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FAQ (Frequently Asked Questions)
Category:
CDS 101/110 Fall 2004
Identifiers: H0 H1 H2 H3 H4 H5 H6 H7 H8 L0.0 L1.1 L1.2 L2.1 L2.2 L2.3 L3.1 L3.2 L4.1 L4.2 L5.1 L5.2 L6.1 L7.1 L9.1
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Can I call Richard Murray "Dick" Murray?
Submitted by: waydo
Submitted on: October 18, 2004
Identifier:
L4.1
You certainly can, although I wouldn't recommend it.
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What is the correct equation for the circuit on slide 11?
Submitted by: waydo
Submitted on: October 18, 2004
Identifier:
L4.1
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Please, sir, may I have another Lyapunov function example?
Submitted by: asa
Submitted on: October 18, 2004
Identifier:
L4.1
For more examples, I encourage you to check out Differential Equations and Dynamical Systems by Lawrence Perko, which is currently being used as a textbook by CDS 140. In particular, secion 2.9 starting on page 129 has several good examples.
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How can we tell from the phase plots if the system is oscillating?
Submitted by: haomiao
Submitted on: October 18, 2004
Identifier:
L4.1
If the phase plot takes the form of a spiral or a circle, then the system is oscillating. Take a look at any of the spiral plots and trace one of the lines around an equilibrium point, looking at either x1 or x2. You'll ntoice that coordinate increasing as you spiral around, then decreasing as you come back towards the equilibrium point, increasing again as you move past it, then decreasing again as you come in on your spiral and so on.
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On slide 8, real eigenvalues & complex eigenvalues look the same, both appear to be asymptotically stable. Why?
Submitted by: haomiao
Submitted on: October 18, 2004
Identifier:
L4.1
The stability of a system is determined by the real part of its eigenvalues, so if the real part of the eigenvalues are negative then the system is asy-stable no matter what the complex parts are. The complex part determines the phase response of the system, and contributes to the oscillatory behavior of the system.
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What is an example of a system with Re(\lambda)=0 that is not stable? What if Im(\lambda) is not zero?
Submitted by: asa
Submitted on: October 18, 2004
Identifier:
L4.1
As we discussed in class, this system:
which has two zero eigenvalues, has this solution:
which is clearly not stable.
For the second question, we have to go to a larger system, such as
.
Here the eigenvalues are
each with multiplicity 2. The solution is quite complicated (it's on p. 36 and 37 of Perko, Differential Equations and Dynamical Systems) but you can see that you're going to have a similar effect as in the first example, and you'll have terms that grow linearly with time.
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