Properties of State Transition Matrix in Modern Control Systems, Slides of Automatic Controls

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Modern Control Systems

Dr. Mohammad Al Janaideh

Properties of State Transition Matrix

Φ ( 0 )  e A^^0  I

1 2 1 2 1 2 Φ ( t  t )  e A ( t^  t^ )  e A t e A^ t

Φ ^1 ( t )  Φ ( t )

[ Φ ( t )] n^  Φ ( nt )

Φ ( t (^) 2  t 1 ) Φ ( t 1  t 0 )  Φ ( t 2  t 0 )

Example



 

 

 

 

  

   s

s s s

sI A 2

3 1 1 2

1 1 ( )( )

( )

























   



   



  

( )( ) ( )( )

( ) ( )( ) ( )( )

1 2 1 2

2

1 2

1 1 2

3 1

s s

s s s

s s s s

s

sI A



 





 

    

        

    t t t t

t t t t t e e e e

e e e e t e 2 2

2 2

2 2 2

A^2 ( )

Nonhomogenous State Equations

x ( t )  ax ( t )  bu ( t ) x ( t )  x ( t )  bu ( t )

[ ( ) ( )] [ e x ( t )] e bu ( t ) dt

d e  at^^ x  t  ax t   at   at

We shall begin by considering the scalar case:

By multiplying the both side e  at

Integrating this equation between 0 and t gives:



 (^)   

t e at^ x t x e a bu d 0

( ) ( 0 ) ^ () 

(^00) 0.5 1 1.5 2 2.5 3 3.5 4

1

2

3

4 x(0)= x(0)= x(0)= -

t=0:0.01:4; for x0=-1:1:1; n=size(t); y=x0*exp(-t)+exp(-t)+ plot(t,y) hold on end

Nonhomogenous State Equations

x ( t )  Ax ( t )  B u ( t ) x ( t )  x ( t )  B u ( t )

[ ( ) ( )] [ e ( t )] e ( t ) dt

d e ^ A t^^ x  t  Ax t   A t x   A t Bu

Considering the vector case:

By multiplying the both side^ e^  A t

Integrating this equation between 0 and t gives:



 (^)   

t e t^ t e u d 0

A x ( ) x ( 0 ) A  (^) B () 

Example

x u

x x

x 

 

 

 

   

  

 

 

 

 

    

 

 

 1

0 2 3

0 1 2

1 2

1 

 u ( t )  1



 

 

 

    

        

    t t t t

t t t t t e e e e

e e e e t e 2 2

2 2

2 2 2

A^2 ( )

    

    d e e e e

e e e e x t e t t t t

t t t t ( ) t^ ( ) [ ] ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) 1 1

0 2 2 2

2 (^0 )

2 2



 

 

 

 

 

 

 

    

 

        A x

x ( t ) ?























   

 

 

 

  (^)  

  t t

t t

e e

e e x t

x t t (^) 2

2

2

(^1 )

1 2

1

( )

( ) x ( 0 )  0 x ( )

(^00 1 2 3 4 5 6 )

x 1 (t) x 2 (t)

t=0:0.01:7; x1=0.5-exp(-t)+0.5exp(-2t); x2=exp(-t)-exp(-2*t); plot(t,x1) hold on plot(t,x2,'red')

Controllability

The system

is controllable if

is a full rank matrix.

The matrix is called

a controllability matrix.

Cx Du

x Ax Bu  

  y



 B AB A^2 B ..... An ^1 B 

 B AB A^2 B ..... An ^1 B 

Observeability

The system

is observable if is a full rank matrix

Cx Du

x Ax Bu  

  y



































CA n^ ^1

CA

C

.

.

Observeability matrix:

Example



 

 

 

  (^0)  1

1 1 A 

 

 

 

  (^0)

1 B



 

 

 

  

 

 

 

 

 

 

 

  (^)  0

1 0

1 0 1

1 1 AB

  

 

 

 

  (^00)

1 1 B AB

Not controllable



 

 

 

  (^)  2  1

1 1 A 

 

 

 

  (^1)

0 B

C   1 0 

CA   1 1  

 

 

 

  

 

 

 

 0 1

1 1 CA

C

  

 

 

 

  (^1)  1

0 1 B AB Controllable

Observable