Chemical Process Dynamics

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Dynamical Systems Analysis I Fixed Points
Linearization By Peter Woolf (pwoolf_at_umich.edu) U
niversity of MichiganMichigan Chemical Process
Dynamics and Controls Open Textbookversion 1.0
Creative commons
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• Problem Given a large and complex system of ODEs
  describing the dynamics and control of your
  process, you want to know
• Where will it go?
• What will it do?
• Is there anything fundamental you can say about
  it?
• E.g. With my control architecture, this process
  will always ________.

Solution Stability Analysis
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Reactions AA --gt B B A --gt C
Example CSTR with cooling jacket and multiple
reactions.
Controls PID on jacket cooling water
What will happen? What is possible? What effect
will my controller have?
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Aside Linear vs. Nonlinear
Linear systems are significantly easier to work
with, and are the basis of stability analysis Few
physical systems are linear, but all can be
locally approximated as linear.
Feature Any nonlinear terms in your model will
render the whole system nonlinear, and as such
harder to analyze.
Example
Goal convert nonlinear system to a simpler
linear system
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Linear System Notation
How do we do this??
Nonlinear system
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Linearization

• Choose a relevant point to make your linear
  approximation.
• Calculate the Jacobian matrix at that point
• Solve to find the unknown constants

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Linearization

• Choose a relevant point to make your linear
  approximation.

Full nonlinear function
Local linear approximation
1X
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Linearization

• Choose a relevant point to make your linear
  approximation.

• Possible relevant points
• Steady state value points where the system does
  not change
• Current location given where I am now, where
  will I go next.

Temp
Time
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Calculating Steady State Values

• Given a system of ODEs, steady state values can
  be found by setting all time derivatives equal to
  zero and solving.

Kinetics example
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Calculating Steady State Values
Mathematica function Solve solves a system
of algebraic expressions analytically or
numerically.
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Linearization

• Choose a relevant point to make your linear
  approximation.
• Calculate the Jacobian matrix at that point

Jacobian matrix is essentially a Tailor series
expansion around a point.
100X
higher order terms
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Calculating a Jacobian matrix
Jacobian shows how every variable changes with
each other variable at a point. Always a square
matrix (rows columns)
Example
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Calculating a Jacobian matrix
A0,B0
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Calculating a Jacobian matrix
A0,B0
Jacobian can also be solved analytically in
Mathematica
New mathematica function Df(x,y,z, z
Example DA2BACCA , A --gt
2ABC
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Calculating a Jacobian matrix
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Calculating a Jacobian matrix
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Linearization

• Choose a relevant point to make your linear
  approximation.
• Calculate the Jacobian matrix at that point
• Solve to find the unknown constants

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Nonlinear model
Linear approximation
??
Jacobian
Approach 1) solve both models at the point A0,
B0
2) Force derivatives of the linear and nonlinear
model to agree by setting unknown constants
k13 0 k23 0
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Nonlinear model
Linear approximation
??
Jacobian
Therefore the full linear approximation at A0,
B0 is
Or in a different format
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Nonlinear model
Linear approximation
??
Jacobian
Note the unknown constants are not always 0.
E.g. linear approximation at steady state A0, B1
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Linear algebra aside How to solve this?
2) Solve k130, k232
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Nonlinear model
Linear approximation
??
Jacobian
Therefore the full linear approximation at A0,
B1 is
Or in a different format
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Take Home Messages

• Nonlinear models are more realistic but harder to
  manipulate
• Any nonlinear model can be approximated as a
  linear one at a point
• The linear approximation is exactly correct at
  the point, but less accurate away from the point.