[<< wikibooks] LMIs in Control/pages/TDSIC
== The System ==
The problem is to check the stability of the following linear time-delay system

  
    
      
        
          
            
              
                
                  
                    {
                    
                      
                        
                          
                            
                              
                                x
                                ˙
                              
                            
                          
                          (
                          t
                          )
                        
                        
                          =
                          A
                          x
                          (
                          t
                          )
                          +
                          
                            A
                            
                              d
                            
                          
                          x
                          (
                          t
                          −
                          d
                          )
                        
                      
                      
                        
                          x
                          (
                          t
                          )
                        
                        
                          =
                          ϕ
                          (
                          t
                          )
                          ,
                          t
                          ∈
                          [
                          −
                          d
                          ,
                          0
                          ]
                          ,
                          0
                          <
                          d
                          ≤
                          
                            
                              
                                d
                                ¯
                              
                            
                          
                          ,
                        
                      
                    
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}x(t-d)\\x(t)&=\phi (t),t\in [-d,0],0
        0
      
    
    {\displaystyle P>0}
  

  
    
      
        
          
            [
            
              
                
                  
                    A
                    
                      T
                    
                  
                  P
                  +
                  P
                  A
                  +
                  S
                
                
                  P
                  
                    A
                    
                      d
                    
                  
                
              
              
                
                  
                    A
                    
                      d
                    
                    
                      T
                    
                  
                  P
                
                
                  −
                  S
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}A^{T}P+PA+S&PA_{d}\\A_{d}^{T}P&-S\end{bmatrix}}}
  
  
    
      
        
          
            
              
                <
                0
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}<0\end{aligned}}}
  This LMI has been derived from the Lyapunov Function for the system.
By Schur Complement we can see that the above matrix inequality is equivalent to the Riccati Inequality

  
    
      
        
          A
          
            T
          
        
        P
        +
        P
        A
        +
        P
        
          A
          
            d
          
        
        
          S
          
            −
            1
          
        
        
          A
          
            d
          
          
            T
          
        
        P
        +
        S
        <
        0
      
    
    {\displaystyle A^{T}P+PA+PA_{d}S^{-1}A_{d}^{T}P+S<0}
  


== Conclusion: ==
We can now implement these LMIs to do stability analysis for a Time delay system on the delay independent condition


== Implementation ==
The implementation of the above LMI can be seen here
https://github.com/yashgvd/LMI_wikibooks


== Related LMIs ==
Time Delay systems (Delay Dependent Condition)


== External Links ==
[1] - LMI in Control Systems Analysis, Design and Applications
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.
D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.


== Return to Main Page: ==