[<< wikibooks] LMIs in Control/pages/LMI for Attitude Control of Nonrotating Missles
LMI for Attitude Control of Nonrotating Missles, Pitch Channel
The dynamic model of a missile is very complicated and a simplified model is used. To do so, we consider a simplified attitude system model for the pitch channel in the system. We aim to achieve a non-rotating motion of missiles. It is worthwhile to note that the attitude control design for the pitch channel and the yaw/roll channel can be solved exactly in the same way while representing matrices of the system are different.


== The System ==
The state-space representation for the pitch channel can be written as follows:

  
    
      
        
          
            
              
                
                  
                    
                      x
                      ˙
                    
                  
                
                (
                t
                )
              
              
                
                =
                A
                (
                t
                )
                x
                (
                t
                )
                +
                
                  B
                  
                    1
                  
                
                (
                t
                )
                u
                (
                t
                )
                +
                
                  B
                  
                    2
                  
                
                (
                t
                )
                d
                (
                t
                )
              
            
            
              
                y
                (
                t
                )
              
              
                
                =
                C
                (
                t
                )
                x
                (
                t
                )
                +
                
                  D
                  
                    1
                  
                
                (
                t
                )
                u
                (
                t
                )
                +
                
                  D
                  
                    2
                  
                
                (
                t
                )
                d
                (
                t
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {x}}(t)&=A(t)x(t)+B_{1}(t)u(t)+B_{2}(t)d(t)\\y(t)&=C(t)x(t)+D_{1}(t)u(t)+D_{2}(t)d(t)\end{aligned}}}
  
where 
  
    
      
        x
        =
        [
        α
        
        
          w
          
            z
          
        
        
        
          δ
          
            z
          
        
        
          ]
          
            T
          
        
      
    
    {\displaystyle x=[\alpha \quad w_{z}\quad \delta _{z}]^{\text{T}}}
  , 
  
    
      
        u
        =
        
          δ
          
            z
            c
          
        
      
    
    {\displaystyle u=\delta _{zc}}
   , 
  
    
      
        y
        =
        [
        α
        
        
          n
          
            y
          
        
        
          ]
          
            T
          
        
      
    
    {\displaystyle y=[\alpha \quad n_{y}]^{\text{T}}}
  , and 
  
    
      
        d
        =
        [
        β
        
        
          w
          
            y
          
        
        
          ]
          
            T
          
        
      
    
    {\displaystyle d=[\beta \quad w_{y}]^{\text{T}}}
   are the state variable, control input, output, and disturbance vectors, respectively. The paprameters 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  , 
  
    
      
        
          w
          
            z
          
        
      
    
    {\displaystyle w_{z}}
  , 
  
    
      
        
          δ
          
            z
          
        
      
    
    {\displaystyle \delta _{z}}
  , 
  
    
      
        
          δ
          
            z
            c
          
        
      
    
    {\displaystyle \delta _{zc}}
  , 
  
    
      
        
          n
          
            y
          
        
      
    
    {\displaystyle n_{y}}
  , 
  
    
      
        β
      
    
    {\displaystyle \beta }
  , and 
  
    
      
        
          w
          
            y
          
        
      
    
    {\displaystyle w_{y}}
   stand for the attack angle, pitch angular velocity, the elevator deflection, the input actuator deflection, the overload on the side direction, the sideslip angle, and the yaw angular velocity, respectively.


== The Data ==
In the aforementioned pitch channel system, the matrices 
  
    
      
        A
        (
        t
        )
        ,
        
          B
          
            1
          
        
        (
        t
        )
        ,
        
          B
          
            2
          
        
        (
        t
        )
        ,
        C
        (
        t
        )
        ,
        
          D
          
            1
          
        
        (
        t
        )
        ,
      
    
    {\displaystyle A(t),B_{1}(t),B_{2}(t),C(t),D_{1}(t),}
   and 
  
    
      
        
          D
          
            2
          
        
        (
        t
        )
      
    
    {\displaystyle D_{2}(t)}
   are given as:

  
    
      
        
          
            
              
                A
                (
                t
                )
                =
                
                  
                    [
                    
                      
                        
                          −
                          
                            a
                            
                              4
                            
                          
                          (
                          t
                          )
                        
                        
                          1
                        
                        
                          −
                          
                            a
                            
                              5
                            
                          
                          (
                          t
                          )
                        
                      
                      
                        
                          −
                          
                            
                              
                                
                                  a
                                  ´
                                
                              
                            
                            
                              1
                            
                          
                          (
                          t
                          )
                          
                            a
                            
                              4
                            
                          
                          (
                          t
                          )
                          −
                          
                            a
                            
                              2
                            
                          
                          (
                          t
                          )
                        
                        
                          
                            
                              
