Flatness (systems theory)

<h2 id="definition">Definition</h2>
<p>A nonlinear system
</p><p>
  
    
      
        
          
            
              
                x
              
              ˙
            
          
        
        (
        t
        )
        =
        
          f
        
        (
        
          x
        
        (
        t
        )
        ,
        
          u
        
        (
        t
        )
        )
        ,
        
        
          x
        
        (
        0
        )
        =
        
          
            x
          
          
            0
          
        
        ,
        
        
          u
        
        (
        t
        )
        ∈
        
          R
          
            m
          
        
        ,
        
        
          x
        
        (
        t
        )
        ∈
        
          R
          
            n
          
        
        ,
        
          Rank
        
        
          
            
              ∂
              
                f
              
              (
              
                x
              
              ,
              
                u
              
              )
            
            
              ∂
              
                u
              
            
          
        
        =
        m
      
    
    {\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {f} (\mathbf {x} (t),\mathbf {u} (t)),\quad \mathbf {x} (0)=\mathbf {x} _{0},\quad \mathbf {u} (t)\in R^{m},\quad \mathbf {x} (t)\in R^{n},{\text{Rank}}{\frac {\partial \mathbf {f} (\mathbf {x} ,\mathbf {u} )}{\partial \mathbf {u} }}=m}

</p><p>is flat, if there exists an output
</p><p>
  
    
      
        
          y
        
        (
        t
        )
        =
        (
        
          y
          
            1
          
        
        (
        t
        )
        ,
        .
        .
        .
        ,
        
          y
          
            m
          
        
        (
        t
        )
        )
      
    
    {\displaystyle \mathbf {y} (t)=(y_{1}(t),...,y_{m}(t))}

</p><p>that satisfies the following conditions:
</p>
<ul><li>The signals 
  
    
      
        
          y
          
            i
          
        
        ,
        i
        =
        1
        ,
        .
        .
        .
        ,
        m
      
    
    {\displaystyle y_{i},i=1,...,m}
  
 are representable as functions of the states 
  
    
      
        
          x
          
            i
          
        
        ,
        i
        =
        1
        ,
        .
        .
        .
        ,
        n
      
    
    {\displaystyle x_{i},i=1,...,n}
  
 and inputs 
  
    
      
        
          u
          
            i
          
        
        ,
        i
        =
        1
        ,
        .
        .
        .
        ,
        m
      
    
    {\displaystyle u_{i},i=1,...,m}
  
 and a finite number of derivatives with respect to time 
  
    
      
        
          u
          
            i
          
          
            (
            k
            )
          
        
        ,
        k
        =
        1
        ,
        .
        .
        .
        ,
        
          α
          
            i
          
        
      
    
    {\displaystyle u_{i}^{(k)},k=1,...,\alpha _{i}}
  
: 
  
    
      
        
          y
        
        =
        Φ
        (
        
          x
        
        ,
        
          u
        
        ,
        
          
            
              
                u
              
              ˙
            
          
        
        ,
        .
        .
        .
        ,
        
          
            u
          
          
            (
            α
            )
          
        
        )
      
    
    {\displaystyle \mathbf {y} =\Phi (\mathbf {x} ,\mathbf {u} ,{\dot {\mathbf {u} }},...,\mathbf {u} ^{(\alpha )})}
  
.</li>
<li>The states 
  
    
      
        
          x
          
            i
          
        
        ,
        i
        =
        1
        ,
        .
        .
        .
        ,
        n
      
    
    {\displaystyle x_{i},i=1,...,n}
  
 and inputs 
  
    
      
        
          u
          
            i
          
        
        ,
        i
        =
        1
        ,
        .
        .
        .
        ,
        m
      
    
    {\displaystyle u_{i},i=1,...,m}
  
 are representable as functions of the outputs 
  
    
      
        
          y
          
            i
          
        
        ,
        i
        =
        1
        ,
        .
        .
        .
        ,
        m
      
