[<< wikibooks] LMIs in Control/LMI for Polytopic Uncertainity/Polytopic Quadratic Stability
LMIs in Control/LMI for Polytopic Uncertainity/Polytopic Quadratic Stability
The System:
Consider the system with Affine Time-Varying 
uncertainty (No input)

  
    
      
        
          
            
              
                
                  
                    
                      x
                      ˙
                    
                  
                
                (
                t
                )
              
              
                
                =
                (
                
                  A
                  
                    0
                  
                
                +
                Δ
                A
                (
                t
                )
                )
                x
                (
                t
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}}
  where

  
    
      
        
          
            
              
                Δ
                A
                (
                t
                )
                =
                
                  A
                  
                    1
                  
                
                
                  δ
                  
                    1
                  
                
                (
                t
                )
                +
                .
                .
                .
                .
                +
                
                  A
                  
                    k
                  
                
                
                  δ
                  
                    k
                  
                
                (
                t
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\Delta A(t)=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\end{aligned}}}
  where 

  
    
      
        
          δ
          
            i
          
        
        (
        t
        )
      
    
    {\displaystyle \delta _{i}(t)}
   
lies in either the intervals

  
    
      
        
          
            
              
                
                  δ
                  
                    i
                  
                
                ∈
                [
                
                  δ
                  
                    i
                  
                  
                    −
                  
                
                ,
                
                  δ
                  
                    i
                  
                  
                    +
                  
                
                ]
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\delta _{i}\in [\delta _{i}^{-},\delta _{i}^{+}]\end{aligned}}}
   or the simplex 

  
    
      
        
          
            
              
                δ
                (
                t
                )
                ∈
                
                  δ
                  :
                  Σ
                  
                    α
                    
                      i
                    
                  
                  =
                  1
                  ,
                  α
                  ≥
                  0
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\delta (t)\in {\delta :\Sigma \alpha _{i}=1,\alpha \geq 0}\end{aligned}}}
   The LMI: Polytopic Quadratic Stability 

The system is Quadratically Stable over 
  
    
      
        Δ
      
    
    {\displaystyle \Delta }
   if there exists a P > 0 

  
    
      
        (
        A
        +
        Δ
        A
        
          )
          
            T
          
        
        +
        P
        (
        A
        +
        Δ
        A
        )
        <
        0
      
    
    {\displaystyle (A+\Delta A)^{T}+P(A+\Delta A)<0}
   for all 
  
    
      
        Δ
        A
        ∈
        Δ
      
    
    {\displaystyle \Delta A\in \Delta }
  

Conclusion: 

Interpretation of the results of the LMI
Implementation 
A link to CodeOcean or other online implementation of the LMI
Related LMIs 
Links to other closely-related LMIs
External Links
A list of references documenting and validating the LMI.

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.Return to Main Page: 
Related LMIs: