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problem6

# problem6 - EE221A Linear System Theory Problem Set 6...

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EE221A Linear System Theory Problem Set 6 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2007 Issued 11/6; Due 11/15 Problem 1: Stiff Differential Equations. In the simulation of several engineering systems we encounter parasitic elements which result in the differential equation becoming “stiff”, for example, parasitic capacitances and inductances in electronic circuits. Consider the elementary circuit of Figure 1 with epsilon1 being a small parasitic capacitance. Write down the state equations for the circuit using x 1 and x 2 as state variables. Note that the A matrix depends on epsilon1 and that some of its elements blow up as epsilon1 0. Show that asymptotically one of the eigenvalues of A is of the order 1 /epsilon1 and the other is of order 1. Now generalize this example to the system ˙ x 1 = A 11 x 1 + A 12 x 2 epsilon1 ˙ x 2 = A 21 x 1 + A 22 x 2 with x 1 R n , x 2 R m , and A 22 nonsingular. Show that m eigenvalues go to as epsilon1

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