Answered: 3. Consider the system a) Use state…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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3. Consider the system

a) Use state feedback ? = ?? to assign the eigenvalues of ? + ?? at −0.5 ± ?0.5. Plot

TT x(t) = [x1(t), x2(t)] for the open- and closed-loop system with x(0) = [−0.6, 0.4] .
b) Design an identity observer with eigenvalues at −? ± ?, where ? > 0. What is the observer gain ? in this case?
c) Use the state estimate ?̂ from (b) in the linear feedback control law ? = ? ?̂, where ? was found in (a). Derive the state-space description of the closed-loop system. If ? = ? ?̂ + ?, what is the transfer function between ? and ??

TT
d) For ?(0) = [−0.6, 0.4] and ?̂(0)= [0, 0] , plot ?(?), ?̂ (?), ?(?), and ?(?) of the closed-

loop system obtained in (c) and comment on your results. Use ? = 1, 2, 5, and 10, and comment on the effects on the system response.

Remark: This exercise illustrates the deterioration of system response when state observers are used to generate the state estimate that is used in the feedback control law.

3. Consider the system
i = R d[_]u, y = [1 0]x
x +
и,
a) Use state feedback u = Kx to assign the eigenvalues of A + BK at –0.5± j0.5. Plot
x(t) = [x1(t), x2(t)1" for the open- and closed-loop system with x(0) = [-0.6, 0.4]" .
b) Design an identity observer with eigenvalues at -a ± j, where a > 0. What is the
observer gain L in this case?
c) Use the state estimate î from (b) in the linear feedback control law u =
was found in (a). Derive the state-space description of the closed-loop system. If u =
Kî +r, what is the transfer function between y and r?
Kî, where K
d) For x(0) = [-0.6, 0.4]" and £(0)= [0, 0]", plot x(t), £ (t), y(t), and u(t) of the closed-
loop system obtained in (c) and comment on your results. Use a
comment on the effects on the system response.
1,2,5, and 10, and

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