Control System Design

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Out response, Poles, and Zeros

• The output response of a system is the sum of two
  responses the forced response (steady-state
  response) and the natural response (zero input
  response)
• The poles of a transfer function are (1) the
  values of the Laplace transform variable, s ,
  that cause the transfer function to become
  infinite, or (2) any roots of the denominator of
  the transfer function that are common to roots of
  the numerator.
• The zeros of a transfer function are (1) the
  values of the Laplace transform variable, s ,
  that cause the transfer function to become zero,
  or (2) any roots of the numerator of the transfer
  function that are common to roots of the
  denominator.

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Figure 4.1a. System showing input and
outputb. pole-zero plot of the systemc.
evolution of a system response.
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1. A pole of the input function generates the form
   of the forced response.
2. A pole of the transfer function generates the
   form of the natural response.
3. A pole on the real axis generates an exponential
   response.
4. The zeros and poles generate the amplitudes for
   both the forced and natural responses.

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Natural response
Forced response
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First-Order Systems
Figure 4.4a. First-order systemb. pole plot
Figure 4.5First-order systemresponse to a unit
step
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• The time constant can be described as the time
  for to decay to 37 of its initial value.
  Alternately, the time is the time it takes for
  the step response to rise to of its final value.
• The reciprocal of the time constant has the units
  (1/seconds), or frequency. Thus, we call the
  parameter the exponential frequency.

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• Rise time Rise time is defined as the time for
  the waveform to go from 0.1 to 0.9 of its final
  value.
• Settling time Settling time is defined as the
  time for the response to reach, and stay within,
  2 (or 5) of its final value.

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Second-Order Systems
Figure 4.7Second-ordersystems, pole plots,and
step responses
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• 1. Overdamped response
• Poles Two real at
• Natural response Two exponentials with time
  constants equal to the reciprocal of the pole
  location
• 2. Underdamped responses
• Poles Two complex at
• Natural response Damped sinusoid with and
  exponential envelope whose time constant is equal
  to the reciprocal of the poles real part. The
  radian frequency of the sinusoid, the damped
  frequency of oscillation, is equal to the
  imaginary part of the poles

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• 3. Undamped response
• Poles Two imaginary at
• Natural response Undamped sinusoid with radian
  frequency equal to the imaginary part of the
  poles
• 4. Critically damped responses
• Poles Two real at
• Natural response One term is an exponential
  whose time constant is equal to the reciprocal of
  the pole location. Another term is the product of
  time and an exponential with time constant equal
  to the reciprocal of the pole location

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Figure 4.10Step responses for second-ordersystem
damping cases
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Figure 4.8Second-order step response
componentsgenerated by complex poles
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• Natural Frequency The natural frequency of a
  second-order system is the frequency of
  oscillation of the system without damping.
• Damping Ratio The damping ratio is defined as
  the ratio of exponential decay frequency to
  natural frequency.
• Consider the general system
• Without damping,

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Figure 4.11Second-order response as a function
of damping ratio
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Underdamped Second-Order Systems
Step response
Taking the inverse Laplace transform
where
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Figure 4.13Second-order underdampedresponses
for damping ratio values
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• Peak time The time required to reach the first,
  or maximum, peak.
• Percent overshoot The amount that the waveform
  overshoots the steady-state, or final, value at
  the peak time, expressed as a percentage of the
  steady-state value.
• Settling time The time required for the
  transients damped oscillations to reach and stay
  within (or ) of the steady-state
  value.
• Rise time The time required for the waveform to
  go from 0.1 of the final value to 0.9 of the
  final value.
• Evaluation of peak time

Setting the derivative equal to zero yields

Peak time
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Figure 4.14Second-order underdampedresponse
specifications
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• Evaluation of percent overshoot (
  )
• Evaluation of settling time
• The settling time is the time it takes for the
  amplitude of the decaying
• sinusoid to reach o.o2, or
• ,
• where is the imaginary part of the pole and
  is called the damped
• frequency of oscillation, and is the
  magnitude of the real part of the
• pole and is the exponential damping frequency.

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Figure 4.17Pole plot for an underdamped
second-order system
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Figure 4.18Lines of constant peak time,Tp ,
settlingtime,Ts , and percent overshoot,
OSNote Ts2 lt Ts1 Tp2 lt Tp1 OS1 ltOS2
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Figure 4.19Step responses of second-orderunderd
amped systems as poles movea. with constant
real partb. with constant imaginary partc.
with constant damping ratio
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Find peak time, percent overshoot, and settling
time from pole location.
,
,
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Figure 4.21Rotational mechanical system
Design Given the rotational mechanical system,
find J and D to yield 20 overshoot and a
settling time of 2 seconds for a step input of
torque T(t).
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Figure 4.23Component responses of a three-pole
systema. pole plotb. component responses
nondominant pole is near dominant second-order
pair (Case I), far from the pair (Case II), and
at infinity (Case III)

• Under certain conditions, a system with more than
  two poles or with zeros can be approximated as a
  second-order system tat has just two complex
  dominant poles. Once we justify this
  approximation, the formulae for percent
  overshoot, settling time, and peak time can be
  applied to these higher-order systems using the
  location of the dominant poles.

Remark If the real pole is five times farther to
the left than the dominant poles, we assume that
the system is represented by its dominant
second-order pair of poles.
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Figure 4.25Effect of addinga zero to a
two-pole system
System response with zeros If the zero is far
from the poles, then is large comared to
and .
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Figure 4.26Step response of a nonminimum-phase
system
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Time Domain Solution of State Equations
where is called the
state-transition matrix.