|
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
-
What is a level set?
Submitted by: aotang
Submitted on: October 13, 2004
Identifier:
L3.2
The level set of a differentiable function f: R^n->R corresponding to a real value c is the set of points
{(x1,...,xn) in R^n : f(x1,...xn)=c}
If n = 2, the level set is a plane curve known as a level curve. If n = 3, the level set is a surface known as a level surface.
By the way, I copy above words from mathworld
[Back to Top]
-
Who do we talk to if we have questions about graded homework?
Submitted by: waydo
Submitted on: October 13, 2004
Identifier:
L3.2
, H1
You should send an email to the ta list (cds101-tas@cds), so the TA who graded the problem you have a question on can respond.
[Back to Top]
-
How do we know how to "twist" our candidate Lyapunov function?
Submitted by: waydo
Submitted on: October 13, 2004
Identifier:
L3.2
This is the difficulty with proving stability with Lyapunov functions - although a Lyapunov function is guaranteed to exist for a stable equilibrium point, finding it may be hard. The energy plus a small "twist" method works well for mechanical systems, and just requires you to add a small cross term (i.e. a term on x*xdot). The best way to find this is to put in such a term with a parameter out front (i.e. alpha*x*xdot), then see if you can adjust the parameter to make Vdot negative definite. The TAs will be doing an example of this in recitation this week.
[Back to Top]
-
Are there lecture notes for today's lecture?
Submitted by: waydo
Submitted on: October 13, 2004
Identifier:
L3.2
Sorry, there are no lecture notes for this lecture. However, all of the material is in the book (AM04, Chapter 3) on the website.
[Back to Top]
-
If every equilibrium point can be transformed to the origin and the operated on w/ Lyapunov, how can a system have both stable and unstable equilibrium points?
Submitted by: haomiao
Submitted on: October 13, 2004
Identifier:
L3.2
The reason that you want to transform points to the origin is to that you can look at local stability and ignore the higher order (O(3) and above) terms. The actual location of the equilibrium terms still exists in the functions you use to do the analysis, however. For example:
Say you have xdot = f(x), with eq. pt. Xe. Then you do the transform:
z = x - Xe, so zdot = xdot = f(x), but x = z + Xe, so
f(x) = f(z + Xe), so the information about the actual location of the eq. point is still in the functions, and when you do the stability analysis for f(z+Xe) that'll show the stability of the actual equilibrium point you're looking at.
[Back to Top]
|