time_analysis - Time Domain Analysis 1. Linear differential...

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Unformatted text preview: Time Domain Analysis 1. Linear differential equation systems 2. Matlab example 3. Linear stability analysis 4. Nonlinear stability analysis 5. Biochemical reactor example Linear Differential Equations Coupled linear differential equations Matrix representation 2 2 1 1 20 2 2 2 22 1 21 2 10 1 1 2 12 1 11 1 ) ( ) ( ) ( n n n nn n n n n n n n x x x a x a x a dt dx x x x a x a x a dt dx x x x a x a x a dt dx = + + + = = + + + = = + + + = = = = = nn n n n n n a a a a a a a a a x x x dt d 2 1 2 22 21 1 12 11 2 1 ) ( A x x x Ax x Matrix Diagonalization Coordinate transformation Columns of M are the eigenvectors of A Transformed equations Solution of decoupled equations ) ( ) ( ) ( ) ( 1 t t t t Mz x x M z = =- 1 2 1 1 ) ( ) ( x M z x Mz z z AMz M z AMz z M-- = = = = = = n dt d dt d n j e z t z z z z dt dz t j j j j j j j j ,..., 2 , 1 ) ( ) ( = = = = Solution Properties Coordinate transformation Solution Solution properties determined by the eigenvalues Eigenvalues in time domain equivalent to poles in Laplace domain ) ( ) ( ) ( 2 1 2 22 21 1 12 11 t m m m m m m m m m t t nn n n n n z Mz x = = 20 2 10 1 2 2 1 1 2 1 ) ( ) (...
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time_analysis - Time Domain Analysis 1. Linear differential...

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