Controllability Gramian

<p>In <a href="/facts/Control_theory/aIuERpn6">control theory</a>, we may need to find out whether or not a system such as

x
                      ˙
                    
                  
                
                (
                t
                )
              
              
                
                =
                
                  A
                  x
                
                (
                t
                )
                +
                
                  B
                  u
                
                (
                t
                )
              
            
            
              
                
                  y
                
                (
                t
                )
              
              
                
                =
                
                  C
                  x
                
                (
                t
                )
                +
                
                  D
                  u
                
                (
                t
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {\boldsymbol {x}}}(t)&={\boldsymbol {Ax}}(t)+{\boldsymbol {Bu}}(t)\\{\boldsymbol {y}}(t)&={\boldsymbol {Cx}}(t)+{\boldsymbol {Du}}(t)\end{aligned}}}

is <a href="/facts/Controllability/p1TLxgLf">controllable</a>, where 
  
    
      
        
          A
        
      
    
    {\displaystyle {\boldsymbol {A}}}
  
, 
  
    
      
        
          B
        
      
    
    {\displaystyle {\boldsymbol {B}}}
  
, 
  
    
      
        
          C
        
      
    
    {\displaystyle {\boldsymbol {C}}}
  
 and 
  
    
      
        
          D
        
      
    
    {\displaystyle {\boldsymbol {D}}}
  
 are, respectively, 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
, 
  
    
      
        n
        ×
        p
      
    
    {\displaystyle n\times p}
  
, 
  
    
      
        q
        ×
        n
      
    
    {\displaystyle q\times n}
  
 and 
  
    
      
        q
        ×
        p
      
    
    {\displaystyle q\times p}
  
 matrices for a system with 
  
    
      
        p
      
    
    {\displaystyle p}
  
 inputs, 
  
    
      
        n
      
    
    {\displaystyle n}
  
 state variables and 
  
    
      
        q
      
    
    {\displaystyle q}
  
 outputs.
</p><p>One of the many ways one can achieve such goal is by the use of the Controllability <a href="/facts/Gramian/SGb2VGTU">Gramian</a>.
</p>

Controllability Gramian open-in-new

Controllability Gramian