HW13 Solutions.pdf - ESE 351-01 Fall 2017 Homework Set#13 due Dec 5 7 Problems 13.1 Consider the system of Mukai 2.6.9(a � ⎤ � ⎤ D x = ⎢ −3

HW13 Solutions.pdf - ESE 351-01 Fall 2017 Homework Set#13...

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ESE 351-01, Fall 2017 Homework Set #13, due Dec. 5 7 Problems 13.1. Consider the system of Mukai 2.6.9(a). D x = 3 4 1 2 x + 1 0 u y = 0 1 x + 5 [ ] u Use state-space methods to find (a) the transfer function and then (b) write the input- output differential equation. 13.2. Consider this discrete system. S x = 1 1 2 1 2 1 3 x + 1 0 u y = 1 1 x + 0 u Use state-space methods to find (a) the transfer function and then (b) write the input- output difference equation. 13.3 . Consider this LTIC continuous-time system. d dt x t ( ) = 1 1 a b x t ( ) + 0 1 u t ( ) y t ( ) = 1 0 0 1 x t ( ) Find a and b to place the poles of the system at -1 + i .
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ESE 351-01, Fall 2017 Homework Set #13, due Dec. 5 13.4. Recall Mukai 1.10.12(a), where the state variables are the angular rates of the corresponding bodies. d dt ω 1 ω 2 = B J 1 B J 1 B J 2 B J 2 A ! " ## $ ## ω 1 ω 2 x !"# + 0 1 J 2 B ! u y = 1 0 C ! " # $ # x + 0 [ ] D ! u Let J 1 = J 2 = B = 1, all compatible units, angles in radians. Use state-space methods to do the following. (a) Find the characteristic equation. (b) Find the eigenvalues (poles) of the system. Is it BIBO stable? (c) Find its mode functions. (d) Find the state transition matrix. (e) Assume the left mass is initially spinning at 1 rad/s and the other is spinning at the same rate, but in the opposite direction: -1 rad/s. No force is applied ( u ( t ) = 0). Write expressions for the two angular rates, x 1 ( t ) and x 2 ( t ), over time. (f) Find the transfer function ( Y/U ). (g) Write its response to a unit step input, starting from rest: u ( t ) = 1( t ). 13.5. Consider this discrete system. x 1 k + 1 ( ) x 2 k + 1 ( ) = 0 1 2 1 1 x 1 k ( ) x 2 k ( ) + 0 1 u k ( ) y = 1 0 x + 0 [ ] u (a) Find the characteristic equation. (b) Find the eigenvalues (poles) of the system. Is it BIBO stable? (c) Find its (real) mode functions. (d) Find the state transition matrix.
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ESE 351-01, Fall 2017 Homework Set #13, due Dec. 5 13.6 . (a) Write the state-space representation. (Chapter 1 stuff.) For parts (b) through (f), set C = R 1 = R 2 = 1 and L = 1/5 (compatible units). (b) Write the characteristic equation, find the poles and list the (real) mode functions. (c) Write the transfer function, H ( s ). (d) Write the system in the form of an input-output differential equation. (e) Write the state transition matrix. (f) Suppose u ( t ) = 0 and the initial current through the inductor is also 0, but the initial voltage on the capacitor (at time t = 0) is 1. Write expressions for the current through the inductor and voltage across the capacitor for t > 0.
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