Title: Chemical Process Dynamics
1Dynamical 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
2- 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
3Reactions 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?
4Aside 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
5Linear System Notation
How do we do this??
Nonlinear system
6Linearization
- Choose a relevant point to make your linear
approximation. - Calculate the Jacobian matrix at that point
- Solve to find the unknown constants
7Linearization
- Choose a relevant point to make your linear
approximation.
Full nonlinear function
Local linear approximation
1X
8Linearization
- 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
9Calculating 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
10Calculating Steady State Values
Mathematica function Solve solves a system
of algebraic expressions analytically or
numerically.
11Linearization
- 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
12Calculating a Jacobian matrix
Jacobian shows how every variable changes with
each other variable at a point. Always a square
matrix (rows columns)
Example
13Calculating a Jacobian matrix
A0,B0
14Calculating 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
15Calculating a Jacobian matrix
16Calculating a Jacobian matrix
17Linearization
- Choose a relevant point to make your linear
approximation. - Calculate the Jacobian matrix at that point
- Solve to find the unknown constants
18Nonlinear 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
19Nonlinear model
Linear approximation
??
Jacobian
Therefore the full linear approximation at A0,
B0 is
Or in a different format
20Nonlinear model
Linear approximation
??
Jacobian
Note the unknown constants are not always 0.
E.g. linear approximation at steady state A0, B1
21Linear algebra aside How to solve this?
2) Solve k130, k232
22Nonlinear model
Linear approximation
??
Jacobian
Therefore the full linear approximation at A0,
B1 is
Or in a different format
23Take 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.