Chemical Process Dynamics

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Dynamical Systems Analysis IV Root Locus Plots
Routh Stability By Peter Woolf
(pwoolf_at_umich.edu) University of
MichiganMichigan Chemical Process Dynamics and
Controls Open Textbookversion 1.0
Creative commons
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Recap

• Qualitatively model your system verbal modeling,
  incidence diagrams, control objectives
• Design your control layout and connectivity on a
  PID
• Make a model, evaluate stability of that model.
• Simulate and visualize dynamics of your model
• Add logical (IF.. THEN..) controllers and PID
  controllers
• Tune controller parameters

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Question of the Day

• Given a controller, how far can you push the
  system before it oscillates or goes out of
  control?
• What is your safety margin?
• Does it really matter what I set some values to?

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Example System
Goal Regulate the level in R003 using the valve
v1 using a P-only controller.
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• Model system

Here, h1, h2, and h3 are the levels of R001,
R002, and R003. The parameters c1, c2, c3 are
valve and pipe constants, and Fo is the feed
2) Model controller
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3) Solve for Steady state
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4) Calculate Jacobian
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4) Calculate Jacobian
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5) Evaluate Stability
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Aside Mathematica is helping us, as the full
solution is not really helpful.
In general, 1st and 2nd order polynomials are
interpretable, 3rd and 4th are analytically
solvable but not easily interpretable, and 5
order polynomials have no analytical solution.
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6) Add in known constants and try again..
A12, A24, A36, c1 c2 c31, Fos1,h3set2
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Observation Solution is still not awfully
useful. What values of Kc are good? How does
the answer change with Kc?
Solution Root Locus Plot
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Root Locus Plot

• Method to visualize the effect of changes to
  control parameters.

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Root Locus Plot

• Method to visualize the effect of changes to
  control parameters.

Real axis
zero
positive
negative
Example Eigenvalue set ?-2, -32i, -3-2i
Imaginary axis
Stable, oscillatory solution
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Root Locus Plot

• Method to visualize the effect of changes to
  control parameters.

Increasing Kc
Real axis
zero
positive
negative
Imaginary axis
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• Root locus plot in Mathematica
• Sample a Kc value
• Solve for roots
• For each root separate the imaginary and real
  components
• Plot

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• Root locus plot in Mathematica
• Sample a Kc value
• Solve for roots
• For each root separate the imaginary and real
  components
• Plot

Kc value at which system becomes unstable
See file lec.17.example.nb
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Another example..
Imagine for another control system you find the
following polynomial describing your eigenvalues
Here k and ti parameterize the P and I part of a
PI controller.
For this system, What are the limits of k that
result in a stable system? What are the limits of
ti that result in a stable system?
SolutionThe limits on k force the first root to
be negative, thus
Thus k must be 2 or greater. Similarly for ti, ti
must be 1 or greater
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Another example..
SolutionThe limits that ki result are ones that
make the first root negative, thus
Thus k must be 2 or greater. Similarly for ti, ti
must be 1 or greater
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Alternative visualization Plot k vs ti and show
regions that are stable vs unstable
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Complications Solutions
1) Sometimes you only care if the solution has
real positive parts (i.e. is unstable) 2)
Sometimes you have too many unknowns to easily
construct and interpret a root locus plot (e.g.
with two PID controllers you have Kc1, Kc2, ?i1,
?i2, ?d1, ?d2)
Solution Routh stability analysis
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Routh Stability

• Routh stability allows us to evaluate the signs
  of the real parts of the roots of a polynomial
  without solving for the roots themselves.

Example In analyzing the stability of your
system, you find the following expression for
your eigenvalues
You dont care what the actual eigenvalues are,
but only care if all of the real parts are
negative. --gt Use Routh Stability
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Routh Stability
Key requirement sign on highest order term must
be positive! If not multiply system by -1 to
make it this way.
Count number of sign changes in first column to
determine the number of positive real roots.
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Routh table Row Entry 1 1 23 2 10 14
3 (1023-141)/1021.6 (100-10)/100 4 (21.614
-100)/21.614 0
Because all entries in the first column are
positive, we can assume that the real components
of the eigenvalues are all negative and the
system is stable. This result does not tell us
if the system spirals and is only as accurate as
the model and possible linear approximation we
made of the model, but it does provide us with a
method.
Note if we did evaluate the roots, we would find
this system has roots of -2, -1, and -7, thus is
stable.
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Back to a previous example
Routh table Row Entry 1 48 12 2 44 1Kc

• (1244-48(1Kc))/44 (440-480)/440
• (12/11)(10-Kc)
• ((12/11)(10-Kc)(1Kc)-140) 0
• / ((12/11)(10-Kc))
• 1Kc

Thus for the first column to be all positive, we
need the following conditions Row 3 Kclt10 Row
4 Kcgt-1
Therefore, for all positive we need -1ltKclt10
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Check and see..
unstable
Borderline stable
stable
stable
Borderline stable
unstable
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Routh Stability Special cases

• One of the coefficients is zero--replace with
  epsilon

Roots s-4 and s2
Therefore row 3 is negative, so expect the system
to have positive real roots -gt unstable
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Take Home Messages

• Adding a controller to a system does not always
  make it stable!
• Root locus plots help you see the effects of
  changing control parameters on system stability.
• Routh stability can help to identify regions of
  parameter space that will be stable.
• Routh stability is particularly useful when there
  are multiple unknown parameters and the system is
  too large handle analytically.