1. Dynamical Systems Analysis IV: Root Locus Plots & Routh Stability
By Peter Woolf
University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0
Creative commons

2. Recap
Qualitatively model your system: verbal modeling, incidence diagrams, control objectives
Design your control layout and connectivity on a P&ID
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

3. 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?

4. Example System
Goal: Regulate the level in R003 using the valve v1 using a
P-only controller.

5. 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

6. 3) Solve for Steady state

7. 4) Calculate Jacobian

8. 4) Calculate Jacobian

9. 5) Evaluate Stability

10. 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.

11. 6) Add in known constants and try again..
A1=2, A2=4, A3=6,
c1= c2= c3=1, Fos=1,h3set=2
Eigenvalues defined by the polynomial expression:

12. Eigenvalues defined by the polynomial expression:
Observation: Solution is still not awfully useful. What values of Kc are good? How does the answer change with Kc?
Solution:
Root Locus Plot

13. Root Locus Plot
Method to visualize the effect of changes to control parameters.
STABLE
UNSTABLE
Real axis
Imaginary
axis
positive
negative
zero
Increasing oscillatory behavior
Increasing stability

14. Root Locus Plot
Method to visualize the effect of changes to control parameters.
Imaginary values always come in pairs
= Eigenvalues for a given value of Kc
Real axis
zero
positive
negative
Example Eigenvalue set:
=-2, -3+2i, -3-2i
Imaginary
axis
Stable, oscillatory solution

15. Root Locus Plot
Method to visualize the effect of changes to control parameters.
Kc value at which system becomes unstable
Increasing Kc
Real axis
zero
positive
negative
Imaginary
axis

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

17. Root locus plot in Mathematica:
Sample a Kc value
Solve for roots
For each root separate the imaginary and real components
Plot
Increasing Kc
Kc value at which system becomes unstable
See file lec.17.example.nb

18. 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.
Conveniently this function can be factored to:
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?
Solution:The 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

19. Another example.. Root Locus Plot Increasing ti Increasing k zero
Solution:The 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
positive
negative
zero
Increasing k
Increasing ti
Root Locus Plot

20. UNSTABLE
STABLE ti k
Alternative visualization: Plot k vs ti and show regions that are stable vs unstable

21. 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

22. 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 don’t care what the actual eigenvalues are, but only care if all of the real parts are negative.
--> Use Routh Stability

23. Routh Stability
Key requirement: sign on highest order term must be positive! If not multiply system by -1 to make it this way.
Picture from controls.engin.umich.edu Routh Stability section
Count number of sign changes in first column to determine the number of positive real roots.

24. Routh table:
Row Entry
3 (10*23-14*1)/10=21.6 (10*0-1*0)/10=0
4 (21.6*14-10*0)/21.6=14 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.

25. Back to a previous example:
Eigenvalues defined by the polynomial expression:
Routh table:
Row Entry Kc
(12*44-48*(1+Kc))/44= (44*0-48*0)/44=0
(12/11)(10-Kc)
((12/11)(10-Kc)(1+Kc)-14*0) 0
/ ((12/11)(10-Kc))=
1+Kc
Thus for the first column to be all positive, we need the following conditions:
Row 3: Kc<10
Row 4: Kc>-1
Therefore, for all positive we need -1<Kc<10

26. Check and see.. Eigenvalues defined by the polynomial expression:
unstable
Borderline stable
stable
stable
Borderline stable
unstable

27. Routh Stability: Special cases
One of the coefficients is zero--replace with epsilon
Example from controls.engin.umich.edu
Roots:
s=-4 and s=2
Therefore row 3 is negative, so expect the system to have positive real roots -> unstable

28. 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.