[<< wikibooks] LMIs in Control/pages/dt mixed H2 Hinf optimal output feedback control
WIP, Description in progress
This part shows how to design dynamic outpur feedback control in mixed 
  
    
      
        
          
            
              H
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathcal {H}}_{2}}
   and 
  
    
      
        
          
            
              H
            
          
          
            ∞
          
        
      
    
    {\displaystyle {\mathcal {H}}_{\infty }}
   sense for the continuous time.


== Problem ==
Consider the discrete-time generalized LTI plant 
  
    
      
        
          
            P
          
        
      
    
    {\displaystyle {\mathcal {P}}}
   with minimal state-space realization 

  
    
      
        
          
            
              x
              ˙
            
          
        
        =
        A
        x
        +
        
          
            [
            
              
                
                  
                    B
                    
                      1
                      ,
                      1
                    
                  
                
                
                  
                    B
                    
                      1
                      ,
                      2
                    
                  
                
              
            
            ]
          
        
        
          
            [
            
              
                
                  
                    w
                    
                      1
                    
                  
                
              
              
                
                  
                    w
                    
                      2
                    
                  
                
              
            
            ]
          
        
        +
        
          B
          
            2
          
        
        u
        ,
      
    
    {\displaystyle {\dot {x}}=Ax+{\begin{bmatrix}B_{1,1}&B_{1,2}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+B_{2}u,}
  

  
    
      
        
          
            [
            
              
                
                  
                    z
                    
                      1
                    
                  
                
              
              
                
                  
                    z
                    
                      2
                    
                  
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  
                    C
                    
                      1
                      ,
                      1
                    
                  
                
              
              
                
                  
                    D
                    
                      1
                      ,
                      2
                    
                  
                
              
            
            ]
          
        
        
          x
          
            k
          
        
        +
        
          
            [
            
              
                
                  
                    D
                    
                      11
                      ,
                      11
                    
                  
                
                
                  
                    D
                    
                      11
                      ,
                      12
                    
                  
                
              
              
                
                  
                    D
                    
                      11
                      ,
                      21
                    
                  
                
                
                  
                    D
                    
                      11
                      ,
                      22
                    
                  
                
              
            
            ]
          
        
        
          
            [
            
              
                
                  
                    w
                    
                      1
                    
                  
                
              
              
                
                  
                    w
                    
                      2
                    
                  
                
              
            
            ]
          
        
        +
        
          
            [
            
              
                
                  
                    D
                    
                      12
                      ,
                      1
                    
                  
                
              
              
                
                  
                    D
                    
                      12
                      ,
                      2
                    
                  
                
              
            
            ]
          
        
        u
        ,
      
    
    {\displaystyle {\begin{bmatrix}z_{1}\\z_{2}\end{bmatrix}}={\begin{bmatrix}C_{1,1}\\D_{1,2}\end{bmatrix}}x_{k}+{\begin{bmatrix}D_{11,11}&D_{11,12}\\D_{11,21}&D_{11,22}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+{\begin{bmatrix}D_{12,1}\\D_{12,2}\end{bmatrix}}u,}
  

  
    
      
        y
        =
        
          C
          
            d
            2
          
        
        x
        +
        
          
            [
            
              
                
                  
                    D
                    
                      21
                      ,
                      1
                    
                  
                
                
                  
                    D
                    
                      21
                      ,
                      2
                    
                  
                
              
            
            ]
          
        
        
          
            [
            
              
                
                  
                    w
                    
                      1
                    
                  
                
              
              
                
                  
                    w
                    
                      2
                    
                  
                
              
            
            ]
          
        
        +
        
          D
          
            d
            22
          
        
        u
      
    
    {\displaystyle y=C_{d2}x+{\begin{bmatrix}D_{21,1}&D_{21,2}\end{bmatrix}}{\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}+D_{d22}u}
  


== Theorem ==
A continuous-time dynamic output feedback LTI controllerwith state-space realization 
  
    
      
