8/3/2019 ENEE 660 HW #7

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Electrical and Computer Engineering Department

University of Maryland College Park

ENEE 660

System Theory

Fall 2008

Professor John S. Baras

Homework Set #7

DUE: Monday, December 15, 2008

Problem 1

A given system is modeled by the two-input three output transfer function matrix:

2

2

2 2

10

( 3)

2( ) 0( 4)

2 1

( 5) ( 5)

s

s

sT ss

s s

s s

+ +

+= + + +

+ +

(a)Find the zeroes and poles of each scalar transfer function between each input andeach output.

(b)Is the union of the zeroes in (a) the zeroes of the system? Is the union of the polesin (a) the poles of the system? If yes justify your answer if not find the zeroes andpoles of the system.

(c)Find a left coprime matrix fraction description of the given matrix transferfunction.

Problem 2

Consider the unstable, linear time invariant system

[ ]

0 1 0( ) ( ) ( )

1 0 1

( ) 0 1 ( )

x t x t u t

y t x t

= +

=

that resulted from an engineering design problem you are considering.

(a) An engineer colleague suggests that you can use constant state feedback tostabilize the system. He indeed argues that by using the control

( ) ( ) ( )u t K x t r t = + , where K is a constant 1 x 2 gain vector, and r(t) a reference

input, the degree of stability of the closed system can be made arbitrary. Do you

agree or not? Justify your answer. Can you make the closed loop systemexponentially stable by such a control? Justify your answer. If you agree find

such a K.

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(b) An engineer colleague suggests that you can stabilize the system with constantoutput feedback. That is, by using the control ( ) ( ) ( )u t Ly t r t = + , where L is a

constant gain and r(t) a reference input, the closed loop system can be stabilized.Do you agree or not? Justify your answer. Can you make the closed loop system

exponentially stable by such a control? Justify your answer. If you agree find

such an L.(c) An engineer colleague suggests that dynamic output feedback compensation can

be used to render the closed loop system exponentially stable with arbitrary

degree of stability. This dynamic compensator is a linear time invariant systemthat accepts as inputs u(t), y(t) and a reference input r(t), and produces the

control u(t). Do you agree or not? Justify your answer. If you agree construct

such a dynamic compensator.

Problem 3

Consider the system with open loop transfer function matrix:

2

2

1

0( )1 1

1

s

sT s

s s s

+

=

Design a dynamic compensator that can place the poles of the closed loop system ats = -1, -1 + j, -1 - j. Use any method you like to accomplish this.

Problem 4

Consider a two input two output system described by the transfer function:

1 1

1

1

12

( ) 2

0

s

s s

s

s

s

T s

++

=

We want to design, using polynomial matrix techniques, an output dynamic compensator

so that all of the closed loop poles of the system are at 1s = . Note that apart from thisrequirement you have quite a number of options.

(a)Construct an appropriate Matrix Fraction Description for ( )T s to help design such acompensator.

(b)Describe the method and develop the equations that determine the desiredcompensator (i.e. its parts from the input and the output of the system to the control).Show the appropriate block diagram to aid your development.

(c)Solve the equations and construct the compensator.