Fase de Sistemas com Deadband

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Problem 9.21 PP
Several systems, such as a spacecraft, a spring-mass system with resonant frequency well 
below the frequency of switching, and a large motor-driven load with very small friction can be 
modeled as just an inertia. For an Ideal switching curve, sketch the phase portraits of the system. 
The switching function is e = 0 + roj. Assume that r = 10 sec and the control signal = 10-3 rad/ 
sec2.
Now sketch the results with
(a) deadband.
(b) deadband plus hysteresis.
(c) deadband plus time delay T,
(c) deadband plus time delay T,
(d) deadband plus a constant disturbance.
Step-by-step solution
step 1 of 8
The given switching function is e ^ 0 + 1 
It is assumed that 
r = 10 s. and
the control signal s 10~^rad/s^.
Find da
de 
d a _ u 
dd a
d a _ lOr^rad/sec* 
d& a
We knowthat — = u&, and $ = 500®*.
2
Thus, we get ^ « 0.05 when a = 10^
(a)
Step 2 of 8
Sketch the phase portrait o f the system with dead band.
0 >)
step 3 of 8
Obtain the phase portrait o f the system where there is deadband as well as 
hysterisis.
Step 4 of 8
The phase portrait o f the system with dead band and the time delay T is sketched.
(c)
(ression for a and a.
We know that 9̂ = - r a j + (d - k).
a = u + D , when t, ^ 9̂ 
a = {u + D )t +Of,
Find 9.
Step 7 of 8
#3
9 s + D'^ — + a ^ + 9j 
Find 9 when t = T.
Step 8 of 8 ^
Sketch the phase portrait o f the sjrstem with dead band and a constant disturbance.