                                
                                  a
                                  ´
                                
                              
                            
                            
                              1
                            
                          
                          (
                          t
                          )
                          −
                          
                            a
                            
                              1
                            
                          
                          (
                          t
                          )
                        
                        
                          
                            
                              
                                
                                  a
                                  ´
                                
                              
                            
                            
                              1
                            
                          
                          (
                          t
                          )
                          
                            a
                            
                              5
                            
                          
                          (
                          t
                          )
                          −
                          
                            a
                            
                              3
                            
                          
                          (
                          t
                          )
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          −
                          1
                          
                            /
                          
                          
                            τ
                            
                              z
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}A(t)={\begin{bmatrix}-a_{4}(t)&1&-a_{5}(t)\\-{\acute {a}}_{1}(t)a_{4}(t)-a_{2}(t)&{\acute {a}}_{1}(t)-a_{1}(t)&{\acute {a}}_{1}(t)a_{5}(t)-a_{3}(t)\\0&0&-1/\tau _{z}\end{bmatrix}}\end{aligned}}}
  

  
    
      
        
          
            
              
                
                  B
                  
                    1
                  
                
                (
                t
                )
                =
                
                  
                    [
                    
                      
                        
                          0
                        
                      
                      
                        
                          0
                        
                      
                      
                        
                          1
                        
                      
                    
                    ]
                  
                
                ,
                
                
                  B
                  
                    2
                  
                
                (
                t
                )
                =
                
                  
                    
                      w
                      
                        x
                      
                    
                    57.3
                  
                
                
                  
                    [
                    
                      
                        
                          −
                          1
                        
                        
                          0
                        
                      
                      
                        
                          −
                          
                            
                              
                                
                                  a
                                  ´
                                
                              
                            
                            
                              1
                            
                          
                          (
                          t
                          )
                        
                        
                          
                            
                              
                                
                                  J
                                  
                                    x
                                  
                                
                                −
                                
                                  J
                                  
                                    y
                                  
                                
                              
                              
                                J
                                
                                  z
                                
                              
                            
                          
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}B_{1}(t)={\begin{bmatrix}0\\0\\1\end{bmatrix}},\quad B_{2}(t)={\frac {w_{x}}{57.3}}{\begin{bmatrix}-1&0\\-{\acute {a}}_{1}(t)&{\frac {J_{x}-J_{y}}{J_{z}}}\\0&0\end{bmatrix}}\end{aligned}}}
  

  
    
      
        
          
            
              
                C
                (
                t
                )
                =
                
                  
                    
                      w
                      
                        x
                      
                    
                    57.3
                  
                
                
                  
                    [
                    
                      
                        
                          57.3
                          g
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          V
                          (
                          t
                          )
                          
                            a
                            
                              4
                            
                          
                          (
                          t
                          )
                        
                        
                          0
                        
                        
                          V
                          (
                          t
                          )
                          
                            a
                            
                              5
                            
                          
                          (
                          t
                          )
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}C(t)={\frac {w_{x}}{57.3}}{\begin{bmatrix}57.3g&0&0\\V(t)a_{4}(t)&0&V(t)a_{5}(t)\end{bmatrix}}\end{aligned}}}
  

  
    
      
        
          
            
              
                
                  D
                  
                    1
                  
                
                (
                t
                )
                =
                0
                ,
                
                
                  D
                  
                    2
                  
                
                (
                t
                )
                =
                
                  
                    1
                    
                      57.3
                      g
                    
                  
                
                
                  
                    [
                    
                      
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          V
                          (
                          t
                          )
                          
                            b
                            
                              7
                            
                          
                          (
                          t
                          )
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}D_{1}(t)=0,\quad D_{2}(t)={\frac {1}{57.3g}}{\begin{bmatrix}0&0\\V(t)b_{7}(t)&0\end{bmatrix}}\end{aligned}}}
  
where 
  
    
      
        
          a
          
            1
          
        
        (
        t
        )
        ∼
        
          a
          
            6
          
        
        (
        t
        )
        ,
        
        
          b
          
            1
          
        
        (
        t
        )
        ∼
        
          b
          
            7
          
        
        (
        t
        )
        ,
        
          
            
              
                a
                ´
              
            
          
          
            1
          
        
        (
        t
        )
        ,
        
          
            
              
                b
                ´
              
            
          
          
            1
          
        
        (
        t
        )
      
    
    {\displaystyle a_{1}(t)\sim a_{6}(t),\quad b_{1}(t)\sim b_{7}(t),{\acute {a}}_{1}(t),{\acute {b}}_{1}(t)}
   and 
  
    
      
        
          c
          
            1
          
        
        (
        t
        )
        ∼
        
          c
          
            4
          
        
        (
        t
        )
      
    
    {\displaystyle c_{1}(t)\sim c_{4}(t)}
   are the system parameters. Moreover, 
  
    
      
        V
      
    
    {\displaystyle V}
   is the speed of the missle and 
  
    
      
        
          J
          
            x
          
        
      
    
    {\displaystyle J_{x}}
  , 
  
    
      
        
          J
          
            y
          
        
      
    
    {\displaystyle J_{y}}
  , and 
  
    
      
        
          J
          
            z
          
        
      
    
    {\displaystyle J_{z}}
   are the rotary inertia of the missle corresponding to the body coordinates.


