Closed-loop pole

<h2 id="closed-loop-poles-in-control-theory">Closed-loop poles in control theory</h2>
<p>The response of a <a href="/facts/Linear_time-invariant_system/kEmj0QDC"> linear time-invariant system</a> to any input can be derived from its <a href="/facts/Impulse_response/l4vaE5er">impulse response</a> and <a href="/facts/Step_response/7pq4VyNt">step response</a>.  The eigenvalues of the system determine completely the <a href="/facts/Natural_response/slDsh1lH">natural response</a> (unforced response).  In control theory, the response to any input is a combination of a <a href="/facts/Transient_response/m4Nr61Px">transient response</a> and <a href="/facts/Steady-state_response/U3ylWEtp">steady-state response</a>.  Therefore, a crucial design parameter is the location of the eigenvalues, or closed-loop poles.
</p><p>In <a href="/facts/Root-locus/PHASdGw7">root-locus design</a>, the <a href="/facts/Gain_(electronics)/CWbehg27">gain</a> <i>K</i> is usually parameterized.  Each point on the locus satisfies the <a href="/facts/Angle_condition/vELuoGcH">angle condition</a> and <a href="/facts/Magnitude_condition/cTWc927F">magnitude condition</a> and corresponds to a different value of <i>K</i>.  For <a href="/facts/Negative_feedback/G9byVgpN">negative feedback</a> systems, the closed-loop poles move along the <a href="/facts/Root-locus/PHASdGw7">root-locus</a> from the open-loop poles to the open-loop zeroes as the gain is increased.  For this reason, the root-locus is often used for design of <a href="/facts/Proportional_control/eeSo345W">proportional control</a>, i.e. those for which 
  
    
      
        
          
            
              G
            
          
          
            c
          
        
        =
        K
      
    
    {\displaystyle {\textbf {G}}_{c}=K}
  
.
</p>
<h2 id="finding-closed-loop-poles">Finding closed-loop poles</h2>
<p>Consider a simple feedback system with controller 
  
    
      
        
          
            
              G
            
          
          
            c
          
        
        =
        K
      
    
    {\displaystyle {\textbf {G}}_{c}=K}
  
, <a href="/facts/Plant_(control_theory)/6lK80ETg">plant</a> 
  
    
      
        
          
            G
          
        
        (
        s
        )
      
    
    {\displaystyle {\textbf {G}}(s)}
  
 and transfer function 
  
    
      
        
          
            H
          
        
        (
        s
        )
      
    
    {\displaystyle {\textbf {H}}(s)}
  
 in the feedback path.  Note that a unity feedback system has 
  
    
      
        
          
            H
          
        
        (
        s
        )
        =
        1
      
    
    {\displaystyle {\textbf {H}}(s)=1}
  
 and the block is omitted.  For this system, the open-loop transfer function is the product of the blocks in the forward path, 
  
    
      
        
          
            
              G
            
          
          
            c
          
        
        
          
            G
          
        
        =
        K
        
          
            G
          
        
      
    
    {\displaystyle {\textbf {G}}_{c}{\textbf {G}}=K{\textbf {G}}}
  
.  The product of the blocks around the entire closed loop is 
  
    
      
        
          
            
              G
            
          
          
            c
          
        
        
          
            G
          
        
        
          
            H
          
        
        =
        K
        
          
            G
          
        
        
          
            H
          
        
      
    
    {\displaystyle {\textbf {G}}_{c}{\textbf {G}}{\textbf {H}}=K{\textbf {G}}{\textbf {H}}}
  
.  Therefore, the closed-loop transfer function is
</p>

T
          
        
        (
        s
        )
        =
        
          
            
              K
              
                
                  G
                
              
            
            
              1
              +
              K
              
                
                  G
                
              
              
                
                  H
                
              
            
          
        
        .
      
    
    {\displaystyle {\textbf {T}}(s)={\frac {K{\textbf {G}}}{1+K{\textbf {G}}{\textbf {H}}}}.}

<p>The closed-loop poles, or eigenvalues, are obtained by solving the characteristic equation 
  
    
      
        
          1
          +
          K
          
            
              G
            
          
          
            
              H
            
          
        
        =
        0
      
    
    {\displaystyle {1+K{\textbf {G}}{\textbf {H}}}=0}
  
.  In general, the solution will be n complex numbers where n is the order of the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a>.
</p><p>The preceding is valid for single-input-single-output systems (SISO). An extension is possible for multiple input multiple output systems, that is for systems where 
  
    
      
        
          
            G
          
        
        (
        s
        )
      
    
    {\displaystyle {\textbf {G}}(s)}
  
 and 
  
    
      
        
          
            K
          
        
        (
        s
        )
      
    
    {\displaystyle {\textbf {K}}(s)}
  
 are matrices whose elements are made of transfer functions. In this case the poles are the solution of the equation
</p>

det
        (
        
          
            I
          
        
        +
        
          
            G
          
        
        (
        s
        )
        
          
            K
          
        
        (
        s
        )
        )
        =
        0.
        
      
    
    {\displaystyle \det({\textbf {I}}+{\textbf {G}}(s){\textbf {K}}(s))=0.\,}

Closed-loop pole open-in-new

Closed-loop pole