        (
        
          A
          
            c
          
        
        ,
        
          B
          
            c
          
        
        ,
        
          C
          
            c
          
        
        ,
        
          D
          
            c
          
        
        )
      
    
    {\displaystyle (A_{c},B_{c},C_{c},D_{c})}
  
is to be designed to minimize the 
  
    
      
        
          
            
              H
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathcal {H}}_{2}}
   norm of the closed-loop transfer matrix 
  
    
      
        
          T
          
            11
          
        
        (
        s
        )
      
    
    {\displaystyle T_{11}(s)}
   from the exogenous
input 
  
    
      
        
          w
          
            1
          
        
      
    
    {\displaystyle w_{1}}
   to the performance output 
  
    
      
        
          z
          
            1
          
        
      
    
    {\displaystyle z_{1}}
   while ensuring the H∞ norm of the closed-loop
transfer matrix 
  
    
      
        
          T
          
            22
          
        
        (
        s
        )
      
    
    {\displaystyle T_{22}(s)}
   from the exogenous input 
  
    
      
        
          w
          
            2
          
        
      
    
    {\displaystyle w_{2}}
   to the performance output 
  
    
      
        
          z
          
            2
          
        
      
    
    {\displaystyle z_{2}}
   is less than 
  
    
      
        
          γ
          
            d
          
        
      
    
    {\displaystyle \gamma _{d}}
  ,
where

  
    
      
        
          T
          
            11
          
        
        (
        s
        )
        =
        
          C
          
            C
            L
            1
            ,
            1
          
        
        (
        s
        I
        −
        
          A
          
            C
            L
          
        
        
          )
          
            −
            1
          
        
        
          B
          
            C
            L
            1
            ,
            1
          
        
        ,
      
    
    {\displaystyle T_{11}(s)=C_{CL1,1}(sI-A_{CL})^{-1}B_{CL1,1},}
  

  
    
      
        
          T
          
            22
          
        
        (
        s
        )
        =
        
          C
          
            C
            L
            1
            ,
            2
          
        
        (
        s
        I
        −
        
          A
          
            C
            L
          
        
        
          )
          
            −
            1
          
        
        
          B
          
            C
            L
            1
            ,
            2
          
        
        +
        
          D
          
            C
            L
            11
            ,
            22
          
        
        ,
      
    
    {\displaystyle T_{22}(s)=C_{CL1,2}(sI-A_{CL})^{-1}B_{CL1,2}+D_{CL11,22},}
  

  
    
      
        
          A
          
            
              d
              
                C
                L
              
            
          
        
        =
        
          
            [
            
              
                
                  A
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    C
                    
                      2
                    
                  
                
                
                  
                    B
                    
                      2
                    
                  
                  (
                  I
                  +
                  
                    D
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    D
                    
                      22
                    
                  
                  )
                  
                    C
                    
                      c
                    
                  
                
              
              
                
                  
                    B
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    C
                    
                      2
                    
                  
                
                
                  
                    A
                    
                      c
                    
                  
                  +
                  
                    B
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    D
                    
                      22
                    
                  
                  
                    C
                    
                      c
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle A_{d_{CL}}={\begin{bmatrix}A+B_{2}D_{c}{\tilde {D}}^{-1}C_{2}&B_{2}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\\B_{c}{\tilde {D}}^{-1}C_{2}&A_{c}+B_{c}{\tilde {D}}^{-1}D_{22}C_{c}\end{bmatrix}}}
  ,

  
    
      
        
          B
          
            C
            L
            1
            ,
            1
          
        
        =
        
          
            [
            
              
                
                  
                    B
                    
                      1
                      ,
                      1
                    
                  
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    D
                    
                      21
                      ,
                      1
                    
                  
                
              
              
                
                  
                    B
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    D
                    
                      21
                      ,
                      1
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle B_{CL1,1}={\begin{bmatrix}B_{1,1}+B_{2}D_{c}{\tilde {D}}^{-1}D_{21,1}\\B_{c}{\tilde {D}}^{-1}D_{21,1}\end{bmatrix}}}
  ,

  
    
