check attach fileq3_ab and q3_c are together!

Question

David · Accepted Answer

Solution: 
The Scalar plant is given as: 
 ̇    
With performance index: 
  
 
 
   ( )
 
∫(       )
The Riccati equation is: 
  ̇    
  
 
     
Solving the above differential equation using separation of variables: 
∫

 ( )
  ( )
 ∫
And this yields: 
√       (
 
√  
)
√ |
|
  ( )
 ( )
     
Alternate form of the above integral is: 
√ 
 √ 
   
  
 
√  
  
 
√  
|
  ( )
 ( )
     
  
 ( )
√  
  
 ( )
√
  
  ( )
√  
  
  ( )
√  
  
 √
 
 
(   )
 
  ( )  
 ( )√  (   (   )   )    (   (   )   )
 ( )(   (   )   )  √  (   (   )   )
 
Where, 
  √
 
 
 
The optimal control is: 
 ( )  
  ( )
 
 
 ( )√
 
 ( 
  (   )   )   (   (   )   )
 ( )(   (   )   )  √  (   (   )   )
 
The steady gain is:

check attach fileq3_ab and q3_c are together!

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