Analysis of limit cycles using describing functions and harmonic balance method
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+. 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 l.jpg

+

G(s)

_

Analysis of Limit Cycles using Describing Functions and Harmonic Balance Method

Consider

Typically


Motivation and the structure of the approach l.jpg
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.

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.


Motivation l.jpg
Motivation

Assume now that

 Filter hypothesis


Motivation4 l.jpg

N.B :

Motivation



Problem formulation main result l.jpg

Given

Problem Formulation & Main Result

As it has been pointed out before, the approach to the solution of the problem is

as follows :


Solution l.jpg

+

G(s)

_

Solution


Solution continued l.jpg
Solution (Continued)

Since

we can write

and


Solution continued9 l.jpg
Solution (Continued)

Still it is an infinite dimensional equation. To make it finite dim., we use the

filter hypothesis :

Then, we have

  • qth order harmonic balance

     could be solves numerically (by computer)

Assume q =1. Then we have


Solution continued10 l.jpg

real

complex

(two real no.)

real

Solution (Continued)

Use one more notation


Solution continued11 l.jpg

(1)

Solution (Continued)

Therefore the first equation is satisfied for all G(0). The second equation can be written

as follows :

and


Solution continued12 l.jpg

(2)

Solution (Continued)

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 :

Reference :

  • Justification of the describing function methodSIAM. J. of Control (vol. 9, no. 4, Nov. 1971, p 568-589)


Alternate method l.jpg
Alternate Method

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.


Example 1 l.jpg

1

-1

1

2

Example 1

1

-1




Example 2 continued17 l.jpg
Example 2 (Continued)

no oscillation

no oscillation

oscillation

oscillation


Example 2 continued18 l.jpg

-1

saturation nonlinearity

1

Example 2 (Continued)


Example 3 4 l.jpg

There exists a nonlinearity

which gives oscillation

Example 3 & 4

 Rayleigh equation


Example 4 continued l.jpg

|G(j)|

1

Example 4 (Continued)


Example non odd feedback l.jpg
Example : Non-Odd Feedback

  • The case of non-odd feedback


Example continued l.jpg

Im

Re

Example (Continued)


Example hysteresis l.jpg
Example: Hysteresis

Ex: Hysteresis


Example continued24 l.jpg
Example (Continued)

Im

Re

a increasing


Stability rules l.jpg
Stability Rules

  • Stability rules

Im

 increasing

a increasing

Re



Harmonic balance method l.jpg
Harmonic Balance Method

  • Justification of the Harmonic Balance Method







Evaluation 133 l.jpg
Evaluation 1

Im 1/G

critical

circle

Re 1/G

error circle





Example l.jpg

1

-1

1

-1

Example

Ex:

+