[<< wikibooks] LMIs in Control/Stability Analysis/Continuous Time/Strong Stabilizability
== The System ==
Consider the continous-time LTI system, 
  
    
      
        G
        :
        
          L
          
            2
            e
          
        
        →
        
          L
          
            2
            e
          
        
      
    
    {\displaystyle G:L_{2e}\rightarrow L_{2e}}
   with state-space realization (A,B,C,0)

  
    
      
        
          
            
              
                
                  
                    
                      x
                      ˙
                    
                  
                
                (
                t
                )
              
              
                
                =
                A
                x
                (
                t
                )
                +
                B
                u
                (
                t
                )
                ,
              
            
            
              
                y
                (
                t
                )
              
              
                
                =
                C
                x
                (
                t
                )
                ,
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t),\end{aligned}}}
  where 
  
    
      
        A
        ∈
        
          
            R
          
          
            n
            ×
            n
          
        
      
    
    {\displaystyle A\in \mathbb {R} ^{n\times n}}
  , 
  
    
      
        B
        ∈
        
          
            R
          
          
            n
            ×
            m
          
        
      
    
    {\displaystyle B\in \mathbb {R} ^{n\times m}}
  , 
  
    
      
        C
        ∈
        
          
            R
          
          
            p
            ×
            n
          
        
      
    
    {\displaystyle C\in \mathbb {R} ^{p\times n}}
  , and it and it is assumed that (A, B) is stabilizable, (A, C) is detectable,
and the transfer matrix 
  
    
      
        G
        (
        s
        )
        =
        C
        (
        s
        1
        −
        
          A
          
            −
            1
          
        
        )
        B
      
    
    {\displaystyle G(s)=C(s1-A^{-1})B}
   has no poles on the imaginary axis.


== The Data ==
The matrices 
  
    
      
        A
        ,
        B
        ,
        C
      
    
    {\displaystyle A,B,C}
  .


== The Optimization Problem ==
The system G is strongly stabilizable if there exist 
  
    
      
        P
        ∈
        
          
            S
          
          
            n
          
        
      
    
    {\displaystyle P\in \mathbb {S} ^{n}}
  , 
  
    
      
        Z
        ∈
        
          
            R
          
          
            n
            ×
            p
          
        
      
    
    {\displaystyle Z\in \mathbb {R} ^{n\times p}}
  , and 
  
    
      
        γ
        ∈
        
          
            R
          
          
            >
            0
          
        
      
    
    {\displaystyle \gamma \in \mathbb {R} _{>0}}
  , where 
  
    
      
        P
        >
        0
      
    
    {\displaystyle P>0}
  , such that

  
    
      
        
          
            
              
                P
                A
                +
                
                  A
                  
                    T
                  
                
                +
                Z
                C
                +
                
                  C
                  
                    T
                  
                
                
                  Z
                  
                    T
                  
                
                <
                0
              
            
            
              
                
                  
                    
                      
                        
                          
                            
                              
                                [
                                
                                  
                                    
                                      −
                                      P
                                      (
                                      A
                                      +
                                      B
                                      F
                                      )
                                      +
                                      (
                                      A
                                      +
                                      B
                                      F
                                      
                                        )
                                        
                                          T
                                        
                                      
                                      P
                                      +
                                      Z
                                      C
                                      +
                                      
                                        C
                                        
                                          T
                                        
                                      
                                      
                                        Z
                                        
                                          T
                                        
                                      
                                    
                                    
                                    
                                      −
                                      Z
                                    
                                    
                                    
                                      −
                                      X
                                      B
                                    
                                  
                                  
                                    
                                      ∗
                                    
                                    
                                    
                                      −
                                      γ
                                      I
                                    
                                    
                                    
                                      0
                                    
                                  
                                  
                                    
                                      ∗
                                    
                                    
                                    
                                      ∗
                                    
                                    
                                    
                                      −
                                      γ
                                      I
                                    
                                  
                                
                                ]
                              
                            
                            <
                            0
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}PA+A^{T}+ZC+C^{T}Z^{T}<0\\{\displaystyle {\begin{aligned}{\begin{bmatrix}-P(A+BF)+(A+BF)^{T}P+ZC+C^{T}Z^{T}&&-Z&&-XB\\*&&-\gamma I&&0\\*&&*&&-\gamma I\end{bmatrix}}<0\end{aligned}}}\\\end{aligned}}}
  


== Conclusion: ==
where 
  
    
      
        F
        =
        −
        
          B
          
            T
          
        
        X
      
    
    {\displaystyle F=-B^{T}X}
   and 
  
    
      
        X
        ∈
        
          S
          
            n
          
        
      
    
    {\displaystyle X\in S_{n}}
   , 
  
    
      
        X
        ≥
        0
      
    
    {\displaystyle X\geq 0}
   is the solution to the Lyapunov equation given by

  
    
      
        
          
            
              
                X
                A
                +
                
                  A
                  
                    T
                  
                
                X
                −
                X
                B
                
                  B
                  
                    T
                  
                
                X
                =
                0
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}XA+A^{T}X-XBB^{T}X=0\end{aligned}}}
  Moreover, a controller that strongly stabilizes G is given by the state-space realization

  
    
      
        
          
            
              
                
                  
                    
                      
                        x
                        ˙
                      
                    
                  
                  
                    c
                  
                
                =
                (
                A
                +
                B
                F
                +
                
                  P
                  
                    −
                    1
                  
                
                Z
                C
                )
                x
                (
                t
                )
                −
                
                  P
                  
                    −
                    1
                  
                
                Z
                u
                (
                t
                )
              
            
            
              
                
                  y
                  
                    C
                  
                
                =
                −
                
                  B
                  
                    T
                  
                
                X
                x
                (
                t
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {x}}_{c}=(A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\y_{C}=-B^{T}Xx(t)\end{aligned}}}
  


== Implementation ==
[1] Example Code


== Related LMIs ==
https://en.wikibooks.org/wiki/LMIs_in_Control/Stability_Analysis/Discrete_Time/DiscreteTimeStrongStabilizability - Discrete Time Strong Stabilizability


== External Links ==
http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control- A course on LMIs in Control by Matthew Peet.https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory] - A List of LMIs by Ryan Caverly and James Forbes.https://web.stanford.edu/~boyd/lmibook/ LMIs in Systems and Control Theory- A downloadable book on LMIs by Stephen Boyd.