Analysis of Control in State Space: Modern Control Systems, Slides of Automatic Controls

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Modern Control Systems

Dr. Mohammad Al Janaideh

Analysis of Control in State Space

• Introduction.
• State-space representation of Transfer

function systems.

• Transformation of systems with Matlab.
• Controllability and observeability
• Solving time Invariant state equation.
• Vector matrix analysis.

Controllable Canonical Form

• Consider a system defined by:
• where

y n^  a 1 yn ^1  ...  an  1 y   an  boun  b 1 un ^1  ... bnu

n n n n UY ss  bsosn  abssn   abn ss  abn  

  1 1 1

Controllable Canonical Form

a 

x xn n 

x

x

1

2

1 . 

a (^) n an an ... a

x xn n 

x

x

1

2

1 . 

  b u

x

x

x y b a b b a b b a b o n

n n o n n o o  

     ... ...  ..^2

1 1 1 1 1

Observable Canonical form

a 

x xn n 

x

x

1

2

1 . 



1

1 0 0 1

a

a

a n

n

...

x xn n 

x

x

1

2

1 . 

  o

n n o

n n o

b a b

b a b

b a b

1 1

1 1 .

  b u

x

x

x y (^) o n

 ... ..^2

1 0 0 0 1

Example (2)

a

s s s

s s U s Y s ( )

bo  0 , b 1  1 , b 2  5 , b 3   9 a 1^ ^ ^6 ,^ a 2 ^4 , a 3 ^12

u x

x

x x

x

x 

3

2

1 3

2

1 

3

2

x

x

x y

Example (3)

(^35172497613460) 2 UY (( ss ) ) ^ s  s  s  s  s  UY (( ss )^ ) ^ s^2 ^1  s ^16  s^ ^410 xxx xxx u   

   

    

   

   

   

       

   

 11

1 00 06 010

1 0 0 32

1 32

1 



  

   

   32

2 1 4 1 xx y x

Jordan Canonical Form

When the denominator polynomial involves multiple roots. (( )) ( ) ( ...)...( n n ) n o n n UY ss ^ b ss  pb s s  p  bs   s p  b

 1 3 2 1 1 1 o (^) s p s cnpn UY (( ss )) ^ b  ( s  cp )  s  cp   c ...  (^1132233)

 



 





 



 



   

 







 

 







 



 





 



 







 

    



 





 



 





1

11

00

0 0

0 00 00 0 1 0 0

(^100021) (^14) 21 1 1 .. ..

..

..... . .. ... .. .. .. . .. .. ... ...

..

.. n n x^ n

xx

p

p p

p p

x

xx



   b u x

y c c c xx o n

n  

 

 

 

  ... ..

1 1 2