Topic #38 Transfer function to state-space Reference textbook : - - PowerPoint PPT Presentation

1

ME 779 Control Systems

Transfer function to state-space

Topic #38

Reference textbook:

Control Systems, Dhanesh N. Manik, Cengage Publishing, 2012

2

Phase-variable and controller canonical forms

 

1 2 1 2 1 1 2 1 2 1

( ) ( )

m m m m m n n n n n

K s b s b s b s b Y s U s s a s a s a s a

       

        

Transfer function to state-space

3

1 2 1 2 1

( ) ( )

n n n n n

Z s K U s s a s a s a s a

   

    

1 2 1 2 1

( ) ( )

m m m m m

Y s s b s b s b s b Z s

   

    

Transfer function to state-space

Phase-variable and controller canonical forms

4

 

1 2 1 2 1

( ) ( )

n n n n n

s a s a s a s a Z s KU s

   

    

1 2 1 2 1 1 2

( )

n n n n n n n n

d z d z d z dz a a a a z Ku t dt dt dt dt

     

          

Transfer function to state-space

Phase-variable and controller canonical forms

5

Phase variable form

1 2 2 3 2 1 1 n n n

x z dz x dt d z x dt d z x dt

 

   

1 2 2 2 3 2 3 3 4 3 1 1 1 n n n n n n n

dz x x dt d z x x dt d z x x dt d z x x dt d z x dt

  

        

Transfer function to state-space

6

 

1 2 1 2 1 1 2 1 2 1 1 2 1

( ) ( )

n n n n n n n n n n n n

d z d z d z dz Ku t a a a a z dt dt dt dt Ku t a x a x a x a x

        

               

Transfer function to state-space

Phase variable form

7

 

1 2 2 3 3 4 1 1 2 1 1 2 0 1

( )

n n n n n n n

x x x x x x x x x Ku t a x a x a x a x

   

        

Transfer function to state-space

Phase variable form

8 1 1 2 2 3 1 1 2 1

1 1 ( ) 1

n n n n

x x x x x u t x x a a a a x K

 

                                                                            Upper companion matrix

Transfer function to state-space

Phase variable form

9

1 2 1 2 1 1 2

( )

m m m m m m m m

d z d z d z dz y t b b b b z dt dt dt dt

     

          

 

1 1 2 1 1 2 0 1

( )

m m m m m

y t x b x b x b x b x

   

    

Transfer function to state-space

Phase variable form

10

 

1 2 3 1 2 1 1 1

( ) 1

m m m n n

x x x y t b b b b x x x x

  

                              

Transfer function to state-space

Phase variable form

11

Transfer function to state-space

Phase variable form

12

EXAMPLE

Obtain the phase-variable representation of the following transfer function

 

2 4 3 2

20 2 5 ( ) ( ) 3 5 6 7 s s Y s U s s s s s       

Transfer function to state-space

Phase variable form

13

EXAMPLE

1 1 2 2 3 3 4 4 1 2 3 4

1 1 ( ) 1 7 6 5 3 20 ( ) [5 2 1 0] x x x x u t x x x x x x y t x x                                                                           

Transfer function to state-space

Phase variable form

14

Controller canonical form

1 1 1 2 1 n n n n n n

d z x dt d z x dt dz x dt x z

  

   

 

1 1 1 2 2 1 1 1 2 1 1 2 1 2 2 1

( )

n n n n n n n n n n n n

d z x Ku t a x a x a x a x dt d z x x dt d z x x dt dz x x dt

       

           

Transfer function to state-space

15

1 1 1 2 3 2 2 3 1

1 ( ) 1 1

n n n n n n

x x a a a a K x x x u t x x x

   

                                                                           

 

1 1 2 2 1 1

( )

n m m n m m n m n n

y t x b x b x b x b x

       

    

Transfer function to state-space

Controller canonical form

16

 

1 2 1 1 2 1 1 2 1

( ) 1

n m n m m m n m n m n n

x x x x y t b b b b x x x x

         

                                  

Transfer function to state-space

Controller canonical form

17

Transfer function to state-space

Controller canonical form

18

Obtain the controller canonical representation

• f the following transfer function

EXAMPLE

 

2 4 3 2

20 2 5 ( ) ( ) 3 5 6 7 s s Y s U s s s s s       

Transfer function to state-space

Controller canonical form

19

EXAMPLE

1 1 2 2 3 3 4 4 1 2 3 4

3 5 6 7 20 1 ( ) 1 1 ( ) [0 1 2 5] x x x x u t x x x x x x y t x x                                                                           

Transfer function to state-space

Controller canonical form

20

Observer canonical form

1 1 1 1 1 2 1 2 1

1 ( ) ( ) 1

m n m n m n n n n n n

b b b K Y s s s s s a a a a U s s s s s

       

             

Transfer function to state-space

21

1 2 1 2 1 1 1 1 1

( ) 1 1 ( )

n n

• n

n m n m n m n n

a a a a Y s s s s s b b b KU s s s s s

       

                   

Transfer function to state-space

Observer canonical form

22

     

1 1 1 1 1 1

1 1 ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( )

n n m n m n m n m

Y s b KU s a Y s b KU s a Y s s s a a KU s a Y s Y s Y s s s s

     

        

Transfer function to state-space

Observer canonical form

23

Transfer function to state-space

Observer canonical form

24

1 1 1 2 2 2 1 3 1 1 1 1 1 1 0 1

( ) ( ) ( )

n n n m m n m n n n

x a x x x a x x x a x x u t K x a x x b u t K x a x b u t K

     

                

Transfer function to state-space

Observer canonical form

25

1 1 1 2 2 2 1 1 1 1

1 1 1 1 1

n n n m m n m n n n n

x a x x a x x a x K x a x b x a x b

     

                                                                                               ( ) u t

Transfer function to state-space

Observer canonical form

26

 

1 2 2 1

( ) 1

n n n

x x y t x x x

 

                      

Transfer function to state-space

Observer canonical form

27

EXAMPLE

Obtain the observer canonical representation

• f the following transfer function

 

2 4 3 2

20 2 5 ( ) ( ) 3 5 6 7 s s Y s U s s s s s       

Transfer function to state-space

Observer canonical form

28

EXAMPLE

1 1 2 2 3 3 4 4 1 2 3 4

3 1 5 1 1 20 ( ) 6 1 2 7 5 ( ) [1 0] x x x x u t x x x x x x y t x x                                                                           

Transfer function to state-space

Observer canonical form

29

Modal form (parallel form)

 

1 2 1 2 1 1 2 1 2 1 1

( ) ( )

m m m m m n n n n n n i i i

K s b s b s b s b Y s U s s a s a s a s a K s p

        

          



Transfer function to state-space

30

1 2 1 2

( ) ( ) ( ) ( )

n n

K K K Y s U s U s U s s p s p s p      

Transfer function to state-space

Modal form (parallel form)

31

U(s)

Transfer function to state-space

Modal form (parallel form)

32

1 1 1 1 2 2 2 2 3 3 3 3 1 2 3

( ) ( ) ( ) ( ) x p x K u t x p x K u t x p x K u t y t x x x         

Transfer function to state-space

Modal form (parallel form)

33

 

1 1 1 1 2 2 2 2 3 3 1 1 1 2 3

( ) ( ) 1 1 1 1

n n n n n n n

x p x K x p x K x K u t x p x p x K x x y t x x

 

                                                                                          

Transfer function to state-space

Modal form (parallel form)

34

EXAMPLE

Obtain the modal (parallel) canonical representation of the following transfer function

 

20 2 ( ) ( ) ( 3)( 4) s Y s U s s s    

Transfer function to state-space

Modal form (parallel form)

35

40 20 ( ) ( ) ( ) 4 3 C s R s R s s s    

1 1 2 2 1 2

4 40 ( ) 3 20 ( ) [1 1] x x u t x x x y t x                                    

EXAMPLE

Transfer function to state-space

Modal form (parallel form)

Transfer function to state-space

Conclusion

36