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SAT Assignment: Observer Design and System Controllability and Observability, Exams of Electrical and Electronics Engineering

Rensselaer Polytechnic Institute (RPI)Electrical and Electronics Engineering

Information about an engineering assignment related to observer design and system controllability and observability. Instructions for designing an observer for a given system, testing the system for complete observability and controllability, and finding the relationship between control gains and pole locations. From the department of electrical, computer, and systems engineering at rensselaer polytechnic institute and is due in fall 2004.

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

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S.A.T.
Due: Thursday, October 28th
SCHOOLOFENGINEERING
DEPARTMENTOFELECTRICAL,
COMPUTER,ANDSYSTEMSENGINEERING
AssIgnment #7 Fall 2004
1. a.) Design an observer of lowest possible order for the following completely observable
system, (10pts)
-4 0 0
1
1
i(t) =[I -4 0x(t) +0
LO 1 0 to
[
0 1 0
]
Y(0 =X(0
001
0
0Iu(t)
1
A =1. = -4 A =0
1"'2 '3
b.) What is the fmal equation that describes the dynamics of the observer? (2 pts)
c.) How do you reconstruct the states from your observer.
2.
(3 pts)
Test the following systems for complete controllability and complete observability,
a.) i(t) =[~
yet) =[~
1
][
1 0 1
]
x(0 +U(0
0 0 0 1
:]X(t)
(i.) First, do the problem by hand. Remember, you only need to find nindependent columns of the
controllability and observability matrices, Le.,you may not need to calculate every column; stop
when you find nindependent columns. (8 pts)
(ii.)Checkyour resultsusing the MATLAB commands ctrb and obsv .You will need to provide the
A, B and C matrices in the usual way. Then the command ctrb(A, B) will return the controllability
test matrix.You canuse the rank command to determineif the system is completely controllable.
In a similar manner, obsv(A, C) returns the observability test matrix. (8 pts)
b.) [
0 4
i(t) =0 20
0 0
y(t) =[- 1 3
3
] [
-1
16 x(0 +3 IU(0
-20 0
0]x(t)
(8 pts)
Rensselaer Polytechnic Institute
110 8th Street I Troy, NY 12180-3590 USA
Fax {518) 276-6261 or (518) 276-4403
pf3

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Download SAT Assignment: Observer Design and System Controllability and Observability and more Exams Electrical and Electronics Engineering in PDF only on Docsity!

S.A.T.

Due: Thursday, October 28th

SCHOOLOFENGINEERING DEPARTMENTOFELECTRICAL, COMPUTER,ANDSYSTEMSENGINEERING

AssIgnment #7 Fall 2004

1. a.) Design an observer of lowest possible order for the following completely observable

system, (10 pts)

1

i(t) =[I -4 0 x(t) + 0

LO 1 0 to

[

]

Y(0 = X(

0 I u(t) 1

A =1. = -41"'2 '3A =

b.) What is the fmal equation that describes the dynamics of the observer? (^) (2 pts)

c.) How do you reconstruct the states from your observer.

(3 pts)

Test the following systems for complete controllability and complete observability,

a.)

i(t) =[ ~

yet) = [ ~

1

] (^) [

1 0 1

]

x(0 + U( 0 0 0 1

: ]X(t)

(i.) First, do the problem by hand. Remember, you only need to find n independent columns of the controllability and observability matrices, Le., you may not need to calculate every column; stop when you find n independent columns. (8 pts)

(ii. )Check your results using the MATLAB commands ctrb and obsv. You will need to provide the

A, B and C matrices in the usual way. Then the command ctrb(A, B) will return the controllability test matrix. You can use the rank command to determine if the system is completely controllable. In a similar manner, obsv(A, C) returns the observability test matrix. (8 pts)

b.) [

i(t) = 0 20

y(t) =[ - 1 3

] [

16 x(0 + 3 IU(

0 ]x(t)

(8 pts)

Rensselaer Polytechnic Institute 110 8th Street I Troy, NY 12180-3590 USA Fax {518) 276-6261 or (518) 276-

Problem 7.2a, page 391.

a.) Is the system completely observable; (1 pt)

b.) Find the L matrix (in I s I - (A - LC~I) that will place the two closed-loop poles at the

specified locations. I (10 pts)

c.) Show the system and the observer OJa block diagram. (3 pis) The system below is not completely obs~rvable.

-4 0 0 1

i(t) =I 0 -4 1 x(t) + olu(t) 0 1 0 1

y(t) =[0 1 0 ]x(t)

0 L 0

In fact, the observability test matrix is eo = I 0 I 4 1

The state, X2(t), is observable at the output y(t). Design an observer to reconstruct xJ(t). Place the eigenvalue ofthe observer at -5. Be sure to spec~fythe complete observer, i.e., make it clear how to reconstruct xJ (t). I (15 pts)

a.)

b.)

c.)

d.)

The system below is not completely con

-4 0 O

J

i(t) =I 0 -4 0 x(t) + olu(t) LO 1 0 1

y(t) =[0 1 0 ]x(t)

What is the rank of the controllability test matrix? (2 pts)

What are the possible pole locations ifJe use output feedback, u(t) = -ky(t)? (8 pts)

For the same system, use full-state feeJ..k, u(t) = -k,x, (t) - 'sX2(t) - k,x,(t), and fmd k1,k2, and kJ to move two of the closediloop poles to -5 and -10. (8 pts)

Where is the third pole? (2 pts)