      
        
          B
          
            C
            L
            1
            ,
            2
          
        
        =
        
          
            [
            
              
                
                  
                    B
                    
                      1
                      ,
                      2
                    
                  
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    D
                    
                      21
                      ,
                      2
                    
                  
                
              
              
                
                  
                    B
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    D
                    
                      21
                      ,
                      2
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle B_{CL1,2}={\begin{bmatrix}B_{1,2}+B_{2}D_{c}{\tilde {D}}^{-1}D_{21,2}\\B_{c}{\tilde {D}}^{-1}D_{21,2}\end{bmatrix}}}
  ,

  
    
      
        
          C
          
            C
            L
            1
            ,
            1
          
        
        =
        
          
            [
            
              
                
                  
                    C
                    
                      1
                      ,
                      1
                    
                  
                  +
                  
                    D
                    
                      12
                      ,
                      1
                    
                  
                  
                    D
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    C
                    
                      2
                      ,
                      1
                    
                  
                
                
                  
                    D
                    
                      12
                      ,
                      1
                    
                  
                  (
                  I
                  +
                  
                    D
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    D
                    
                      22
                    
                  
                  )
                  
                    C
                    
                      c
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle C_{CL1,1}={\begin{bmatrix}C_{1,1}+D_{12,1}D_{c}{\tilde {D}}^{-1}C_{2,1}&D_{12,1}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\end{bmatrix}}}
  ,

  
    
      
        
          C
          
            C
            L
            1
            ,
            2
          
        
        =
        
          
            [
            
              
                
                  
                    C
                    
                      1
                      ,
                      2
                    
                  
                  +
                  
                    D
                    
                      12
                      ,
                      2
                    
                  
                  
                    D
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    C
                    
                      2
                      ,
                      2
                    
                  
                
                
                  
                    D
                    
                      12
                      ,
                      2
                    
                  
                  (
                  I
                  +
                  
                    D
                    
                      c
                    
                  
                  
                    
                      
                        
                          D
                          ~
                        
                      
                    
                    
                      −
                      1
                    
                  
                  
                    D
                    
                      22
                    
                  
                  )
                  
                    C
                    
                      c
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle C_{CL1,2}={\begin{bmatrix}C_{1,2}+D_{12,2}D_{c}{\tilde {D}}^{-1}C_{2,2}&D_{12,2}(I+D_{c}{\tilde {D}}^{-1}D_{22})C_{c}\end{bmatrix}}}
  ,

  
    
      
        
          D
          
            C
            L
            11
            ,
            22
          
        
        =
        
          D
          
            11
            ,
            22
          
        
        +
        
          D
          
            12
            ,
            2
          
        
        
          D
          
            c
          
        
        
          
            
              
                D
                ~
              
            
          
          
            −
            1
          
        
        
          D
          
            21
            ,
            2
          
        
      
    
    {\displaystyle D_{CL11,22}=D_{11,22}+D_{12,2}D_{c}{\tilde {D}}^{-1}D_{21,2}}
  ,
and 
  
    
      
        
          
            
              D
              ~
            
          
        
        =
        I
        −
        
          D
          
            22
          
        
        
          D
          
            c
          
        
      
    
    {\displaystyle {\tilde {D}}=I-D_{22}D_{c}}
  .


== Synthesis Method ==
Solve for 
  
    
      
        
          A
          
            n
          
        
        ∈
        
          
            R
          
          
            
              n
              
                x
              
            
            ×
            
              n
              
                x
              
            
          
        
        ,
        
          B
          
            n
          
        
        ∈
        
          
            R
          
          
            
              n
              
                x
              
            
            ×
            
              n
              
                x
              
            
          
        
        ,
        
          C
          
            n
          
        
        ∈
        
          
            R
          
          
            
              n
              
                u
              
            
            ×
            
              n
              
                x
              
            
          
        
        ,
        
          D
          
            n
          
        
        ∈
        
          
            R
          
          
            
              n
              
                u
              
            
            ×
            
              n
              
                y
              
            
          
