Mathematical+concepts

# Mathematical+concepts - L 7 Linear Systems and Metabolic...

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L 7: Linear Systems and Metabolic Networks Reading: Chapter 3.1,3.2 Integration of Ordinary Differential Equations dy’/dt = f(t,y) Marco Lattuada

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System as a mathematical mapping System H x(t) x[n] y(t) y[n]
Linear Equations Form System 1 1 2 2 n n a x a x a x b + + + = K 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 n n n n m m mn n m a x a x a x b a x a x a x b a x a x a x b + + + = + + + = + + + = K K M K

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Linear Systems 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 n n n n m m mn n m a x a x a x b a x a x a x b a x a x a x b + + + = + + + = + + + = K K M K = Ax b 11 12 1 1 21 22 2 2 1 2 n n m m mn m a a a b a a a b a a a b = = A b L L M M O M M L
Matrices in review Notation Identity Matrix [ ] 11 12 1 1 21 22 2 2 1 2 n n ik m m mn m a a a b a a a b a a a a b = = = A b L L M M O M M L 1 0 0 0 0 1 ; 0 0 0 0 0 1 0 0 0 0 0 1 n = = I AI A O O O O O

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Matrix operations Transpose: interchange all the rows and columns Sum and difference [ ] [ ] 11 21 1 12 22 2 1 2 n T n T ik ki m m nm a a a a a a a a a a a = = = A L L M M O M L [ ] [ ] / / ik ik a b + - = + - A B
Linear Time-invariant Systems Linear: weighted sum of signals lead to output Superposition holds If input is x 3 (t)=ax 1 (t)+bx 2 (t), then output is y 3 (t)=ay 1 (t) +by 2 (t) Time-Invariant: If you put in the same input at any time, then you get the same output. If input is x 3 (t)=x(t-t 0 ), then output is y 3 (t)=y(t-t 0 )

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Example LTI or Not? ( 29 2 2 0 K r r r r r θ θ θ - = - + = & && && & &
Why is this important? Most enzyme kinetic systems are not linear or even time-invariant So why?

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Row Echelon Form A matrix is in row-echelon form if All zero rows are at the bottom of the matrix If two successive rows are non-zero, the first one starts with more zeros than the one before it Row-equivalent: if matrix A can be obtained from matrix B using basic row operations Implies that A and B have the same solution sets.
Vocabulary If b =0, then system is homogeneous If a solution (values of x that satisfy equation) exists, then system is consistent, else it is inconsistent.

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Solve linear system using Gaussian Elimination Form Augmented Matrix, Row equivalence, can scale rows and add and subtract multiples to transform matrix 11 12 1 1 1 21 22 2 2 2 1 2 1 1 1 2 2 2 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 n n m m mn m n n n n a a a b d a a a b d m n a a a b d d x d d x d d x d = = = = = x L L L L M M O M M M M M O M L L L L M M M M O M L
11 12 1 1 1 21 22 2 2 2 1 2 1 0 0 0 1 0 0 1 1 n n m m mn m m a a a b d a a a b d m n a a a b d < L L L L M M O M

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