HW02_solutions

# HW02_solutions - EECS 460 Control System Analysis and...

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EECS 460: Control System Analysis and Design Homework #2 Solutions: 1. (a) We’ll use Nodal Analysis (could also write loop equations). 11 2 2 1 2 2 12 2 00 xu xx x x Cx RR R −− ++ = + = && Therefore, 1 2 2 1 22 1 2 2 1 1 x x u R R x x yx ⎛⎞ =− + + + ⎜⎟ ⎝⎠ = & & so our state matrices are: [] 2 2 1 1 2 11 1 1 1 0 01 CR R R C RC u x y x ⎡⎤ + ⎢⎥ =+ ⎣⎦ = & &

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(b) Note: No amplifier current into the Ideal Amplifier. 12 1 11 2 2 1 22 1 2 2 00 xu x x Cx RR x u x x yx −− += + = =+ =− = && & & From this you can calculate the state space for the system. [] 1 2 1 0 1 0 01 RC xx u x y x ⎡⎤ ⎢⎥ ⎣⎦ = & & 2. (a) There are several ways to do this; I’ll use a different method on each network for purposes of illustration. Brute Force: Take the Laplace Transform of equations (1), (2), and (3) from Problem #1. 1 2 2 1 1 2 2 1 1 (1) ( ) (2) ( ) ( ) ( ) (3) ( ) ( ) C s Xs Us R R C s Ys X s ⎛⎞ + + + ⎜⎟ ⎝⎠ = In (2), solve for X 1 (s) and substitute into (1). Then solve for X 2 (s) and apply (3).
() 12 2 2 22 1 1 11 2 2 1212 1 1 1 ## 1 Xs RC s RR s Us Ys X s Ys RRCCs RC RC RC s =+ ⎡⎤ ⎛⎞ + + ⎢⎥

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HW02_solutions - EECS 460 Control System Analysis and...

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