        
        ,
        
          X
          
            1
          
        
        ,
        
          Y
          
            1
          
        
        ∈
        
          
            S
          
          
            
              n
              
                x
              
            
          
        
        ,
        Z
        ∈
        
          
            S
          
          
            
              n
              
                
                  Z
                  
                    1
                  
                
              
            
          
        
        ,
      
    
    {\displaystyle A_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},B_{n}\in \mathbb {R} ^{n_{x}\times n_{x}},C_{n}\in \mathbb {R} ^{n_{u}\times n_{x}},D_{n}\in \mathbb {R} ^{n_{u}\times n_{y}},X_{1},Y_{1}\in \mathbb {S} ^{n_{x}},Z\in \mathbb {S} ^{n_{Z_{1}}},}
   and 
  
    
      
        μ
        ∈
        
          
            R
          
          
            >
            0
          
        
      
    
    {\displaystyle \mu \in \mathbb {R} _{>0}}
   that minimizes 
  
    
      
        
          
            J
          
        
        (
        μ
        )
        =
        μ
      
    
    {\displaystyle {\mathcal {J}}(\mu )=\mu }
   subjects to 
  
    
      
        
          X
          
            1
          
        
        >
        0
        ,
         
        
          Y
          
            1
          
        
        >
        0
         
        Z
        >
        0
        ,
      
    
    {\displaystyle X_{1}>0,\ Y_{1}>0\ Z>0,}
  

  
    
      
        
          
            [
            
              
                
                  
                    N
                    
                      11
                    
                  
                
                
                  A
                  +
                  
                    A
                    
                      n
                    
                    
                      T
                    
                  
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      n
                    
                  
                  
                    C
                    
                      2
                    
                  
                
                
                  
                    B
                    
                      1
                      ,
                      1
                    
                  
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      n
                    
                  
                  D
                  
                    21
                    ,
                    1
                  
                
              
              
                
                  ∗
                
                
                  
                    X
                    
                      1
                    
                  
                  A
                  +
                  
                    A
                    
                      T
                    
                  
                  
                    X
                    
                      1
                    
                  
                  +
                  
                    B
                    
                      n
                    
                  
                  
                    C
                    
                      2
                    
                  
                  +
                  
                    C
                    
                      2
                    
                    
                      T
                    
                  
                  
                    B
                    
                      n
                    
                    
                      T
                    
                  
                
                
                  
                    X
                    
                      1
                    
                  
                  
                    B
                    
                      1
                      ,
                      1
                    
                  
                  +
                  
                    B
                    
                      n
                    
                  
                  
                    D
                    
                      21
                      ,
                      1
                    
                  
                
              
              
                
                  ∗
                
                
                  ∗
                
                
                  −
                  I
                
              
            
            ]
          
        
        <
        0
      
    
    {\displaystyle {\begin{bmatrix}N_{11}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1,1}+B_{2}D_{n}D{21,1}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1,1}+B_{n}D_{21,1}\\*&*&-I\end{bmatrix}}<0}
  ,

  
    
      
        
          
            [
            
              
                
                  
                    N
                    
                      11
                    
                  
                
                
                  A
                  +
                  
                    A
                    
                      n
                    
                    
                      T
                    
                  
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      n
                    
                  
                  
                    C
                    
                      2
                    
                  
                
                
                  
                    B
                    
                      1
                      ,
                      1
                    
                  
                  +
                  
                    B
                    
                      2
                    
                  
                  
                    D
                    
                      n
                    
                  
                  D
                  
                    21
                    ,
                    1
                  
                
                
                  
                    Y
                    
                      1
                    
                    
                      T
                    
                  
                  
                    C
                    
                      1
                      ,
                      2
                    
                    
                      T
                    
                  
                  +
                  
                    C
                    
                      n
                    
                    
                      T
                    
                  
                  
                    D
                    
                      12
                      ,
                      2
                    
                    
                      T
                    
                  
                
              
              
                
                  ∗
                
                
                  
