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problem5

# problem5 - EE221A Linear System Theory Problem Set 5...

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EE221A Linear System Theory Problem Set 5 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2011 Issued 10/18; Due 10/27 Problem 1: Dynamical systems, time invariance. Suppose that the output of a system is represented by y ( t ) = integraldisplay t −∞ e ( t τ ) u ( τ ) Show that it is a (i) dynamical system, and that it is (ii) time invariant. You may select the input space U to be the set of bounded, piecewise continuous, real-valued functions defined on ( −∞ , ). Problem 2: Jacobian Linearization I. Consider the now familiar 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 (1) 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 coefficient, and T is a constant torque. Considering T

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