[<< wikibooks] LMIs in Control/Click here to continue/Controller synthesis/Minimum Decay Rate in State Feedback
LMIs in Control/Click here to continue/Controller synthesis/Minimum Decay Rate in State Feedback


== The System: ==
The number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix 
  
    
      
        B
      
    
    {\displaystyle B}
  . This can be achieved by adding an Apkarian filter in the input of the system.


== The Optimization Problem: ==
Apkarian Filter 
Let consider our TS-LIA model. This can be re written in linear form as:

  
    
      
        
          
            
              x
              ˙
            
          
        
        =
        A
        (
        z
        (
        t
        )
        )
        x
        +
        B
        (
        z
        (
        t
        )
        )
        u
      
    
    {\displaystyle {\dot {x}}=A(z(t))x+B(z(t))u}
  The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown , where 
  
    
      
        
          A
          
            F
          
        
      
    
    {\displaystyle A_{F}}
   = −100
  
    
      
        
          I
          
            4
          
        
      
    
    {\displaystyle I_{4}}
  , 
  
    
      
        
          B
          
            F
          
        
      
    
    {\displaystyle B_{F}}
   = 100
  
    
      
        
          I
          
            4
          
        
      
    
    {\displaystyle I_{4}}
   and 
  
    
      
        
          I
          
            4
          
        
      
    
    {\displaystyle I_{4}}
   ∈ 
  
    
      
        
          R
          
            4
            ×
            4
          
        
      
    
    {\displaystyle R^{4\times 4}}
   is the identity matrix.

  
    
      
        
          
            
              
                x
                
                  F
                
              
              ˙
            
          
        
        =
        
          A
          
            F
          
        
        
          x
          
            F
          
        
        +
        
          B
          
            F
          
        
        
          u
          
            F
          
        
        ;
        
          y
          
            f
          
        
        =
        
          x
          
            f
          
        
      
    
    {\displaystyle {\dot {x_{F}}}=A_{F}x_{F}+B_{F}u_{F};y_{f}=x_{f}}
  .When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. 
  
    
      
        u
      
    
    {\displaystyle u}
   = 
  
    
      
        
          y
          
            F
          
        
      
    
    {\displaystyle y_{F}}
   ). Then, the extended model is:

  
    
      
        
          
            
              
                x
                
                  c
                
              
              ˙
            
          
        
        =
        
          
            [
            
              
                
                  A
                  (
                  z
                  (
                  t
                  )
                  )
                
                
                  B
                  (
                  z
                  (
                  t
                  )
                  )
                
              
              
                
                  0
                
                
                  
                    A
                    
                      F
                    
                  
                
              
            
            ]
          
        
        
          x
          
            c
          
        
        +
        
          
            [
            
              
                
                  0
                
              
              
                
                  
                    B
                    
                      F
                    
                  
                
              
            
            ]
          
        
        
          u
          
            F
          
        
        =
        
          A
          
            e
          
        
        (
        z
        (
        t
        )
        )
        
          x
          
            e
          
        
        +
        
          B
          
            e
          
        
        
          u
          
            f
          
        
        ;
        
          x
          
            e
          
        
        =
        
          
            [
            
              
                
                  x
                
              
              
                
                  
                    x
                    
                      F
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\dot {x_{c}}}={\begin{bmatrix}A(z(t))&B(z(t))\\0&A_{F}\\\end{bmatrix}}x_{c}+{\begin{bmatrix}0\\B_{F}\end{bmatrix}}u_{F}=A_{e}(z(t))x_{e}+B_{e}u_{f};x_{e}={\begin{bmatrix}x\\x_{F}\end{bmatrix}}}
  This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.
State Feedback Controller Design
Let consider the state feedback control law  for the extended TS-LIA model:
  
    
      
        
          
            
              
                x
                
                  e
                
              
              ˙
            
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            32
          
        
        
          h
          
            i
          
        
        (
        z
        (
        t
        )
        )
        [
        
          A
          
            e
            i
          
        
        
          x
          
            e
          
        
        +
        
          B
          
            e
            i
          
        
        
          u
          
            F
          
        
        ]
      
    
    {\displaystyle {\dot {x_{e}}}=\sum _{i=1}^{32}h_{i}(z(t))[A_{ei}x_{e}+B_{ei}u_{F}]}
  , where the state feedback control laws are :
  
    
      
        
          u
          
            F
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            32
          
        
        
          h
          
            i
          
        
        (
        z
        (
        t
        )
        )
        
          K
          
            i
          
        
        x
        (
        t
        )
      
    
    {\displaystyle u_{F}=\sum _{i=1}^{32}h_{i}(z(t))K_{i}x(t)}
  , we get the closed loop system  :
  
