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time_analysis

# 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 Λz AMz M z AMz z M-- = ⇒ = = = = ⇒ = n dt d dt d λ λ λ n j e z t z z z z dt dz t j j j j j j j j ,..., 2 , 1 ) ( ) ( = = ⇒ = = λ λ 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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