hw1_09 - ) , T > x (0) = 1 (1) • What is the...

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ECE521 Linear Systems Fall 2009 Homework 1: Due September 10, in class To the extend possible please type your homeworks. Problem 1 : The example of a mechanical system given in lecture notes for Lecture 1 can be generalized to M ¨ q ( t ) + L ˙ q ( t ) + Kq ( t ) = f ( t ) , where q R k is a vector of positions, f R k is a vector of forces, M R k × k is an invertible mass matrix, L R k × k is a damping matrix, and K R k × k is a stiFness matrix. Choose a state vector x ( t ) for this system and derive a state-space model for this system in the standard, that is a model of the form b ˙ x = A ( t ) x ( t ) + B ( t ) u ( t ) y = C ( t ) x ( t ) + D ( t ) u ( t ) Assume that the output is y ( t ) = q ( t ). Specify matrices A,B,C,D and the input u(t). Problem 2 : In class we proved the existence and uniqueness ofthe solution to the folowing form of the (vector) IVP: b ˙ x = f ( t,x ) x ( t 0 ) = x 0 Now consider the following (scalar) equation b ˙ x = 2 t · x ( t ) for t [0 , T
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Unformatted text preview: ) , T > x (0) = 1 (1) • What is the form of the function f in (1)? • Is your f Lipshitz continuous with respect to x ? Explain. • In order to solve (1) apply the Picard iteration. Does it converge and if yes to what? Problem 3 : Let A = p a 1 a P for some scalar a n = 0. ±ind e tA . September 3, 2009 2 Problem 4 : Consider the system ˙ x ( t ) = A ( t ) x ( t ) where A ( t ) = λ λt λ λt λ • What are the eigenvalues of A ( t )? • How many eigenvectors are associated with the eigenvalues of A ( t )? • Find the state transition matrix for the system. Problem 5 : Let T be a positive scalar and A a square n × n matrix. Find a concise expression fre i T e tA dt ....
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hw1_09 - ) , T > x (0) = 1 (1) • What is the...

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