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FAQ (Frequently Asked Questions)
Category:
CDS 101/110 Fall 2003
Identifiers: FN H0 H1 H2 H3 H4 H5 H6 H7 H8 L0.0 L1.1 L1.2 L10.1 L2.1 L2.2 L3.1 L3.2 L4.1 L4.2 L5.1 L5.2 L6.2 L8.1 L9.1 L9.2
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In stable ISL, why does the system have to go back to the circle with radius < δ?
Submitted by: waydo
Submitted on: October 15, 2003
Identifier:
L3.2
In fact it does not - the definition simply says that for any ε there is a δ such that if the system starts inside a disc of radius δ around the origin it never leaves a disc of radius ε. It doesn't matter if the system leaves the disc of radius δ and never returns (although you could find a still smaller δ' that would ensure the system would never leave the δ-disc).
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There were a couple of minor errors in todays lecture 3.2.
Submitted by: lars
Submitted on: October 15, 2003
Identifier:
L3.2
Yes, in the cruise control example I think there was a missing coefficient in front of the yd term on the RHS of the second order equation in y, and also one (a 1/m) in front of the the x1 and x2 terms in the state
space formulation.
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What did you mean by "V(x) is an energy-like function"?
Submitted by: demetri
Submitted on: October 15, 2003
Identifier:
L3.2
Our simple physical intuition suggests that stability of a system is in some way related to energy inputs and outputs. We would expect a system which perpetually loses energy to be asymptotically stable (as long as there are no energy inputs).
The Lyapunov stability theory is essentially a generalization of this idea. We would like have a function which measures how much "energy" there is in the system. This is why we require it to be locally positive definite: as the size of the x-vector increases, so should the "energy". This is also why we check to see if the Lyapunov function decreases (technically this is an abuse of terminology, since we do not call it a Lyapunov function unless it satisfies the decrease condition).
In physical systems, the total energy can be used as a Lyapunov function (certainly try this one before trying any others). However, you should take this with a grain of salt; using the physical energy as a Lyapunov function may not give the most straightforward analysis, and it is often possible to find Lyapunov functions which facilitate the manipulations better than the actual physical energy.
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A note on the alpha parameter in HW3.
Submitted by: demetri
Submitted on: October 15, 2003
Identifier:
L3.2
, H3
There was some confusion regarding the parameter alpha involved in the suggested Lyapunov functions. This is NOT the bounding function discussed in class. It is just a scalar. We apologize for any possible confusion.
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For LAST WEEK... we need some harder Lyapunov function examples.
Submitted by: lars
Submitted on: October 21, 2003
Identifier:
L4.1
, L3.2
, H3
Noted. We acknowledge that the Lyapunov function homework problem requires more effort than that presented in class. We'll be more conscientious about teaching the class at a consistent level.
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