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ps1sol_6243_2003

# ps1sol_6243_2003 - Massachusetts Institute of Technology...

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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 1 Solutions 1 Problem 1.1 Behavior set B of an autonomous system with a scalar binary DT output consists of all DT signals w = w ( t ) ≤ { , 1 } which change value at most once for ∀ t < → . (a) Give an example of two signals w 1 , w 2 ≤ B which commute at t = 3 , but do not define same state of B at t = 3 . To answer this and the following questions, let us begin with formulating necessary and suﬃcient conditions for two signals z 1 , z 2 ≤ B to commute and to define same state of B at a given time t . For w ≤ B , t ≤ [0 , → ) let w [ t ] = lim t, <t w ( ) , w (0) , if t > , if t = be the left side limit value of w at t . Let , if w ( t ) = lim w ( ) , N + ( w, t ) = 1 , otherwise be the number of discontinuities of w ( ) between = t and = → . Similarly, let , if w (0) = w [ t ] , N − ( w, t ) = 1 , otherwise be the number of discontinuities of w ( ) between = and = t − 0. 1 Version of October 2, 2003 2 Lemma 1.1 Signals z 1 , z 2 ≤ B commute at time t ≤ [0 , → ) if and only if z 1 ( t ) = z 2 ( t ) and N − ( z 1 , t ) + N + ( z 2 , t ) + | z 2 ( t ) − z 1 [ t ] | ∀ 1 (1.1) and N − ( z 2 , t ) + N + ( z 1 , t ) + | z 1 ( t ) − z 2 [ t ] | ∀ 1 . (1.2) Proof First note that the “hybrid” signal z 12 , obtained by “gluing” the past of z 1 (before time t ) to the future of z 2 (from t to → ), is a discrete time signal if and only if z 1 ( t ) = z 2 ( t ). Moreover, since he discontinuities of z 12 result from three causes: discontinuities of z 1 ( ) before = t , discontinuities of z 2 between = t and = → , and the inequality between z 1 [ t ] and z 1 ( t ), condition (1.1) is necessary and suﬃcient for z 12 ≤ B (subject to z 1 ( t ) = z 2 ( t )). Similarly, considering the discontinuities of the other “hybrid” obtained by “gluing” the past of z 2 to the future of z 2 yields (1.2)....
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