Chapter 4 The Stability of Linear Feedback Systems

Illustrations

Chapter 4

The Stability of Linear Feedback Systems

4 - 1

Illustrations

4 - 3

Absolute Stability, Relative Stability

2 4 1 3

1 3 1

1 3 1 5 1 3

1 1 1

1 1 1

, ,

n n n n n n

n n n

n n n n n n

n n n

a a a a a a

b b c

a a a a b b

a a b

   

  

     

  

  

  

Routh array

2 4

1

1 3 5

2

1 3 5

3

1 3 5

0

1 n

n n n

n

n n n

n

n n n

n

n n n

n

s a a a

s a a a

s b b b

s c c c

s h

 

   

   

   



   

   

   

4 - 4

Illustrations

0 0

1 0

1 1

0 2

b s

a s

a a

s

4 - 5

Illustrations

4 3 2

1 1

1 0

2 4 10 6 0 10 0 0 0 10 0 0 s

s

s c

s d

s



4 - 8

Illustrations

0 0 2

0

2 1 8

K s

s

K s

K

Stable

4 - 10

𝑑𝑈

𝑑𝑠 = 4𝑠 + 0 0

8

0 0

8 2

4 1

0 1 2 3

s s s s

4 0

Illustrations

1

( )

2

( 2) ( ) ( 2)( 2 )( 2 ) q s  s  U s  s  s  j s  j

4 - 11

( ) 2 1 ( 1) U s  s  s   s 

4 - 12

𝑈 𝑠 = 𝑠² + 1

𝑑𝑈

𝑑𝑠 =2𝑠 0

1

0 1

0 2

1 0 1

0 1

0 1

2 3

s s s s s

𝑑𝑈

𝑑𝑠 =4𝑠³ + 4𝑠 4 4

2

Illustrations

4 - 13

𝑑𝑠 =42𝑠 42

Illustrations

observing the relative real part of each root. In this diagram r₂ is relatively more stable than the pair of roots labeled r₁.

4 - 16

Illustrations

4 - 17

𝑠 = 𝑠_𝑛 − 1

vehicle. Select K and a so that the system is stable. The system is modeled below.

4 - 18

Illustrations

Ka s

c s

Ka b

s

K s

Ka s

0

3 1

3 2

3

8 ( 10 ) 0

17 1



4 - 19

Illustrations