[<< wikibooks] LMIs in Control/Stability of Lure's Systems
== The System ==

  
    
      
        
          
            
              
                
                  
                    
                      x
                      ˙
                    
                  
                
                (
                t
                )
              
              
                
                =
                A
                x
                (
                t
                )
                +
                
                  B
                  
                    p
                  
                
                p
                (
                t
                )
                +
                
                  B
                  
                    w
                  
                
                w
                (
                t
                )
                ,
              
            
            
              
                z
                (
                t
                )
              
              
                
                =
                
                  C
                  
                    z
                  
                
                x
                (
                t
                )
              
            
            
              
                
                  p
                  
                    i
                  
                
                (
                t
                )
              
              
                
                =
                
                  ϕ
                  
                    i
                  
                
                (
                
                  q
                  
                    i
                  
                
                (
                t
                )
                )
                ,
                i
                =
                1
                ,
                …
                ,
                
                  n
                  
                    p
                  
                
              
            
            
              
                q
              
              
                
                =
                
                  C
                  
                    q
                  
                
                x
                ,
              
            
            
              
                0
              
              
                
                ≤
                σ
                
                  ϕ
                  
                    i
                  
                
                (
                σ
                )
                ≤
                
                  σ
                  
                    2
                  
                
                 
                ∀
                σ
                ∈
                
                  R
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}}
  


== The Data ==
The matrices 
  
    
      
        A
        ,
        
          B
          
            p
          
        
        ,
        
          B
          
            w
          
        
        ,
        
          C
          
            q
          
        
        ,
        
          C
          
            z
          
        
      
    
    {\displaystyle A,B_{p},B_{w},C_{q},C_{z}}
  .


== The LMI: The Lure's System's Stability ==
The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.

  
    
      
        
          
            
              
                
                  Find
                
                
              
              
                P
                >
                0
                ,
                Λ
                =
                d
                i
                a
                g
                (
                
                  λ
                  
                    1
                  
                
                ,
                …
                ,
                
                  λ
                  
                    
                      n
                      
                        p
                      
                    
                  
                
                )
                ⪰
                0
                ,
                T
                =
                d
                i
                a
                g
                (
                
                  τ
                  
                    1
                  
                
                ,
                …
                ,
                
                  τ
                  
                    
                      n
                      
                        p
                      
                    
                  
                
                )
                ⪰
                0
                :
              
            
            
              
              
                
                  
                    [
                    
                      
                        
                          
                            A
                            
                              ⊤
                            
                          
                          P
                          +
                          P
                          A
                        
                        
                          P
                          
                            B
                            
                              p
                            
                          
                          +
                          
                            A
                            
                              ⊤
                            
                          
                          
                            C
                            
                              q
                            
                            
                              ⊤
                            
                          
                          Λ
                          +
                          
                            C
                            
                              q
                            
                            
                              ⊤
                            
                          
                          T
                        
                      
                      
                        
                          
                            B
                            
                              p
                            
                            
                              ⊤
                            
                          
                          P
                          +
                          Λ
                          
                            C
                            
                              q
                            
                          
                          A
                          +
                          T
                          
                            C
                            
                              q
                            
                          
                        
                        
                          Λ
                          
                            C
                            
                              q
                            
                          
                          
                            B
                            
                              p
                            
                          
                          +
                          
                            B
                            
                              p
                            
                            
                              ⊤
                            
                          
                          
                            C
                            
                              q
                            
                            
                              ⊤
                            
                          
                          Λ
                          −
                          2
                          T
                        
                      
                    
                    ]
                  
                
                ≺
                0
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\text{Find}}\;&P>0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0:\\&{\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\prec 0\\\end{aligned}}}
  


== Implementation ==
https://codeocean.com/capsule/0232754/tree


== Conclusion ==
If the feasibility problem with LMI constraints has solution, then the Lure's system is stable.


== Remark ==
The LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system . It is also necessary when there is only one nonlinearity, i.e., when 
  
    
      
        
          n
          
            p
          
        
        =
        1
      
    
    {\displaystyle n_{p}=1}
  .


== External Links ==
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.


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