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Cyberphysical systems assignment, Exams of Electronics

Assignment 1 for Cyberphysical systems assignment offered by Coursera online

Typology: Exams

2017/2018

Uploaded on 10/15/2018

nomimalik15
nomimalik15 🇵🇰

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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).
The solution initially flows until τs = T s . When τs reaches T s , the solution jumps to zero.
Since τs(0, 0) = T s 2 , the first jump occurs at t = T s 2 .
m˙ s = 0 and ms(0, 0) = 0 = ms stays at zero during the time [0, T s 2 ].
At j = 1, the timer τs is reset to zero and vs is used to update ms (i.e., τ + s = 0, m+ s = vs).
b1)
What are the values of the ms components of the solution from τs(0, 0) = 0 and ms(0, 0) = 0 at
the following specific hybrid times:
(t, j) = (0.5, 0),(1, 0),(1, 1)
when t = 0.5, j = 0, ms(0.5, 0) = 0
when t = 1, j = 0, ms(1, 0) = 0
when t = 1, j = 1, ms(1, 1) = vs(1, 1) = sin(2πt) = 0.
b2):
Plot as a function of (t, j) the component τs of the
execution from τs(0, 0) = 0 and ms(0, 0) = 0
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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).

  • The solution initially flows until τs = T ∗ s. When τs reaches T ∗ s , the solution jumps to zero.
  • Since τs(0, 0) = T ∗ s 2 , the first jump occurs at t = T ∗ s 2.
  • m˙ s = 0 and ms(0, 0) = 0 =⇒ ms stays at zero during the time [0, T ∗ s 2 ].
  • At j = 1, the timer τs is reset to zero and vs is used to update ms (i.e., τ + s = 0, m+ s = vs). b1) What are the values of the ms components of the solution from τs(0, 0) = 0 and ms(0, 0) = 0 at the following specific hybrid times: (t, j) = (0.5, 0),(1, 0),(1, 1)
  • when t = 0.5, j = 0, ms(0.5, 0) = 0
  • when t = 1, j = 0, ms(1, 0) = 0
  • when t = 1, j = 1, ms(1, 1) = vs(1, 1) = sin(2πt) = 0. b 2 ): Plot as a function of (t, j) the component τs of the execution from τs(0, 0) = 0 and ms(0, 0) = 0

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):