== The Optimization Problem ==
The optimization problem is to find a state feedback control law 
  
    
      
        u
        =
        K
        x
      
    
    {\displaystyle u=Kx}
   such that:
1. The closed-loop system:

  
    
      
        
          
            
              
                
                  
                    
                      x
                      ˙
                    
                  
                
              
              
                
                =
                (
                A
                +
                
                  B
                  
                    1
                  
                
                K
                )
                x
                +
                
                  B
                  
                    2
                  
                
                d
              
            
            
              
                z
              
              
                
                =
                (
                C
                +
                
                  D
                  
                    1
                  
                
                K
                )
                x
                +
                
                  D
                  
                    2
                  
                
                d
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\dot {x}}&=(A+B_{1}K)x+B_{2}d\\z&=(C+D_{1}K)x+D_{2}d\end{aligned}}}
  
is stable. 
2. The 
  
    
      
        
          H
          
            ∞
          
        
      
    
    {\displaystyle H_{\infty }}
   norm of the transfer function:

  
    
      
        
          G
          
            z
            d
          
        
        (
        s
        )
        =
        (
        C
        +
        
          D
          
            1
          
        
        K
        )
        (
        s
        I
        −
        (
        A
        +
        
          B
          
            1
          
        
        K
        )
        
          )
          
            −
            1
          
        
        
          B
          
            2
          
        
        +
        
          D
          
            2
          
        
      
    
    {\displaystyle G_{zd}(s)=(C+D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2}+D_{2}}
  
is less than a positive scalar value, 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  . Thus:

  
    
      
        
          |
        
        
          |
        
        
          G
          
            z
            d
          
        
        (
        s
        )
        
          |
        
        
          
            |
          
          
            ∞
          
        
        <
        γ
      
    
    {\displaystyle ||G_{zd}(s)||_{\infty }<\gamma }
  


== The LMI: LMI for non-rotating missle attitude control ==
Using Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:

  
    
      
        
          
            
              
              
                
                  min
                
                
                γ
              
            
            
              
              
                
                  s.t.
                
                
                X
                >
                0
              
            
            
              
              
                
                  
                    [
                    
                      
                        
                          (
                          A
                          X
                          +
                          
                            B
                            
                              1
                            
                          
                          W
                          
                            )
                            
                              T
                            
                          
                          +
                          A
                          X
                          +
                          
                            B
                            
                              1
                            
                          
                          W
                        
                        
                          
                            B
                            
                              2
                            
                          
                        
                        
                          (
                          C
                          X
                          +
                          
                            D
                            
                              1
                            
                          
                          W
                          
                            )
                            
                              T
                            
                          
                        
                      
                      
                        
                          
                            B
                            
                              2
                            
                            
                              T
                            
                          
                        
                        
                          −
                          γ
                          I
                        
                        
                          
                            D
                            
                              2
                            
                            
                              T
                            
                          
                        
                      
                      
                        
                          C
                          X
                          +
                          
                            D
                            
                              1
                            
                          
                          W
                        
                        
                          
                            D
                            
                              2
                            
                          
                        
                        
                          −
                          γ
                          I
                        
                      
                    
                    ]
                  
                
                <
                0
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&{\text{min}}\quad \gamma \\&{\text{s.t.}}\quad X>0\\&{\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{2}&(CX+D_{1}W)^{T}\\B_{2}^{T}&-\gamma I&D_{2}^{T}\\CX+D_{1}W&D_{2}&-\gamma I\end{bmatrix}}<0\end{aligned}}}
  


== Conclusion: ==
As mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
   is the disturbance attenuation level. When the matrices 
  
    
      
        W
      
    
    {\displaystyle W}
   and 
  
    
      
        X
      
    
    {\displaystyle X}
   are determined in the optimization problem, the controller gain matrix can be computed by:

  
    
      
        K
        =
        W
        
          X
          
            −
            1
          
        
      
    
    {\displaystyle K=WX^{-1}}
  


== Implementation ==
A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Non-rotating-Missle-Attitude-Control


== Related LMIs ==
LMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel


== External Links ==
[1] - LMI in Control Systems Analysis, Design and Applications


== Return to Main Page ==
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