the system for varying . That is, is there some value of for which the system loses stability?

3. Consider the two nonminimum phase systems:

G1(s) = 3(s 2)(s+ 2)(s+ 3) ,

G2(s) =12(s 1)(s 2)

(s+ 2)(s+ 3)(s+ 4).

(a) Use the Matlab step response command to obtain unit step responses for both transferfunctions.

(b) Explain the difference in the behavior of the two responses as it relates to the zerolocations.

(c) Consider a stable, strictly proper system with m zeros, n poles, where n > m. Showthat the step response has an undershoot (i.e. it initially starts in the wrong direction)if and only if the transfer function has an odd number of real RHP zeros.