5. Dynamic behavior of First-order and Second-order Systems

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5. Dynamic behavior of First-order and
Second-order Systems

• Contents
• 1. Standard Process Inputs.
• 2. Response of First-Order Systems.
• 3. Response of Integrating Process.
• 4. Response of Second-Order Systems.

• In this chapter, we learn how process respond to
  typical changes in some of input changes.
• Two categories in process inputs.
• 1. Inputs that can be manipulated to control the
  process.
• 2. Inputs that are not manipulated, classified as
  disturbance or load variable.

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5.1 Standard Process Inputs

• - Six important types of input changes.
• 1. Step input.
• 2. Ramp input.

Figure 5.1. Step input.
Figure 5.2. Ramp input.
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3. Rectangular Pulse. or Where
is unit step function. 4. Sinusoidal input.
Figure 5.3. Rectangular Pulse.
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• 5. Impulse input.
• It has the simplest Laplace transform, but it is
  not a realistic input signal. Because to obtain
  an impulse input, it is necessary to inject
  amount of energy or material into a process in an
  infinitesimal length of time.
• 6. Random inputs.
• Many process inputs change with time in such a
  complex manner that it is not possible to
  describe them as deterministic functions of time.
  If an input exhibits apparently random
  fluctuation, it is convenient to characterize it
  in statistical terms.

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5.2 Response of First-Order Systems

• - General first-order transfer function.
• Where is the process gain and is the
  time constant.
• Find and for some particular
  input .
• 1. Step response.

Figure 5.4. Step response.
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• A first-order system dose not respond
  instantaneously to a sudden change in its input
  and that after a time interval equal to the
  process time constant ( ), the process
  response is still only 63.2 complete.
• Theoretically the process output never reaches
  the new steady-state value it dose approximate
  the new value when t equals 3 to 5
  process time constants.

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• 2. Ramp response.
• Interesting property for large values of time(
  ).

• After an initial transient period, the ramp
  input yields a ramp output with slope equal to
  , but shifted in time but the process time
  constant .

Figure 5.5. Ramp response. - comparison of input
and output.
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• 3. Sinusoidal response.
• By trigonometric identities.
• Where .
• Trigonometric identities.
• Where .

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• Remarks
• 1. In both (5.21) and (5.22), the exponential
  term goes to zero as leaving a pure
  sinusoidal response.
• Frequency Response ! (will be discussed later).
• 2. Since , amplitude is less
  than output f or input . ? Amplitude
  attenuation !

Figure 5.6. Typical sinusoidal response.
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5.3 Response of Integrating Process Units

• What is an Integrating Process?
• The process which has integrating unit( )
  in its transfer function.
• Open-loop unstable process(Non-self-regulating
  process).

• A process that cannot reach a new steady state
  when subjected to step changes in inputs is
  called Open-loop unstable process or
  Non-self-regulating process.

• Which process is an integrating process?

Figure 5.7. Liquid level system with a pump(a) or
valve(b).
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• Answer ) (a) is the integrating process!
• The flowrate of the effluent stream (b) increase
  automatically if the level increase. Therefore,
  the influent flowrate is increased then the level
  will increase and the effluent flowrate also
  increased up to the influent flowrate so the
  level will converge.
• Liquid level system with a valve is a stable(or
  self-regulating) process.
• But, in (a), regardless of the level, the
  effluent flowrate is constant due to the pump.
  So, if the influent flowrate is bigger than the
  effluent stream the level always increase, vice
  versa. That is, the difference between the
  influent flowrate and the effluent flowrate is
  integrated to the process output(the level).
• Liquid level system with a pump is a unstable(or
  non-self-regulating) process.

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• Example

Where is independent of .
Integrating process !
Figure 5.8. Liquid level system with a pump.
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5.4 Response of Second-Order Systems

• A second order transfer function can arise
  physically,
• Two first-order processes are connected in
  series.

Figure 5.9. Two first-order systems in series
yield an overall second-order system.

• A second-order differential equation process
  model is transformed.

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• Standard form of the second-order transfer
  function.

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• Three important subcases.
• Denominator of (5.28)

• Roots

• unstable second-order system that
  would have an unbounded response to any input.

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• 1. Step response.

Case c. , complex root
Underdamped.
Where
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Figure 5.10. Step response of critically-damped
and overdamped(a), and underdamped(b)
second-order processes.

• Remarks
• Responses exhibiting oscillation and overshoot(
  ) are obtained only for values
  of less than one.
• Large value of yield a sluggish response.
• The faster response without overshoot is obtained
  for critically damped case( ).

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• A number of terms that describe the dynamics of
  underdamped processes.
• 1. Rise time( ) is the time the process output
  takes to first reach the new steady-state value.
• 2. Time to first peak( ) is the time required
  for the output to reach its first maximum value.
• 3. Settling time( ) is defined as the time
  required for the process output reach and remain
  inside a band whose width is equal to
  of the total change in .
• 4. Overshoot.
• 5. Decay ratio.
• 6. Period of Oscillation( ) is the time
  between two successive peaks or two successive
  valleys of the responses.

Figure 5.11. Performance characteristics for the
step response.
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• Rise time.

• Time to first peak.

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• Decay ratio.

• Period of oscillation.

Figure 5.12. Relation between some performance
characteristics of an underdamped second-order
process and the process damping coefficient.
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• 2. Sinusoidal response.

As , the first and second terms
vanish. Thus the output for large values of time
is obtained as follows.
Where

• Amplitude ratio

• Normalized amplitude ratio

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• The maximum value of can be found by
  differentiating (5.45) with respect to .

For , there is no maximum.

• At high frequency, the output is well damped.
• At low frequency, the output is not damped well.

Figure 5.13. Sinusoidal response amplitude of a
second-order system after exponential terms have
become negligible.