                    X
                    
                      1
                    
                  
                  A
                  +
                  
                    A
                    
                      T
                    
                  
                  
                    X
                    
                      1
                    
                  
                  +
                  
                    B
                    
                      n
                    
                  
                  
                    C
                    
                      2
                    
                  
                  +
                  
                    C
                    
                      2
                    
                    
                      T
                    
                  
                  
                    B
                    
                      n
                    
                    
                      T
                    
                  
                
                
                  
                    X
                    
                      1
                    
                  
                  
                    B
                    
                      1
                      ,
                      1
                    
                  
                  +
                  
                    B
                    
                      n
                    
                  
                  
                    D
                    
                      21
                      ,
                      1
                    
                  
                
                
                  
                    C
                    
                      1
                      ,
                      2
                    
                    
                      T
                    
                  
                  +
                  
                    C
                    
                      2
                    
                    
                      T
                    
                  
                  
                    D
                    
                      n
                    
                    
                      T
                    
                  
                  
                    D
                    
                      12
                      ,
                      2
                    
                    
                      T
                    
                  
                
              
              
                
                  ∗
                
                
                  ∗
                
                
                  −
                  
                    γ
                    
                      d
                    
                  
                  I
                
                
                  
                    D
                    
                      11
                      ,
                      22
                    
                    
                      T
                    
                  
                  +
                  
                    D
                    
                      21
                      ,
                      2
                    
                    
                      T
                    
                  
                  
                    D
                    
                      n
                    
                    
                      T
                    
                  
                  
                    D
                    
                      12
                      ,
                      2
                    
                    
                      T
                    
                  
                
              
              
                
                  ∗
                
                
                  ∗
                
                
                  ∗
                
                
                  −
                  
                    γ
                    
                      d
                    
                  
                  I
                
              
            
            ]
          
        
        <
        0
      
    
    {\displaystyle {\begin{bmatrix}N_{11}&A+A_{n}^{T}+B_{2}D_{n}C_{2}&B_{1,1}+B_{2}D_{n}D{21,1}&Y_{1}^{T}C_{1,2}^{T}+C_{n}^{T}D_{12,2}^{T}\\*&X_{1}A+A^{T}X_{1}+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&X_{1}B_{1,1}+B_{n}D_{21,1}&C_{1,2}^{T}+C_{2}^{T}D_{n}^{T}D_{12,2}^{T}\\*&*&-\gamma _{d}I&D_{11,22}^{T}+D_{21,2}^{T}D_{n}^{T}D_{12,2}^{T}\\*&*&*&-\gamma _{d}I\end{bmatrix}}<0}
  ,

  
    
      
        
          
            [
            
              
                
                  
                    Y
                    
                      1
                    
                  
                  I
                  
                    Y
                    
                      1
                    
                  
                  
                    C
                    
                      1
                      ,
                      1
                    
                    
                      T
                    
                  
                  +
                  
                    C
                    
                      n
                    
                    
                      T
                    
                  
                  
                    D
                    
                      12
                      ,
                      1
                    
                    
                      T
                    
                  
                
              
              
                
                  ∗
                
                
                  
                    X
                    
                      1
                    
                  
                
                
                  
                    C
                    
                      1
                      ,
                      1
                    
                    
                      T
                    
                  
                  +
                  
                    C
                    
                      2
                    
                    
                      T
                    
                  
                  
                    D
                    
                      n
                    
                    
                      T
                    
                  
                  
                    D
                    
                      12
                      ,
                      1
                    
                    
                      T
                    
                  
                
              
              
                
                  ∗
                
                
                  ∗
                
                
                  Z
                
              
            
            ]
          
        
        >
        0
      
    
    {\displaystyle {\begin{bmatrix}Y_{1}IY_{1}C_{1,1}^{T}+C_{n}^{T}D_{12,1}^{T}\\*&X_{1}&C_{1,1}^{T}+C_{2}^{T}D_{n}^{T}D_{12,1}^{T}\\*&*&Z\end{bmatrix}}>0}
  ,

  
    
      
        
