Final Exam Solutions.pdf - ESE 351-02 Spring 2017 Final Exam Due in homework bin by 2 pm May 8 1(20 points On Planet Mukai there are just two species

Final Exam Solutions.pdf - ESE 351-02 Spring 2017 Final...

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ESE 351-02, Spring 2017 Final Exam, Due in homework bin by 2 pm May 8 1. (20 points) On Planet Mukai, there are just two species: Pixies (prey) and Trolls (predators). Alien Overlords visit Planet Mukai at the beginning of every Mukaian year to perform a census of the Pixies and Trolls. If there are no Trolls, Pixies will reproduce such that there will be 50% more Pixies from one year to the next. If there are no Pixies, Trolls will die off such that there will be 50% fewer Trolls from one year to the next. But each Troll will eliminate enough Pixies so that there will be 5 fewer Pixies (than there otherwise would have been) in year k +1 for each Troll in year k . Also, there will be one more Troll (than there otherwise would have been) in year k +1 for each 20 Pixies in year k . After taking each census, the Alien Overlords have a choice of adding or removing a number of Pixies and/or Trolls from the planet. Define the state variables as x 1 ( k ) = number of Pixies in the year k census; x 2 ( k ) = number of Trolls in the year k census. Define the inputs as u 1 ( k ) = number of Pixies added by the Overlords in year k (after the census); u 2 ( k ) = number of Trolls added by the Overlords in year k . The output is the total number of Pixies plus Trolls in year k (at census time). In doing the following, ignore the facts that populations can’t be negative or in fractional units. That is, assume normal linear analysis. (a) Write the state-space model. (Clearly identify the A , B , C and D matrices.) (b) Write the characteristic equation. Is the system BIBO stable? Write the mode functions. (c) Write the state transition matrix. (d) If there are 100 Pixies in year 0, what number of Trolls in year 0 would keep both populations stable (the same number from year to year), given no Overlord intervention? (Such a number exists.) (e) Suppose there are 100 Pixies and 8 Trolls in year 0. How many Pixies and Trolls would there be in year 20, given no Overlord intervention? (f) Write the transfer function, H ( z ).
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ESE 351-02, Spring 2017 Final Exam, Due in homework bin by 2 pm May 8 a ( ) x 1 k + 1 ( ) = 1.5 x 1 k ( ) 5 x 2 k ( ) + u 1 k ( ) x 2 k + 1 ( ) = 1 20 x 1 k ( ) + 0.5 x 2 k ( ) + u 2 k ( ) y k ( ) = x 1 k ( ) + x 2 k ( ) x 1 k + 1 ( ) x 2 k + 1 ( ) = 3 2 5 1 20 1 2 A ! " # $ # x 1 k ( ) x 2 k ( ) + 1 0 0 1 B ! " # $ # u 1 k ( ) u 2 k ( ) y k ( ) = 1 1 C !" # $ # x k ( ) + 0 0 D ! " # $ # u k ( ) b ( ) zI A = z 3 2 5 1 20 z 1 2 = z 3 2 ( ) z 1 2 ( ) + 1 4 = z 2 2 z + 3 4 + 1 4 = z 2 2 z + 1 zI A = z 2 2 z + 1 = z 1 ( ) 2 = 0 not BIBO stable , mode functions : 1, k c ( ) zI A ( ) 1 z = z z 1 ( ) 2 z 1 2 5 1 20 z 3 2 z z 1 ( ) 2 k 1 k ( ) z 2 z 1 ( ) 2 = z z z 1 ( ) 2 k + 1 ( ) 1 k + 1 ( ) = k + 1 ( ) 1 k ( ) φ k ( ) = k + 1 ( ) 1 2 k 1 k ( ) 5 k 1 k ( ) 1 20 k 1 k ( ) k + 1 ( ) 3 2 k 1 k ( ) φ k ( ) = 1 2 k + 1 5 k 1 20 k 1 2 k + 1 1 k ( ) d ( ) x 1 k ( ) = 1 2 k + 1 ( ) x 1 0 ( ) 5 kx 2 0 ( ) x 2 k ( ) = 1 20 kx 1 0 ( ) + 1 2 k + 1 ( ) x 2 0 ( ) 100 = 1 2 k + 1 ( ) 100 5 kx 2 0 ( ) = 50 k + 100 5 kx 2 0
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