Feedback linearization

<p>Feedback linearization is a common strategy employed in <a href="/facts/Nonlinear_control/KScKca1A">nonlinear control</a> to control <a href="/facts/Nonlinear_systems/4aYItALC">nonlinear systems</a>. Feedback linearization techniques may be applied to nonlinear control systems of the form
</p>
<table><tbody><tr><td>                                                        x              ˙                                      (        t        )        =        f        (        x        (        t        )        )        +                  ∑                      i            =            1                                m                                            g                      i                          (        x        (        t        )        )                          u                      i                          (        t        )              {\displaystyle {\dot {x}}(t)=f(x(t))+\sum _{i=1}^{m}\,g_{i}(x(t))\,u_{i}(t)}  </td> <td></td> <td>1</td></tr></tbody></table>
<p>where 
  
    
      
        x
        (
        t
        )
        ∈
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle x(t)\in \mathbb {R} ^{n}}
  
 is the state, 
  
    
      
        
          u
          
            1
          
        
        (
        t
        )
        ,
        …
        ,
        
          u
          
            m
          
        
        (
        t
        )
        ∈
        
          R
        
      
    
    {\displaystyle u_{1}(t),\ldots ,u_{m}(t)\in \mathbb {R} }
  
 are the inputs. The approach involves transforming a nonlinear control system into an equivalent linear control system through a change of variables and a suitable control input. In particular, one seeks a change of coordinates 
  
    
      
        z
        =
        Φ
        (
        x
        )
      
    
    {\displaystyle z=\Phi (x)}
  
 and control input 
  
    
      
        u
        =
        a
        (
        x
        )
        +
        b
        (
        x
        )
        
        v
        ,
      
    
    {\displaystyle u=a(x)+b(x)\,v,}
  
 so that the dynamics of 
  
    
      
        x
        (
        t
        )
      
    
    {\displaystyle x(t)}
  
 in the coordinates 
  
    
      
        z
        (
        t
        )
      
    
    {\displaystyle z(t)}
  
  take the form of a linear, controllable control system,
</p>
<table><tbody><tr><td>                                                        z              ˙                                      (        t        )        =        A                z        (        t        )        +                  ∑                      i            =            1                                m                                    b                      i                                  v        (        t        )        .              {\displaystyle {\dot {z}}(t)=A\,z(t)+\sum _{i=1}^{m}b_{i}\,v(t).}  </td> <td></td> <td>2</td></tr></tbody></table>
<p>An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.
</p>

Feedback linearization open-in-new

Feedback linearization