          D
          
            11
            ,
            11
          
        
        +
        
          D
          
            12
            ,
            1
          
        
        
          D
          
            n
          
        
        
          D
          
            21
            ,
            1
          
        
        =
        0
        ,
      
    
    {\displaystyle D_{11,11}+D_{12,1}D_{n}D_{21,1}=0,}
  

  
    
      
        
          
            [
            
              
                
                  
                    X
                    
                      1
                    
                  
                
                
                  I
                
              
              
                
                  ∗
                
                
                  
                    Y
                    
                      1
                    
                  
                
              
            
            ]
          
        
        >
        0
        ,
      
    
    {\displaystyle {\begin{bmatrix}X_{1}&I\\*&Y_{1}\end{bmatrix}}>0,}
  
tr
  
    
      
        Z
        <
        μ
        ,
      
    
    {\displaystyle Z<\mu ,}
  
where 
  
    
      
        
          N
          
            11
          
        
        =
        A
        
          Y
          
            1
          
        
        +
        
          Y
          
            1
          
        
        
          A
          
            T
          
        
        +
        
          B
          
            2
          
        
        
          C
          
            n
          
        
        +
        
          C
          
            n
          
          
            T
          
        
        
          B
          
            2
          
          
            T
          
        
      
    
    {\displaystyle N_{11}=AY_{1}+Y_{1}A^{T}+B_{2}C_{n}+C_{n}^{T}B_{2}^{T}}
  .
The controller is recovered by 

  
    
      
        
          A
          
            c
          
        
        =
        
          A
          
            K
          
        
        −
        B
        
          c
        
        (
        I
        −
        
          D
          
            22
          
        
        
          D
          
            c
          
        
        
          )
          
            −
            1
          
        
        
          D
          
            22
          
        
        
          C
          
            c
          
        
        ,
      
    
    {\displaystyle A_{c}=A_{K}-B{c}(I-D_{22}D_{c})^{-1}D_{22}C_{c},}
  

  
    
      
        
          B
          
            c
          
        
        =
        
          B
          
            K
          
        
        (
        I
        −
        
          D
          
            c
          
        
        
          D
          
            22
          
        
        )
        ,
      
    
    {\displaystyle B_{c}=B_{K}(I-D_{c}D_{22}),}
  

  
    
      
        
          C
          
            c
          
        
        =
        (
        I
        −
        
          D
          
            c
          
        
        
          D
          
            22
          
        
        )
        
          C
          
            K
          
        
        ,
      
    
    {\displaystyle C_{c}=(I-D_{c}D_{22})C_{K},}
  

  
    
      
        
          D
          
            c
          
        
        =
        (
        I
        +
        
          D
          
            K
          
        
        D
        
          22
        
        
          )
          
            −
            1
          
        
        
          D
          
            K
          
        
        ,
        w
        h
        e
        r
        e
        <
        m
        a
        t
        h
        >
        
          
            [
            
              
                
                  
                    A
                    
                      K
                    
                  
                
                
                  
                    B
                    
                      K
                    
                  
                
              
              
                
                  
                    C
                    
                      K
                    
                  
                
                
                  
                    D
                    
                      K
                    
                  
                
              
            
            ]
          
        
        =
        
          
            
              [
              
                
                  
                    
                      X
                      
                        2
                      
                    
                  
                  
                    
                      X
                      
                        1
                      
                    
                    
                      B
                      
                        2
                      
                    
                  
                
                
                  
                    0
                  
                  
                    I
                  
                
              
              ]
            
          
          
            −
            1
          
        
        (
        
          
            [
            
              
                
                  
                    A
                    
                      n
                    
                  
                
                
                  
                    B
                    
                      n
                    
                  
                
              
              
                
                  
                    C
                    
                      n
                    
                  
                
                
                  
                    D
                    
                      n
                    
                  
                
              
            
            ]
          
        
        −
        
          
            [
            
              
                
                  
                    X
                    
                      1
                    
                  
                  
                    A
                    
                      d
                    
                  
                  
                    Y
                    
                      1
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
              
            
            ]
          
