This preview shows pages 1–13. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 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 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 1 ; 1 1 n = = I AI A O O O O O 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 Timeinvariant 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) • TimeInvariant: – If you put in the same input at any time, then you get the same output. – If input is x 3 (t)=x(tt ), then output is y 3 (t)=y(tt ) Example • LTI or Not? ( 29 2 2 K r r r r r θ θ θ =  + = & && && & & Why is this important? • Most enzyme kinetic systems are not linear or even timeinvariant • So why? Row Echelon Form • A matrix is in rowechelon form if – All zero rows are at the bottom of the matrix – If two successive rows are nonzero, the first one starts with more zeros than the one before it • Rowequivalent: 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. 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 1 1 1 1 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...
View
Full
Document
This note was uploaded on 03/08/2010 for the course BCB 570 taught by Professor Juliedickerson during the Spring '10 term at Iowa State.
 Spring '10
 JulieDickerson

Click to edit the document details