    
    {\displaystyle y_{i},i=1,...,m}
  
 and of its derivatives with respect to time 
  
    
      
        
          y
          
            i
          
          
            (
            k
            )
          
        
        ,
        i
        =
        1
        ,
        .
        .
        .
        ,
        m
      
    
    {\displaystyle y_{i}^{(k)},i=1,...,m}
  
.</li>
<li>The components of 
  
    
      
        
          y
        
      
    
    {\displaystyle \mathbf {y} }
  
 are differentially independent, that is, they satisfy no differential equation of the form 
  
    
      
        ϕ
        (
        
          y
        
        ,
        
          
            
              
                y
              
              ˙
            
          
        
        ,
        
          
            y
          
          
            (
            γ
            )
          
        
        )
        =
        
          0
        
      
    
    {\displaystyle \phi (\mathbf {y} ,{\dot {\mathbf {y} }},\mathbf {y} ^{(\gamma )})=\mathbf {0} }
  
.</li></ul>
<p>If these conditions are satisfied at least locally, then the (possibly fictitious) output is called <i>flat output</i>, and the system is <i>flat</i>.
</p>
<h2 id="relation-to-controllability-of-linear-systems">Relation to controllability of linear systems</h2>
<p>A <a href="/facts/LTI_system_theory/kEmj0QDC">linear system</a>

x
              
              ˙
            
          
        
        (
        t
        )
        =
        
          A
        
        
          x
        
        (
        t
        )
        +
        
          B
        
        
          u
        
        (
        t
        )
        ,
        
        
          x
        
        (
        0
        )
        =
        
          
            x
          
          
            0
          
        
      
    
    {\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t),\quad \mathbf {x} (0)=\mathbf {x} _{0}}

with the same signal dimensions for 
  
    
      
        
          x
        
        ,
        
          u
        
      
    
    {\displaystyle \mathbf {x} ,\mathbf {u} }
  
 as the nonlinear system is flat, if and only if it is <a href="/facts/Controllability/p1TLxgLf">controllable</a>. For <a href="/facts/LTI_system_theory/kEmj0QDC">linear systems</a> both properties are equivalent, hence exchangeable.
</p>
<h2 id="significance">Significance</h2>
<p>The flatness property is useful for both the analysis of and controller synthesis for nonlinear dynamical systems. It is particularly advantageous for solving trajectory planning problems and asymptotical setpoint following control.
</p>
<h2 id="literature">Literature</h2>
<ul><li>M. Fliess, J. L. Lévine, P. Martin and P. Rouchon: Flatness and defect of non-linear systems: introductory theory and examples. <i>International Journal of Control</i> 61(6), pp. 1327-1361, 1995 <a href="https://www.researchgate.net/profile/Philippe-Martin-9/publication/230872857_Flatness_and_defect_of_non-linear_systems_introductory_theory_and_examples/links/02e7e52338e7fa738d000000/Flatness-and-defect-of-non-linear-systems-introductory-theory-and-examples.pdf">[1]</a></li>
<li>A. Isidori, C.H. Moog et A. De Luca. A Sufficient Condition for Full Linearization via Dynamic State Feedback. 25th CDC IEEE, Athens, Greece, pp. 203 - 208, 1986 <a href="https://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=4048739&contentType=Conference+Publications&matchBoolean%3Dtrue%26searchField%3DSearch_All%26queryText%3D%28%28p_Authors%3Aisidori%29+AND+p_Authors%3Amoog%29">[2]</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Control_theory/aIuERpn6">Control theory</a></li>
<li><a href="/facts/Control_engineering/5ckfXmB8">Control engineering</a></li>
<li><a href="/facts/Controller_(control_theory)/aIuERpn6">Controller (control theory)</a></li>
<li><a href="/facts/Flat_pseudospectral_method/x8R7TL3k">Flat pseudospectral method</a></li></ul>

Flatness (systems theory) open-in-new

Flatness (systems theory)