Transfer Function Matrix   전달함수 행렬

1. 상태방정식과 전달함수 간의 관계

  ㅇ (상태방정식 표현)
       
[# \mathbf{\dot{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} \\
          \mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u} #]
       
[# s\mathbf{X}(s) = \mathbf{A}\mathbf{X}(s) + \mathbf{B}\mathbf{U}(s) \\
          \mathbf{Y}(s) = \mathbf{C}\mathbf{X}(s) + \mathbf{D}\mathbf{U}(s) #]

  ㅇ (`상태방정식` → `전달함수 행렬` 형태로 변환)
      - 상태방정식으로부터 전달함수가 유일하게(Uniquely) 결정됨
         
[# (s\mathbf{I}-\mathbf{A})\mathbf{X}(s) = \mathbf{B}\mathbf{U}(s) \\
             \mathbf{X}(s) = (s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}\mathbf{U}(s) \\
             \mathbf{Y}(s) = \mathbf{C}\mathbf{X}(s) + \mathbf{D}\mathbf{U}(s) \\
             \quad\quad    = \mathbf{C}[(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}\mathbf{U}(s)]
                             + \mathbf{D}\mathbf{U}(s) \\
             \quad\quad    = [\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}
                             + \mathbf{D}] \mathbf{U}(s) #]
      - 전달함수 행렬
         
[#  \mathbf{T}(s) = \frac{\mathbf{Y}(s)}{\mathbf{U}(s)} 
                          = \mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B} + \mathbf{D} \\
             \qquad\qquad\qquad  = \mathbf{C}\frac{\text{adj}(s\mathbf{I}-\mathbf{A})}
                                         {\det(s\mathbf{I}-\mathbf{A})}\mathbf{B} + \mathbf{D} #]
      - 한편, 특성방정식은, 전달함수 분모를 0으로 놓은 것
         
[# \det(s\mathbf{I}-\mathbf{A}) = |s\mathbf{I}-\mathbf{A}| = 0 #]

  ㅇ (블록선도 표현)

      

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