ps2_soln - EE 530 Reading assignment: Ch 3 PROBLEM SET 2...

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EE 530 PROBLEM SET 2 DUE: 6 DEC 2006 Reading assignment: Ch 3 The goal of the second problem set is to: 1. Gain experience using stability analysis tools. 2. Design and study the stability properties of elementary MRAC systems. Problem 5: (16 points) 1. (8 points) Consider a nonlinear system with state-space model ˙ x 1 = - x 1 + x 2 ( x 1 + c ) ˙ x 2 = - x 2 1 - x 1 c where c is a nonzero constant. Investigate the stability properties of the equilibrium state at the origin using Lyapunov’s direct method. In particular, determine whether or not the origin is an asymptotically stable equilibrium state. 2. (8 points) Repeat part 1 when c ( t ): ± + →± is a bounded time-varying function. Problem 6: (25 points) 1. (6 points) Consider the transfer function G ( s )= b 1 s + b 0 s 2 + a 1 s + a 0 . (a) (3 points) For what conditions on b 0 ,b 1 ,a 0 and a 1 is G ( s ) strictly positive real ? (b) (3 points) Suppose that b 1 = 0 and b o ² = 0, can G ( s ) be positive real ? Justify your answer.
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ps2_soln - EE 530 Reading assignment: Ch 3 PROBLEM SET 2...

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