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Problem 7.57PP
Assume that the linearized and time-scaled equation of motion for the ballbearing levitation 
device is x - x = w + w. Here w is a constant bias due to the power amplifier. Introduce integral 
error control, and select three control gains K = [K^ K2 K3]so that the closed-loop poles are at 
-1 and -1 ± j and the steady-state error to w and to a (step) position command will be zero. Let y 
= X and the reference input ^ ^ be a constant. Draw a block diagram of your design 
showing the locations of the feedback gains Ki. Assume that both x and x can be measured. Plot 
the response of the closedloop system to a step command input and the response to a step 
change in the bias input. Verify that the system is Type 1. Use Matlab (Simulink) software to 
simulate the system responses.
Step-by-step solution
Step-by-step solution
step 1 of 2
The equations o f motion are given by
x - x = u + W 
0 = 0
A realization of these equation is
»i[:]
= y - r
z =Xi X X
'0 1 O' O'
K = 0 0 1 0
_0 1 0 1_
Step 2 of 2
The design ofthe state feedback vector, K 
For closed loop poles o f 5 = -1 , - l ± j t 
det(51-^'^+G S ,^) = 0 
When
K , / : , ]= [2 5 3]
The closed loop system is given by
■-1"
Z = {F^-a^K )Z+0^(S >+ I
y ^ M , z
For the closed loop system we have
0 1 0
0 0 1
1 - ^ j - ^ 3
This immediately gives Zz—0 
AndZz—y — 0