Homework 1 Answer

1

Homework 1 Answer

Problem 1

Known the differential equations of a system

( )r t denotes input. ( )c t denotes output.

1( )x t , 2 ( )x t and 3( )x t are intermediate variables .

1K , 2K , 3K , 1T , 2T are all constants .

All initial conditions are zero.

Draw the block diagram and find transfer function of the system.

Answer

1

11 2 2 1 1 2 1

1

3 2 3

22 2 3 3

2

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) 1

( ) ( ) ( )

( ) ( ) ( ) C( ) ( ) 1

X s R s C s

KT sX s X s K X s X s X s

T s

X s X s K C s

KT sC s C s K X s s X s

T s

1

21 2 1 1

3 2 3

2 2 3

( ) ( ) ( )

( )( ) ( )

( ) ( ) ( )

( )( ) ( )

x t r t c t

dx tT x t K x t

dt

x t x t K c t

dc tT c t K x t

dt

Homework 1 Answer

2

Problem 2

Find the transfer function of the system shown in Fig.problem2 (block

diagram reduction)

Fig.problem2

Answer

G1 G2 G3 G4

RC

1 4

1

GG

_3 4

3 41

G G

G G1 2

1 21

GG

GGR C

1 4

1

GG

1 2 3 4

1 2 3 4 2 3 1 2 3 4

( )

( ) 1

GG G GC s

R s GG G G G G GG G G

21221

21

)1)(1()(

)(

3KKKKsTsT

KK

sR

sC

G1 G2 G3 G4

R C