10. State Feedback and Pole Placement

(1)

System Control System Control

Professor Kyongsu Yi

Vehicle Dynamics and Control Laboratory Seoul National University

Seoul National University

(2)

State Feedback

• State Feedback

x Ax Bu u Kx

 

 



( )

x   A  BK x

( , ) A B

Controllable : arbitrary pole assignment

In general, full states are not available Observer.g ,

ˆ ˆ ( ˆ )

ˆ y Cx

x Ax Bu L y Cx



   



ˆ

( )

e x x e A LC e

 



( , ) A C

Observable : arbitrary pole assignment

x ˆ  x

ˆ 2n system x x

u Kx

x Ax Bu



  

 



 ˆ ˆ ( ˆ ) ˆ

x Ax Bu L y Cx y Cx

u Kx

   



 



2 eigenvalues. n

(3)

State Feedback Control and Pole Placement

K(s)

) (

s A

s

u

B

e +

-





 k



Location of closed loop poles are restricted in root locus.

Q: Is it possible to assign arbitrary closed loop poles?

A: Yes, If the system is controllable & all state variables are feed back single input system in controllable canonical form.

(4)

State Feedback Control and Pole Placement

0 1 0 0 0

 0 1 0 0    0

0 0 1 0 0 0

0 0 0

x x u

   

   

     

   



   



1 1

0 1 0

n n

1

a a

_

a

   

      

   

  

 

1 1

det( sI  A )  s

ⁿ

 a s

1 ⁿ^

   a

_n

 0

x u

r ^u x  AxBu ^x ^c

r

K K

1

u Kx r

x

  

 

1

2

1 2 3 3

[

n

]

x

k k k k x r

   

      

   



 x

   

 



(5)

Closed loop System

[ ]

x Ax B Kx r A BK x Br

   

  



[ ]

0 0

 

 

  

 



  

1

0 1 0 0

k k

n

 

 

 

 

  

 



0 0 1 

0

0 1

x Br

 

   

 



 

1 1

( a

_n

k ) ( a k

_n

)

 

     

    

(6)

Closed loop Ch. Eq

1

1 1 2 1

( ) ( ) ( ) 0

n n

n n n

s  a  k s

^

   a

_

 k s  a  k 

1 2

{ , , } : Desired closed loop poles

: symmetric with respect to the real axis of s-plane

( )( ) ( ) 0

n

s s s

  



  

    



1 2



1 2

1 2 1 2 1 2

( )( ) ( ) 0

[ ( )] [ ] [( )(

m

n n n

n

s s s

  

^

 

^

 



           

1 2

) ( )] 0

0

n

n n n

c c cn

s a s a s a



 

 

     



1 ( 1 )

: State feed back control gain

n i i c n i

i

a k a

k

 

 

 

(7)

Single-Input Controllable System not in controllable canonical form

1

2 1

,

, [

ⁿ

]

M M exist

x Ax Bu rank B AB A B A B n





  

 

1

a a a



1 2 1



1 2

1

1 0

0 0

n n

a a a

a

W a

 

 

 

  

 





 

  

1

1 1 1

1 0 0

c

a

T MW T

^

W M

^ ^

x Tx

 

 

  

  

1 1

: Controllable Canonical form

c

c c

Tx ATx Bu

Tx T ATx T Bu A x B u

 

 



1

c c c

K

A x B u u K x V

K T

^

x V



  

Controllability  arbitrary

 pole assignment.

Necessary and sufficient condition

(8)

State Feedback Control and Pole Placement

Discrete Time Case

( 1) ( ) ( ) , ( ) ( )

( ) ( )

x k Gx k Hu k u k Kx k

G HK x k

    

 

0 1 0 0

0 0 1

0 x k ( )

 

 

  





 ( )

1

0 0 0 0

Nilpotent matrix

 

  

Nilpotent matrix

- Characteristic Equation

( 0)( 0) ( 0) 0

z

n

  ( z 0)( z  0)  ( z   0) 0

( )

ⁿ

0

z z z z

G HK

    

 



- Dead beat responseDead beat response (Finite Settling Time control)(Finite Settling Time control)

( ) ( )

ⁿ

(0) 0 (0)

x n  G  HK x  for any x

In general, Dead beat response should not be attempted In general, Dead beat response should not be attempted with a small sampling time.

