Double integrator

<h2 id="differential-equations">Differential equations</h2>
<p>The differential equations which represent a double integrator are:
</p>

q
              ¨
            
          
        
        =
        u
        (
        t
        )
      
    
    {\displaystyle {\ddot {q}}=u(t)}

y
        =
        q
        (
        t
        )
      
    
    {\displaystyle y=q(t)}

<p>where both 
  
    
      
        q
        (
        t
        )
        ,
        u
        (
        t
        )
        ∈
        
          R
        
      
    
    {\displaystyle q(t),u(t)\in \mathbb {R} }

Let us now represent this in state space form with the vector 
  
    
      
        
          
            x(t)
          
        
        =
        
          
            [
            
              
                
                  q
                
              
              
                
                  
                    
                      
                        q
                        ˙
                      
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\textbf {x(t)}}={\begin{bmatrix}q\\{\dot {q}}\\\end{bmatrix}}}

</p>

x
              
              ˙
            
          
        
        (
        t
        )
        =
        
          
            
              d
              
                
                  x
                
              
            
            
              d
              t
            
          
        
        =
        
          
            [
            
              
                
                  
                    
                      
                        q
                        ˙
                      
                    
                  
                
              
              
                
                  
                    
                      
                        q
                        ¨
                      
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\dot {\textbf {x}}}(t)={\frac {d{\textbf {x}}}{dt}}={\begin{bmatrix}{\dot {q}}\\{\ddot {q}}\\\end{bmatrix}}}

<p>
In this representation, it is clear that the control input 
  
    
      
        
          
            u
          
        
      
    
    {\displaystyle {\textbf {u}}}
  
 is the second derivative of the output 
  
    
      
        
          
            x
          
        
      
    
    {\displaystyle {\textbf {x}}}
  
. In the scalar form, the control input is the second derivative of the output 
  
    
      
        q
      
    
    {\displaystyle q}
  
.
</p>
<h2 id="state-space-representation">State space representation</h2>
<p>The normalized state space model of a double integrator takes the form 
</p>

x
              
              ˙
            
          
        
        (
        t
        )
        =
        
          
            [
            
              
                
                  0
                
                
                  1
                
              
              
                
                  0
                
                
                  0
                
              
            
            ]
          
        
        
          
            x
          
        
        (
        t
        )
        +
        
          
            [
            
              
                
                  0
                
              
              
                
                  1
                
              
            
            ]
          
        
        
          
            u
          
        
        (
        t
        )
      
    
    {\displaystyle {\dot {\textbf {x}}}(t)={\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}0\\1\end{bmatrix}}{\textbf {u}}(t)}

y
          
        
        (
        t
        )
        =
        
          
            [
            
              
                
                  1
                
                
                  0
                
              
            
            ]
          
        
        
          
            x
          
        
        (
        t
        )
        .
      
    
    {\displaystyle {\textbf {y}}(t)={\begin{bmatrix}1&0\end{bmatrix}}{\textbf {x}}(t).}

<p>According to this model, the input 
  
    
      
        
          
            u
          
        
      
    
    {\displaystyle {\textbf {u}}}
  
 is the second derivative of the output 
  
    
      
        
          
            y
          
        
      
    
    {\displaystyle {\textbf {y}}}
  
, hence the name double integrator.
</p>
<h2 id="transfer-function-representation">Transfer function representation</h2>
<p>Taking the <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> of the state space input-output equation, we see that the <a href="/facts/Transfer_function/slDsh1lH">transfer function</a> of the double integrator is given by
</p>

Y
              (
              s
              )
            
            
              U
              (
              s
              )
            
          
        
        =
        
          
            1
            
              s
              
                2
              
            
          
        
        .
      
    
    {\displaystyle {\frac {Y(s)}{U(s)}}={\frac {1}{s^{2}}}.}

<p>Using the differential equations dependent on 
  
    
      
        q
        (
        t
        )
        ,
        y
        (
        t
        )
        ,
        u
        (
        t
        )
      
    
    {\displaystyle q(t),y(t),u(t)}
  
 and 
  
    
      
        
          
            x(t)
          
        
      
    
    {\displaystyle {\textbf {x(t)}}}
  
, and the state space representation:
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Venkatesh G. Rao and Dennis S. Bernstein (2001). "Naive control of the double integrator" (PDF). IEEE Control Systems Magazine. Retrieved 2012-03-04. <a href="http://www-personal.umich.edu/~dsbaero/others/25-DoubleIntegrator.pdf" target="_blank">http://www-personal.umich.edu/~dsbaero/others/25-DoubleIntegrator.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Double integrator open-in-new

Double integrator