EEL 4657 : LINEAR CONTROL SYSTEMS

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EEL 4657 LINEAR CONTROL SYSTEMS
LECTURE 16 February 17 1999
INSTRUCTOR Dr. LATCHMAN UNIVERSITY OF FLORIDA
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DISCUSSION HOMEWORK 3, Problem 1
Consider the following system

• Is the system a SISO or MIMO System?
• Find the transfer function given by Y(s) G(s)
  U(s)
• Is the system a) Internally (asymptotically)
  stable?
• b) BIBO stable?
• What is the zero input response?

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REVIEW
A system is BIBO stable if a)
b)
Weaker conditions Marginal Stability
Non-repeated poles
Repeated poles
Given G(s) 1/s, Is the system BIBO stable?
No, as
has no bound
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REVIEW (continued)
Asymptotic Stability
Asymptotic stability always implies BIBO
stability. BIBO stability implies Asymptotic
stability if there are no cancellations and the
system is observable, controllable and is
minimal. In the sequel we will work with transfer
functions assuming that the system is internally
stable ( i.e. no cancellations) in forming G(s)
C(sI-A)-1B D
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PERFORMANCE

• We know that the poles of a system should be in
  the left hand plane for stability.
• Performance is dependent on the precise location
  of the poles in the left hand plane.
• Measures of Performance
• Rise time
• Percent Overshoot
• Steady state Error
• Settling time

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PERFORMANCE (continued)
It turns out that in classical controls, it is
useful to consider the second order model
Usually this pole is far enough away from the jw
axis, that it can be ignored.
This means that we try to approximate the more
complex systems by this second order system model.
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PERFORMANCE (continued)
An example
k
R(s)
Y(s)
Simplifying the feedback loop
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PERFORMANCE (continued)
Comparing with the original second order model
equation
Consider the standard second order system
The poles are given by
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PERFORMANCE (continued)
Plotting these poles
z
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PERFORMANCE (continued)
The step response for the system is
As part of homework 3, problem 2, show that
this is
where
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PERFORMANCE (continued)
Ystep(t)
t
At t 0, ystep(t) 0 as