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Assignment 1 for Cyberphysical systems assignment offered by Coursera online
Typology: Exams
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a) Derive an analytical expression for the τs component of the solution (equivalently, execution or trajectory) starting from τs(0, 0) = T ∗ s 2 and ms(0, 0) = 0. Write it in terms of hybrid time (t, j).
c):Is every solution to (1)-(2) of dwell-time type? Justify. By definition, if there exists a positive constant c such that, for each j > 0 such that (t, j) ∈ dom φ for some t, there exists (t 0 , j),(t 00, j) ∈ dom φ with t 00 − t 0 ≥ c; then, every such solution is dwell time. For every solution to (1),(2) with τs(0, 0) < T∗ s , after the first jump (j > 0), there exists a positive constant c. Therefore, every such solution is of dwell- time type. d):
a nontrivial solution from every point of C ∪ D. - Furthermore, since G(D) ⊂ C ∪ D, item 5(c) does not occur, and due to the properties F, item 5(b) does not hold either. Thus, every maximal solution is complete. b 1 ):
b 2 ): b 3 ):
within C. This implies that (VC) holds. - By item 2, if (VC) holds for every ξ ∈ C \ D, then there exists a nontrivial solution from every point of C ∪ D. - Furthermore, since G(D) ⊂ C ∪ D, item 5(c) does not occur, and due to the properties F, item 5(b) does not hold either. Thus, every maximal solution is complete. b): HyEQ_Toolbox_V2_04/Examples/CPS_examples/ContinuousPlant/ContinuousPlant_example.mdl which after installation of the toolbox, it should be in your local MATLAB folder (see the lecture "Simulations of Cyber-Physical Systems" and beyond for instructions and examples), inspect the Simulink file and perform the following tasks: b.1) Define the associated functions F and G, and the sets C and D of a hybrid equation/inclusion (without inputs) that models that Simulink block. b.2) Sketch the execution from the initial conditions and the input used in the Simulink model. c):