        
        )
        
          
            
              [
              
                
                  
                    
                      Y
                      
                        2
                      
                      
                        T
                      
                    
                  
                  
                    0
                  
                
                
                  
                    
                      C
                      
                        2
                      
                    
                    
                      Y
                      
                        1
                      
                    
                  
                  
                    I
                  
                
              
              ]
            
          
          
            −
            1
          
        
      
    
    {\displaystyle D_{c}=(I+D_{K}D{22})^{-1}D_{K},where{\begin{bmatrix}A_{K}&B_{K}\\C_{K}&D_{K}\end{bmatrix}}={\begin{bmatrix}X_{2}&X_{1}B_{2}\\0&I\end{bmatrix}}^{-1}({\begin{bmatrix}A_{n}&B_{n}\\C_{n}&D_{n}\end{bmatrix}}-{\begin{bmatrix}X_{1}A_{d}Y_{1}&0\\0&0\end{bmatrix}}){\begin{bmatrix}Y_{2}^{T}&0\\C_{2}Y_{1}&I\end{bmatrix}}^{-1}}
  , and the matrices 
  
    
      
        
          X
          
            2
          
        
      
    
    {\displaystyle X_{2}}
   and  
  
    
      
        
          Y
          
            2
          
        
      
    
    {\displaystyle Y_{2}}
   satisfy 
  
    
      
        
          X
          
            2
          
        
        
          Y
          
            2
          
          
            T
          
        
        =
        I
        −
        
          X
          
            1
          
        
        
          Y
          
            1
          
        
      
    
    {\displaystyle X_{2}Y_{2}^{T}=I-X_{1}Y_{1}}
  . If 
  
    
      
        
          D
          
            22
          
        
        =
        0
      
    
    {\displaystyle D_{22}=0}
  , then 
  
    
      
        
          A
          
            c
          
        
        =
        
          A
          
            K
          
        
        ,
        
          B
          
            c
          
        
        =
        B
        
          K
        
        ,
        
          C
          
            c
          
        
        =
        
          C
          
            K
          
        
      
    
    {\displaystyle A_{c}=A_{K},B_{c}=B{K},C_{c}=C_{K}}
   and 
  
    
      
        
          D
          
            c
          
        
        =
        
          D
          
            K
          
        
      
    
    {\displaystyle D_{c}=D_{K}}
  .
Given 
  
    
      
        
          X
          
            1
          
        
      
    
    {\displaystyle X_{1}}
   and 
  
    
      
        
          Y
          
            1
          
        
      
    
    {\displaystyle Y_{1}}
  , the matrices 
  
    
      
        
          X
          
            2
          
        
      
    
    {\displaystyle X_{2}}
   and 
  
    
      
        
          Y
          
            2
          
        
      
    
    {\displaystyle Y_{2}}
   can be found using a matrix decomposition, such as
a LU decomposition or a Cholesky decomposition.
If 
  
    
      
        
          D
          
            11
            ,
            11
          
        
        =
        0
        ,
        
          D
          
            12
            ,
            1
          
        
        ≠
        0
        ,
        
           and 
        
        
          D
          
            21
            ,
            1
          
        
        ≠
        0
        ,
      
    
    {\displaystyle D_{11,11}=0,D_{12,1}\neq 0,{\text{ and }}D_{21,1}\neq 0,}
   then it is often simplest to choose 
  
    
      
        
          D
          
            n
          
        
        =
        0
      
    
    {\displaystyle D_{n}=0}
   in order
to satisfy the equality constraint 
  
    
      
        
          D
          
            11
            ,
            11
          
        
        +
        
          D
          
            12
            ,
            1
          
        
        
          D
          
            n
          
        
        
          D
          
            21
            ,
            1
          
        
        =
        0
        ,
      
    
    {\displaystyle D_{11,11}+D_{12,1}D_{n}D_{21,1}=0,}
  .
WIP, additional references to be added


== External Links ==
A list of references documenting and validating the LMI.


== Return to Main Page: ==