Text: P.J. Antsaklis and A.N. Michel, Linear Systems. Birkhauser, Boston, 2006, 2nd . coected...

Question

Text: P.J. Antsaklis and A.N. Michel, Linear Systems. Birkhauser, Boston, 2006, 2nd . coected printing (ISBN XXXXXXXXXXProblem-1Problem-2Problem-3Problem-4Problem-5

Robert · Accepted Answer

Solution 1: 
  [
   
   
   
] 
   [
   
   
   
] [
   
   
   
] 
   [
   
   
   
] 
   [
     
   
   
] 
Hence generalizing: 
   [
    (   )
   
   
] 
Hence A^100 is: 
     [
    (     )
   
   
] 
     [
     
   
   
] 
Solving second part  
       [(    )  ] 
      [
   
   
   
]  [
   
   
   
] 
     [
      
    
     
] 
|    |  (   )(  (   )) 
[    ]   
 
(  (   ) )
[
 (   )  (   )  
 (   )  (   )
   (   )
] 
[    ]   
[
 
 
 
 
 
 

 (   )
 
 (   ) 
 

 (   )
  
 
   ]
 
 
 
 
 

   [(    )  ]  [
                 
       
    
] 
Hence  
    [
                 
       
    
] 
Solution 2: 
a)   [
  
       
] 
[    ]   [
  
  
]  [
  
       
] 
[    ]  [
   
    (   )
] 
|    |  ( (  (   ))     
|    |     (   )     
[    ]   
 
(   )(   )
[
      
    
] 
[    ]   
[
 
 
 
     
(   )(   )
 
(   )(   )
   
(   )(   )
 
(   )(   )]
 
 

[    ]   
[
 
 
 
 
   

   
[
 
(   )

   
]
 
   
[
 
(   )

   
]
   
   
[
 
(   )

   
]
 
   
[
 
(   )

   
]
]
 
 

   [    ]   [

   
(         )
 
   
(         )
   
   
(         )
 
   
(           )
]
   [    ]   [
 
    

   
(    )
 
   
(         )
   
   
(         )
 
   
(           )
] 
    [
 
    

   
(    )
 
   
(         )
   
   
(         )
 
   
(           )
] 
b) Converting to state space: 
Taking state variables as:
 ̇     
 ̈       ̇   ̇    
This fails the state space. Hence we modify our state variables
 ̇       
 ̇       
  ̇       
  ̇   ̈   ̇ 
 ̈   ̇       ̇    
Substituting   ̇ 
 ̈   ̇       (    )    
 ̈   ̇            
Hence representing in state space 
[
  ̇
  ̇
]  [
  
    
] [
  
  
]  [
 
   
]   
Now taking laplace of state space equation to find x(t) 
  ( )   ( )    ( )    ( ) 
Where A and B are as shown: 
  [
  
    
] 
  [
 
   
] 
And x(0) is initial condition:

Text: P.J. Antsaklis and A.N. Michel, Linear Systems. Birkhauser, Boston, 2006, 2nd . corrected printing (ISBN XXXXXXXXXX Problem-1 Problem-2 Problem-3 Problem-4 Problem-5

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