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State Feedback Control and Pole Placement

Ex

m x

u ₂

2

2 2( 1)

) 1 ( 1



 

 z

z T ms

u x

If T=0.0000001 sec u

Q: Is X directly accessible for state feed back?

A: No in general A: No, in general.

(10)

State Feedback Control and Pole Placement

Observer / State estimator

ˆ;

ˆ

x state estimator x  x in some sense

Looks reasonable to replace

ˆ u   Kx by u   Kx

,

ˆ ˆ

x Ax Bu y Cx x Ax Bu

  

 



A B

+ -

e

x   Ax  Bu

Bu x A x ˆ   ˆ  e   x x ˆ

ˆ ˆ

( ) ( )

e x x

e x x A x x Ae



     

 

(11)

State Feedback Control and Pole Placement

If A is asymptotically stable, then If A is not stable, then e  are t 

lim ( ) 0

t

e t





 

x   Ax  Bu

+ e

x

c u

) ˆ ( ˆ ˆ

y y L

Bu x A x





 

-

x ˆ

c

ˆ ˆ ( ˆ ) ( ) ( ) ˆ

x   Ax  Bu  L cx cx   A  LC  A  LC x  Bu  LCx

( )

e   A  LC e

(12)

System in obs. Canonical form

1 2

1 0 0

0 1 a

a

  

  

 



l

1

   

2 3

0 1

1

0 0

a

A a

a

 

  

 

 





 1 0 0 

C  

²

n

L l

l

   

  

   



0 0

a

n

  

 

1 1

( a l ) 1 0 0

 

 

  

2 2

( ) 0 1

1 a l

A LC

   

 

   

 

 



0 0

a

n

 

  

 

(A,C) observable Arbitrary pole assignment for (A-LC)

(13)

State Feedback Control / Observer

x Ax Bu y Cx

 





ˆ A ˆ B L ( ˆ )

 ( )

ˆ ˆ

ˆ

x Ax Bu L y y y Cx

u Kx r

   



  

x   Ax  Bu

+ e

x C u +

-

) ˆ ( ˆ ˆ

y y L

Bu x A x







xˆ C

k

ˆ

( )

x   Ax   B Kx  r

 ˆ ˆ ( ˆ ) ( ˆ )

x  Ax   B Kx   r L Cx Cx 

(14)

State Feedback Control / Observer

The order of the overall system is 2n  2n eigenvalues

ˆ _ ˆ

x A BK x B

d r

x LC A BK LC x B

dt

        

 

        

       

ˆ 0

ˆ e x x

x I x

e I I x

 

      

      

      ˆ

( )

( ) ( )

( )

x Ax B Kx BKx A BK x BK e e A LC e

     

   

  





 ( )

0 0

e A LC e

x A BK BK x B

d r

e A LC e

dt

 

          

        

       

(15)

State Feedback Control / Observer

- Eigenvalues are form - Eigenvalues are form

( )

det( ) det

0 ( )

d t( ( )) d t( ( ))

sI A BK BK

sI A

sI A LC

I A BK I A LC

  

 

       

det( ( )) det( ( ))

0

sI A BK sI A LC

observerpoles

    



- n eigenvalues = regulator poles

x Ax Bu u Kx

 

 

^ separation property

- Selection of Best

L

ˆ ˆ ( ˆ )

ˆ ( ) ˆ

x Ax Bu L y y x A LC x

   



 ( )

( )

x A LC x e A LC Ln

 

  



(16)

State Feedback Control / Observer

- In general

i) For fast error convergence Large

ii) If y is not reliable, contaminated by disturbance, and measurement small

L

-Thus,

i) Select several , is choose several pole locations of the observer ii) Simulation

iii) Ch th b t f th i i t f ll t f

L L

iii) Choose the best , from the view point of overall system performance.

