+. G ( s ). _. Analysis of Limit Cycles using Describing Functions and Harmonic Balance Method. Consider. Typically. Motivation and the structure of the approach. Given this system – first we would like to study its stability using the circle, Popov, and small gain theorems.
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_Analysis of Limit Cycles using Describing Functions and Harmonic Balance Method
Given this system – first we would like to study its stability using the circle, Popov,
and small gain theorems.
Assume that none of the methods show the stability of the origin. Thus it may be
unstable. The next thing to look for – oscillation or limit cycles.
In general, we can apply the Poincare-Bendixson or the index theory – but this is
only for the planar system. The harmonic balance method allows one to study
the limit cycle in the feedback system shown above for any nonlinearity.
Assume now that
As it has been pointed out before, the approach to the solution of the problem is
as follows :
we can write
Still it is an infinite dimensional equation. To make it finite dim., we use the
filter hypothesis :
Then, we have
could be solves numerically (by computer)
Assume q =1. Then we have
(two real no.)
Use one more notation
Therefore the first equation is satisfied for all G(0). The second equation can be written
as follows :
Introduce the describing function
Then equation (1) can be written as
If (2) has a solution a, w, then the original system “probably” has
a periodic solution close to asinwt, if (2) does not have a solution,
then the original system probably has no periodic solution.
Main result :
The problem could also be approached analytically
So find w’s where G(jw) intersects with real axis. Then find a from the first
Equation for each point of intersection.
1Example 2 (Continued)
which gives oscillationExample 3 & 4
1Example 4 (Continued)