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hw1_6243_2003

# hw1_6243_2003 - Massachusetts Institute of Technology...

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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 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 ) → { 0 , 1 } which change value at most once for 0 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. (b) Give an example of two different signals w 1 , w 2 → B which define same state of B at t = 4. (c) Find a time-invariant discrete-time finite state-space “difference inclusion” model for B , i.e. find a finite set X and functions g : X ∈� { 0 , 1 } , f : X ∈� S ( X ), where S ( X ) denotes the set of all non-empty subsets of X , such that a sequence w (0) , w (1) , w (2) , . . . can be obtained by sampling a signal w → B if and only if there exists a sequence x (0) , x (1) , x (2) , . . . of elements from X such that x ( t + 1) f ( x ( t )) and w ( t ) = g ( x ( t )) for t = 0 , 1 , 2 , . . . .

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• Spring '09
• AlexandreMegretski
• Dynamics, Constant of integration, Empty set, Vector Motors, Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science

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