L

Compromise between speedy response and sensitivity to disturbances and noise

Kalman Filter

(17)

State Feedback Control / Observer

ˆ x Ax Bu u Kx r

 

  



 ˆ ˆ ( ˆ ) x  Ax  Bu  L y  y

obsever ^controller r

y u

Cx y

Bu Ax x







xˆ 

y compenstor

ˆ ˆ ( ˆ ) ( ˆ )

x   Ax   B ( Kx   r ) LC y Cx (  ) ˆ

( ) ( ) ( ( ))

ˆ( )

x Ax B Kx r LC y Cx

sI A BK LC x s L Y S

U K

   

     

1

ˆ( )

( ) ( )

U Kx s r

K sI A BK LC

^

Y

  

     

(18)

Discrete Time Observer

( 1) ( ) ( )

( ) ( )

x k G x k H u k y k Cx k

    



Observer

 

ˆ ( 1) ˆ ( ) ( ) ( ) ˆ ( )

ˆ ( ) ( ) ( )

( 1) ( ) ( )

e

x k G x k H u k K y k Cx k e k x k x k

e k G K C e k

      

 

  

 

( 1) ( ) ( )

ˆ( 1) is given in terms of (0), (1),..., ( )

e

y

e k G K C e k

x k y y y k



  

ˆ( 1| ) : estimate of ( 1) based on

k

y

x k  k x k  y

  

 x k ˆ(  1| k  1) : estimate of ( x k  1) ba 



sed on

1

ˆ ( 1| 1) ˆ ( 1| ) [ ( 1) ˆ ( 1| )] : Current Observer.

k

e

y

x k k x k k K y k Cx k k







 

        

(19)

Error Dynamics

( 1| ) ( 1) ˆ ( 1| )

( ) ( ) [ ( | ) ( )]

e k k x k x k k

Gx k Hu k Gx k k Hu k

    

   

ˆ ˆ

( ) [ ( | 1) [ ( ) ( | 1)]]

( ) ( | 1)

( 1| 1) ( 1) ˆ ( 1| 1)

e e

Gx k G x k k K Cx k Cx k k G I K C e k k

e k k x k x k k

     

   

      

*

(

_e

) ( | )

e C

G K C G

I K C G e k k

 

  



 : ( ,

^*

)Observable

arbitrary pole assignment G C



( : measurement noise)

y Cx n n

x Ax Bu

 



ˆ ˆ ( ˆ )

( )

y Cx n

x Ax Bu L y Cx e A LC e L n

 

    

   



: Optimal compromise between convergence speed and noise effects Kalman Filter.



  min [( E x  x ˆ ) (

^T

x  x ˆ )]

u   Kx

(20)

Vector Differential

(1)   f ( )  

1 1 1

(1) ( ) :

n n n

f x x y

  



1 2

( ) ( ) ( )

f x f x f x

 

 

   

n

( ) f x

 

 



1 1 1

f f f

  

 

  x

₁

 x

₁

  x

₁

 f

x

    

 

   

  

 



1 1

f f

x x

   

 

   

(2) R u R n n :  u n :  1 (2) R u  R n n :  , u n :  1

Ru R u

 



T T

u R R u

R R

u u

 

              

(21)

Vector Differential

  

^T

1

(3)

n n n n n

u R u



 



T

u Ru

T



_T

u Ru 2 u u R

 



 

T

x u

T

x u x

u

 

1_n n_1

  u

T

u x

T

u x x

u

 



(22)

Review

1. Key Concepts - Modeling

- Control System Design - Controller Design

- Linear, Nonlinear Systems - Design Approach

2. System Representations

- Laplace Transform, Z-Transform

Graphical representation block diagram signal flow graph - Graphical representation, block diagram, signal flow graph - Transfer Function

- State Equation ( cont. discrete. ) - State Eq. T.F.

- Canonical forms - Canonical forms

- Solution of Linear state Eq. : cont. discrete.

- Discrete Representation of cont. time systems

- Relationships between cont. and discrete time eigenvalues - Sampling of cont time systems

- Sampling of cont. time systems

(23)

Review

3. Dynamic Properties of the L.T.I Systems - Stability. Definitions

- Routh stability Criterion - Bilinear Transformation - Lyapunov function

- Stability / In-stability theorems

- Controllability / Observability definitions : cont. discrete Sufficient Conditions for controllability and observability - Pole – Zero cancellations

4 Analysis and Design 4. Analysis and Design

- Root locus. Design concepts. Pole-Zero additions - Frequency response

- Nyquist stability Criterion - Lead – Lag compensators

- Robustness to modeling errors – Nyquist stability Criterion - MIMO Nyquist stability Criterion

- Singular values

- Pole placement – state feedback - Observer / State estimation - Discrete time current observer - Separation property

- LQ optimal control