problem4 - T is a constant torque. Let B r = { x ∈ R 2 :...

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EE221A Linear System Theory Problem Set 4 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2010 Issued 10/1; Due 10/8 Problem 1: Existence and uniqueness of solutions to linear diFerential equations. Let A ( t ) and B ( t ) be respectively n × n and n × n i matrices whose elements are real (or complex) valued piecewise continuous functions on R + . Let u ( · ) be a piecewise continuous function from R + to R n i . Show that for any Fxed u ( · ), the di±erential equation ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) (1) satisFes the conditions of the ²undamental Theorem. Problem 2: Existence and uniqueness of solutions to nonlinear diFerential equations. Consider the pendulum equation with friction and constant input torque: ˙ x 1 = x 2 ˙ x 2 = - g l sin x 1 - k m x 2 + T ml 2 (2) where x 1 is the angle that the pendulum makes with the vertical, x 2 is the angular rate of change, m is the mass of the bob, l is the length of the pendulum, k is the friction coe³cient, and
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Unformatted text preview: T is a constant torque. Let B r = { x ∈ R 2 : || x || < r } . ²or this system (represented as ˙ x = f ( x )) Fnd whether f is locally Lipschitz in x on B r for su³ciently small r , locally Lipschitz in x on B r for any Fnite r , or globally Lipschitz in x (ie. Lipschitz for all x ∈ R 2 ). Problem 3: Perturbed nonlinear systems. Suppose that some physical system obeys the di±erential equation ˙ x = p ( x, t ) , x ( t ) = x , ∀ t ≥ t where p ( · , · ) obeys the conditions of the fundamental theorem. Suppose that as a result of some perturbation the equation becomes ˙ z = p ( z, t ) + f ( t ) , z ( t ) = x + δx , ∀ t ≥ t Given that for t ∈ [ t , t + T ], || f ( t ) || ≤ ǫ 1 and || δx || ≤ ǫ , Fnd a bound on || x ( t )-z ( t ) || valid on [ t , t + T ]. 1...
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This note was uploaded on 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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