    
      
        
          
            
              
                x
                
                  e
                
              
              ˙
            
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            32
          
        
        
          ∑
          
            j
            =
            1
          
          
            32
          
        
        
          h
          
            j
          
        
        (
        z
        (
        t
        )
        )
        [
        
          A
          
            e
            i
          
        
        
          x
          
            e
          
        
        +
        
          B
          
            e
            i
          
        
        
          K
          
            j
          
        
        ]
        
          x
          
            e
          
        
      
    
    {\displaystyle {\dot {x_{e}}}=\sum _{i=1}^{32}\sum _{j=1}^{32}h_{j}(z(t))[A_{ei}x_{e}+B_{ei}K_{j}]x_{e}}
  


== The LMI: ==
The design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of  LMI constraints are needed:

  
    
      
        L
        ⊗
        P
        +
        M
        ⊗
        P
        (
        
          A
          
            e
            i
          
        
        +
        
          B
          
            e
          
        
        
          K
          
            i
          
        
        )
        +
        M
        T
        ⊗
        (
        
          A
          
            e
          
        
        i
        +
        
          B
          
            e
          
        
        
          K
          
            i
          
        
        
          )
          
            T
          
        
        P
        <
        0
      
    
    {\displaystyle L\otimes P+M\otimes P(A_{ei}+B_{e}K_{i})+MT\otimes (A_{e}i+B_{e}K_{i})^{T}P<0}
   ∀i = 1, . . . , 32.A pair of conjugate complex poles s of the closed loop system can be written as 
  
    
      
        s
      
    
    {\displaystyle s}
   = − 
  
    
      
        
          ξ
        
        
          ω
          
            n
          
        
        ±
        j
        
          ω
          
            d
          
        
      
    
    {\displaystyle {\xi }\omega _{n}\pm j\omega _{d}}
  where 
  
    
      
        ξ
      
    
    {\displaystyle \xi }
   is the damping ratio,
  
    
      
        
          ω
          
            n
          
        
      
    
    {\displaystyle \omega _{n}}
   is the undamped natural frequency and 
  
    
      
        
          ω
          
            d
          
        
      
    
    {\displaystyle \omega _{d}}
   is the frequency response defined as 
  
    
      
        
          ω
          
            d
          
        
        =
        
          ω
          
            n
          
        
      
    
    {\displaystyle \omega _{d}=\omega _{n}}
   
  
    
      
        
          
            1
            −
            
              ξ
              
                2
              
            
          
        
      
    
    {\displaystyle {\sqrt {1-\xi ^{2}}}}
  .Three different LMI regions have been considered, each one related with a performance specification regarding 
  
    
      
        α
        =
        ξ
        
          ω
          
            n
          
        
        ,
        
          ω
          
            n
          
        
      
    
    {\displaystyle \alpha =\xi \omega _{n},\omega _{n}}
   and 
  
    
      
        ξ
      
    
    {\displaystyle \xi }
  : 
Minimum Decay Rate:
If we want to set a minimum decay rate α in the closed loop system response, the poles should be inside the LMI region defined in : 
  
    
      
        
          S
          
            α
          
        
      
    
    {\displaystyle S_{\alpha }}
   = [s = x + j y | x < − 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  ].where 
  
    
      
        α
      
    
    {\displaystyle \alpha }
   > 0. In this case L
  
    
      
        
          
          
            α
          
        
      
    
    {\displaystyle _{\alpha }}
   = 
  
    
      
        2
        α
      
    
    {\displaystyle 2\alpha }
   and M
  
    
      
        
          
          
            α
          
        
      
    
    {\displaystyle _{\alpha }}
   = 1.
Applying condition  to the closed-loop system , the LMI condition associated to this LMI region is: 

  
    
      
        2
        α
        P
        +
        (
        
          A
          
            e
            i
          
        
        +
        
          B
          
            e
          
        
        
          K
          
            i
          
        
        
          )
          
            T
          
        
        P
        +
        P
        (
        
          A
          
            e
            i
          
        
        +
        
          B
          
            e
          
        
        
          K
          
            i
          
        
        )
        <
        0
      
    
    {\displaystyle 2\alpha P+(A_{ei}+B_{e}K_{i})^{T}P+P(A_{ei}+B_{e}K_{i})<0}
   ∀i = 1, . . . , 32.


== Conclusion: ==
The LMI is feasible.


== Related LMIs ==
Apkarian Filter and State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Apkarian_Filter-and_State_Feedback
Maximum Natural Frequency in State Feedback https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Natural_Frequency_in_State_Feedback#The_LMI%3A


== References ==
Control, A. (2016). Gain-scheduling Control of a Quadrotor Using the Takagi-Sugeno Approach.